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Sustainable intervention for learning gaps in middle school mathematics: a gap analysis

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Sustainable intervention for learning gaps in middle school mathematics: a gap analysis

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Content Running head: LEARNING GAPS IN MIDDLE SCHOOL MATH

Sustainable Intervention for Learning Gaps in Middle School Mathematics: A Gap Analysis

by

Monica M. San Jose

A Dissertation Presented to the
FACULTY OF THE USC ROSSIER SCHOOL OF EDUCATION
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF EDUCATION

May 2018

Copyright 2018 Monica M. San Jose
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DEDICATION
This dissertation is dedicated to my mom and dad, Eva I. Schrodt and Dr. Verle N.
Schrodt, for their love and commitment to each other in all aspects of life, for their no-nonsense
approach to life’s bigger challenges, and for their model of the finer qualities I hope to exude as
a spouse, parent and professional educator. I am truly grateful that you are my parents.

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ACKNOWLEDGEMENTS
Many people ask me why I chose to pursue a doctorate in education and sometimes
through this process, I also asked myself the same question. I have come to understand that I
have chosen to do this work for myself, to become an expert in my chosen profession and to
attain a level of distinction of which I could be proud. I am deeply indebted to all those friends,
family members, colleagues, and advisors who supported my efforts over the past three years to
achieve this personal goal.
To Dr. Larry Picus, thank you for your time and advice to help improve my practice and
to think more critically about the choices I made and will continue to make in the education of
others. Our dissertation journey together began with a walk through the Orchid Garden in the
Botanic Gardens of Singapore and the delicate beauty and variety of orchids will always remind
me of the care and mentorship you gave to me along the dissertation process.
To Dr. Ruth Chung, thank you for your critical review and advice as I started gathering
ideas for my dissertation and as I continued in my writing. Your specific guidance and high
expectations grounded my work.
To Dr. Rebecca Darrough, an expert in mathematics education in the United States and
my sister, thank you for the conversations and your advice on how my work would apply to
middle school mathematics teachers outside of the international school context, where I have
spent most of my teaching years. I feel so fortunate for you to professionally review my work,
and I appreciate your kind words of encouragement along the way.
To Dr. Chip Kimball at the Singapore American School, thank you for the vision to bring
the University of Southern California doctoral program to our campus. The value of this
education and the learning that I have gained from the other members of the cohort is
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4
immeasurable. I am indebted to the entire cohort for the work we have completed together, the
moral support and the laughs we have shared as we completed the marathon together. I am most
grateful to Jennifer Sparrow for leading the class and organizing the logistics for our cohort, the
logistics for the teachers who came to visit, the course materials for us, and so much more. I am
also extremely grateful to the special support I received from my writing partners, Dennis
Steigerwald and Susan Shaw, who became even greater moral support as the months of writing
went on and on and on… I am thankful for Cris Ewell and Martha Began for always having a
kind word and for the talks we shared.
To the middle school mathematics team, thank you so much for freely sharing your time,
your documents, the work you spent hours creating, your expertise, your ideas, your inspirations
and hopes and concerns for your students. I have learned so much from having the opportunity
to work with you. Thanks especially to those who gave extra time to support my dissertation -
my math teaching partners, Matt Medina and Jason Windust, who helped to make teaching 7
th

graders fun!, Isagani Celzo, who inspired me with his ideas and expertise every time we talked
and Terri McComb, who inspired me by her creative methods, the care she showed her students
and her brilliant mind. Finally, I am forever grateful for the knowledge I have gained about
teaching mathematics and the friendship I have formed with Melissa Trainor, one of the most
gifted educators that I have ever met. I will remember my time teaching with this team as the
best teaching experience in my career.
To the 7
th
grade A-side team, Kate Bucknall, Joann Olsen, Sharon Carroll, Rebecca
Watters, Heather Rodocker, Molly Stefano, Chris Strance, Crystal Madsen and Jason Martin,
thank you for living the day-to-day life of teaching middle schoolers and for making memories in
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5
every aspect of life we experienced together. Many memories fill my heart and often, when I am
teaching, I think of all that I have learned from you.
To my support network of friends in Singapore, whom I consider as close as family,
thank you for always being there to enjoy a glass of wine, a run around Marina Bay, a coffee or a
workout. Thank you to the running crew - Beth and Bob Helmer and Jon and Erica Simons
Hansen, and to Katie Walthall, for afternoon walks and talks, and to the wine ladies - Lisa Ball,
Teresa Smith and Gloria Strydhorst-Piers for helping me to maintain sanity and enjoy life. I look
forward with great anticipation of every chance to see you.
To Shannon Doak, my new friend and colleague in Hawaii, who shared in my doctoral
journey, and was always one step ahead. Thank you for the motivation to persevere and your
support at work along the way. I am glad to have shared the final leg of the marathon with you.
Most of all, I thank my family for supporting me in my personal goal. To my children,
Nicolas, Alyssa and Elena, thank you for giving me the reason for becoming involved in the field
of education. When you were first born, I wanted only the best for you and studied everything I
could find on how to provide the best education for all of you. This dissertation is the
culmination of those studies, which began 24 years ago. Thanks to Nick for sharing afternoons
with me after work, discussing my dissertation in its final stages and giving me your thoughts,
over a cup of tea. Thanks to Alyssa for continuing to inspire me as she worked on her own
doctoral pursuit. Many special thanks to Elena, who LIVED this journey with me. Thank you
for being there with me for the Masters Degree, spending a summer in Bangkok and sitting in
with my study groups even into the late hours of the night, and for being there every step of the
way in the doctoral journey. We worked hard together, relaxed together, and I appreciated so
much that you were there with me.
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Most of all, Mahalo to Tony, my partner in life. Thank you for continuing to support my
personal goals and my second career, and for stepping in to take on more duties around the
house. The family is forever grateful for you taking over dinner duties! Thank you for listening,
or even pretending to listen, when I rambled on and on about what I was writing or thinking
about when it came to helping kids who struggle to learn math. You have been my thought
partner in our family life but also in my professional life. Thank you for helping me to practice
science experiments on many weekends so I could get through my day job and still be able to
finish my doctoral studies or dissertation writing. Thank you for waking up with me early in the
morning so I could have coffee and company as I wrote before going in to work. Thank you
mostly for being there, to share this experience and to see where it might take us. I love you,
always and forever.

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TABLE OF CONTENTS
LIST OF TABLES ……………………………………………………………………………. 12
LIST OF FIGURES ……………………………………………………………………………. 14
ABSTRACT …………………………………………………………………………………..... 16
CHAPTER 1: INTRODUCTION TO THE PROBLEM ………………………………………. 18
Background of the Problem ………………………………………………………………… 19
Importance of Addressing the Problem …………………………………………………….. 21
Organizational Context and Mission ………………………………………………………...22
Organizational Performance Status ………………………………………………………… 23
Organization Performance Goal ………………………………………………………….… 24
Description of Stakeholder Groups and Performance Goals ……………………………….. 26
Stakeholder Group for the Study and Stakeholder Performance Gap …………………….... 28
Purpose of the Project and Questions ………………………………………………………. 28
Conceptual and Methodological Framework ………………………………………………. 29
Definitions ………………………………………………………………………………….. 29
CHAPTER 2: REVIEW OF THE LITERATURE …………………………………………….. 33
How Do Adolescents Learn and What Are They Expected to Know ……………………… 34
Learning and Motivation Theory ……………........………...…………………………... 34
Learning Progression for Middle School Mathematics ……………………………………...39
Purpose of Education …………………………………………………………………… 39
Learning Progression for Mathematics …………………………………………………. 53
Successful Strategies in Improving Mathematics Learning at the Middle School Level.. 60
Organization Theory ………………………………………………………………………... 75
Factors Contributing to Effective Organizations ……………………………………….. 75
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Knowledge, Motivation and Organizational Factors………………………………………... 78
Summary of Literature Review ……………………………………………………………... 81
Conclusion ………………………………………………………………………………….. 82
CHAPTER 3: METHODOLOGY ……………………………………………………………... 84
Purpose of the Project and Questions ………………………………………………………. 84
Conceptual and Methodological Framework ……………………………………………….. 85
Assessment of Performance Influences …………………………………………………….. 86
Knowledge Assessment …………………………………………………………………. 87
Organization/Culture/Context Assessment ……………………………………………… 88
Participating Stakeholders and Sample Selection …………………………………………... 90
Sampling ………………………………………………………………………………… 90
Recruitment ……………………………………………………………………………… 92
Data Collection ……………………………………………………………………………... 93
Surveys …………………………………………………………………………………... 93
Interviews ………………………………………………………………………………... 94
Document Analysis ……………………………………………………………………… 94
Data Analysis ……………………………………………………………………………….. 95
Trustworthiness of Data …………………………………………………………………….. 97
Role of Investigator …………………………………………………………………………. 97
Limitations and Delimitations ………………………………………………………………. 98
CHAPTER 4: RESULTS AND FINDINGS ………………………………………………..... 100
Participants ………………………………………………………………………………… 101
Report of the Findings …………………………………………………………………….. 101
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Results of Research Question One: Effective Existing Strategies ………………………… 102
Document Analysis Results …………………………………………………………… 102
Interview Results ………………………………………………………………………. 116
Summary of RQ1 Findings …………………………………………………………….. 135
Results of Research Question Two: Knowledge and Skills, Motivation and Organization . 136
Influences ………………………………………………………………………………………136
Knowledge and Skills ………………………………………………………………….. 136
Motivation ……………………………………………………………………………… 168
Organization .. ………………………………………………………………………….. 172
Summary of Gaps Found ………………………………………………………………….. 177
Results of Research Question Three: Possible New Programs ……………………………. 178
Additional Problem Solving Time……………………………………………………… 178
Teacher Collaboration ……………………………………………………………….…. 180
Summary …………………………………………………………………………………... 181
Conclusion ………………………………………………………………………………… 181
CHAPTER 5: SOLUTIONS, IMPLEMENTATION AND EVALUATION ………………... 185
Validated Effective Strategies for Middle School Mathematics Instruction ……………… 186
Validated Influences ………………………………………………………………………. 186
Solutions …………………………………………………………………………………... 189
Research Question Two – Recommendations …………………………………………. 189
Knowledge and Skills ………………………………………………………………….. 189
Motivation ……………………………………………………………………………… 198
Organization Barriers …………………………………………………………………... 204
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Integrated Implementation and Evaluation Plan …………………………………………... 215
Implementation and Evaluation Framework …………………………………………… 215
Organizational Purpose, Need and Expectations ………………………………………. 216
Level 4: Results and Leading Indicators ……………………………………………….. 217
Level 3: Behavior ………………………………………………………………………. 219
Level 2: Learning ………………………………………………………………………. 227
Level 1: Reaction ………………………………………………………………………. 232
Evaluation Tools ………………………………………………………………………,, 234
Data Analysis and Reporting …………………………………………………………... 234
Summary ……………………………………………………………………………….. 237
Future Research ………………………………………………………………………….... 237
Conclusion ………………………………………………………………………………… 238
Gap Analysis Framework ……………………………………………………………… 239
Proposed Solutions ………………………………………………………………….…. 241
Proposed Implementation ……………………………………………………………… 243
Proposed Evaluation Plan ……………………………………………………………… 244
Implications for Wider Mathematics Community ……………………………………... 244
REFERENCES ……………………………………………………………………………….. 246
APPENDICES………………………………………………………………………………… 260
Appendix A: “Getting Started with RTI in MS Mathematics” Survey …………………… 260
Appendix B: “First Quarter Pulse Check in RTI for MS Mathematics” Survey ………….. 262
Appendix C: Grade 7 Mathematics Planning and Pacing Calendar for September 2017 … 263
Appendix D: Grade 7 Math PLC Agendas from September 2017 ………………………... 264
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Appendix E: Grade 8 Lesson Demonstrating Differentiation ……………………………. 266
Appendix F: Grade 8 Learning Target Sheet Sample ……………………………………. 267
Appendix G: Grade 7 Concept Checklist Sample ………………………………………... 269
Appendix H: Grade 6 “I Can” Statements Sample ……………………………………….. 270
Appendix I: Samples of Tiered Problems for MS Mathematics ………………………….. 271
Appendix J: Examples of MS Mathematics PLC Agendas Showing Student Thinking …. 273
Appendix K: Grade 6 Math PLC “Unit Summative Assessment Itemized Math Practice
Standard Alignment …………………………………………………………………. 275
Appendix L: Summary of Existing, Effective Strategies Currently in Use in MS Math at
ISSEA ……………………………………………………………………………….. 283
Appendix M: Eight Effective Mathematics Teaching Practices …………………………. 287
Appendix N: MS Math Articulation of Standards – Sample …………………………….. 288
Appendix O: Power Standards Math #SBG ……………………………………………… 292
Appendix P: Prime Time Unit Summative Assessment (Grade 6 Math PLC) …………… 304
Appendix Q: Recommended Clusters of Study by PARCC ……………………………… 307
Appendix R: Grade 6 Mathematics Planning and Pacing Calendar – Sample …………… 310
Appendix S: Grade 8 Mathematics Planning and Pacing Calendar – Sample …………… 312
Appendix T: Suggested Individual Learning Plan (ILP) …………………………………. 313
Appendix U: Suggested Checklists for Evaluation of MS Math Teachers’ Efforts ……… 314
Appendix V: Grade 7 Student Data Collection Spreadsheet ……………………………... 320
Appendix W: Grade 8 Unit Plan Sample …………………………………………………. 325
Appendix X: Supervisor “Look For” Observation Protocol Example ……………………. 326
Appendix Y: Second Quarter Pulse Check in the RTI Effort for MS Mathematics Survey. 327
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LIST OF TABLES

Table 1. Organizational and Stakeholder Goals 25
Table 2. CCSS Prioritized 6 – 8 Topics 58
Table 3. PARCC’s Prioritized Clusters of 6-8 Standards 59
Table 4. Assumed Knowledge Needs of Successful Mathematics Teachers 60
Table 5. Assumed Motivational Needs of Successful Mathematics Teachers 75
Table 6. Assumed Organizational Needs for Mathematics Teaching Effectiveness 76
Table 7. Summary of Influences on Middle School Mathematics Teachers 79
Table 8. Assessments of Assumed Influences on Middle School Mathematics Teachers 89
Table 9. Summary of Assumed Influences and Validation 96
Table 10. Effective Strategies from Literature Review 117
Table 11. Validated Assumed Knowledge Influences 136
Table 12. Validated Assumed Motivational Influences 168
Table 13. Validated Assumed Organizational Influences 173
Table 14. Summary of Validated Influences 187
Table 15. Summary of Influences Validated in Part 188
Table 16. Knowledge Gaps and Solutions 191
Table 17. Motivation Gaps and Solutions 199
Table 18. Organization Barriers and Solutions 204
Table 19. Roles and Responsibilities of Intervention Team 208
Table 20. Intervention Plan 212
Table 21. Expected Outcomes, Metrics and Methods 218
Table 22. Critical Behaviors and Metrics and Methods for Assessing Progress 221
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Table 23. Required Drivers, which support Critical Behaviors 226
Table 24. Components of Learning for the RTI Implementation Program 231
Table 25. Components to Measure Reactions to the RTI Implementation Program 233
Table 26. Wordle Showing Signs of Success and Challenges 235
Table 27. A Sample RTI Effort Dashboard based on Survey Results 236

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LIST OF FIGURES
Figure 1. Anderson and Krathwohl’s (2001) taxonomy for learning, teaching and assessing 31
Figure 2. Ebbinghaus forgetting curve 38
Figure 3. Ebbinghaus remembering curve 38
Figure 4. Cognitive theory of multimedia learning 40
Figure 5. Nodes of information 42
Figure 6. An example from the Grade 6 daily skills review 103
Figure 7. Sixth grade MAP test results 104
Figure 8. Seventh grade unit summative assessment and reassessment results 107
Figure 9. Seventh grade math PLC reflection 110
Figure 10. Eighth grade quantitative data for all 8
th
grade students 112
Figure 11. Algebra 1 lab recommendations 113
Figure 12. Eight effective teaching practices 119
Figure 13. A sample portion of 17.18 seventh grade students of concern spreadsheet 139
Figure 14. A sample portion of the MS Mathematics Articulation of Standards 143
Figure 15. A sample portion of the Power Standards Math #SBG 145
Figure 16. A sample portion of the Grade 6 Unit Summative Assessment Itemized Math Practice
Standard Alignment 147
Figure 17. Year overview from Grade 6 Unit Summative Assessment Itemized Math Practice
Standard Alignment 148
Figure 18. A sample portion of 7
th
grade concept checklist, alignment with math practices 149
Figure 19. A sample portion of 7
th
grade concept checklist, summary of student objectives 150
Figure 20. A sample of PARCC’s recommended clusters of study for Grade 8 151
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Figure 21. A sample unit plan from Grade 8 Math PLC 155
Figure 22. TOFU test correction analysis document 160
Figure 23. An example of a Grade 6 TOFU test correction document with error analysis 162
Figure 24. The Grade 7 Math PLC template for test corrections and reflections 163
Figure 25. A sample portion of Grade 7 Math PLC planning and pacing calendar 164
Figure 26. M7 student daily update 165
Figure 27. M7 math instructions for completing test correction document 166
Figure 28. Suggest organization chart for RTI intervention team 207
Figure 29. Learning Cycle, a framework for instruction 230

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ABSTRACT

At the start of every new school year, there are always middle school students who arrive
with learning gaps in mathematics. Middle school mathematics teachers want to support these
students in their collective goal of achieving grade-level standards in mathematics by the end of
the school year. However, sometimes teachers do not possess the confidence or ability to
support these struggling students. The intention of this study was to understand what teaching
strategies were effective in teachers’ ability to support students learning mathematics. Through a
gap analysis, this study also looked at the knowledge, motivational and organizational influences
that affected at teachers ability to guide students in closing their learning gaps and understanding
mathematics at a high level. This qualitative case study explored the middle school mathematics
department at a high-performing international school in South East Asia to find what strategies
they effectively used that were proven in research. Data was collected through document
analysis, survey and semi-structured interviews. Data was triangulated by data collection
methods and by interviews with teachers from each grade level in the middle school. Data was
then coded and checked against a conceptual framework generated from an in-depth literature
review to validate the influences on middle school mathematics teachers’ ability to guide 100
percent of students to achieve grade level standards by the end of the school year. The findings
revealed the middle school mathematics teachers knew the factors causing gaps in learning,
knew how to use effective strategies to deliver instruction, believed that the Response to
Intervention contributed to students filling learning gaps and had the resources to achieve the
goal of 100 percent of students achieving grade level proficiency. The findings also revealed
there was more work that could be done to help close the gap that currently existed in students
who were unable to achieve grade-level standards and those who were able to achieve at high
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levels by the end of the school year. This study documents the strategies that work to support
teachers’ ability in supporting all students goals to reach grade level standards, suggests several
improvements to an already high-performing middle school mathematics department and offers
some new possibilities to consider for any middle school mathematics program interested in
creating a sustainable intervention program.

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CHAPTER 1: INTRODUCTION
At the beginning of every school year, classroom teachers are challenged to identify the
specific learning needs of each student and to differentiate instruction so each child has the
opportunity to achieve at high levels. Most schools, however, continue to struggle to meet the
needs of all their students, especially the ones at risk (Buffum, Mattos & Weber, 2012). Schools
have embraced interventions for reading and behavior but lag in implementation of interventions
for mathematics. (Buffum, Mattos, & Weber, 2009, 2010, 2012). Written and spoken language
abilities are valued over mathematics. At the start of the 21
st
century, two national organizations
evaluated their subject areas for the purpose of improving curriculum and instruction, but they
had different results. The National Institute of Child Health and Human Development’s Report of
the National Reading Panel (2000) impacted reading instruction to a greater extent than the
National Mathematics Advisory Panel (NMAP, 2008), which had little effect on mathematics
instruction (Weber, Crane & Hierck, 2015). The identification of 80 percent of learning
disabilities in literacy in the Report of the National Reading Panel (2000) influenced early
adoption of the Response to Intervention (RTI) for reading. On the other hand, Weber, Crane
and Hierck (2015) stated many mathematics teachers across North America were unaware of any
significant recommendations made by the NMAP (2008). Many schools have invested more for
professional development in areas of reading and behavior than mathematics, and assessment
and intervention strategies are not as accessible to math teachers (Weber, Crane & Hierck, 2015).
According to the National Council of Teachers of Mathematics (NCTM), the classroom
teacher is deemed the most important ingredient to student success (NCTM, 2014: Kanold-
McIntyre, Larson, & Briars, 2015). To meet the needs of all students, math teachers must have
access to high quality professional development, to research based curriculum and instructional
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tools, and to administrators who will advocate for these resources when they are not available
(Windram, Bollman, & Johnson, 2012). In addition, math teachers need to work collaboratively
towards a system that provides a more systemic, directive and timely response when a child does
not learn adequately (Buffum, Mattos & Weber, 2009). Because some students need more time
and support to ensure their learning in mathematics, it is imperative that math teachers have
access to and invest in the knowledge and skills, motivation and organizational support structures
necessary to ensure high levels of learning for all of their students.
In every middle school mathematics classroom, teachers greet students at the start of the
year who possess learning gaps for a variety of different reasons and these mathematics teachers
need to be able to identify the learning gap, to understand a pathway to a solution and to create
and document interventions to the successful attainment of proficiency for all students by the end
of the school year. To successfully address all academic, behavioral and social needs that affect
learning in mathematics, teachers need the support of an intervention team, including the
administrative leadership team, learning support specialists, a psychologist, a speech and
language pathologist, nurse, librarian and counselors. Teachers need resources and professional
development as well as structures that allow for collaboration between intervention team
members. Every mathematics teacher needs to feel they have the support they need to give them
the confidence to believe that they have the ability to support all of their students to achieve
grade level standards by the end of the school year.
Background of the Problem

To have confidence in their ability to support all students, middle school mathematics
teachers need to know why learning gaps exist and they need to know how to address the variety
of gaps exhibited by their students. In a historical review of literature, there are common strands
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20
of thought on why some middle school math students exhibit learning gaps and why some
students learn at a slower pace. Mathematics teachers need to identify common problems and
create common pathways to ensure all students have high quality instruction and the opportunity
to learn. Mathematics teachers also need to identify those students who are most at risk. Some of
the causes for learning gaps and slower processing time are related to a student’s information
processing system where limitations exist in accessing sensory data, retaining and processing
organization, prior education background and math foundation, development of neural
connections, and organization of long-term memory (Ambrose et al. 2010; Mayer, 2011; Schraw
& McCrudden, 2013). Based on awareness of processing deficiencies, teachers can create
explicit instructional materials to improve students’ organization and executive functioning
skills. Learning gaps also result from students’ passage through adolescence, one of the most
challenging periods in life due to occurrence of concurrent developmental processes. Middle
school teachers help students navigate through these academic, social and biological changes
(Romero et al. 2014). The middle school students that arrive in math classrooms today learn
differently from each other and differently from the teachers who attended middle school twenty
to thirty years ago (Sousa & Tomlinson, 2011). Middle school teachers need to be responsive to
the students of the 21
st
century.
Research shows how some teachers have been more successful than others in addressing
the needs of all their learners (Slavin, 1990). In meta-analyses of almost 100 studies, Slavin
(1990) found that collaborative teaching and teacher effectiveness are cornerstones of the most
effective mathematics programs for middle schools. Other high impact strategies include
teachers’ explicit instruction to develop student motivation and attitude toward mathematics,
self-efficacy of students, and metacognitive strategies for students (Ambrose et al. 2010; Mayer,
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21
2011; Friedel et al. 2010; Romero et al. 2014). Student learning outcomes also improve when
teachers create a proactive classroom environment, communicate with parents, teach math
content through the lens of math practices (Slavin, 1990; Hill & Tyson, 2009; Wilder, 2014;
Standards for Mathematical Practice, 2017), and use heterogeneous grouping with differentiated
instruction for individual or small groups (Slavin, 1990). Heterogeneous within-class grouping
allows for time for the teacher to work with students who are struggling to understand the
material. With small groups, teachers have time to modify instruction for those students who do
not understand the core instruction as delivered. Higher ability students have the opportunity to
practice communicating their understanding and others benefit from peer models. Math practice
is encouraged at home each night to solidify individual understanding of knowledge gained in
the collaborative classroom (Hill & Tyson, 2009; Wilder, 2014) and students are found to be
most successful when they have set academic and emotional goals that align with something in
the future (Ambrose et al. 2010; Mayer, 2011; Friedel et al. 2010; Romero et al. 2014). The
practice of “personalizing learning” is also a new trend that is emerging (Gates, 2014;
Personalized Learning “Look Fors”, 2017) where students co-create a learning plan that
describes what they know, what they need to know and how they are going to learn. Middle
school mathematics teachers need to know how to guide students in personalizing their learning.
Twenty first century students require different skills and new ways of learning and it is the
responsibility of the teacher to adjust pedagogy to improve outcomes for all students (Weber et
al., 2015).
Importance of Addressing the Problem

The problem of filling math learning gaps for middle school students while ensuring they
achieve proficiency for the grade level standards is important to solve for a variety of reasons. At
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22
most schools, the only systemic intervention process is to provide previously identified students
extra time and support to resolve their learning deficiencies (Buffum, Mattos, & Weber, 2012).
Other students, however, often fall far behind before they qualify for help. Once a child reaches
this level of discrepancy, it is nearly impossible for a child to catch up. The ‘wait to fail’ model
causes the achievement gap to widen and self-esteem suffers (Fuchs & Young, 2006; Buffum et
al., 2012). Consequently, early identification of deficiencies and follow-on interventions are
essential to filling the math-learning gap (Buffum & Mattos, 2011).
Middle school students are beginning to navigate increased responsibilities in their
learning behaviors as well as their personal and social responsibilities. Prevention and early
intervention for academic needs are keys to fostering resiliency and mitigating the effects of
barriers to adolescent learners (Windram, et.al, 2011). The consequences of not addressing the
needs of students will result with unmotivated students, not prepared for the future. The lifelong
effects of poor achievement are too great to not continue intervening and problem solving at the
middle school level (Windram & Bollmann, 2011). The work to create sustainable interventions
is urgent because it offers a positive, proactive alternative to the deficit model, which often leads
to poor self-esteem and continued failure. Today’s youth need a support network encouraging
resilience, confidence and attainment of goals (Buffum & Mattos, 2011).
Organizational Context and Mission
“A world leader in education, cultivating exceptional thinkers, prepared for
the future.”
- Vision of International School South East Asia (2017)

The International School South East Asia is a large, urban international school located in
South East Asia, one degree north of the equator. The school educates nearly 4,000 students
from Prekindergarten to Grade 12 and is the largest single-campus international school in the
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23
world (ISSEA, 2017). In order to achieve the vision outlined above for every student, the
International School South East Asia (ISSEA) is committed to a three-prong approach, which
includes a focus on excellence, where every student learns at high levels, a focus on possibilities
where every student personalizes their learning, and a focus on extraordinary care, which
signifies that every student is known and cared for (ISSEA Strategic Focus, 2017). The ISSEA
Strategic Plan for 2020 involves five task forces to help achieve the vision, including pastoral
care, professional learning communities, standards-based grading, high impact instructional
practices and systems supporting learning for four thousand students (ISSEA Strategic Focus,
2017). The mission of the International School South East Asia is to provide each student an
exemplary American educational experience with an international perspective (ISSEA Strategic
Focus, 2017).
Organizational Performance Status
The organizational performance problem at the root of this study is the lack of a
sustainable intervention program for students that exhibit math deficiencies in the middle school.
A quarter of middle school students start each year with a math content knowledge gap as
compared to their peers, and of that 25 percent, approximately 11 percent start each year below
grade level in mathematics. In the 2016 - 2017 school year, there were 77 out of 317 students in
Grade 7 that achieved less than the 70
th
percentile on the Measures of Academic Progress
(MAP) at the start of the year. Only 12 of the 77 students were enrolled in the learning support
program, which provides extra time and instruction. The remaining 65 students stayed in the
mainstream classes, struggling to keep up with the other students. While math teachers
differentiate instruction for students in class, there are few systemic structures in place to support
students that experience greater challenge than their peers. The ability of the middle school math
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24
department to contribute to ISSEA’s strategic plan for every student to learn at high levels is
compromised, teachers are discouraged because they feel powerless to help all of their students
and student learning gaps expand while students’ self-confidence plummets.
Organizational Performance Goal
To achieve the mission of the International School South East Asia, there are three pillars
to guide success: excellence (every student learns at high levels), extraordinary care (every
student is known and advocated for), and possibilities (every student personalizes their learning).
Success is achieved when emphasis is placed on the development of content knowledge,
character development, communication, creativity, cultural competence, critical thinking, and
collaboration. Middle School math teachers in Professional Learning Communities (PLC) will
work collaboratively with the Learning Support (LS) department to create a sustainable program
to ensure that all students who exhibit learning gaps in mathematics will correct their deficit and
achieve grade level standards by the end of the school year. Through the math PLCs and LS
teams, interventions will be planned and executed to assist students with identified learning
deficiencies that are not already in the learning support program. Specifically, one hundred
percent of middle school students will achieve a score of at least 70 percent on the MAP
standardized test and will meet expectations in all standards for their grade level by the end of
the 2018-2019 school year. The organizational and stakeholder goals are highlighted in Table 1.

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25
Table 1

Organizational and Stakeholder Goals

Mission
The mission of the International School South East Asia is to provide each student with an exemplary
American education with an international perspective. The strategic focus of the International School South East
Asia is that every student is known and advocated for, learns at high levels and personalizes their own learning.

Organizational Goal
By the end of the school year 2018-2019, the middle school math Professional Learning Community (PLC)
team members and Learning Support (LS) teachers will co-create and implement a sustainable intervention program
to support 100 percent of students achieving proficiency in grade level power standards and 70 percent or greater on
the math portion of the Spring Measure of Academic Progress (MAP) standardized assessment.

Stakeholder Goal:
Middle School Math PLC team members
and
Learning Support Teachers

By the end of school year 2018-2019, the
middle school math PLC team members
and LS teachers will have completed the
following…

Create an inclusive, collaborative
environment that includes flexible grouping
for the purpose of intervention and
extension and.

Document what mastery looks like in
prioritized grade level standards.

Document and analyze how they identified
student learning deficiencies.

Co-create digital learner profiles for all
students identified with a learning
deficiency and communicate with parents
on student progress toward a goal.

Create and implement intervention plans,
including common pre-assessments.

Show evidence of students using
metacognition in their learning plan.

Show documented evidence of student
achievement of goals.
Stakeholder Goal:
Middle School Leadership
Team

The middle school leadership
team will have completed the
following…

Demonstrate knowledge of
math content and practices,
specifically the prioritized
standards for each grade level.

Observe interventions and
provide positive and
constructive feedback to
middle school PLC members
and LS teachers, allowing time
and space for constructive
conversation.

Assess intervention program in
the second semester of the
school year 2016-2017 PGE
(Professional Growth
Evaluation) interview
conducted with each teacher.

Provide adequate professional
development and instructional
tools necessary to support
implementation of an
intervention program.
Stakeholder Goal:
Middle School students

By the end of each unit, students
will ...

Independently demonstrate with
confidence their knowledge of
applicable grade level standards.

Demonstrate automatic
procedurally fluency of grade level
appropriate foundational math
concepts and skills.

Co-create a learning plan,
demonstrate knowledge of they
already know, what they need to
learn and how to get to a goal they
set.

Demonstrate use of metacognitive
thinking in their learning plan.

Demonstrate self-efficacy and
positive attitude toward
mathematics.

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26
Description of Stakeholder Groups and Performance Goals
At the International School South East Asia ‘s Middle School, there are three
stakeholders that contribute to the success of the middle school students’ math achievement. The
stakeholders include the administrative leadership team, middle school math teachers, and the
middle school students.
The administrative leadership team will demonstrate knowledge of math content and
practice, specifically the prioritized topics for each grade level. They will observe lessons and
provide positive and constructive feedback to middle school math teachers, allowing time and
space for constructive conversation. To allow this time and space, the administrative leadership
team will conduct Professional Growth Evaluation (PGE) interviews with teachers at the start
and end of the school year. The administrative team will also ensure adequate professional
development and instructional tools are available to support middle school math teachers and
learning support team.
To contribute to the success of all students learning at high levels, the middle school math
teachers will create an inclusive, collaborative classroom environment that supports comfortable
yet challenging learning. Teachers will engage in high impact instructional strategies, showing
evidence of great teaching, every day for every student (ISSEA Strategic Focus, 2017). They
will document the participation of students in class discussions, show evidence of nurturing the
growth mindset in students, ensure and make known their high expectations for all students, and
believe that they along with colleagues possess the ability to ensure all students learn at high
levels. They will participate in professional learning communities and clearly articulate a
sequence of content progressions, both horizontally and vertically, and determine collectively
what mastery will look like in prioritized topics. They will show evidence of collaboratively
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27
analyzing student work to identify targeted needs and of opportunities where students received
immediate feedback and were given time to correct their understanding. Middle school math
teachers will also show evidence of guiding students to mastery using metacognition.
In collaboration with the LS team, middle school math teachers will co-create digital
learner profiles for all students identified with a learning deficiency. Together they will collect
and analyze assessment data to identify needs and they will set clearly defined goals for student
achievement. Math and LS teachers will create and conduct interventions while communicating
with parents and tracking student progress. Student achievement and digital record of
intervention for all identified students will be evident by the end of the school year 2018-2019.
Middle school students will practice and demonstrate automatic procedural fluency in
foundational math concepts and skills, relative to their grade level. Students will make
conjectures and build a logical progression of statements to explore the truth of their conjectures.
Students will justify their own thinking and their classmates’ thinking by critiquing the reasoning
of others. Students will show evidence of understanding for previously identified gaps in math
learning. By the end of each unit of study, students will independently demonstrate, with
confidence, their knowledge and analysis of different possible solutions for a given situation.
Also by the end of each unit, students will show active participation in class discussions using
descriptive math vocabulary. Students will co-create a learning plan, demonstrating knowledge
of what they know about given concepts, standards and skills, what they need to learn and what
they need to know to get there. Students will believe they can learn math at high levels. Students
will score 70 percent or greater on the spring MAP and they will achieve grade level standards
for each unit of study.

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28
Stakeholder Group for the Study and Stakeholder Performance Gap
The organization’s mission is achieved when all middle school students achieve at high
levels in mathematics. Therefore, the stakeholders of focus for this study are all middle school
math teachers. While the joint efforts of all stakeholders contribute to the achievement of the
overall organizational goal of 100 percent of all middle school students filling learning gaps and
achieving proficiency in grade level mathematics, it is most important to understand why
students have deficiencies and to identify strategies that will work to create a systemic program
of support for each student with a demonstrated deficiency or learning gap.
Purpose of the Project and Questions
The purpose of this project is to conduct a gap analysis to examine the knowledge,
motivation and organizational influences that provide support for middle school teachers creating
a sustainable intervention program where 100% of students receive support and are able to
achieve proficiency in all grade level standards. The analysis will begin by generating a list of
possible or assumed influences that will be examined systematically to focus on actual or
validated support for mathematics teachers’ ability to meet the goal of 100 percent of middle
school students meeting the ISSEA math standards.
As such, the questions that guide this study are the following:
1. What strategies exist at the International School South East Asia middle school to help
students improve their performance so they all meet grade level standards for math
proficiency/achievement?
2. What are the knowledge and skills, motivation, and organizational causes that exist to
provide support to the middle school teachers’ ability to fill their students’ learning gaps
while at the same time guide them to achieve grade level standards by the end of the
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29
school year?
3. What programs might ISSEA implement to help middle school math students fill learning
gaps and achieve proficiency for math grade level standards?
Conceptual and Methodological Framework
Clark and Estes’ (2008) gap analysis will be used as the conceptual framework for this
study. Gap analysis offers a systematic, analytical method that helps clarify organizational goals
and identify the gap between the actual performance level and the preferred performance level
within an organization. The methodological framework is a qualitative case study with
descriptive statistics. Assumed knowledge, motivation and organizational influences that provide
support for teachers and lead to organizational goal achievement will be generated based on
personal knowledge and related literature. These influences will be assessed by using surveys,
document analysis, interviews, classroom observations, literature review and content analysis.
Research-based solutions will be recommended and evaluated in a comprehensive manner.
Definitions
Academic Socialization: Linking education to future success and scaffolding independence (Hill
& Tyson, 2009).
Cultural Model: Shared mental schema or normative understandings of how the world works or
ought to work, which incorporates behavioral (activity) as well as cognitive and affective
components to foster valued goals (Gallimore & Goldenberg, 2001).
Deficiency: A learning gap on the mathematics learning progression outlined by K-12 Common
Core State Standards (CCSS) and refined by the curricular goals established locally by the
middle school mathematics department (CCSSI, 2010).
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30
Domain: The building blocks of learning that become increasingly more complex and
interrelated as students progress through elementary, middle and high school. For example,
fractional awareness that is developed in K-5 connects to proficiency in proportional reasoning,
algebra, geometry, personal finance and measurement (Weber, Crane, & Hierck, 2015).
Intervention: A formal set of steps to prevent academic failure through universal screening, early
action by the teacher with specific, differentiated instruction and materials, frequent progress
monitoring and increasingly intensive research-based instruction (Buffum, Mattos & Weber,
2009).
Learning Gap: The difference between what a student has learned and what the student was
expected to learn at particular age or grade level.
Math Anxiety: Math anxiety is a form of tension and a student may exhibit worry, helplessness
and mental disorganization (a cognitive feature) and fear (an emotional feature) (Spielberger,
1972, p. 1).
Math Practices: There are eight habits of mind called the Standards for Mathematical Practice,
which were identified by the CCSS (NGA & CCSSO, 2010). To be successful as 21
st
century
learners and to graduate ready for college and a skilled career, students must:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
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31
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning. (pp. 6-8).
Motivation: A theoretical construct that explains behavior and gives reasons for people’s actions,
desires and needs. Motivation involves goals that drive physical or mental action requiring effort
and persistence to sustain an activity.
Parental Involvement: Parents’ interactions with school and with their children to promote
academic success (Hill et al., 2004).
Personalized learning: Personalized learning develops leaders, cultivates exceptional thinkers,
and prepares students for their futures. Personalized learning is student-centered, grounded in
each learner’s profile, and characterized by competency-based progressions, customized
pathways, and flexible learning environments. Students take ownership of their learning, while
also developing meaningful relationships with each other, teachers, and members of the local and
global communities (Gates, 2014; Personalized Learning “Look Fors”, 2017).
Power standard: When focusing content and curriculum and contextualizing standards for the
school, grade level teachers collaboratively select standards (power standards) that student must
know rather than standards that are nice to know (supporting standards) (Weber, Crane &
Hierck, 2015).
Response to Intervention: The practice of providing high-quality instruction and interventions
matched to students’ needs, monitoring progress frequently to make changes in instruction or
goals, and applying child response data to important educational decisions (Buffum, Mattos, &
Weber, 2009).
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32
Self-efficacy: A belief that students are capable of doing well on a particular learning task
(Multon, Brown & Lent, 1991). .
Social Cognitive Theory: The Social Cognitive Theory (SCT) refers to a psychological model of
behavior that emerged primarily from the work of Albert Bandura. SCT emphasizes that learning
occurs in a social context and that much of what is learned is gained through observation
(Denler, Wolters, & Benzon, 2006).
Standard: A proficiency target in a subject area. (Kanold-McIntyre et al., 2015).
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33
CHAPTER 2: REVIEW OF THE LITERATURE
The purpose of this study is to examine current pedagogical practices at the International
School South East Asia’s Middle School that are in place to support students who indicate a
deficiency in mathematics content knowledge. Learning gaps are revealed through a combination
of local diagnostic testing and the Measures of Academic Progress (MAP) mathematics test
results. According to Thum and Matta (2015), middle school students are likely to be college
ready if they perform between the 70
th
to 84
th
percentiles in mathematics. This study investigates
alternative approaches and makes recommendations that the International School South East
Asia may want to implement to improve the ability of middle school mathematics teachers to
affect the performance of middle school mathematics students who score below the 70
th

percentile on the Measures of Academic Progress (MAP) mathematics test.
This chapter reviews learning and motivation theory, focusing on knowledge and skills,
motivation, and organizational factors, to help middle school mathematics teachers understand
why a student may be experiencing learning gaps. With improved understanding of what may
cause the gap, a teacher can better pinpoint a more accurate solution path for the student to
follow to fill this area of need. This chapter also provides a general review of grade level
expectations in mathematics and the learning progression for students as they progress from
elementary through middle school mathematics for the purpose of helping middle school
mathematics teachers identify exactly where student specific learning gaps exist in the learning
progression. This chapter then reviews strategies that have shown to be successful by teachers in
improving mathematics learning at the middle school level. Finally, the chapter concludes with a
description of the Clark and Estes (2008) gap analysis model that is used to address the
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34
performance gap in mathematics learning that exists in the middle school at International School
South East Asia.
How Do Adolescents Learn and What Are They Expected to Know
Learning and Motivation Theory
Knowledge and skills. Knowledge, in the broad sense, includes facts, procedures,
concepts, strategies and beliefs and these concepts, procedures and applications are mutually
reinforcing (Mayer, 2011; Weber, Crane & Hierck, 2015). How do we know if a learner is
acquiring knowledge of mathematics principles and skills? First, learning, by definition, is a
change in what a learner knows and is caused by the learner’s experience in their environment
(Mayer, 2011). According to behaviorism, a change in knowledge cannot be directly detected,
but it can be inferred that learning occurs when a change is observed in a learner’s behavior
(Weber et al., 2015). However, learning involves not just the acquisition of new behaviors, but
also the acquisition of knowledge, cognitive skills, concepts, values, abstract rules, and other
cognitive constructs (Denler, Wolters, & Benzom, 2014).
The Common Core State Standards (CCSS) for Mathematical Practice advocate for
students to become problem solvers who can reason, justify and effectively use appropriate
mathematics vocabulary to demonstrate their understanding of mathematics concepts (CCSSI,
2010). However, when the teacher does most of the talking, provides direct instruction or
encourages students to use “tricks” to make mathematics easier or to memorize facts, students
learn without thought (Boaler, 2009). Middle school students who struggle to learn mathematics
often try to force rules rather than trying to make sense of new concepts and skills (Karp, Bush &
Dougherty, 2015). Middle school mathematics teachers need to be aware and conscious of the
reliance on rules and tricks so they don’t unwittingly send students down the wrong path.
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35
What should students know and be able to do? To determine the correct path for
students who are learning mathematics, the above question guides teacher’s preparation of
lessons and can be answered with the help of references such as Bloom’s Taxonomy (Bloom,
Englehart, Furst, Hill, & Krathwohl, 1956) and the more recent elaboration on the original
taxonomy (Anderson & Krathwohl, 2001). The original work by Bloom included three
categories, but the cognitive domain proved to be the most useful to understanding what a
student should know and to developing strategies to support classroom instruction. Bloom’s
taxonomy explained the cognitive domain to include knowledge, comprehension, application,
analysis, synthesis and evaluation, with knowledge being the most basic level and evaluation
being the most complex (Rueda, 2011). The evolution of Bloom’s taxonomy is revealed in
Anderson & Krathwohl’s (2001) identification of four types of knowledge (factual, conceptual,
procedural, and metacognition) intersecting with six different levels of cognitive processes
(remember, understand, apply, analyze, evaluate and create) in which knowledge might be
applied (see Figure 1).
For application in the mathematics classroom, teachers use the knowledge dimension of
the revised taxonomy to plan how to facilitate instruction of the standards, previously prioritized
from the K-12 Common Core State Standards (CCSS). The cognitive process domain, seen in
Figure 1, is used to identify the level and depth of understanding of each standard (Weber et al.,
2015) and would represent a measure of assessment through mastery level for what a student
knows and is able to do.
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36

Figure 1. Anderson and Krathwohl’s (2001) taxonomy for learning, teaching and assessing. A
revision of Bloom’s taxonomy.
The pinnacle of the taxonomy framework is to create an innovative learning portfolio, which
documents learning over a period of time. To build knowledge, skills and self-efficacy in
students, middle school mathematics teachers can use the taxonomy structure as a guide that
culminates with instruction and facilitation of an innovative learning portfolio for students who
are personalizing their learning.
An active learner makes sense of the environment. Over the past century, there have
been common elements in the theories of how learning takes place in learners and how teachers
can affect that learning process. In the early 1900s, theorists believed that response
strengthening based on a system of rewards and punishment aided in the development of
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37
cognitive skills. The mid 1900s brought about the theory that teachers dispensed knowledge and
students were active recipients, a theory that perpetuated the direct instruction of factual
knowledge. The most predominant and current theory was developed in the late 1900s and
focused on knowledge construction with an active learner making sense of their environment.
This theory is most relevant to the learning progression of concepts and strategies, also known as
conceptual and procedural knowledge (Anderson & Krathwohl, 2001; Mayer, 2011). More
specifically, the theory where a learner makes sense of their environment is applicable to the
current middle school mathematics teacher who guides students to understand their place on their
personal learning progression and their next steps to acquiring more knowledge. One of the most
recent studies by the Rand Corp, funded by the Gates Foundation, found that 11,000 students at
62 schools using personalized-learning approaches made greater gains in math than their peers at
more traditional schools (Herold, 2017). When teachers highlight the level and depth of
knowledge for students as they maneuver through a learning progression of mathematics,
students begin to understand and are able to advocate for their own needs in learning, thus
personalizing their own learning process.
Learning factors and challenges that affect achievement. Middle school mathematics
teachers need to know how students acquire information to better help them gain new knowledge
that they can store and use in the future. To be able to construct knowledge depends on the
students’ ability to acquire information. In 1885, Herman Ebbinghaus was the first to study the
science of learning and memory. The results of his studies demonstrated the first quantitative
relationship between the amount of practice and the amount of learning (See Figure 2) and the
time of learning to the amount remembered (See Figure 3) (Ambrose, Bridges, DiPietro, Lovett,
LEARNING GAPS IN MIDDLE SCHOOL MATH

38
& Norman, 2010). Most notably, Ebbinghaus showed in a Forgetting Curve, that within one
hour of learning, 50 percent of new knowledge was forgotten.

Figure 2. Ebbinghaus’ Forgetting Curve. The amount of practice vs. the time of learning

(Ambrose et al., 2010).

Figure 3. Ebbinghaus’ Remembering Curve. The amount remembered vs. the time of

Learning (Ambrose et al., 2010).
LEARNING GAPS IN MIDDLE SCHOOL MATH

39

Through controlled experiments, Ebbinghaus identified factors that increased the amount of
learning and showed that as the number of repetitions increased, the amount of learning also
increased. Four repetitions in six days are necessary for remembering about 90 percent of new
knowledge after one week. Understanding the timing correlation with remembering and learning
helps to guide mathematics teachers’ presentation of learning targets and confirms the necessity
of mathematics practice to be completed at home.
Making connections with prior knowledge. Frederick Bartlett conducted an early study
into knowledge construction in the early 1930s. His study proposed that learning is impaired
when the learner lacks the appropriate prior knowledge that is necessary to assimilate the new
knowledge being presented (Bartlett, 1932). He theorized that learning is a constructive process
of assimilation to schema rather than adding presented information to memory (Mayer, 2011). In
short, the learner makes sense of or rationalizes new information by reconstructing prior
knowledge. Bartlett’s study highlighted that information is added to memory during learning and
retrieved during remembering. Consequently, in the context of the mathematics classroom, the
need for mathematics teachers to help build a strong foundation of mathematics skills is
imperative for their students to be able to acquire new knowledge by making connections with
prior knowledge.
Information presented in both audio and visual form. Middle school mathematics
teachers may wonder how to present new knowledge so that all students can access information
in a way that they will be able to remember it. By looking at the human processing system,
middle school mathematics teachers can develop lessons that are tailored to students’ ability to
remember. The human information processing system is based on three scientific principles: dual
channels, limited capacity and active processing and is structured in three types of memory:
LEARNING GAPS IN MIDDLE SCHOOL MATH

40
sensory, working and long-term (Mayer, 2011). The cognitive theory of multimedia learning
relates these variables to describe how the information processing system works (See Figure 4).

Figure 4. Cognitive theory of multimedia learning (Mayer, 2011).
In 1971, Allan Paivo explained that people have two separate channels to process information – a
verbal channel to process words and a visual channel to process pictures. According to Paivo,
people learn better when information is presented in both visual and verbal form (Paivo, 1971).
Spoken words are represented as sounds in auditory sensory memory and printed words and
pictures are represented as images in the visual sensory memory. When the learner attends to
this information, some of it can be transferred into working memory. In the middle school
mathematics classroom, teachers must design lessons that incorporate both visual and auditory
means to ensure all students have every opportunity to access and store new knowledge.
Limited capacity in working memory. In 1956, George Miller suggested that people
couldn’t be tape recorders and take in and record vast amounts of information because they have
limited processing capacity in working memory (Mayer, 2011; Miller, 1956). Limited amounts
of material can be processed in working memory at any one time. When the mathematics teacher
presents new information through verbal and pictorial models, the learner mentally connects the
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41
models in working memory and then combines them with prior knowledge in long-term memory.
The mathematics teacher provides an avenue where new information can be received and stored
in long-term memory. A study by Wittrock, showed that people learn more deeply when they are
actively processing the material by focusing on new knowledge, mentally organizing new
information and integrating it with prior knowledge activated from long term memory (as cited
in Mayer, 2011). Consequently, for a mathematics classroom, the teacher needs to chunk skills
and new knowledge into manageable amounts of information and needs to help students organize
the chunks of information with their prior understanding of the topic.
Connecting nodes of information with organizational structures. Middle school
mathematics teachers welcome to their classroom students with different backgrounds and
experiences, different genetic makeup of information processing systems and different
foundations of skills. Because of the differences, the limitations of working memory and the
ability to organize and to build meaningful, structures for information storage in long-term
memory vary among individual learners. The way that learners organize their knowledge
influences how they learn and apply what they know (Ambrose et al., 2010). Mathematics
teachers need to learn about their students’ history to provide guidance on how to connect new
information to prior knowledge. In some cases, students with limited experiences and
mathematics foundations create only sparse connections between information and the result is
superficial knowledge structures (See Section A in Figure 5).
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42

Figure 5. Nodes of information (Ambrose et al., 2010).
In other cases, students are able to make meaningful relationships across varying pieces of
knowledge, forming a firm foundation for further, more advanced learning. The hierarchal,
organization structure seen in Section C of Figure 5 indicates a system of foundational skills that
builds upon previous knowledge and where the learner can effectively access information. The
denser the connections between nodes of information (Section D in Figure 5) indicate additional
associations in mathematics or cross referencing with other subject areas, which result in the
most efficient and effective long-term memory storage.
The goal of education is achieved through the positive transfer of knowledge (Mayer,
2011). The positive transfer is evident when the application of skills (or knowledge, strategies,
and practices) learned in one context is transferred to a new and novel context (Ambrose et al.,
2010). The ability of a learner to transfer basic knowledge for specific tasks to the more
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43
complex task of transferring and applying knowledge within a new context may vary greatly
among individual learners but all learners can develop more sophisticated knowledge
organizations over time and with the support and guidance of their mathematics teacher. In the
mathematics classroom, students apply their understanding of math to real world scenarios, but
students who struggle to solve word problems, may have a learning gap if not specifically taught
how to organize their thinking. Research-based evidence found that teachers who study the
science of learning are better able to understand the challenges and possibilities for different
learners and their ability to access and apply information retrieved from long-term memory
storage.
Challenges specific to adolescent developmental years. Middle school mathematics
teachers need to know how the challenges during adolescent years affect students’ ability to store
information and process new knowledge. As evidenced by previous literature and research,
learning challenges can result from: a lack of prior knowledge, a limited ability to process
information in short term working memory, and little experience with organizing information in
long-term memory (Ambrose et al., 2010; Mayer 2011). Learning challenges can also result
from the student’s experiences as they progress through the adolescent years. Learning is a
developmental process that intersects with other age-related processes in a learner’s life
(Ambrose et al., 2010). Adolescence is widely regarded as one of the most challenging stages of
life (Steinberg & Morris, 2001) and therefore, this time of development impacts the ability of a
teacher to support all students in achieving proficiency in mathematics.
Cultural differences in the range of adolescent years. The adolescent period is defined
to include physical, cognitive, and social maturation between childhood and adulthood (Learner
& Steinberg, 2004; Sisk & Foster, 2004). Different cultures express different age ranges for
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44
puberty, starting between nine and twelve years old and lasting until the early twenties
(Steinberg, 2005). Consequently, the span for adolescence can be a two to four year range in
some cultures and up to fourteen years in others. For schools with diverse populations, there may
be additional cultural issues as students from different backgrounds pass through the adolescent
period. These cultural differences can present themselves in the middle school mathematics
classroom as students work together in small groups to collaboratively solve problems. Teachers
need to be aware of potential cultural or diversity conflicts in order to be prepared for any
discussion that might present itself in the classroom.
A myriad of developmental processes during adolescent period. Although cultural
differences exist for adolescents, at the onset of puberty, all teenagers experience dramatic
changes in hormone levels and in physical appearance (including physical growth, changes in
facial structure, and the appearance of secondary sexual characteristics). At the same time,
adolescents experience numerous additional challenges in social, academic (Alspaugh, 1998;
Midgely, Anderman, & Hicks, 1995) and other emotional (Buchanan, Eccles, & Becker, 1992;
Kasdin, 1994) and environmental influences (Blakemore, Burnett & Dahl, 2010). Adolescents
experience a myriad of developmental processes during this period of time and their ability to
cope with these academic and emotional challenges impacts the amount of learning that takes
place during middle school years. A students’ passage through puberty may cause a learning gap
in mathematics to develop. Middle school mathematics teachers need to understand that
biological changes occur in their students so they can adjust lesson structure or the support
needed for students who struggle to gain new knowledge in a regular classroom lesson.

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45
Positive impact of supportive social structures. Despite the academic and emotional
challenges, mathematics teachers may find that some middle school students become more
engaged and motivated during the period of adolescence. Students may have a supportive social
and developmental context that helps to minimize the specific effects of aggressive behavior
(Blakemore et al., 2010). However, other students may not come from a positive, supportive
environment and rely on the classroom for their safe place during the day. Middle school
mathematics teachers should consider the classroom environment so that it is welcoming to all
students.
Teachers should be aware of the other factors that impact how students fare during
adolescence, one of which includes socio-economic status (SES). Those from higher SES homes
tend to have parents that value education (Eccles, Vida, & Barber, 2004), higher quality teachers
in their school districts, and neighborhood norms (Harding, 2003). On the other hand, lower SES
homes predict lower adolescent well-being and increased risk of adolescent depression
(Goodman, Slap, & Huang, 2003). Providing interventions during the adolescent time period is
imperative for all students who need support, as the effectiveness of such interventions may be
much greater than if performed later in adulthood.
Brain development continues despite stage of puberty. Middle school teachers need to
be aware of the stages of brain development during the adolescent years as it effects students’
choice and learning. Despite common thought that puberty and brain development progress at the
same rate, most elements of cognitive development show a trajectory that follows the age and
experience of an adolescent rather than the timing of puberty (Dahl, 2004). As time passes,
middle school students’ brains continue to develop and grow with continued experiences, even if
these students are in different stages of puberty. Often as maturation changes, an adolescent is
LEARNING GAPS IN MIDDLE SCHOOL MATH

46
described as being on the upstream or downstream of puberty. There are six primary changes
that are exhibited during puberty that can affect the ability to learn in school: “romantic
motivation, sexual interest, emotional intensity, changes in sleep/arousal regulation, appetite, risk
for affective disorders in females, increase in risk taking, novelty seeking, reward-seeking”. (p.
17) The period of adolescence is developmentally one of great strength and resilience, but also
one where behavioral, familial, and social influences interact in ways that may or may not
always positively impact their learning in school. Trajectories of positive or negative choices are
set in adolescence that can have a major effect later in life. Because mathematics is one of the
more demanding subjects for student at this age, there will be students who make poor choices.
The advice of a teacher during the middle school years can help guide adolescent students onto
the correct path.
Goal setting positively impacts challenges. Academic and emotional challenges are the
most important challenges facing students during the middle school years and teachers can help
by advising students on goal setting and by providing support when needed. Middle school math
teachers become one of many teachers for adolescent students as they begin to change classes for
each subject. In general, middle school teachers have higher expectations and higher grading
standards, more difficult work, and more pressure for students to perform well than previous
experience in elementary school (Romero, Master, Paunesku, Dweck, & Gross, 2014). Because
mathematics is more rigorous than other subjects, it presents the opportunity for mathematics
teachers to help students develop goals that lead to good study and work habits which are
beneficial in all areas.
Students’ peer relationships are changing and their biological changes during puberty
powerfully alter their emotional experiences. Adolescent behavior is more emotionally
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47
influenced than any other age group and adolescents generally have difficulty controlling their
behavior and emotions (Dahl, 2004). Adolescents haven’t yet developed skills to harness these
strong feelings and they still need to develop self-regulation of emotion and complex behavior
aligned to long-term goals. Goal setting and support by teachers in mathematics classes can help
to overcome some of the academic and emotional challenges experienced by middle school
students.
Motivation as a prerequisite for learning. Hattie (2009) evaluated over 850 studies
and determined four primary factors that influence learning: a willingness to engage in learning,
a degree of reputation enhancement that a student gets from being engaged in learning, an
understanding that success is based on effort rather than ability, and a positive attitude towards
learning. If a mathematics teacher can engage a student in the learning, then the student will
experiment, attend to the lesson, participate in the discussion, ask questions, answer questions,
and take notes (Hattie, 2009). These actions indicate a positive attitude is present and the student
is demonstrating the motivation to learn. Likewise, when a teacher supports a student in their
environment and they make long-term goals, the student will be able to retain more information
in long-term memory, will learn more, and will be more motivated. Motivation is highest when
students are competent and independently set worthwhile goals (Dornvei, 2001). In the totality
of the research on motivation and learning, motivation is defined as “the process whereby goal-
directed activity is instigated and sustained” (Schunk, Pintrich & Meece, 2009, p. 4). However,
during the adolescent period, students have competing goals that vie for their attention, time, and
energy and these will increase or decrease motivation to pursue a learning goal (Ambrose et al,
2010). Understanding the other competing goals that a student has made, a teacher can better
support and guide a student in creating their individual mathematics goals.
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48
Motivation’s impact on long-term growth in learning. The goal of education for the 21
st

century is to facilitate sustainable learning and create life-long learners, rather than focus on the
learning that is occurring at one point in time (Murayam, Pekrun, Lichtenfeld, & Vom Hofe
(2011). To ensure mathematics achievement during the adolescent developmental years, long-
term growth relative to each individual, rather than current achievement is more desirable.
NCTM’s Principles to Action: Ensuring Mathematical Success for All (2014) states that
instruction should be focused on sense making and reasoning. The Common Core’s Standards
for Mathematical Practice (SMP) further comments that mathematics teachers should encourage
precision, including the appropriate use of mathematics vocabulary and notation and the
reasoned application of “rules” (CCSSI, 2010) rather than rote memorization or using “tricks” to
make math easier. To ensure students are ready for the 21
st
century, mathematics teachers need
to instill in students the emphasis on process and reasoning over correct answers.
During early adolescence, students begin to develop higher order thinking and they learn
the more difficult difference between intrinsic (self-motivated) and extrinsic (motivated by
rewards) motivation (Wigfield & Eccles, 1992). As the student progresses through middle
school, growth in learning can be positively predicted by perceived control, intrinsic motivation
(positive emotion), and deep learning strategies, which often occur later in a student’s
developmental stage and likewise, later in a student’s academic career. Growth in learning is
negatively impacted by extrinsic motivation (a reward based strategy that is effective only at the
initial levels of learning) and surface learning strategies (important only for skill tasks like rote
memorization and initial stages of learning). Research has shown that children and adolescents’
emotions are linked to their academic achievement. Typically, positive emotions such as
enjoyment of learning show positive links with achievement (Goetz & Hall, 2013; Pekrun &
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49
Linnenbrink-Garcia, 2014; Zeidner, 1998). While studies show that intelligence is a predictor of
current mathematics achievement, they also reveal that intelligence is not a predictor of growth.
Motivation and deeper learning strategies have proven to be more important than intelligence
when facilitating adolescents’ development of mathematical competencies (Murakayma et al.,
2013). Subsequently, the critical factor of growth in achievement is not how smart you are, but
how motivated you are and how you study.
While positive emotions and motivation lead some students toward a goal, negative
emotions may also be present in other students who are less motivated. Negative emotions can
be seen as anxiety and the prominent factors are test anxiety and mathematics anxiety.
Mathematics anxiety is a form of tension and a student may exhibit worry, helplessness and
mental disorganization (a cognitive feature) and fear (an emotional feature). When a student
experiences test anxiety, he or she is unable to focus and therefore has limited ability to store or
access information in their working memory. Anxiety is the outcome of a “chain reaction
consisting of a stressor, perception of threat, state reaction, cognitive reappraisal and coping”
(Spielberger, 1972, p. 1). The effects of anxiety and a poor attitude towards mathematics are
substantial, but they are also amenable to teacher influence (Ma & Kishor, 1997). To prevent
future learning gaps, teachers should invest time to ensure students minimize test anxiety before
tests by teaching them techniques on how to control and manage their anxiety.
Controllable motivation factors. There are some motivation factors that can be
controlled; therefore, a teacher has the ability to influence a student’s positive or negative
attitude towards mathematics. Current research shows the teacher’s greatest impact is on student
learning of metacognition skills, self-efficacy in students, and classroom environment.
Metacognition is an awareness of one’s own cognitive processing and knowing how to monitor
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50
and control one’s own learning. A goal of teachers is to help students to become self-regulated
learners (Mayer, 2011). In addition, a teacher can help guide a student’s development of self-
efficacy, a belief that they are capable of doing well on a particular learning task. The
relationship between self-efficacy and achievement is among the strongest of self-measures
(Multon, Brown & Lent, 1991). In a middle school mathematics classroom, a teacher can guide
students to understand how to reach mastery in prioritized topics and what goal to set next in
their learning progression.
Influence of self-efficacy in students. The influence of internal beliefs as associated with
achievement is much greater for adolescents than children or adults (Findley & Cooper, 1983;
Kalechstein & Nowicki, 1997). Intelligence theories, individuals’ beliefs about whether
intelligence is fixed or malleable, have been shown to predict students’ academic achievement
and engagement (Aronson, Fried, & Good, 2002), particularly in challenging subjects like
mathematics (Blackwell, Trzesniewski, & Dweck, 2007). Consequently, a teacher’s guidance to
help ensure a high level of confidence can help an adolescent to get through a myriad of
roadblocks that may appear in the mathematics classroom and in middle school (Hattie, 2009).
Likewise, students who are more self-aware and confident about their learning capacities try
harder and persist in the face of challenges (Aronson et al., 2002). Students who set high
academic goals, have self-discipline, motivate themselves, manage their stress, and organize
their approach to work, learn more and get better grades (Duckworth & Seligman, 2005; Elliot &
Dweck, 2005). Also, students who use problem-solving skills to overcome obstacles and make
responsible decisions about studying and completing homework do better academically (Zins &
Elias, 2006).
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51
Importance of a proactive classroom environment. Students work harder when they
view their instructor as a social partner (Mayer, 2011). Researchers have highlighted how
interpersonal, instructional, and environmental supports produce better school performance
through the following means: (a) peer and adult norms that convey high expectations and support
for academic success, (b) caring teacher–student relationships that foster commitment and
bonding to school, (c) engaging teaching approaches such as proactive classroom management
and cooperative learning, and (d) safe and orderly environments that encourage and reinforce
positive classroom behavior (Blum & Libbey, 2004; Hamre & Pianta, 2006; Hawkins, Smith, &
Catalano, 2004; Jennings & Greenberg, 2009).
Uncontrollable motivation factors. There are some motivation factors that cannot be
controlled and two of these are present at International School South East Asia, gender and
ethnicity differences. Gender is something that occurs at birth and a student has no control over
the result. Research shows that this matters very little to achievement in mathematics. Hattie’s
(2009) meta-analyses of over 850 different studies showed low to nonstandard effect of gender.
Cohn (1991) found that adolescent girls achieved developmental milestones earlier than boys and
Else- Quest, Hyde, Goldsmith and Hulle (2006) indicated girls have slightly higher scores on
attention and persistence and very large differences in effort control and inhibitory control. Girls
“display a stronger ability to manage and regulate their attention and inhibit their impulses”. (p.
61)
Like gender, a student does not control ethnicity, although they have to deal with some of
the ramifications. For example, there is a stereotype that Asians are better at mathematics and
this positive stereotype sometimes rears negative results for the Asian student and community.
In addition, stereotypes are generally more pronounced in middle school rather than in high
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52
school (Picho, 2013), leaving the middle school adolescent to deal with yet another social
pressure. Despite the existence of stereotypes, Cooper and Door (1995) show that the effects of
classroom engagement and mathematics achievement are similar across ethnic groups.
Furthermore, Hattie (2009) found no difference between African American and white students in
their need for achievement, personal expectations, feelings of hopelessness, denial of importance
of individual effort, or lack of persistence. In fact, there are many gaps in achievement for
students of all ethnicities. As of Hattie’s 2009 study, there was a great deal of research on
stereotype threat, but there were no meta-analyses exploring ethnicity and achievement. In
general, the research showed that it was important for students to maintain a positive image of
their cultural background (Hattie, 2009). In addition, “accepting that students come to school
with different cultural heritage and that they can be allowed and encouraged to have a positive
image of their own racial or cultural heritage is an acknowledgement of the importance of culture
and can show students they are accepted and welcomed into the learning environment” (as cited
in Hattie, 2009, p. 58). Any evidence of ethnic stereotypes related to mathematics, should be
addressed immediately in the classroom so that students understand the research-based evidence
proving that ethnicity is not the reason for achievement and that there are negative effects of
stereotypes, even the positive ones.
In conclusion, motivation and learning strategies, more so than current intelligence, are
important for students’ academic growth over the school years and teachers should facilitate the
long-term learning process to include development of a collaborative classroom environment,
promotion of self-efficacy beliefs, and metacognitive skills.
“Achievement is more likely to increase when students invoke learning rather than
performance strategies, accept rather than discount feedback, benchmark to difficult
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53
rather than easy goals, compare themselves to subject criteria rather than other students,
possess high rather than low efficacy for learning and effect self regulation and personal
control rather than learned helplessness in the academic situation” (Hattie, 2009, p. 47).
The key ingredients to long-term growth in learning are based in motivation and they include a
willingness to invest in learning, gaining a reputation as a learner (Carrol, Hattie, Durkin &
Houghton, 2001; Goff & Ackerman, 1992; Goff & Ackerman, 1994), building a sense of self,
and showing openness to experiences (Robers, Walton, & Viechtabauer, 2006).
Learning Progression for Middle School Mathematics
Purpose of Education
Students need different skills for the 21
st
century, and mathematical practices are as
important as mathematics content. Teachers need to effectively implement mathematical
practices without lowering the cognitive demand of the task (Kanold-McIntyre, Larson & Briars,
2015). Students need to know how to solve problems (Pollack, 1987 Bell Lab Study). “Decision
making in a complex world requires consideration of a vast amount of data and the ability to
construct logical arguments with supporting data” (Brahier, 2016, p. 90). For college graduates
in today’s world, jobs only stay the same for about five years, so retraining is inevitable and
students need to be lifelong learners (as cited in Brahier, 2016). So while learning academic
content is important to achievement in education, there are a myriad of other factors, such as
problem solving, collaboration, the ability to reason, the ability to support logical arguments with
data, and a growth mindset, that must be included to define success for a student’s overall
education.
Common goals and objectives for mathematics. In addition to the shared, overarching
philosophies of education that contribute to how a teacher conducts instruction in the classroom,
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54
there are also general goals and objectives for education that are similar across the United States.
For example, for the subject of mathematics, most agree that all students should know the
fundamental skills, such as being able to add and subtract, multiply and divide, and operate on
fractions, and for college–bound, high school students, most agree that algebra, geometry,
statistics and probability, and discrete mathematics and calculus should be courses included in a
high school curriculum (College Board, 2012).
History of standards. While some countries, such as Singapore and Japan, mandate
national mathematics curriculum, the United States does not require states to follow one set of
standards. However, in 2012, a set of standards, called the Common Core State Standards
(CCSS) was accepted and is being used by 42 of 50 states as a model of standards to be taught
from Kindergarten through 12
th
grade (Brahier, 2016). The roots of the CCSS date back to the
mid-1980s when the National Council of Teachers of Mathematics (NCTM) started writing a set
of standards that were eventually endorsed by educators, business leaders, other professionals
and organizations throughout the United States (NCTM, 2017). By 1989, the NCTM published
an initial book called Curriculum and Evaluation Standards for School Mathematics. This
publication was revised and by 2000, NCTM published Principles and Standards for School
Mathematics, which presented a common foundation of mathematics to be learned by all
students and included content standards in number sense, algebra, geometry, discrete
mathematics, statistics and probability, and process standards in problem solving, reasoning and
proof, communication, connections, and representations. Principles and Standards for School
Mathematics served as a guide for rigorous, college and career readiness standards for the 21
st

century (NCTM, 2017).
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55
In the early 2000s, however, the Trends in International Mathematics and Science Study
(TIMSS), described the US mathematics curriculum as being “a mile wide and an inch deep”
(Schmidt, Houang, & Cogan, 2002). Analysis of the report was conducted by education
professor and author William Schmidt and revealed a poorer relative performance of eighth-
grade students from fourth-grade students (Weber et al., 2015). Schmidt concluded that a
significant number of mathematics topics, an average of 78 topics, were covered in US
classrooms compared to countries that were outperforming the United States, such as Germany
and Japan, which covered an average of 23 and 17 topics per year, respectively (Weber et al.,
2015). This comparative study spurred ongoing revisions and reform measures which ultimately
led to the NCTM’s 2006 release of Curriculum Focal Points for PreK to Grade 8 Mathematics:
A Quest for Coherence (NCTM, 2016). The focal points were the initial steps towards more
coherent, focused mathematics curriculum in the United States, but the curriculum still had a
long way to go.
Learning Progression for Mathematics
Prioritization of standards. Carefully defining key core content is essential to ensure
that all students have the opportunity to learn at high levels and receive the support they need.
Even in light of the National Governors Association (NGA) and the Council of Chief State
School Officers (CCSSO) press release in June 2010 which signaled the adoption and
implementation of the standards in each state, too many standards still exist (DuFour & Marzano,
2011; Schmoker, 2011). Although the NCTM provided focal points, further prioritization at the
school level is necessary and will contextualize the learning to the local environment, thus
leading to deeper understanding and greater sense of ownership (Weber et al., 2015). In
addition, more precise unpacking of the standards leads to better understanding of content and
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56
consequently, better development of assessments that demonstrate mastery of student learning.
Educators need to know what mastery looks like so that instruction can match these expectations
and teams of educators can plan backwards (McTighe & Wiggins, 2012). By prioritizing
standards, more focus can be placed on creating better assessments. Assessments not only
demonstrate mastery, but also help to identify students in need of extra support, allow for
diagnostic analyses of specific areas of need, and provide more targeted interventions (Buffum,
Mattos, & Weber, 2009, 2010, 2012).
Vertical articulation with elementary mathematics. The purpose of middle school
mathematics is to ensure all students have experiences with each content area so they are
prepared for more advanced high school mathematics courses, which typically begin with
Algebra I. However, the foundation for algebra begins in early elementary school years where
students garner conceptual understanding of number systems, competence with basic number
properties, and an understanding and application of mathematics operations (adding, subtracting,
multiplying and dividing). Of all the mathematics content covered during the elementary years,
the most valid predictor of success in algebra and more advanced mathematics courses is the
ability to understand representations of the magnitude of whole numbers (Booth & Siegler, 2008)
and fractions (Siegler, Thompson, & Schneider, 2011). Early knowledge of fractions, whole
number division and the concept that numbers possess magnitudes is consistently related to
success in algebra and later overall mathematics proficiency (Ma et al., 1997; Siegler et al., 2011,
2012).
Place value and magnitude are central concepts that support students’ understanding of
number systems (Wu, 2001). Students begin to realize all numbers have magnitudes and can be
assigned specific locations on number lines (Siegler et al., 2011). Once students can locate and
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57
order numbers on a number line, they can see that multiple numbers may exist within a system,
such as fractions and decimals (Siegler et al, 2012). While students are learning to use a number
line, they are also learning to use whole number operations. Once they have achieved
proficiency with whole numbers operations, students can then extend their understanding to
performing operations with fractions and decimals. Eventually, developing a mental number line
will help students to solve operations with rational numbers.
This developmental sequence can be found in many curricular resources, including the
CCSSM (NGA & CCSSO, 2010). The importance of this sequence lies in the young student’s
first opportunity to see that properties of operations with whole numbers are not true for all
numbers (Siegler, 2012). For example, when multiplying by fractions, numbers do not increase
in magnitude as they do with whole numbers. Understanding how to represent fractions and
decimals and achieving procedural fluency in mathematics operations with them is critical to
laying the foundations of algebra. If a student does not gain adequate understanding of fractions
and whole number division in elementary school, when these topics are taught explicitly, they
will be faced with a deficiency to correct, while still working to achieve understanding the grade-
level topic being taught. Vertical articulation is required to ensure this developmental sequence
happens in a way that all students can achieve at high levels in middle school.
Middle school knowledge expectations. The purpose of middle school mathematics is
to provide experience in all content areas (number systems and equations, ratio and proportional
relationships, geometry, statistics and probability, linear relationships, and functions) so that
students can be successful in their high school mathematics courses (Brahier, 2016). As a student
progresses through middle school, there are certain expectations of the knowledge that should be
attained at each grade level. Because there is not a mandatory national curriculum in the United
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58
States, it is up to the state and the school to decide what should be taught at each grade level.
While it is necessary to contextualize and prioritize standards at each local school system,
several examples of prioritized topics are presented to help guide the decision of what could be
taught at each grade level, while maintaining vertical alignment between middle school grades
six to eight. The basis of the nationally recommended mathematics standards lies in the work of
the NGA and CCSSO, so their prioritized topics are presented first in Table 2 (Weber et al.,
2015).
Table 2

CCSS Prioritized 6-8 Topics

Grade 6 Grade 7 Grade 8
1. Interpret ratios and solve
rate problems.
2. Multiply and divide with
fractions.
3. Compute with rational
numbers (except
integers).
4. Make sense of integers
on a number line and in
the coordinate plane.
5. Evaluate and interpret
expressions.
6. Solve and interpret
simple equations and
inequalities.
7. Find and interpret
measure of center or
variation.
8. Compute surface areas
using nets.
9. Compute volumes of
rectangular prisms.
1. Solve proportion and
percent problems.
2. Find and interpret unit
rates, including those
that describe linear
relationships in the first
quadrant of the
coordinate plane.
3. Represent quantities in
various forms –
decimals, fractions, and
percents.
4. Compute with all
rational numbers.
5. Solve and interpret
multistep equations and
inequalities.
6. Compute surface areas
and volumes of prisms.
7. Generate and draw
inferences from data
sets.
1. Recognize, generate,
interpret, and analyze
linear equations and
their graphs.
2. Solve and interpret
equations and systems
of linear equations.
3. Understand and
describe functions.
4. Interpret and generate
transformations.
5. Solve problems
involving transversals
and parallel lines.
6. Interpret and apply the
Pythagorean Theorem.

The next example of prioritized clusters of standards is shown in Table 3 and comes from
the Partnership for Assessment of Readiness for College and Careers (PARCC), an organization
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59
that created assessments based on the CCSSM and have prepared frameworks that define the
relative weight of grade level standards (Weber et al., 2015).
Table 3
PARCC’s Prioritized Clusters of 6-8 Standards

Grade 6 Grade 7 Grade 8
1. Interpret ratios and
solve rate problems.
2. Multiply and divide
fractions.
3. Compute with rational
numbers (except
integers).
4. Evaluate expressions.
5. Interpret and solve
one-variable equations
and inequalities.
6. Interpret and write
equations that directly
relate two variables.
1. Interpret and solve
proportions.
2. Compute with rational
numbers.
3. Manipulate
expressions.
4. Evaluate expressions
and solve equations.
1. Evaluate radicals
and expressions
with integer
exponents.
2. Interpret, produce,
and solve linear
equations and
systems of linear
equations.
3. Interpret, compare,
and model with
functions.
4. Interpret and
produce congruent
and similar shapes.
5. Interpret and apply
the Pythagorean
Theorem

Prioritization of standards is necessary to balance content between middle school grade levels,
while at the same time accommodating students of varying ability levels. Prioritization is a
fundamental step to identify students with deficiencies, as well as a way to enrich higher ability
students more deeply. The desired outcome is achieved when all students learn at high levels.
So far, this literature review has looked at the factors that may cause gaps in student
learning, the grade level expectations for mathematics and the learning progression from
elementary to middle school mathematics to help pinpoint exactly where the learning gap exists
on the mathematics learning progression. This analysis of literature was to help middle school
mathematics teachers choose the most accurate solution path for a student to fill their learning
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60
gap. Next, the literature reveals some of the most effective teaching strategies in an effort to
help teachers deliver the appropriate intervention. See Table 4 for a summary of the assumed
knowledge needs of a successful mathematics PLC team member.
Table 4
Assumed Knowledge Needs of Successful Mathematics Teachers
Knowledge

Declarative

Procedural

Metacognitive

Teachers know the myriad of factors that could possibly cause gaps in
learning.

• Teachers know grade level expectations in mathematics and the learning
progression from elementary through middle school.

Teachers know the goal for all students is to achieve 100% proficiency in
grade level power standards.

Teachers know how to identify learning difficulties and choose the
correct solution path.

Teachers know how to use effective strategies to deliver the appropriate
instruction.

Teachers know how to guide students in awareness of their own thinking
and learning process and help them to self-regulate their learning.

In mathematics problem solving, teachers guide students to ask each
other to articulate the main problem, categorize it, choose an appropriate
solution strategy, and identify similarities and differences with other
problems they have seen.

Successful strategies in improving mathematics learning at the middle school level
Teacher effectiveness through cooperative learning. One question educators have
pondered over the years is how to ensure student learning in a classroom with students of varying
ability levels. To address this issue there are three types of instruction are prevalent in
mathematics classrooms today: whole group instruction; between-class grouping; and within-
class grouping (Slavin, 2009). Whole class instruction is the easiest for the teacher because he or
she delivers the same content with the same materials to one group of students. However,
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61
studies have shown the benefits of grouping students as opposed to this method of direction
instruction to a whole classroom (Lou, Abrami, Spence, Poulsen, Chambers, & d’Apollonia,
1996; Kulik & Kulik , 1982; Slavin, 1990). Between-class grouping is the most common type of
grouping and is seen when high achievers are assigned to higher sections and lower achievers are
assigned to lower sections of the same course or high and low level ability students are assigned
to completely different courses (Slavin, 2009). Within-class groupings are seen occasionally in
simple models where students are grouped within the same class or in more complex models
where students are flexibly grouped across grade lines or for particular topics (Slavin, 2009).
While grouping students for enhanced instruction has advantages over whole class
instruction, grouping by ability has had varied results when different variables have been
controlled for during experiments. Slavin has researched the effectiveness of mathematics
programs in both elementary (1987) and middle school and high school years (1990) and has
found with controlled variables “including randomized experiments of a quality rarely seen in
educational research, find no positive effects of ability grouping in any subject or at any grade
level, even for the high achievers most widely assumed to benefit from grouping” (Slavin, 1990,
p. 484). In Slavin’s (1987) meta-analysis of studies on the effectiveness of ability grouping in
elementary years, when variables such as ethnicity and socioeconomic status were considered,
the results showed some minimal positive effects with ability grouping for high achievers but
greater negative impact on motivation and self-esteem of students in the lower groups. The
argument over the past 90 years remains the same with proponents of ability grouping seeing the
slight effectiveness for high achieving individuals as worthwhile and the contrarians questioning
the equitable distribution of opportunities for those in the lower achieving groups. Slavin’s
(1990) equally robust research on the effectiveness of mathematics programs in middle school
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62
and high school programs showed that ability grouping had little to no effect on achievement in
the middle school years. Neither study duration, location nor when the study had any impact on
the overall findings and Slavin (1990) concluded that ability grouping was not the desirable
method for middle school years.
Although ability grouping has been proven ineffective for middle school, student
performance on mathematics assessments still seems to improve for those adolescent students
working in a heterogeneous group. One study showed that students learning mathematics in
small groups within the same classroom achieved more (i.e., 57
th
percentile) than students who
were learning in the classroom without a group (i.e., 50
th
percentile) (Lou et al., 1996). In
addition, research has shown that students learning in a group had significantly more positive
attitudes towards mathematics (Kulik & Kulik, 1982; Lou et al., 1996). Other advantages of
grouping students were the social aspects of cognitive growth (Piaget, 1954; Vygotsky, 1978),
whereby students have the opportunity to develop social and communication skills because of
the need and the opportunity to work with others. Peer learning allows the teacher time to
remediate and adjust learning objectives and pacing to meet the needs of the class (Lou et al.,
1996). Many studies over the years have looked at the effectiveness of grouping students for
instruction weighed against the effectiveness of curricular change. However, more recent studies
have noted that the positive effects of with-in class grouping are maximized when student
groupings are accompanied by variations to teaching methods and instructional materials (Lou et
al., 1996; Tieso, 2005).
The most striking revelation from a review of almost 100 studies on effective middle and
high school mathematics programs was that curricular differences appeared to be less
consequential than instructional differences (Slavin et. al, 2009). Evidence supported that
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63
instructional process strategies, especially cooperative learning, had more impact than teaching
mathematics content alone (Slavin et al., 2009). Cooperative learning is group work that is
characterized by positive interdependence, where students perceive that it is better to work
together to maximize their own and each other’s learning (Johnson, Johnson, & Smith, 2014).
Specific examples of cooperative learning, which were found to show the strongest evidence of
effective middle and high school mathematics programs, were IMPROVE and Student Teams
Achievement Divisions (STAD). The only program to show moderate evidence of effectiveness
was the Prentice Hall Course 2 and all other programs that were reviewed in almost 100 studies
showed limited or no effectiveness (Slavin et al. 2009).
In both of the two highly effective cooperative learning programs, IMPROVE and STAD,
students work in small (three to four), heterogeneous groups. In STAD, teachers follow a
schedule of teaching, teamwork and individual assessment leading to students’ mastery of
academic content. Teams receive recognition based on average scores of all team members on
weekly quizzes (Slavin et al., 2009). Slavin (1990) states that recognition and individual
accountability are essential for positive effects of cooperative learning.
IMPROVE is slightly different from STAD because it combines cooperative learning,
metacognitive instruction and mastery learning (Slavin et al., 2009). Developed by Mevarech
and Kramarski (1997) in Israel, IMPROVE starts like STAD with the teacher introducing the
concepts and then students separating into small groups. However, in the small groups, students
ask and answer metacognitive questions. For example students ask each other to articulate the
main problem, categorize it, choose an appropriate solution strategy, and identify similarities and
differences with other problems they have seen. Students take a formative assessment on content
after about 10 lessons and are given corrective instruction if they score less than 80 percent.
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64
Others move on to enrichment activities and those who received corrective instruction take a
parallel test of content knowledge. While both cooperative learning models IMPROVE and
STAD were found to be the most effective programs for middle school and high school
mathematics instruction, Slavin recognized that the standardized test results that were used to
guide his research may fail to detect more sophisticated skills such as problem solving or
concepts and applications. Different approaches to mathematics that were seen as having limited
effects may have additive effects if used together with cooperative learning (Slavin et al., 2009).
Teaching methods. Slavin’s (2009) research highlighted the importance of teachers’
instructional process as having more effect than curricular design alone. The National Council
of Teachers of Mathematics (NCTM) defines an effective teacher with eight effective
mathematics-teaching practices to ensure mathematical success for all. The teaching practices
are as follows:
1) Establish mathematics goals to focus learning.
2) Implement tasks that promote reasoning and problem solving.
3) Use and connect mathematical representations.
4) Facilitate meaningful mathematical discourse.
5) Pose purposeful questions.
6) Build procedural fluency from conceptual understanding.
7) Support productive struggle in learning mathematics.
8) Elicit and use evidence of student thinking.
Likewise, the Common Core State Standards (CCSS) has eight K-12 standards for mathematical
practice that teachers seek to develop in their students (Mathematics Standards, 2018). The
CCSS mathematics practices were developed from a combination of sources, including the
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65
NCTM process standards of problem solving, reasoning and proof, communication,
representation, and connections as well as the strands of mathematical proficiency from the
National Research Council’s report: adaptive reasoning, strategic competence, conceptual
understanding, procedural fluency, and productive disposition. The combined work resulted in
eight CCSS K-12 mathematics practices, which are listed below:
1) Make sense of problems and persevere in solving them.
2) Reason abstractly and quantitatively.
3) Construct viable arguments and critique the reasoning of others.
4) Model with mathematics.
5) Use appropriate tools strategically.
6) Attend to precision.
7) Look for and make use of structure.
8) Look for and express regularity in repeated reasoning.
The standards to mathematical practice are connected to the standards for mathematics content
for the purpose of balancing a student’s procedural knowledge and understanding of content.
For the mathematics educator, the effective teaching practices are closely aligned with the
mathematics practices that students hope to achieve during their schooling.
Explicit instruction of mathematical practices in 21
st
century instruction. There are
eight mathematical practices identified in the CCSS learning progression, commonly used in the
majority of schools following an American curriculum, that integrate with mathematical content
knowledge and instructional strategies to improve and provide highly effective mathematics
instruction (Weber et al., 2015). These eight practices include: making sense of problems and
persevering to solve them, reasoning abstractly and quantitatively, constructing viable arguments
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66
and critiquing the reasoning of others, modeling, using appropriate tools, making sense of
structure, attending to precision and looking for and expressing regularity in repeated reasoning.
To achieve the goal of 21
st
century mathematics instruction, these eight mathematical practices
should be explicitly woven into K-12 learning experiences to increase student engagement,
confidence, and achievement in mathematics (Weber et al., 2015).
Mathematical practice at home and tuition (peers or hired). For mathematics, the
educator must work closely with parents to obtain the best possible results for their child. In
middle school, students are beginning to go to different classes with different teachers and it
becomes more difficult for parents to know what is going on in each subject. Communication is
necessary to help guide parents to support their education from home. At the time a child is
becoming more independent, the courses at school are becoming more rigorous with more
practice necessary at home.
Research showed the best strategies to improve growth in mathematics showed positive
correlation with homework in middle and high school but in elementary school, homework has
limited impact. At the elementary level, Hattie (2012) states the effect size is 0.10 while middle
school is 0.3 and high school is 0.55. Homework that is another chance to practice something
can be effective in the later years. Sheldon and Epstein’s (2005) research revealed the
effectiveness and importance of learning-at-home activities. Activities that support mathematics
learning included (a) homework assignments that required students and parents to interact and
talk about mathematics and (b) mathematics materials and resources provided for families to use
at home. Activities that encouraged parent and child interaction proved to be the most successful.
Some examples include discussing ways students can use mathematics skills in their everyday
life, student demonstrating mastery of a new skill and then discussing the use of mathematics in
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67
every day life (Epstein, Salinas & Van Voorhis, 2001), and providing games and materials to
borrow and to be used by parent and child (Sheldon & Epstein, 2005). The relationship between
these learning-at-home activities and mathematics achievement were strong and positive
(Sheldon & Epstein, 2005) because of the parent interaction with the child.
Parent involvement. When parents start the middle school years with their child, they
often wonder how much to help their child in mathematics either personally or through the use of
a paid tutor. Middle school mathematics teachers need to set the expectation of support at the
start of the school year. This section reviews literature that defines the types of parental support
that have proven to be most successful to student achievement.
Extent of parent involvement and the effect of parent involvement on achievement. To
discover the effect of parent involvement on mathematics achievement specifically for middle
school students, this literature review covered five meta-analyses of over 100 individual studies
and the overwhelming research revealed that parent involvement in education is positively
associated with academic outcomes during middle school (Hill & Tyson, 2009; Jeynes, 2007;
Wang, Hill & Hofkens, 2014; Wilder, 2014). The impact of parent involvement has more
significance in elementary years than in middle and high school (Jeynes, 2007; Pataal, Cooper &
Robinson, 2008) and there are several possible reasons for the decreasing impact of parent
involvement. For example, parents have a better mastery of mathematics in earlier grades and a
higher chance of affecting still relatively undeveloped study habits and skills (Patall, Cooper, &
Robinson, 2008). Also, the middle school years coincide with key changes in adolescent
development, including biological and cognitive growth, social development and renegotiation of
the parent-adolescent relationship (Adams & Berzonsky, 2003; Keating, 2004, Lerner &
Steinberg, 2004; Steinberg & Silk, 2002). Furthermore, the move to middle school is usually a
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shift to a larger school environment with more teachers, more peers, and more choices of courses
(Dauber & Epstein, 1989; Eccles & Harold, 1996; Hill & Chao, 2009). Students also want to be
more autonomous and independent (Hill & Tyson, 2009). In the context of these changes,
academic performance often declines for the adolescent student and the need for parent
involvement remains. Students who exhibit deficiencies during the middle school years are at
risk of not achieving success and setting out on a trajectory that is hard to return from so the need
for continued parent support and teachers’ guidance on how much support is paramount.
Three types of parental involvement. Across the numerous studies, there were certain
aspects of parent involvement that proved to have more impact than others. In general, there are
three types of parent involvement: home-based involvement, school-based involvement and
academic socialization. Of these different approaches to parent involvement in the middle
school years, academic socialization emerged as the strongest positive association with
achievement (Hill & Tyson, 2009). School-based involvement presented positive results in
relation to achievement and home-based involvement revealed mixed results.
Home-based involvement: homework help. Home-based involvement showed some
positive association with achievement but it also revealed one predominantly negative or nil
effect on achievement resulting from homework help from parents. The synthesized findings of
nine meta-analyses implied there was no positive relationship between homework assistance and
student academic achievement (Wilder, 2014). In some instances, homework assistance even
negatively correlated with student achievement (Hill & Tyson, 2009; Jeynes, 2005; Cooper,
2007). Research showed that the negative effects of homework help could result from parents
not trained to teach concepts or parents are not familiar with the appropriate or current teaching
methods (Hill & Tyson, 2009; Wilder, 2014). Also, help with homework might be an outcome
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69
from poor school performance thus creating a negative relation to homework help and
achievement (Reynolds & Gill, 1994; Hill & Tyson, 2009). In addition, the negative relation
could be attributed to parental interference with students’ autonomy or to excessive parental
pressure (Hill & Tyson, 2009). Students need some mathematics practice at home but research
clearly shows that parental involvement with homework negatively affects achievement.
Home-based involvement – creating structure and space for homework. Where
homework assistance from a parent is negatively associated with achievement, there were more
themes that emerged from parental involvement that fostered academic success. One aspect that
related to a student’s achievement was the act of parents to create structure at home with
guidelines for homework and a space for doing homework that included the educational
materials necessary to be successful (e.g. supplies, books, newspapers, etc.) (Wang, Stanton,
Deveaux, Li, Koci, & Lunn, 2011; Hill & Tyson, 2009). Structure provides scaffolding for
adolescents and gives them tools to feel competent to be able to succeed and to have
opportunities to fulfill their goals without compromising their developmental needs for
increasing autonomy and independence (Skinner & Zimmer-Gembeck, 2009). Middle school
mathematics teachers can help guide parents in setting up the space and structure for homework
and practice at home.
School-based involvement – communication with school and participation in school
community. Another aspect of home-based parental involvement that showed positive relation to
achievement was creating and maintaining quality communication with the school. This type of
parental involvement at home bridges with school-based involvement and is essential to parents’
successful support of their child. Wang (2014) research showed that the average parent declined
in both preventative and quality of communication (sharing ideas about how a child can
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improve) between school and home from Grades 7 to 11. Maintaining effective communication
between home and school becomes especially challenging in middle school due to the increased
school size and number of teachers that students have over the course of the day. It becomes
difficult for teachers to initiate interactions with all the parents of the students they teach (Hill &
Chao, 2009) and likewise, it becomes difficult for parents to develop productive relationships
with their child’s teacher. This is the time for adolescents to begin to take responsibility for their
schoolwork, to develop solutions to problems on their own and to build a new relationship with
their parent (Sheldon & Epstein, 2005). For the middle school teacher, this is the time to teach
parents how to navigate in the new middle school context, providing communication between the
school and home and by providing communications through electronic means (as cited in Hill &
Tyson, 2009).
As parent involvement becomes more indirect in the middle school years, parents become
involved in activities such as visits to school events (e.g., PTA meetings, open houses, etc.),
volunteering at school, and participating in school governance (Hill & Tyson, 2009). School-
based parental involvement shows more positive relation to academic achievement than home-
based parental involvement but academic socialization demonstrates the greatest impact.
Academic socialization shows greatest impact on student achievement. Academic
socialization is a term that was coined by Hill and Tyson (2009) but captures many common
themes across the literature on the effects of parental involvement on academic achievement.
Academic socialization includes communicating parental expectations for achievement and value
for education (Hill & Tyson, 2009; Wilder, 2014), linking schoolwork and homework to current
events, discussing learning strategies with children (Hill & Tyson, 2009; Wang et al., 2014), and
linking material discussed at school with students interests and goals. Academic socialization
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71
helps to scaffold adolescents’ growing autonomy, independence and problem-solving skills. It
also provides young adolescents the tools to make semi-autonomous decisions about their
academic pursuits (Hill & Tyson, 2009). Consequently, students are increasingly experiencing
the ability and the opportunity to engage in logical and analytic thinking, problem solving,
planning and decision-making, skills they will need in the 21
st
century (Halpern-Felsher &
Cauggman, 2001; Keating, Lerner & Steinberg, 2004). As brain structures develop over the
course of adolescence, young adults are beginning to learn from their own mistakes and think
about future outcomes of their decisions (Keating, et al., 2004). Therefore, the parental
expectation on the value of education critically links to the adolescent’s future goals, helps to
support their emerging self-identity (Bandura, Barbaranelli, Caprara, & Pastorelli 2001) and
gives purpose to the adolescents work. In addition, parental expectations help to pass on parental
values and beliefs and foster a transition from external control from parents to an internal
motivation for students to act on social values and to achieve future goals (Sheldon & Epstein,
2005; Wigfield, Byrnes, & Eccles, 2006; Jeynes, 2007).
Personalized learning. Up until this point, the literature reviewed has viewed
mathematics education during the middle school years from the viewpoint of the supporting
network, comprised of the educator, parent, and school community. At the International School
South East Asia, the vision and goal of the support network is to develop leaders and cultivate
exceptional thinkers that are prepared for the future (ISSEA Focus, 2017). However, the most
valuable component of the network is the student and their ownership of their own learning.
Recent research (Gates, 2014) has shown that when learning is personalized, the student will be
able to develop more meaningful relationships with their peers, teachers and members of their
local and global community. “Personalized learning is student-centered, grounded in each
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learner’s profile, and characterized by competency-based progressions, customized pathways,
and flexible learning environments” (ISSEA Personalized Learning, 2017). In order to achieve
personalized learning for every student, the school needs to have structures in place, such as
schedule, digital platform and space utilization protocols, to respond and adapt to support
students’ learning goals. The curriculum for each course needs to be based on standards that are
written in a way that students understand and are able to determine what they have already
learned, what they are currently learning, what they will learn next and why they are learning it
(ISSEA Personalized Learning, 2017). Educators will collaborate to differentiate learning,
provide feedback and modify instruction based on formative assessment. Students will be
responsible to co-create and articulate their learning plans, apply feedback and set future goals
based on evidence.
Metacognition and goal setting. Literature review and learning theory has shown the
primary importance of metacognition and setting goals to the success of a student in school and
in their daily life. “Metacognition is commonly associated with an awareness of one’s own
thinking and learning process and is closely associated with both the assessment as learning
process and self-regulated learning” (Weber et al., 2015, p. 71). When a student is self-regulating
their learning, they follow the following steps. First, they know what they have already learned.
With the help of their teacher, they decipher what they need to know and how to achieve
proficiency in that area. Finally, they co-create a plan and they where to go to ask for help.
Personal success is realized when a learning plan ends with achievement of a specific goal.
Hattie (2012) reports that students’ self-reporting of their achievement had the greatest influence
on student learning of any variable studied.
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73
Having a plan for success follows a routine, prescribed list of steps. However, in the
specific domain of mathematics, some of the metacognitive strategies that are most useful in
problem solving include having a plan for approaching the task, using appropriate skills and
strategies to solve a problem, monitoring and noticing when a problem doesn’t make sense, self-
assessing and self-correcting, evaluating progress toward the completion of a task, and becoming
aware of distractions (Rosenweig, Krawec, & Montague, 2011; Throndsen, 2011). This process
may be natural for some students but some students may need to be taught these metacognitive
strategies explicitly (Weber et al., p. 71, 2015). The use of metacognition is the key to moving
from surface to deeper learning.
Response to intervention. Response to Intervention (RTI) is about using the collective
knowledge and skills of an organization to benefit all students (Buffum et al., 2012). To address
the need for sustainable interventions for student with learning deficiencies, the International
School South East Asia has adopted the PLC model for collaborative work and the RTI model to
ensure every student achieves at high levels. The decision to work through PLCs using a RTI
model was based on careful consideration of the variables that emerged as impactful to student
learning. RTI provides an organizational framework to meet the needs of all learners across a
multitiered model (three tiers) that includes data-driven decision-making and empirically
supported instruction (Windram et al., 2012). RTI is not a new instructional technique but
instead, it offers a structure and common language to practices that already exist. The goal of
RTI implementation is to emphasize what practices are highly effective while eliminating any
ineffective or unnecessary practices.
If a child can’t learn the way we teach, maybe we should teach the way they learn. –
Ignacio Estrada (Windram et al., 2012, p. 3).
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RTI begins with assessment. Educators use universal screening measures, usually in the
fall and spring, to identify students at risk of not meeting intended grade level targets. Student
progress is monitored and teachers provide feedback to students to improve their understanding.
Instruction can be defined in three tiers where Tier 1 is the primary level and is core instruction
that every student receives. Tier 2 is strategic small group intervention in response to an
identified weakness to close the achievement gap between them and their peers. Tier 3 is
intensive one-on-one intervention to address a specific area of weakness. Multi-tiered
instruction is designed so teachers have the right tools at the right time to address the different
learning needs of their students (Windram et al., 2012).
ISSEA has adopted the RTI program as an organizational framework using data driven
decision making to support instructional practices. As highlighted in Table 5, the RTI program
success depends on the teacher placing a value on the program and on it’s ability to bring
students from a deficit to a growth in learning. It also depends on the teacher’s belief and
confidence in their own ability to implement the program. Finally success depends on the
collective belief that the PLC team members along with the LS teachers can reach their goal of
100 percent of students achieving proficiency for grade level power standards. Middle school
mathematics teachers’ motivation is necessary to persist through a year of challenges and success
experience in the first phase of RTI implementation.

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Table 5
Assumed Motivation Needs of an Effective Teacher
Motivation Teachers value the goal of every student learning achieving proficiency in
grade level power standards by the end of the school year.

Teachers believe that engaging in a RTI program will contribute to students
filling their learning gap.

Teachers are confident in their ability to execute the RTI program to achieve
100 % student on grade level power standards.

Teachers are confident in their PLC team and with the LS teachers’ ability to
reach the goal.

Organization Theory

Factors Contributing to Effective Organizations

The International School South East Asia promotes a pantheon of great teachers and a
collaborative cultural model. Middle school mathematics teachers work together in a
Professional Learning Community (PLC) towards a common vision where every child is
provided extraordinary care and achieves at high levels. Teachers collaborate to prioritize
essential learning standards, to create formative assessments and share feedback, to share
assessment scoring, and to analyze data to ensure student learning and growth. Learning support
teachers, instructional coaches, technical coaches, and the librarian support the middle school
mathematics teacher team. An administrative team of three provides guidance and leadership to
each of the three grade levels in middle school. The mathematics faculty, support staff, and
administrative leadership team are committed to the six institutional commitments (a common
guaranteed viable curriculum, evidence of learning, great teaching in every classroom, every day
for every student, the integration of technology, a healthy organizational culture, and
participation in a PLC) and they work together to ensure positive response to challenges and
changing conditions.
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Organizations require resources, an effective process and procedures in order for a new
program implementation to be successful. There needs to be a feedback loop where feedback is
given in a timely manner and teachers are able to adjust their instructional strategies to become
more effective. The general needs for teachers to be successful are highlighted in Table 6 below
and the needs specific to the context of the International School South East Asia Middle School
Mathematics department are described in the following paragraphs.
Table 6
Assumed Organizational Needs for Mathematics Teacher Effectiveness
Organizational Teachers have the resources needed to achieve the goal.

The organization has an effective communication process to share
necessary information with the teachers.

The organization has procedures for RTI to support the teachers in
achieving the goal.

There is a process that ensures the teachers get timely, concrete feedback
about their intervention effectiveness.

Improving mathematics performance at the middle school level. Policies and
procedures are in place to help provide clear and formalized coordination and accountability
(Clark & Estes, 2008). In middle school mathematics, it was important for the entire department
to have shared agreements and protocols regarding homework assignment, reassessment of unit
tests, the transition between grade levels and between standard and advanced courses, placement
testing and process, and vertical articulation of mathematics standards and practices. These
agreements are revisited and refined throughout the year.
Common resources and materials are provided to the middle school mathematics faculty.
The Connected Mathematics Project 3 is the primary resource and is an inquiry based middle
school mathematics program focusing on problem solving, reasoning and proof, communication,
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representation and connections. Mathematics manipulatives such as counters, unifix blocks, three
dimensional geometric shapes, algebra tiles, mirrors, and measuring tools support hands-on
learning in the classroom. The IXL an immersive learning experience that provides online skills
practice engages students with curriculum-aligned, standard based questions with real-world
scenarios. Along with IXL practice, KUTA Software LLC creates and distributes customized,
differentiated worksheet practice problems for students in every skill from Pre-Algebra through
Geometry and Algebra 2. IXL online practice combined with KUTA worksheets provides
students with the repetition and practice needed to ensure long-term memory storage. Finally,
Interactive White Boards (IWBs) are being added to classrooms for multimedia learning and to
help students to keep up with classes when they are unable to be in the classroom.
One of the structures that are in place to support high-impact mathematics instructional
strategies is a class schedule that is aligned by grade level, so teachers are able to share and
differentiate between classes that meet at the same time. Common time is provided every other
day for PLCs to meet to plan together. A learning support teacher is assigned to one third of the
students from each grade level and assists the classroom teacher in the mathematics instruction
of approximately 110 students. The learning support teacher helps to facilitate inclusion classes
by focusing on those who need interventions when they are not able to grasp standards at the
same pace as other students. The learning support teacher also provides additional structure and
lessons for students with identified learning disabilities. A common digital shared folder
contains all department documents for all mathematics teachers to access and organize planning
and policy documents. Teachers also receive monetary support for extension materials for
students who already know the material being covered in class. Finally, teachers are given
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financial support for their own professional development in support of personal professional
goals or in support of the school’s initiatives and vision.
Knowledge, Motivation, and Organizational Factors
International School South East Asia can be identified as an effective organization as
described in the above paragraphs. However, as with any organization, problems exist where the
actual performance level is not at the preferred performance level. According to Clark and Estes
(2008), one of the “big three” causes of a performance gap is people’s knowledge and skills.
The purpose of the knowledge analysis is to identify and discuss whether people know how (and
when, what, why, where and who) to achieve their performance goals. Rueda (2011) states that
solutions to problems fail because we attempt to apply solutions before the problem is fully
understood. To help identify and analyze causes of performance gaps, Clark and Estes (2008)
suggest assessment should include listening openly and fully to stakeholders, reexamining
learning, motivation and organization theory and rethinking the literature of the subject matter.
The performance problem that exists at the Middle School at International School South
East Asia where students enter a grade level with a learning deficit and leave the grade level still
at a deficit in their mathematics understanding. The stakeholders of focus for this study are
middle school mathematics teachers because it is their responsibility to identify students with
learning deficiencies and plan successful interventions that help students to achieve their
academic goals. An assessment of teachers’ knowledge and skills of mathematics interventions
and RTI implementation, their motivation to persist in the implementation effort and the support
from the policies, procedures and characteristics of the environment is imperative to the success
of the organizational goal of 100 percent of students achieving grade level standards, see Table
7.
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Table 7
Summary of Influences on Middle School Mathematics Teachers
Assumed Knowledge Needs of
Successful Mathematics Teachers

General Literature
Knowledge
Declarative

Procedural

Metacognitive
Teachers know the
myriad of factors
that could possibly
cause gaps in
learning.

Teachers know
grade level
expectations in
mathematics and
the learning
progression from
elementary through
middle school.

Teachers know the
goal for all students
is to achieve 100%
proficiency in grade
level power
standards.

Teachers know
how to identify
learning difficulties
and choose the
correct solution
path.

Teachers know how
to use effective
strategies to deliver
the appropriate
instruction.

Teachers know how
to guide students in
awareness of their
own thinking and
learning process
and help them to
self-regulate their
learning.

Many conditions, or the lack thereof, can lead to learning gaps. The middle school
mathematics student experiences an intersection of developmental processes
during the adolescent years (biological, cognitive and social). Social structures,
such as a higher SES, err to a higher value placed on education as well as offers
access to higher quality teachers and positive neighborhood norms. The
interconnection or lack of relationship between goal setting, attitude and
motivation can affect long-term growth in learning. Scientific experiments on
learning show how forgetting causes the need for remembering through repetition
and practice. The human information processing system takes in sensory data and
organizes it in short term working memory. Information is organized in long-term
memory when connections are made with prior knowledge. Students’ anxiety can
limit working memory capacity. Effective teachers can help guide self-efficacy
and motivation and a proactive classroom can provide interpersonal, instructional
and environmental supports.

In middle school, students should have experience in each content area and should
identify power standards to balance learning through grade levels and
accommodate for different abilities. CCSS and PARCC can help identify
standards. Elementary knowledge of the magnitude of whole numbers and
fractions and the order they are placed on the number line is a predictor of future
success in mathematics.

Teachers and students need to believe they can have 100% proficiency in grade
level mathematics standards by the end of the school year.

The most effective strategy in learning is the teacher and his or her instructional
process. Cooperative learning with flexible within class grouping is next on the
list. Other effective strategies include following the mathematics effective
teaching strategies, explicit instruction of mathematics practices, and practice at
home. Parent involvement in the form of academic socialization has the most
positive effect on learning but it is still not as effective as other strategies.
Academic socialization is when parents set expectations for achievement, show the
value of education, link schoolwork and home practice with interests and goals.
Personalized learning, where students co-create their learning plan, apply feed
back and set goals is a new theme that is emerging in the literature as having
positive effects.

In mathematics problem solving, teachers guide students to ask each other to
articulate the main problem, categorize it, choose an appropriate solution strategy,
and identify similarities and differences with other problems they have seen.

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80
Assumed Motivational Needs of
Successful Mathematics Teachers

General Literature
Motivational Teachers value the
goal of every student
achieving proficiency
in grade level power
standards by the end
of the school year.

Teachers believe that
engaging in a RTI
program will
contribute to students
filling their learning
gap.

Teachers are
confident in their
ability to execute the
RTI program to
achieve 100 % student
on grade level power
standards.

Teachers are
confident in their PLC
team and with the LS
teachers’ ability to
reach the goal.

The indicators for motivation are active choice, persistence and effort.
RTI depends on the collective choice of teachers to implement this
program and to do so in an energizing way that demonstrates intense
effort. The components of motivation are first that RTI implementation is
personal; it has value to the teacher. It is activating; it instigates behaviors
positive to RTI implementation. It is energizing; it fosters persistence and
intensity in interventions. And RTI implementation is directed; it is aimed
at a goal.

Motivation can also be seen from the students’ perspective as they are
guided by teachers but self-regulating their learning. By creating a
personalized learning plan, they also experience the four components of
motivation. The learning plan is inherently personal as it is directly
related to a personal deficiency and goal. It is activating as it outlines a
path to achieve the chosen goal and it is energizing to receive and apply
immediate feedback to work towards achieving the goal.

Teacher and student alike experience a rise in confidence and self-efficacy
when the RTI program is implemented to integrity and students achieve
their goal of proficiency.
Assumed Organizational Needs of
Successful Mathematics Teachers

General Literature
Organizational Teachers have the
resources needed to
achieve the goal.

The organization has
structures in place
where roles and
responsibilities have
been delineated and
an effective
communication
process exists to share
necessary information
between stakeholders.

The organization has
policies and
procedures for RTI to
support the teachers in
achieving the goal.

There is a process that
ensures the teachers
get timely, concrete
feedback about their
intervention
effectiveness.
Getting people what they need to their work is important in maximizing
efficiency, in demonstrating to employees that their work is valued and in
showing that the company is supporting them in what they are asked to do.

Asking for the employee’s input and considering that input can often lead
to better decision-making. In addition, when employees feel they are
involved in decisions, they take greater ownership for the outcomes.

Policies and procedures are in place to help provide clear and formalized
coordination and accountability.

Employees need constant feedback to know if what they are doing
matters. Providing a structured time to discuss an employee’s progress,
achievements and goals is important for both managers and employees.

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Summary of Literature Review
This study has reviewed the literature around the science of learning and more
specifically how students learn mathematics to discover all possible variables that impact student
learning. The goal of this literature review was to better understand the challenges that some
students face in order to choose the appropriate strategy to best support students that exhibit
learning gaps. First, this study looked at what a student should know and be able to do. Some of
the variables that emerged as possible causes for gaps in understanding were the inability to
remember resulting from differences in the student’s information processing center. Each
person’s system works in different ways resulting in a need for complex instruction strategies.
For example, some students have different ways to process information (visual and auditory), or
a limited working memory storage capacity, or a limited ability to organize information or find
connections between different nodes of information, or a combination of these different
variables. Some students have a lack of prior knowledge on which they can build long-term
memory storage. And all middle school students have the enormous challenge of navigating
through the biological, emotional and academic developmental processes that occur during the
time of adolescence. These variables are often uncontrollable in their existence but strategies
can be learned to accommodate for the challenges they present.
Other variables that emerged in this study were more controllable and dependent on the
environment in which the student existed. For example, cultural identities play a part in the
challenges of being a minority in a diverse community as well as a lack of support structures in
place. Motivation is a controllable variable and is dependent on internal and external factors. A
student’s self-efficacy is their belief on whether they will or will not be successful in intrinsic but
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can be modified with support. In addition, a supportive, proactive classroom environment is
essential for the study of mathematics.
Educators can help to mitigate the negative effects of some variables by starting with a
learning progression based on standards that have been prioritized to fit the context of the school.
Teachers, through collaborative PLC teams, can ensure vertical alignment with elementary and
high schools to identify areas of possible weakness and also to identify future goals. At the
grade level, they can guide students in their understanding of what they need to know for their
specific grade level mathematics content and practice. The effectiveness of a teacher’s ability to
help guide their students has been found to be the most impactful variable on student learning.
Other variables such as heterogeneous grouping combined with curricular design changes, can
make differentiation for individual students even more powerful. The use of mathematics
practices is a strategy to ensure students are getting to a deeper understanding of their
mathematics learning. Educators must work closely with students to co-create students’ personal
learning plans, plans that are set with a goal of achievement in an identified area in mind. Parent
communication and involvement emerged as an important variable in student success and one
that should be nurtured as part of the support system for the child. The International School
South East Asia has adopted the RTI model to best use the structures that exist in the middle
school so that every student has the opportunity to learn and achieve at high levels.
Conclusion
This study is important because currently there are middle school students who have been
identified with learning deficiencies that are not receiving support in the learning support system.
Without systemic differentiation of instructional support, struggling learners may continue to fall
further behind their peers, and achievement gaps will continue to widen (Gersten, Chard,
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Jayanthi, Baker, Morphy & Flojo, 2009). The middle school does not yet have a systemic
intervention program although they are beginning to implement RTI. The RTI approach to
determining educational needs, as opposed to relying on labels categorizing learning disabilities,
keeps the focus of professional resources on student learning outcomes. It is a necessary step to
help students close the gap with their peers. Providing sustainable interventions to students with
identified learning deficiencies is the responsibility of educators to ensure every student has
equal opportunity to achieve at high levels.
Therefore the purpose of this study is to provide ISSEA with strategies and/or programs
to ensure all students are able to exit grade level proficient in grade level expectations. Middle
school teachers should evaluate whether existing strategies help students improve their
performance and keep those that do while purging ineffective approaches. The administrative
leadership team should consider the knowledge and skills, motivation, and organizational causes
that exist to provide support to the middle school teachers’ ability to fill their students’ learning
gaps. Finally, the framework of the RTI program at ISSEA should be evaluated and suggestions
should be made to improve implementation. RTI is a coordinated effort to ensure that no one
teacher is responsible for intervention. And more importantly, implementation of an intervention
program means that a school acknowledges their students learn in different ways and data can be
used to make decisions about who needs what, how much and when.

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CHAPTER 3: METHODOLOGY
Purpose of the Project and Questions
The purpose of this project is to use the Clark and Estes (2008) gap analysis model to
investigate the knowledge and skills, motivation and organizational influences on middle school
mathematics teachers and their ability to create a sustainable intervention program that identifies,
addresses and tracks progress of students’ mathematics learning deficiencies. While a complete
gap analysis would focus on all stakeholders, for practical purposes the stakeholders of focus for
this study are the middle school mathematics teachers. The goal of the mathematics teachers is
to ensure 100 percent of students achieve grade level prioritized standards and students achieve
results at or above the 70
th
percentile of the math portion of the Measure of Academic Progress
(MAP) standardized test. This analysis will systematically investigate knowledge and skills,
motivation and organizational barriers and suggest strategies to support the stakeholders’ goal.
The specific questions that guide this project are as follows:
1. What strategies exist at the International School South East Asia middle school to help
students improve their performance so they all meet grade level standards for math
proficiency/achievement?
2. What are the knowledge and skills, motivation, and organizational causes that exist to
provide support to the middle school teachers’ ability to fill their students’ learning gaps
while at the same time guide them to achieve grade level standards by the end of the
school year?
3. What programs might ISSEA implement to help middle school math students fill learning
gaps and achieve proficiency for math grade level standards?

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Conceptual and Methodological Framework
This study utilized Clark and Estes’ (2008) gap analysis, a systematic, analytic method of
examining an organization’s performance by clarifying organizational goals, identifying the gap
between the actual performance level and the preferred performance level as defined by the
organization’s goals, and suggesting research-based strategies to achieve these goals. In the gap
analysis method, organizations first define clear organizational goals that are flexible enough to
reflect changing conditions yet specific enough to direct day-to-day tasks (Clark & Estes, 2008).
From the organizational goals, the gap analysis method examines stakeholder goals ensuring
these goals are concrete, challenging, and current and connected to the higher-level
organizational goal. Analyzing the current status and comparing it to the desired performance
level determines the performance gap. Finally results are analyzed to find the cause of the gap,
improvement programs or strategies are suggested, and organizational goals are revised for
continued improvement (Clark & Estes, 2008).
The gap analyses method states there are three human causes of performance gaps:
people’s knowledge and skills, motivation to achieve the goal (particularly compared with other
work goals they mush also achieve) and organizational barriers such as lack or resources or
missing or inadequate structures and processes. All three of these factors must be aligned or
work together to successfully achieve performance goals (Clark & Estes, 2008). By
investigating each cause of performance gaps separately, the nature of the problem is understood
and solutions are targeted to the roots of the problems resulting in greater organizational
productivity (Rueda, 2011).
This study accordingly examined the performance gap in the organization’s actual
implementation of a sustainable intervention program for students demonstrating learning gaps
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in mathematics at the beginning of the school year and the organization’s desired performance of
every student learning at high-levels. It then assessed assumed influences on the organization’s
performance gap based on personal knowledge and related literature in the context of the
knowledge, motivation and organizational factors. Different methods were used to validate these
assumed influences, including surveys, document analysis, interviews, and literature review.
Research-based solutions were recommended and evaluated in a comprehensive manner.
Assessment of Performance Influences
Gap analysis is a systematic, problem-solving approach to improve performance and
achieve organizational goals (Clark & Estes, 2008). In the gap analysis model, measurable goals
are defined at three levels: long-term (global goals of 1 to 5 years or more), intermediate goals
(weeks to months), and day-to-day performance goals. Organization goals and stakeholder goals
provide direction and a tangible mechanism for determining when to change the present course
of action. As an alternative to making assumptions about causes of performance gaps, the gap
analysis method uses research to validate the causes of the gaps that exist between actual levels
of performance and the organization and stakeholder goals (Rueda, 2011). Research includes
three components: active listening or surveys with relevant stakeholders; an analysis of
knowledge and skills, motivation and organization theory; and a review of related literature on
how students learn, what they need to know and successful strategies as demonstrated in other
schools. This study reviewed influences on mathematics teachers from general literature in
Chapter 2 and will include the influences in the tables in Chapter 3 and attached appendices. The
following is a discussion of the assumed needs and influences in the areas of knowledge,
motivation, and organization for mathematics teachers’ ability to create a sustainable
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intervention program to help students fill learning gaps in mathematics and achieve proficiency
in grade level power standards.
Knowledge Assessment
Rueda (2011) asked, “What do teachers need to know as they struggle with design and
implementation of accountability programs, standards based curricula, and authentic
assessments?” (p. 27). The literature revealed six possible knowledge influences on middle
school mathematics teachers, as displayed in Table 6. Three of these (possible factors of
learning gaps, mathematics learning progression and grade level expectations, and knowledge of
organization goal) are representative of declarative knowledge and will be assessed through
survey. Two possible knowledge influences are characteristic of procedural knowledge
influences and will be assessed through interview questions asking participants to describe how
they identify learning difficulties, how they choose the correct solution path and how they know
and use effective strategies to deliver the appropriate instruction. Finally, there is one knowledge
influence that is typical of metacognitive knowledge and will also be assessed through an
interview questions about how teachers guide students in awareness of their own thinking and
learning process and how the teachers help students self-regulate their learning.
Motivation Assessment
“Individuals with higher self-efficacy, greater belief in their own competence, and higher
expectancies for positive outcomes will be more motivated to engage in, persist at and work hard
at a task or activity” (Rueda, 2011, p. 41). The literature revealed four possible motivation
influences, as displayed in Table 6. Motivation influences included teacher value of the goal of
every student achieving proficiency in grade level power standards by the end of the school year
and teacher belief that engaging in a RTI program will contribute to students filling their learning
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gap. Motivation influences on middle school mathematics teachers also included teacher
confidence in their ability to execute the RTI program and confidence in the ability of the PLC
math team and learning support teacher to collaboratively achieve the organization goal.
Motivational influences were assessed through survey and interviews.
Organization/Culture/Context Assessment
Organizational structures, policies and practices can influence whether the performance
goals of schools or organizational units are met (Rueda, 2011). The literature revealed four
organizational influences on middle school math teachers’ ability to achieve their goal.
Organizational influences include resources necessary to achieve the stakeholder’s goal,
structures in place so roles and responsibilities are clear, and an effective communication process
exists to share necessary information between stakeholders. In addition, organizational
influences include policies and procedures for RTI implementation and a process that ensures
teachers receive timely, concrete feedback about their intervention effectiveness. Organizational
influences were assessed through survey and interviews.

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Table 8
Assessments of Assumed Influences on Middle School Mathematics Teachers
Assumed Influence

How Will It Be Assessed?

KNOWLEDGE
Teachers know factors that could cause gaps in
learning.

Teachers know grade level expectations in
mathematics and the learning progression from
elementary through middle school.

Teachers know how to identify learning difficulties
and choose the correct solution path.

Teachers know how to use effective strategies to
deliver the appropriate instruction.

Teachers know how to guide students in awareness of
their own thinking and learning process and help them
to self-regulate their learning.

Survey Questions for “Getting Started with RTI”

Short answer:
1. List factors that cause learning gaps.
2. Describe your grade level expectations (your
grade level power standards).
3. What do students need to know to before you
begin a lesson on your grade level power
standards?
4. List possible learning gaps (deficiency) you have
experienced in your grade level.
5. What strategies have you used to help students to
fill their learning gap (deficiency)?
6. Describe how you have incorporated
metacognition in your lessons. (How have you
guided students in their awareness of their own
thinking and learning process and help them to
self-regulate their learning).

MOTIVATION
Teachers value the goal of every student achieving
proficiency in grade level power standards by the end
of the school year.

Teachers believe that engaging in a RTI program will
contribute to students filling their learning gap.

Teachers are confident in their ability to execute the
RTI program to achieve 100 % student on grade level
power standards.

Teachers are confident in their PLC team and with the
LS teachers’ ability to reach the goal.

Survey Questions for “Getting Started with RTI”

Likert Scale:
1. I am confident that I know how to identify
misconceptions on pre-assessments.
2. I am confident that I can flexibly group students
within classes that meet at the same time.
3. How confident do you feel about starting the
process of RTI implementation in conjunction
with the learning support teachers?

Check the boxes that apply:
o My confidence is not high because…
o I do not have the necessary knowledge and
skills.
o I do not have a clear picture of what is
expected of me.
o I have other, higher priorities.
o I do not have the necessary resources to
begin an intervention.
o I do not have the support to begin an
intervention.
o I don’t think what I have learned thus far will
work.
o There is not an adequate system of
accountability to ensure application of what I
have learned about RTI.
o Other (please explain):
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Assumed Influence

How Will It Be Assessed?

ORGANIZATION
Teachers have the resources needed to achieve the
goal.

The organization has structures in place where roles
and responsibilities have been delineated and an
effective communication process exists to share
necessary information between stakeholders.

The organization has policies and procedures for RTI
to support the teachers in achieving the goal.

There is a process that ensures the teachers get timely,
concrete feedback about their intervention
effectiveness.

Survey Questions for “Getting Started with RTI”

Likert scale:
1. The degree to which the team received
helpful information before the
implementation phase.
2. The way the program was rolled out
contributed positively to my learning
experience.
3. The degree to which the mathematics team
was involved in the data collection.
4. The degree to which the mathematics team
was involved in the creation of the RTI
framework.

Open-ended questions:
1. What resources do you need to plan and
conduct an intervention?
2. What structures (roles and responsibilities)
need to be in place for you to conduct an
intervention?
3. What policies and procedures need to be in
place for you to conduct an intervention?
4. What barriers to you anticipate that could
limit your success in executing the
intervention?

.
Participating Stakeholders and Sample Selection
Sampling
The stakeholders of focus for this study are middle school mathematics teachers.
Purposeful sampling was used to discover, understand, and gain insight from the stakeholder
from whom the most can be learned (Merriam & Tisdale, 2016). The respondents were chosen
from the middle school mathematics faculty, which is comprised of nine teachers, because of
their special experience and competence in teaching mathematics to students with learning
deficiencies. For this study, sampling criteria included current mathematics teachers that were in
the middle school math department for at least one year to ensure each participant understood the
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learning support culture in the middle school at International School South East Asia. In
addition, the sampling criteria included returning teachers only, because they had contributed to
the work that was included in the document analysis. Teachers that were new to International
School South East Asia were not included in the respondents group because they did not
participate in or have knowledge about the documents to be analyzed for the purpose of this
study. Because newly hired teachers were not yet active in the learning support culture that
existed at ISSEA, only six teachers were eligible to be included in the sample. An adequate
number of participants was necessary to answer the questions posed in this study and all
teachers, who fit the sampling criteria, voluntarily elected to participate in the study. Of the six
teachers, three were chosen from different grade levels to ensure the results of this study were
representative of all middle school grades.
According to the constructivist worldview, the researcher aimed “to establish the
meaning of the phenomenon from the view of the participants”, in this case, the review of the
RTI implementation and the teacher ability to achieve the goal of all students achieving
proficiency in grade level power standards by the end of the school year (Creswell, 2014, p. 19).
The data collection strategies used in this study employed the same sampling criteria and
included document analysis, survey, and interviews. Surveys were conducted through email
request, which included a link to the survey form. A survey was conducted at the beginning of
the school year, in the week of 18-22 September, to gauge mathematics teachers’ knowledge and
skills, motivational and organizational support as they began implementing RTI collaboratively
with the learning support teachers. This week was chosen because Measure of Academic
Progress (MAP) standardized testing would be complete, grade-level, diagnostic assessment
would be complete, and the start-of-the-year, relationship-building field trip would be complete.
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Interviews were subsequently scheduled, after the end of the first quarter, in the week of 23-29
October, for no more than 30 minutes. These dates were chosen after parent-teacher conferences
and first quarter assessments to look at the data of interventions and to reflect on what worked
and what didn’t work. The interviews were conducted online through Skype for convenience
and accessibility.
Recruitment
Following a probability sampling technique, the participants were invited to take part in
the study through a personal email request at the end of the school year 2016-2017. To be
eligible, middle school mathematics teachers had to be currently employed to ensure participants
understood the context of the learning support program as well as the use of PLCs and RTI at
International School South East Asia. All six teachers that were eligible were emailed with a
request to be included in the sample to ensure all had an equal chance to be included in the
sample. The participants’ response was by reply to the email. The sampling design was single-
stage because access to the people for this population was easily accessible (Creswell, 2014).
The middle school mathematics teachers were selected as the stakeholder population for
this study because they are directly responsible for the growth in student mathematics learning.
They assess and identify students that demonstrate deficiencies and plan interventions for their
personal learning and eventual growth. Additionally, this group collaborates with the learning
support teachers to ensure all students receive the level of support necessary to correct the
identified deficiency. All eligible middle school math teachers are familiar with the school’s
culture and RTI methods through the use of PLCs.

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Data Collection
To validate knowledge, motivation, and organizational assumed causes, this study
collected data through surveys, online interviews using Skype, and document reviews. By using
surveys, interviews and document analysis, results were triangulated to reduce the risk of chance
associations and of biases due to specific methods (Maxwell, 2013). Inductive data analysis
using a constant comparative method was used. Results were coded and categories were created
for the data that was deemed most important. Organizational categories were based on broad
areas or issues, similar to the headings in the previous chapter in the literature review.
Substantive categories were also used to describe the participants’ concepts and beliefs
(Maxwell, 2013).
Surveys
A survey was created using the Kirkpatrick & Kirkpatrick (2016) four levels of training,
including results, behavior, learning and reaction. The survey was administered online via an
email request to the participants. Respondents were given one week to complete the survey and
were sent a gentle reminder when there were only three days left to send in their answers. There
were seven Likert scale questions, one checklist of nine items, six short answer questions and
four open-ended questions. All questions were mandatory. The respondents were asked to allow
20-30 minutes to complete the survey and were reminded that this survey was anonymous data
collection. No identifiable demographics were included in the survey and were not referenced
specifically in the data collection or results.
The survey was conducted to evaluate the perception of middle school mathematics
teachers as they started the implementation of RTI. It was valuable to see how well the
participants felt at the start of the implementation and to address initial concerns. Too often,
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potential barriers are not carefully thought out and planned for and crisis results. This survey
helped prevent crisis and shifted teams to a more pro-active prevention of trouble spots. See
Appendix A for the survey titled “Getting Started with RTI in MS Mathematics.”
Interviews
Sample group participants were interviewed individually online through Skype.
Interviews consisted of approximately ten questions and lasted no more than thirty minutes.
Interviews were scheduled at a time and location convenient to the participant. Using an
emergent design, the interview began with a semi-structured template. Questions were based on
the six types of questions, as described by Patton (2015): experience and behavior, opinion and
value, feeling, knowledge, sensory, and background demographic questions. In addition, the
interview protocol also followed the typology of Strauss, Schatzman, Bucher, and Sabshin’s
(1981) four major categories of questions: hypothetical, devil’s advocate, ideal position and
interpretive questions (Merriam & Tisdall, 2016). The interview participants’ comments were
recorded on the interview protocol for later transcription and coding.
The primary purpose of the interviews was to determine successes and challenges
experienced by middle school mathematics teachers at the end of the first quarter and first phase
of RTI implementation. The interviewer listened to the audio recording, made notes based on
teachers’ responses and revised the interview guide. The interview protocol “First Quarter Pulse
Check in RTI for MS Mathematics” is included in Appendix B.
Document Analysis
This study analyzed multiple documents to find the meanings, symbolic qualities, and
expressive contents they have and the communicative roles they play in the lives of the data
sources (Merriam & Tisdell, 2016). With a request to the deputy principal, Spring MAP test data
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from the math portion of the assessment was gathered in a Google spreadsheet to identify
students under the 50% compared to their local peers. The investigator requested from the
sample population access to documents, such as PLC meeting agendas, lesson slide shows, unit
planning documents, online websites or blogs, learning support student progress spreadsheets
and student learning plans, which were then gathered and reviewed to triangulate results.
Content was analyzed using measures of frequency and variety of messages and confirming
hypothesis.
Data Analysis
Descriptive statistical analysis of middle school mathematics teachers involvement in
RTI was conducted based on teachers’ implementation of interventions and evidence of student
progress. To validate the assumed knowledge, motivation and organizational performance
issues, this study collected data from relevant documents and online resources and then from
surveys. Additional data was collected through interviews to identify the challenges and
successes of the initial interventions designed to support students who were trying to overcome
their learning deficiency in mathematics. Table 9 summarizes influences on middle school
mathematics teachers and shows how each influence was validated. Each of the knowledge,
motivation, and organizational influences were validated using qualitative and quantitative data
to the extent that such data was available.

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Table 9
Summary of Assumed Influences and Validation
Assumed Knowledge Influence Survey Interview Document
Analysis
o Teachers know factors that could cause gaps in learning.
o Teachers know grade level expectations in mathematics and
the learning progression from elementary through middle
school.
o Teachers know how to identify learning difficulties and
choose the correct solution path.
o Teachers know how to use effective strategies to deliver the
appropriate instruction.
o Teachers know how to guide students in awareness of their
own thinking and learning process and help them to self-
regulate their learning.
X

X

X
X

X

X

X
X
X
X

X

X
Assumed Motivation Influence Survey Interview Document
Analysis
o Teachers value the goal of every student achieving proficiency
in grade level power standards by the end of the school year.
o Teachers believe that engaging in a RTI program will
contribute to students filling their learning gap.
o Teachers are confident in their ability to execute the RTI
program to achieve 100 % student on grade level power
standards.
o Teachers are confident in their PLC team and with the LS
teachers’ ability to reach the goal.
X
X

X

X
X
X

X

X

X
Assumed Organizational Influence Survey Interview Document
Analysis
o Teachers have the resources needed to achieve the goal.
o The organization has structures in place where roles and
responsibilities have been delineated and an effective
communication process exists to share necessary information
between stakeholders.
o The organization has policies and procedures for RTI to
support the teachers in achieving the goal.
o There is a process that ensures the teachers get timely,
concrete feedback about their intervention effectiveness.
X
X

X
X
X

X

X

X

X

X

Results from the surveys of the sample group helped to address confidence issues of the
respondents as well as set the stage for pro-active treatment or prevention of possible problem
areas. The results from the interviews of the same sample group were transcribed and
subsequently coded into themes related to knowledge, motivation and organizational categories.
Document analysis was conducted to triangulate survey and interview results.
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The data collected from surveys, interviews and document analysis was analyzed in the
context of knowledge, motivation, and organization. For the knowledge category, data was
assessed using Krathwohl’s (2002) description of knowledge as factual, procedural, conceptual
and metacognitive. To assess motivation, data was examined for the presence of data in the form
of active choice, persistence and mental effort as seen in the students’ participation in the
intervention to support their need in learning. Finally, the analysis assessed the organization by
looking into the structures, policies and procedures, resources, stakeholder values, and the
culture of the middle school mathematics department.
Trustworthiness of Data
Qualitative research can never capture an objective truth or reality, but there are a number
of strategies that can increase the credibility of the findings (Merriam & Tisdell, 2016). This
study used four ways to ensure the data was trustworthy: 1) triangulation of data between
surveys, interviews and document analysis, 2) the use of multiple methods of data collection to
cross-check, 3) assurance of anonymity in the survey and confidentiality in the interviews and 4)
peer examination to assess whether the findings are plausible, based on the data.
Role of Investigator
I was a middle school mathematics teacher at the International School South East Asia
Middle School at the start of the study. I was a Professional Learning Community Leader for
Grade 7 mathematics and filled the roll of the Middle School Math Department Chair for one
semester. I was responsible to ensure K-12 alignment of school-wide initiatives, including the
implementation of Response to Intervention (RTI). During the data collection phase of the
study, I left the school for employment elsewhere and I continued the research from an off-site
location. Because of the change in my position, I was knowledgeable about the process of work
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being done in the middle school mathematics department but I no longer had an affect on the
day-to-day work of the other employees. They were more willing to share information me
without the threat or pressure of my previous supervisory role. As the principal investigator of
this project, my roles were to conduct a gap analysis of the performance problem stated above
and propose strategies for teachers to help their students to fill their mathematics learning gaps
and for 100 percent of students to achieve proficiency in grade level power standards. Meeting
the organizations academic goal helped ensure that the school achieved its objective for every
student to learn at high levels.
For this project, stakeholders were informed of my role as the investigator, and they were
reassured that steps were taken to ensure anonymity in the surveys and confidentiality in the
interviews. As part of the interview protocol, stakeholders were reminded of the primary
purpose of the study to learn more about works well already in the Middle School at
International School South East Asia and to offer research based suggestions for continuous
improvement. Stakeholders were reminded that surveys and interviews were voluntary and there
would be no negative consequences if they decided not to participate. All stakeholders were
informed that the study was for the purpose of completing a doctoral dissertation and findings
would be presented to the leadership team. The leadership team has the final authority to act on
the findings of this project.
Limitations and Delimitations

This study acknowledges several limitations. First, the sample size was small, consisting
of only six teachers. In addition, the self-reported data provided by interview participants was
limited because it could not be independently verified. Self-reported data also contained several
sources of bias; participants could have selective memory and choose what to remember from
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events that occurred from the past and they could exaggerate the outcome so the event appears
more significant than the data suggests. Participants may have feared a negative impact from
their responses or comments in surveys or interviews and withheld answers. Finally, participants
may have interpreted the questions differently and provided answers that were not connected to
other participants in desired manner.
The focus of this project was to conduct a gap analysis to study how to help teachers
support students with learning deficiencies in mathematics. The primary delineation of this
study was that it was specific to the middle school mathematics department at the International
School South East Asia. This study may have applicable portions that would be a benefit in
other contexts but for the most part it was not generalizable to other schools. Other institutions,
however, may benefit from the application of this project’s use of the gap analysis method to
improve the use of interventions to help students achieve grade level proficiency in power
standards by the end of the school year.

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CHAPTER 4: RESULTS AND FINDINGS
The purpose of this study was to conduct a gap analysis to examine the knowledge,
motivation and organizational influences that interfere with middle school teachers’ ability to
create a sustainable intervention program where 100% of students receive support and are able to
achieve proficiency in all grade level standards. The study focused on strategies that middle
school math faculty currently employ to support struggling students, organizational structures
that are in place to enhance mathematics teachers’ ability to provide support to struggling
students, and suggestions of new programs that could be implemented to enhance the current
intervention program. While International School South East Asia (ISSEA) has achieved great
success in offering a culture of excellence, extraordinary care, and possibilities, the challenge
exists in whether middle school math teachers have the knowledge and skills, motivation and
organization supports to ensure that every student, especially those who exhibit learning gaps,
learns at high levels and personalizes their learning.
This chapter presents the data collected from a review of school documents and math
class websites or blogs, a survey based on the Kirkpatrick & Kirkpatrick (2016) four levels of
training, and standardized, open-ended interviews conducted with a purposeful sampling of
ISSEA’s middle school math teachers. Research Question One examined the strategies that
currently exist at ISSEA middle school to help students improve their performance so they can
all achieve math grade level standards. Research Question Two explored the knowledge and
skills, motivation and organizational causes that exist to provide support to middle school math
teachers’ ability to fill students’ learning gaps while at the same time guiding them to achieve
grade level standards by the end of the year. Research Question Three explored new programs
that ISSEA could implement to support struggling middle school math students. The Clark and
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Estes Gap Analysis Framework (2008) was used to help identify whether knowledge, motivation
or organizational factors were the greatest barriers for teachers ability to support struggling
students.
The researcher will first describe the participants in the study, followed by the discussion
of the results.
Participants
This qualitative study collected data by surveying and interviewing faculty from the three
grade levels of the middle school: Grade 6, Grade 7 and Grade 8. A purposeful sample of six
teachers participated in the survey and four of those six teachers participated in interviews. They
were selected as returning teachers to explain the intervention program in the mathematics
department.
There was at least one teacher from each grade level and each teacher that participated in
the interview process held a leadership role as the middle school math department head or the
Professional Learning Community Coordinator (PLC) for their grade level team of three
teachers. The following codes are used to ensure anonymity for the participants: MS6.1, MS6.2,
MS7.1, MS7.2, MS7.3, MS8.1.
Report of the Findings
This chapter presents the research findings as they relate to the three areas of this study:
1) results of Research Question One - effective, existing strategies, 2) results of Research
Question Two – knowledge and skills, motivation and organization influences, and 3) results of
Research Question Three – possible new programs. The chapter concludes with a summary of
the findings from the research results, which will be further discussed in Chapter 5.

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Results of Research Question One: Effective, Existing Strategies
What strategies exist at the International School South East Asia middle school to help students
improve their performance so they all meet grade level standards for math proficiency and/or
achievement?
International School South East Asia has adopted the Response to Intervention (RTI)
method for implementation of interventions for students who struggle in mathematics. RTI is a
multi-tier approach for the early identification and support of students with learning and behavior
needs. The RTI process begins with high-quality instruction and universal screening of all
children in the general education classroom (Buffum et al., 2009). At each problem-solving
level, the process is to determine the magnitude of the deficiency, analyze the causes, design a
goal-directed intervention, conduct the intervention, monitor student progress, modify the
intervention based on student responsiveness, evaluate the intervention’s effectiveness and plan
future actions (Fuchs and Fuchs, 2006).
Document Analysis Results
In order to explore the strategies that currently exist to support struggling students, this
researcher investigated digital documents for grades six, seven and eight to help identify what
strategies could be identified through review of meeting agendas, lesson slides, unit planning
documents, online websites and blogs, learning support progress spreadsheets and student
learning plans.
Grade six. In document analysis of 6
th
grade files, there was evidence from the fall MAP
standardized test results that basic skills were lower than expected for the student cohort for the
2016-2017 school year and an intervention, a daily spiral skills review, was executed for the
entire year. A sample of the daily spiral skills review can be seen in Figure 6.
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Figure 6. An example from the Grade 6 daily skill review and intervention.
The Grade 6 Math PLC created a SMART goal for the school year where 90 percent of students
who scored below 220 would achieve the targeted growth of eight points in the Number Systems
category from fall to spring. The MAP test results from the fall and spring assessments show
collective improvement across the grade. The SMART goal and MAP results are shown in
Figure 7.

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A B C D E F G
Goal: 90% of sub 220 scorers on MAP Number Systems will achieve target growth in
NS from Fall to Spring. (Currently achieving 70%)
Student
Name
Number
System
Fall
Number
System
Spring
Number
System
Growth
Ave growth
RIT points
among these
same
students
S1 189 215 26
Number
Systems 16
S2 199 231 32 Ops & Alg 11
S3 205 233 28 Geometry 9
S4 206 219 13 Statistics 8
S5 208 234 26
S6 210 232 22
S7 210 222 12
S8 211 206 -5
S9 211 217 6 Yes 24
S10 212 227 15 No 5
S11 212 249 37 83%
S12 213 232 19
S13 213 237 24
S14 213 234 21
S15 213 223 10
S16 213 234 21
S17 214 219 5
S18 214 217 3
S19 216 212 -4
S20 217 240 23
S21 217 230 13
S22 218 226 8
S23 218 230 12
S24 218 238 20
S25 219 240 21
S26 219 228 9
S27 219 231 12
S28 219 227 8
S29 219 244 25

Figure 7. Sixth Grade MAP test results showing collective improvement in the number systems
category.
In Figure 7, the sixth grade MAP scores from the fall testing period were collected for
students who were below the 70
th
percentile, or alternatively, achieved a score less than the
targeted score of 220 in the Number Systems category (Column B). Although the daily spiral
review of basic skills could not be directly linked to individual student learning of basic skills,
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the improvement in the Number System category from the fall MAP test (Column B) to the
Number System category on the spring MAP test (Column C) showed that 24 out of 29 students,
or 83 percent of students, achieved the desired eight point growth rate or better. Other factors
could be involved in student growth in basic skills, but the grade six Spring MAP test results
highlight overall improvement and pinpoint specific students that still need more support. The
Grade 6 Math PLC used data to inform a common intervention and collectively tracked student
progress from fall to spring of one school year, specifically identifying which individual students
still needed help.
One problem noted during research was that this spreadsheet, including the grade six,
student growth analysis, was not stored in the digital files of the mathematics department. The
MAP data analysis was found as an attachment in an email. The inefficient storage of data
resulted with information that was not accessible to all who need it in the scope of their position.
In addition, in Figure 1, Row 1, the Grade 6 Math PLC stated their goal where 90 percent of sub
220 scorers on the MAP Numbers and Systems portion of the test would achieve the target
growth rate by the Spring MAP test. Using 90 percent rather than 100 percent indicated a lack of
confidence in the collaborative ability of the Grade 6 Math PLC to support all students in
learning at high levels. Sixth grade mathematic teachers show documented evidence of student
growth in learning, but they did not believe they could achieve 100 percent and it was not clear
whether the Learning Support teachers participated in the intervention or had access to the names
of students who still needed support. Document analysis revealed possible problems with data
storage and with communication within the intervention team that prevented all students from
achieving grade level standards by the end of the school year.
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Grade seven. Further document analysis into 7th grade files revealed more evidence that
the middle school math teachers were using effective strategies to support students with learning
gaps. In grade 7, Tier 1 support was evident in common formative and summative assessments
that could be easily found in shared assessment folders in every unit of study. In addition, unit
summative assessment results were recorded and retake assessment results were also recorded
for the entire class for the purpose of data analysis, reflection, and continual improvement of
instruction. Interview responses revealed that all grade levels in the middle school at ISSEA
have adopted this data collection method for assessments, starting in the 2017 - 2018 school
year.
Figure 8, shows a sample of test results for one unit assessment, the first unit assessment
called ‘Shapes and Design,’ which was based on the Geometry strand for Grade 7. The top left
frame shows results for students’ first attempt at a unit assessment. Performance levels, such as
Exemplary, Meeting, Approaching, and Below, are listed in Column A and the number of
students reaching each performance level are categorized according to the students’ classroom
teacher (Columns B, C, and D). The totals column in the top left frame (Figure 8, Column F)
shows total results for the entire 7
th
grade class.
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Figure 8. Seventh grade unit summative assessment results and reassessment results.
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The first summative assessment result, in the top left frame of Figure 8, reveal that 88
students scored in the approaching or below level which means they did not meet grade level
expectations (Column F) after the first initial assessment attempt. Examination of the 7
th
grade
shared digital files showed evidence of Tier 1 support where this common unit assessment was
used to identify students’ weak areas. Class lessons were adjusted on the Grade 7 Math planning
and pacing calendar to provide differentiation for the whole class in September 2016 (See
Appendix C). The bottom left frame of Figure 8 shows test results after the class wide
intervention and subsequent retest. The class as a whole showed some improvement by the
upward movement between performance levels, but there were no specific students tracked to
see if the improvement was a direct result of the intervention and 48 students still did not meet
expectations for the grade level standard in Geometry.
The two, green frames on the right side of Figure 8 analyze student test scores by teacher.
While there is information to show that all teachers exhibited improvement in their students’
retest results, Teacher MS7.2 had a 71 percent increase in the number of students ‘meeting
expectations’ after conducting the intervention while Teachers MS7.1 and MS7.3 showed only a
31 to 40 percent increase, respectively. In the Grade 7 Math PLC agendas before and after the
‘Shapes and Designs’ intervention, there was no evidence of this analysis and there was no
documented discussion on the different approaches to the intervention (See Appendix D). There
is also no documentation in agendas to show involvement by a Learning Support teacher in the
intervention.
Data was collected for each unit assessment to track and monitor students’ collective
results and improvement after interventions and reassessment. Data was also collected to ensure
consistent results for all grade level teachers and to inform and improve instructional practice.
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However, there does not appear to be data analysis and collaborative revision of instruction in
the Grade 7 Math PLC agendas, which document planning meetings.
In addition to review of the Grade 7 Math PLC documentation showing effective use of
interventions based on unit test results data, effective strategies were also discovered in a
preplanned intervention for a major content area for seventh grade, simplifying expressions and
solving equations. In the Grade 7 Math PLC files, a reflection document was found revealing a
goal created based on the previous year’s fall MAP standardized test data. See Figure 9 on the
following page. A program was created and implemented to introduce the use of a manipulative
(algebra tiles) for use in simplifying expressions and solving equations. Teachers noted that the
plan originally included a collaborative intervention for 51 students but the number dropped to
only 30 total students at the start of the next school year, due to the transition of students to
another school. Rather than change the goal, the Grade 7 Math PLC decided to give the
intervention to the whole class because they had not yet been exposed to this teaching method.
In Figure 9, limited improvement was recorded and 10 out of 30 students needing support
were able to achieve the grade level standard after this intervention. In the far right column, the
total number of students who needed the intervention was recorded at 30. Beneath the starting
row was another row of numbers and the totals column showed 20, the number of students after
the intervention. Twenty students still did not yet meet the target ‘Average’ category on the
MAP or the grade level performance standard, which was assessed in class. Despite this lack of
achievement, the red, blue and black qualitative comments by Teachers MS7.1, MS7.2 and
MS7.3, in the below figure, indicated that 15 of the 20 still struggling students grew in their
learning and reached the projected targeted growth rate for the MAP testing period from fall to
spring.
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Figure 9. Seventh grade Math PLC reflection on SMART goal to improve students
understanding of operations and algebraic thinking through the use of algebra tiles.
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The Grade 7 Math PLC used data to inform their intervention and to help identify
specific students needing extra support. Teachers tracked and monitored a general group of
struggling students and gave additional time and support to those who needed it.
Grade eight. The 8th grade files showed extensive data collection from standardized
tests, school assessments and annotated comments that identify the weak areas of every student
in 8
th
grade (See Figures 10 and 11). In columns F, G and H of Figure 10, scores are recorded
for an initial diagnostic test at the start of the year. Categories on the initial diagnostic test are as
follows: Number Systems (Column G), Expressions and Equations (Column H), and Linear
Relationships (Column I). The total score is in Column I and the box is highlighted in red for
students of great concern and a darker yellow for students that need to be watched for potential
support needs. Basic skills and fluency are tracked using the Hooda Math online timed tests and
results are recorded in Columns J and K. Fall MAP test scores are documented in totality
(Column L) and by percentile as compared to peers (Column M). The students’ range for each
category of the test is recorded in Columns O through R. Again, boxes are highlighted in red or
darker yellow if students do not achieve the targeted performance level or range.

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Figure 10. Eighth grade quantitative data for all 8
th
grade students during the 2016-2017 school
year.
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In Figure 11, the Grade 8 Math PLC Teachers commented on students that presented a
potential concern and make recommendations for Tier 3 support, where students would be pulled
out of the classroom for a specific skill. In this case, the Tier 3 support was called ‘Algebra 1
Lab.’ Student data was recorded in the third column titled ‘Fall 16 MAP %ile/Semester 1 Math
Grades.’ The first number in the respective box in this column is the MAP percentile, a ranking
against peers. The three words below this percentage value represent the performance level
achieved on the three unit assessments that occurred during the first semester. The next column
is titled ‘Math Needs’ and describes what problems were noted during the first semester and
what strategies had been successful for the student. Finally, accountability for parent
communication was evident in the last column. There was no further evidence of interventions
in the math department files resulting from placement into the Algebra 1 Lab intervention
program.

Figure 11. A sample portion of the 8
th
Grade qualitative data for struggling students and their
recommendation for Algebra Lab Intervention.
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Despite the lack of documentation after the gaps were identified, the Grade 8 Math PLC
showed strength in both quantitative and qualitative data collection and analysis that guided their
understanding of which students needed interventions. Specific students needing support were
tracked and monitored and parents were notified.
The Grade 8 Math PLC also demonstrated other effective strategies directed towards
students who exhibited a need for extra help. For example, differentiated instruction for the
whole group was evident in shared daily lessons after formative (ungraded) assessment revealed
students who didn’t achieve the desired standard. An example of Grade 8 class slides showing
differentiation can be seen in Appendix E. Class slide shows revealed a formative assessment,
which starts the lesson, followed by student choice of small group instruction. Small groups
were leveled by the amount of instruction needed to complete the task. Some groups worked
independently, following lessons in the class slide shown on their computer. Some worked with
the classroom math teacher to understand the daily lesson. Some who felt confident and
achieved a high performance level on the formative assessment were offered opportunities to
work in small groups on extension projects. While eighth grade mathematics teachers used
common formative assessments to find students with gaps in learning, there was no evidence that
these differentiated sessions directly impacted student learning. It does not mean that
intervention was not successful; there was just no documentation linking the formative
assessment data to improved scores between the pre-assessment and post-assessment of student
knowledge per unit of study and during the course of instruction.
Another problem noted during research was the lack of procedures in place for MAP data
to be efficiently entered into a spreadsheet and disseminated to teachers for their analysis and
use. Although the fall test results were sorted into a spreadsheet, the spring results were not
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disseminated, until requested by teachers. When received, the spring MAP test results were not
in the same spreadsheet format as the fall results so it was difficult to enter the data in a coherent,
helpful way. In Figure 10, which records 8
th
grade quantitative data, on the far right column, the
spring MAP assessment results were not entered. The comparison of fall and spring MAP test
data, by student and by category, was impossible without the spring MAP test results entered in
columns S through X.
The MAP testing online website reports data in graphical form as well as listing results
by student names (NWEA, 2017). The data is accessible but does not show individual student
scores by categories, such as Number System, Operations and Algebraic Thinking, Statistics and
Probability, Geometry and Solving Equations. While students’ overall growth in learning can be
seen by results of the MAP test data on the online MAP website, it is difficult to pinpoint
specific areas of weakness to guide future interventions. In addition, the information flow from
administration to faculty regarding the MAP test results was not consistent each testing cycle,
nor was the spreadsheet format to display the specific data of the MAP test results. The result
was an inability to use data effectively to plan focused interventions for students in need.
The middle school math department faculty showed evidence of using summative and
formative data to identify learning gaps and to inform decision-making regarding interventions.
They use standardized MAP test results, a start of the school year diagnostic inventory,
qualitative comments from previous teachers and the previous school year’s standards
achievement as initial summative data to start understanding who may need support during the
school year. During the school year, they use formative data, such as pre-assessment, mid-unit
checks, student social and behavioral concern meetings, and practice tests to ensure that
students are receiving immediate and daily, tiered support. All teachers showed evidence that
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they track and monitor individual results and some teachers have documented parent
communication. Math teachers also showed evidence that collective groups of students have
improved their understanding of the targeted math content through unit assessments and retake
opportunities on unit tests (as noted in the sample record of test scores in Figure 8) and through
standardized MAP testing in the spring and fall. Although there was no direct evidence
showing that individual students are benefitting or not benefitting from the interventions, the
impact of using the data to inform their instruction in significant and positive gains across
grade-levels were evident.
Interview Results
The interview response data was gathered from four mathematics teachers spanning each
grade level of middle school. The following data was collected and revealed strategies
currently used by middle school mathematics teachers at ISSEA to help students improve their
performance so they all meet grade level standards for math proficiency and/or achievement.
The existing strategies utilized by middle school mathematics teachers were categorized into 13
areas, designated by letters A through M.
A. Differentiated instruction: three-tiered instruction
B. Push-in model with Learning Support (LS) teacher for Tier 1 and 2 support
C. Pull-out model with LS teacher for Tier 3 support
D. Flexible grouping between grade level teachers
E. Professional Learning Communities (PLC) for teacher collaboration
F. Start of the year mathematics foundational skills inventory diagnostic assessment

G. Practice exams prior to unit of study final summative assessment
H. Review opportunities or additional support
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I. Parent involvement and support
J. Student concern meetings to address academic, social and behavioral issues
K. Formative assessments to provide timely feedback
L. Teacher effectiveness in the classroom
M. Metacognition
The literature review in Chapter 2 revealed the most effective teaching strategies to help teachers
to deliver the appropriate intervention, and these eight strategies are listed in Table 10.
Table 10
Effective strategies from literature review to help teachers improve mathematics learning at the
middle school level
Validated Not
Effective Strategy Validated in Part Validated
1. Teacher effectiveness through cooperative
learning √

2. Teaching Methods – Math teaching practices √

3. Explicit instruction of mathematical practices
in 21
st
century instruction √

4. Math practice at home and tuition
(peers or hired) √

5. Parent involvement √

6. Personalized learning √
7. Metacognition and goal setting √
8. Response to intervention √

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Further analysis of interview responses revealed that the teaching strategies currently in use at
ISSEA’s middle school, designated by letters A through M, ‘validated’ or ‘validated in part’ the
eight effective strategies previously outlined in the review of the literature in Chapter 2.
Table 10 outlines results from the review of literature from Chapter 2 for the eight
strategies found to be most effective in improving math learning in middle school.
Strategy 1: Teacher effectiveness through cooperative learning. The first strategy,
highlighted in the review of literature as the most effective strategy, was ‘Teacher Effectiveness
through Cooperative Learning.’ Middle school mathematics teachers noted in interviews that
they employ similar methods to those methods also used in research-proven, successful models
of cooperative learning: flexible grouping between grade level teachers (Letter D above),
collaborative planning in PLCs to support the focus on cooperative learning (Letter E above),
and the use of practice tests prior to unit of study assessment (Letter G above).
Strategy 2: Teaching methods – math teaching practices. Second, the second most
effective strategy in the review of literature was the eight effective mathematics-teaching
practices, which were formalized by the NCTM (NCTM, 2017), as outlined in Figure 12.
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Figure 12. Eight effective teaching practices as stated by the NCTM (2017).

While the middle school teachers did not explicitly discuss the use of math teaching practices as
guiding their classroom instruction, their interview responses and planning documents
demonstrated evidence of these eight mathematics-teaching practices. For example, the first
practice is to establish goals to focus learning. MS8.1 noted that “at the beginning of each unit,
students write one or two goals and then they reflect on these goals at the end of the unit.”
MS8.1 further stated that students were given learning target sheets, which they glue into their
notebook and then revisit as the unit progresses (See Appendix F). Other grade levels in middle
school also use goals to focus learning in the form of concept checklists in 7
th
grade (Appendix
G) or ‘I can…’ statements templates in 6
th
grade (Appendix H). Evidence of math teaching
practice number 2 was seen in the implementation of tasks that promote reasoning and problem
solving. Teacher interview responses from all grade levels described the process of using a
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three-tiered, problem-solving method in their classrooms and samples of tiered problems for a
variety of middle school math topics were found in document analysis (Appendix I). The three-
tiered, problem-solving method also supports mathematics teaching practice number 7 where
students engage in productive struggle in learning mathematics. Teacher MS7.1 noted that this
method was motivating for students and they pushed themselves to try the harder questions. All
participants in this study demonstrated mathematics teaching practice number 4 by facilitating
meaningful mathematical discourse and number 5 by posing purposeful questions in their
classrooms by the use of small groups in their classroom design. Teacher MS8.1 said that he
wanted to make students comfortable in asking questions so he created his classroom as a
Positive Learning Zone. He provided opportunities to work with partners, teams or individually
and changed his groupings to give opportunities to work with other students. Teacher MS8.1
specifically stated, “Discussing with a partner or sharing answers with the whole class helps to
build confidence and students can learn from each other.” In addition, agendas for each math
PLC revealed the math teaching practice number 8 where teachers show some evidence of using
student thinking to assess student progress and to adjust instruction (Appendix J).
Strategy 3: Explicit instruction of mathematical practices in 21
st
century instruction.
The third most effective strategy outlined in the review of literature in Chapter 2 was ‘Explicit
instruction of mathematical practices in 21
st
century instruction.’ The eight math practices are
standards of achievement for students, which is different than the eight effective mathematics-
teaching practices for teachers. The ‘explicit instruction of mathematical practices’ was
‘validated in part’ by interview responses and document analysis. Planning and instruction
through the lens of the math practices could be seen in documents in the math department files,
such as the Grade 6 Math PLC’s ‘Unit Summative Assessment Itemized Math Practice Standard
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Alignment’ (Appendix K) and the Grade 7 Math PLC’s ‘Concept Checklists’ per unit (Appendix
G), and could also be seen in teachers’ interview responses describing differentiated instruction
for three-tiered problem-solving (Letter A above). However, the standards for math practices do
not appear to be taught or assessed explicitly at this time.
Strategy 4: Math practice at home and tuition. The fourth most effective strategy
discovered in the literature review in Chapter 2 was completing mathematics practice at home
with support from peers and tutors. While this strategy was discussed during literature review,
the research showed varying levels of effectiveness of homework and from support of tutors.
Homework and the use of tutors was not mentioned by any teachers during interviews and
received the lowest number of priority on the survey question asking to rank the most important
factors for students that need support. However, a need for time to practice mathematics
problem-solving outside the classroom was evident in all stakeholder interviews, as documented
in Appendix L.
Strategies 5 – 8: Parent involvement; Personalized learning; Metacognition and goal
setting; and Response to intervention. The remaining strategies for effective mathematics
instruction during middle school that were discovered during the review of the literature,
numbered five to eight in Table 10 and included parent involvement (Letter I above),
metacognition and goal setting (Letter M above), and Response to Intervention (Letters B, C, F,
J, and K above) were all validated through document analysis, survey results and interview
results. Personalized learning was validated in part (Letter A above) and was also seen in
overlapping categories shared with Response to Intervention (Letters B, C, F, J, and K above).
Each of the strategies from Letter A to M that are currently being used by middle school
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mathematics faculty is described in more detail below. Appendix L provides a detailed summary
of existing, effective strategies currently in use by middle school mathematics teachers at ISSEA
Teacher Effectiveness through Cooperative Learning (Literature Review - Table 10,
Strategy 1 and Current ISSEA Strategy - Letter A)
The middle school math faculty in each grade level (grades 6, 7 and 8) employed a
complex model of cooperative learning. Of the four teachers interviewed, all believed students
learning in a group would achieve more than students learning without a group, supporting Lou’s
findings (1996). In addition, Slaven (2009) found, after a review of more than 100 studies that
cooperative learning had more impact than teaching mathematics content alone. In every math
class across grade levels in the middle school at ISSEA, students are in classes of no more than
24 students and they are grouped with one to three other students. Classroom furniture in every
middle school math classroom is set up with table groups. As Teacher MS8.1 points out,
“Groupings are changed to give students the opportunity to work with other students.” Teacher
MS8.1 describes a classroom based on the ideas of Piaget (1954) and Vygotsky (1978) where
other positive advantages of grouping students are the social aspects of cognitive growth where
students develop social and communication skills because of the need an opportunity to work
with. MS8.1 explains, “Grouping students helps with student communication of math. They are
discussing with a partner or sharing answers with the whole class to build confidence and learn
from each other.” All teachers confirm the idea presented by Johnson, Johnson, & Smith (2014)
whereby all students work together to maximize their own and each other’s learning. They
specifically report that students are problem solving with other students and sharing strategies
with each other in partners or in small groups. Communication in groups as well as discussions
in class occurs often in the middle school at ISSEA.
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For the majority of the math teachers in middle school, the benefit of grouping allows
time for teachers to work more closely with students struggling to understand a concept. MS6.1
had this to say about the benefits of small group work. “Small group intervention work happens
when some students work more independently with challenging stuff that engages and motivates
them. This allows more time that I can work one-on-one within the classroom environment with
certain groups.” MS6.1 specifies, “You have time to work with struggling kids.” In addition,
peer learning allows the teacher time to remediate and adjust learning objectives and pacing to
meet the needs of the class (Lou et al., 1996). At ISSEA, middle school math teachers
collaborate to meet the needs of all students in the grade level by grouping students across
classes.
Flexible grouping between grade level teachers. (Literature Review - Table 10,
Strategy 1 and Current ISSEA Strategy - Letter D) The 6
th
and 8
th
grade teachers describe a
complex structure for within class groupings where students are flexibly grouped across the
grade level for particular topics based upon formative pre-assessment. Teacher MS6.1 notes:
We have different classes in the same grade level and we group students according to
formative assessment results and based on per lesson need. One of us takes the students
at or above grade level expectations and the other one will take the struggling students.
We swap sides or we co-teach in the library where one teacher delivers instructions.
Some students work independently while we have two of us helping those in need.

Teacher MS6.2 confirms by stating:
Sometimes, even at the last minute, we team-teach and go to the library. I can be
working with kids that need support with the lower level problems while MS6.1 works
with students who need support with the higher-level problems.
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While some grouping happens at the last minute, some within grade level grouping is planned
ahead of time. MS6.2 explains, “For each unit, kids who need support go to one teacher. Next,
will be my turn. We try to swap the low kids and high kids so they don’t always feel like one
teacher is this and one teacher is that.” Likewise, in the grade 8 math classes, interventions are
conducted between grade-level classes. Teacher MS8.1 explains a similar situation to that which
is happening in grade 6 classes.
Students who struggle stay in my classroom and the ones who already mastered the
standards go to the other classroom for extensions. Average kids go to the LS teacher.
We try to change the teacher so students don’t go to the same teacher for extensions or
interventions each time.

As Lou (1996) and Tieso (2005) suggest, the positive effects of within class grouping are
maximized when groupings are accompanied by variations to teaching methods and the middle
school math teachers, through their willingness to attempt different approaches, are creative in
their to efforts to ensure all students are learning at high levels.
Professional Learning Community (PLC). (Literature Review - Table 10, Strategy 1
and Current ISSEA Strategy - Letter E) Teacher instructional process has more effect than
curricular design alone. At the ISSEA, teachers work in PLC models where teachers are grouped
into one team by discipline and another team by grade-level. For the math PLC, the schedule
supports common planning time with grade-level math teachers and the Learning Support
teacher and sometimes there are times to meet as an entire math department. The teachers who
were interviewed highlighted five major areas where they felt the PLC model helped to support
and enhance their ability to affect student learning: 1) time to share teaching methodology; 2)
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time to create common curriculum with consistent learning targets; 3) time to create common
assessment tools and to review and analyze data recording documents; 4) planning time with the
Learning Support specialist to support needs of those who don’t understand the material as it is
presented and to support those who need an extension; and 5) observations to improve and
calibrate instruction and grading.
Teacher MS6.1 states,
We work in a PLC model. We talk with the previous grade level so we don’t overlap and
we are consistent in the methods we teach foundational skills. Common planning time
allows time for teachers to discuss teaching methods to ensure that each grade level
supports the instruction of the following year.

Teacher MS6.2 further elaborates,
We are tight as a PLC. We are very in-line and we share a planning and pacing calendar
and daily lessons. We have a common blog with resources for students and parents and we
create common assessments. We calibrate our grading by grading our tests together.

Similarly in grades 7 and 8, teachers decide together on common standards that will be taught
during the year and they create common assessments for each unit. Teacher MS7.1 highlighted
the following; “We create common assessments and collect data on each assessment, keeping a
spreadsheet to note patterns that we can use to better our instruction.” Teacher MS8.1 agrees “by
recording assessment results in each class by each different teacher, we can see what is different
in the classroom that’s working.” The common planning time with the Learning Support (LS)
teacher is new to the 7
th
grade this year and Teacher MS7.1 commented, “The LS teacher is able
to come to all of our PLC meetings and she can work across all sides (groupings with-in a grade
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level) and in each teacher’s classroom. Because of two of three teachers being new to 8
th
grade
this year, the time to observe each other and to have others observe their teaching has been
important to create consistent teaching throughout the grade level. Teacher MS8.1 states:
Teachers observe each other and the LS teacher helps by substituting in the classroom to
allow teachers time to observe. The observations help to improve and calibrate our
instruction and ensure it is consistent between the three different grade-level teachers and
to ensure students receive the same quality of instruction. Principals observe, share
observations and provide feedback.

Teachers in different grade levels had different needs but all teachers that were interviewed
highlighted the importance of this PLC structure because of shared planning time in the schedule
which contributes to their ability to teach consistently to all students in a large school
environment and to provide the support struggling students need to be successful.
Personalized Learning (Literature Review - Table 10, Strategy 6 and Current ISSEA
Strategy – Letters A, B, C, F, J, and K)
The most valuable component of the network of student, educator, parent and school
community is the student and their ownership of their learning. When learning is personalized,
students will be able to develop more meaningful relationships with their peers, teachers and
members of the larger communities (Gates, 2014). The curriculum needs to be developed so that
students understand and are able to determine what they have already learned, what they are
currently learning, what the will learn next and why they are learning it (ISSEA Personalized
Learning, 2017). In the middle school, math teachers across grade levels are working towards a
more personalized program for each individual student. Document analysis revealed that all
grade level PLCs created concept checklists or “I can” statements based on the learning targets
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for each unit so student understand what they need to know and where they are in the process.
Samples of learning target documents for each grade level can be seen in Appendix F (Grade 8),
Appendix G (Grade 7) and Appendix H (Grade 6). Through the tiered problems solving model
of instruction, students have choice in their own learning and they can visually see the next level.
Educators are collaborating in PLCs to differentiate learning, provide feedback and modify
instruction based on formative assessment. At this time, middle school students are guided in the
process of articulating their own learning plans, applying feedback and setting future goals based
on evidence.
Three-Tiered Problems for Differentiation Instruction. (Literature Review - Table
10, Strategy 6 and Current ISSEA Strategy - Letter A) This year, at the start of the school
year, all middle school math teachers attended a professional development training to learn a
new method of teaching problem solving in mathematics. The method, called “Challenge by
Choice”, was created and refined by the teachers from the Jakarta Intercultural School in
Indonesia from 2003 until present time. “Challenge by Choice” has three tiers of problems with
the level of challenge coordinated to the colors on a ski slope – Green (Standard), Blue
(Intermediate) and Black (Challenge). Students are presented all three levels of problems on a
similar topic and they are guided in choosing the problem that they are ready to work on given
their current understanding (Suarez, 2017). All middle school mathematics teachers have agreed
to implement this new method and have stated that there have been initial positive results.
All teachers who were interviewed agreed that the tiered method of instruction was
advantageous to higher-level students who would work independently with problems that
engaged and motivated them. They also all agreed that this new method helped to identify
weaknesses and offered additional time to work with certain groups. Teacher MS7.1
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commented, “It’s motivating for high-level kids and they push themselves and some of the lower
end students got up to try the black questions and they could totally do it.” Teacher MS6.1
explained a process that seemed similar across grade levels. “ The problems are on the board
with solutions so students can see the problem solving steps of it. They physically move around
the classroom to solve different problems.” Students have a choice of which level they will work
on. “The students objective is to think of their optimal learning level and they have a choice of
what they want to work on. They know that green is the standard and what will be on the test,”
stated Teacher MS6.2. Teacher 7.1 reiterated, “Kids choose the level the are going to work on.
The standard level is what most of the test is going to be like.” In class periods of 83 minutes,
students have approximately one hour to work on problems, an amount of time they have never
had before due to a more direct teaching style. Teacher MS6.2 described,
Students get an hour to work, they never had an hour to work. They are super into it
because the problems still make them think, but they feel like the are all challenging
themselves so they tackle the problems at their own pace. Students are engaged in learning
for a whole hour because they want to and that’s amazing.
While grade 8 teachers follow the same process and use the same teaching methodology of three-
tiered instruction, they have chosen the blue, middle level to be their standard level and the
green, lowest level to be their intervention level. The other grade levels do not have a level for
struggling students. They continue to employ flexible grouping to provide additional support to
students who are unable to meet the standard level given the regular instruction.
The National Council for Teachers of Mathematics (NCTM) defines an effective teacher
with eight effective mathematics-teaching practices to ensure success for all students (Appendix
M). The teaching practices that are most emphasized by the middle school mathematics teachers
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in their newly adopted three-tier instruction are: 2) Implement task that promote reasoning and
problem solving, 3) Use and connect mathematical representations, 4) Pose purposeful questions,
and 7) Support productive struggle in learning mathematics.
Start of the year inventory assessment. (Literature Review - Table 10, Strategy 1
and Current ISSEA Strategy - Letter F) All middle school math classes take a diagnostic
assessment to help identify areas of weakness as students enter a new grade level. There is a
common focus across grade levels to ensure foundational skills are known and understood. The
grade 6 diagnostic focuses more on basic skills and conceptual understanding of foundational
topics. Teacher MS6.1 summarizes that the grade 6 PLC uses the diagnostic results to identify
students that struggle with basic skills and to put students on radar to keep their eye on during the
year. “There seems to always be a core group who struggle throughout the year and it is good to
keep our eyes on them.” Teacher MS6.2 further elaborates, “The inventory gives us an idea
which kids don’t have their basic skills, such as fractions. Before we started using this
inventory, we were doing skill drills and now we are focusing more on the kids who need
additional strategies to master basic skills.” Grade 7 and 8 inventories focus some on identifying
students with gaps in basic skills but the focus for these grade levels is more on the skills needed
to be successful at the new grade level. In the tracked system in the middle school, the inventory
diagnostic is another data point to verify correct placement into the standard level math class or
the advanced level math class.
Formative Assessments. All teachers commented on using formative assessments to
guide instruction and this year, common pre-assessments per unit have been introduced in all
grade levels. Teacher MS6.1 describes formative assessments for grade 6. “There is a pre-test
for each unit so we have an idea of the skill set. There are also check-ups and pre-unit check-ups
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to re-sort and re-group students. Mid-point master checks (MMC’s) determine which group
students will go into the next day.” Teacher MS7.1 also describes, “a pre-test for each unit that
identifies areas the lower-end kids really struggle with.” In grade 7, Teacher MS7.1 reports,
“The LS teacher analyzes pre-assessment data to determine areas where students are lacking and
pulls out students to work on those underdeveloped skills.” While grade 8 using pre-assessments
per unit, their formative assessments primarily drive their groupings from one lesson to the next.
Teacher MS8.1 explains, “Small group intervention is conducted daily, based on results from
previous lesson.
Practice exams. (Literature Review - Table 10, Strategy 1 and Current ISSEA
Strategy - Letter G) Another change from last year is the addition of practice exams prior to the
actual assessment. Grades 6 and 7 have adopted this practice while Grade 8 has decided not to
use practice test. Teacher MS8.1 explains, “We decided not to do practice assessment because
students are getting ready for high school where they will not be afforded this opportunity.”
In the review of the literature, one of the most effective cooperative learning programs,
IMPROVE, used a system that incorporated practice tests. When students were initially
assessed, those that achieved 80 percent or better went on to enrichment activities while others
received corrective instruction and then take an additional test on parallel material of content
knowledge. At the ISSEA’s middle school, a similar system is followed. Students are given a
practice test that is very similar to the actual test. Teacher MS6.2 states, “It allows students to see
what the test will look like and it puts it all together in a way that makes sense to them now.”
Sixth and seventh grade math teachers consistently described that if a student met expectations
on the practice test, they did not have to take the actual assessment and they could work on an
extension project or enrichment activities. Both sets of grade level teachers acknowledged the
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benefit of an extra one to three days to direct, re-teach concepts to smaller groups. Teacher
MS7.1 excitedly shared, “Kids were excited about the extension projects and it’s motivating
them.” Teacher MS7.1 also shared other benefits where “some students need re-teaching and
some also get multiple opportunities to try and get it.”
Review opportunities or additional support. (Literature Review - Table 10, Strategy
and Current ISSEA Strategy - Letter H) While practice tests were a new addition this year,
review opportunities and additional support did not wane from years before. All teachers who
were interviewed mentioned re-teaching during recess, lunch and breaks, review days before the
summative assessment and multiple re-take opportunities. For all grades, re-takes are required
for two to three key units per grade-level and are offered for those who wish to improve their
understanding in the other units. Grade 7 and 8 teachers offer after school help sessions or
individual appointments. Data is recorded to track students who need multiple opportunities to
meet expectations.
Parent Communication and Involvement
Parent communication and involvement was noted as an effective strategy to help student
in their mathematics learning. It was not one of the most effective strategies and cooperative
learning coupled with teacher effectiveness has proven to contribute to greater student success.
However, parent communication does play a part and can contribute to greater success for a
student. In middle school, students want to be more autonomous and independent and parents
are beginning to feel left out due to new schools with many new teachers and new forms of
communication. Adolescents should start to begin to take responsibility for their schoolwork, to
develop solutions to problems on their own and to build a new relationship with their parent
(Sheldon & Epstein, 2005). But the school can assist in this process by creating and maintaining
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quality communication with the school. For the middle school teacher, this is the time to teach
parents how to navigate in the new middle school context, providing communication between the
school and home and by providing communications through electronic means. Email
communication with parents is consistent in every teacher’s approach for student support. When
students are not performing as expected, teachers reach out to parents through email. Teacher
MS7.1 explains, “Conversations with parents usually happen when there is something pressing,
like when a student didn’t meet expectations and requires a retest.” In addition, there are three
way conferences in October, just before the school’s scheduled fall break. Students are able to
share their learning progress with their parents, noting their goals for the future with the teacher
there to help facilitate the process.
Response to Intervention (Literature Review - Table 10, Strategy 8 and Current ISSEA
Strategy – Letters B, C, F, J, and K above)
At the ISSEA’s middle school, the administration has adopted the model of Response to
Intervention (RTI) to support students who struggle to achieve grade level standards. In a broad
definition of the program, educators use universal screening measures to identify students at risk
of not meeting intended grade level targets. Student progress is monitored and teachers provide
feedback to students to improve their understanding. There are three tiers of support for students.
Tiered support by a specialist. In the middle school of ISSEA, there are three tiers of
instruction that somewhat follow the official RTI program. Instruction is defined in three tiers
where Tier 1 is the primary level and is comprised of the core instruction that every student
receives. Curriculum is determined by the grade level math PLCs and is aligned with the middle
school math department to ensure each standard of learning is covered without overlap.
Together the math department has determined the learning targets deemed most important in the
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context of the overall school program. Tier 2 is strategic small group intervention in response to
an identified weakness to close the achievement gap between them and their peers. This is
designed by the classroom teacher and supported by a Learning Support specialist. In the middle
school math department, Tier 2 is approached in different ways for different reasons.
In Grade 6, two of three math teachers share a multi-discipline LS teacher and the other
teacher has a different and designated math specialist LS teacher due to a pilot program for
interdisciplinary studies in shared spaces. In the sixth grade, the two teachers who share the
support of the multi-disciplinary LS teacher lead the Tier 2 interventions for their classes.
According to Teacher MS6.2, they direct a pull-out model where “the LS teacher usually takes
the lowest kids after the practice test.” The LS teacher no longer comes to designated support
classes so students who may need support but are not officially in the Learning Support program,
will not have access to the LS teacher.
In Grades 7 and 8, the LS teacher leads Tier 2 interventions. Teacher MS7.1 notes, “The
LS teacher analyzes pre-assessment data to determine areas where students are lacking and pulls
out students to work on those skills in small groups.” Teacher MS8.12 concurs, “The LS
teacher pulls out students for necessary skills that students should have before the unit begins.”
Both 7
th
and 8
th
grade have a specific LS teacher who is a math specialist assigned to their grade
level.
Tier 3 is intensive one-on-one intervention to address a specific area of weakness. Multi-
tiered instruction is designed so teachers have the right tools at the right time to address the
different learning needs of their students (Windram et al., 2012). While ISSEA’s middle school
has a pull-out program, it exists as a separate class, in place of an elective, where students get
additional support to keep up with current grade level standards. It is not evident that students
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receive specific instruction to address the learning gaps that these students bring with them to the
new grade level.
Student concern meetings. RTI not only includes academic learning gaps but it
addresses the social emotional and behavior issues that students may also be experiencing. At
the ISSEA’s middle school, in every grade level, there are student concern meetings, identified
as ‘Kids Chats’, to discuss an potential student concern in any of the three categories: academic,
social-emotional, or behavioral. Document analysis revealed shared documents with grade level
team teachers, counselors, administrators and school psychologist. Comments are collected prior
to the meeting on the concern and what has already been done to address the concern. At the
meeting, teachers discuss what is working and what is not working for this student and an action
plan is made to ensure next steps to support the student of concern. This important component of
a successful RTI program is a well-developed, pro-active component of the daily operation of the
middle school at ISSEA.
One problem noted during document analysis was that the student concern documents
were created, updated, stored and shared by the grade level counselor. Each counselor had a
different method for documenting meetings and storing files student information. Because
counselors advance to the next grade level with the students each year, the grade level
mathematics faculty need to learn new systems each year that are not integrated into their regular
mathematics department files. The counseling department, the Learning Support department and
the mathematics department all collect data, but store it separately with different sharing rights to
documents. The lack of effective communication between departments could result with
information not reaching the person that made need it in the scope of their responsibilities.

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Summary of Research Question One Findings
The mathematics teachers at the International School South East Asia middle school
employ a variety of strategies to help students improve their performance so they all meet grade
level standards for math proficiency and/or achievement. They have adopted RTI as the method
to help students who are struggling to achieve grade level standards in the regular classroom.
RTI requires a foundation of high quality instruction and an early universal screening. The
middle school mathematics teachers have demonstrated through document analysis, survey and
interview responses that they use 13 strategies, which have been proven in research, as presented
in the literature review in Chapter 2. They have shown ability to identify learning gaps through
data collection and analysis from standardized MAP test results, previous years qualitative and
quantitative data, and a diagnostic assessment at the start of the year. Despite these positive
findings, the middle school teachers have noted that they themselves could use a universal
screening assessment that has been proven by research to capture a more comprehensive group
of students with learning gaps.
RTI requires a process to determine the magnitude of deficiency, which could also be
captured by a proven universal screening instrument. To analyze the causes of the deficiency,
middle school mathematics teachers kept track of and monitored what problems students
experienced in the past and what strategies worked with an annual watch list and with a weekly
kids’ chat for academic, behavioral and/or social challenges. During daily instruction, formative
assessments guided regular, small group interventions but data was not consistently collected on
the daily interventions so it was difficult to evaluate the effectiveness of the interventions or to
plan future actions.

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Results of Research Question Two: Knowledge and Skills, Motivation and Organization
Influences
What are the knowledge and skills, motivation, and organizational causes that exist to provide
support to the middle school teachers’ ability to fill their students’ learning gaps while at the
same time guide them to achieve grade level standards by the end of the school year?
Knowledge and Skills
Data was assessed using Krathwohl’s (2002) description of knowledge as factual, conceptual,
procedural, and metacognitive. The five assumed knowledge influences, outlined in Chapter 3
were validated or validated in part, as summarized in Table 11. Following the table, there is a
discussion of the knowledge influences on middle school mathematics teachers’ ability to create
sustainable interventions for students that present learning gaps.
Table 11

Validated Assumed Knowledge Influences
Category Assumed Influence Validated Validated
in Part
Factual Teachers know factors that could cause
gaps in learning. √

Procedural Teachers know how to identify learning
difficulties and choose the correct solution √
path.

Procedural Teachers know how to use effective strategies √
to deliver the appropriate instruction.

Conceptual Teachers know grade level expectations
in mathematics and the learning progression √
from elementary through middle school.

Metacognitive Teachers know how to guide students in
awareness of their own thinking and learning √
process and help them to self-regulate their
learning.

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To assess the validation of the assumed knowledge influences, the results were
triangulated and compared across grade levels and through different data collection sources. It
was determined that an influence was validated when a majority of the participating stakeholders
confirmed the findings or if the findings were consistent across two or more sources. The results
showed that the five assumed knowledge influences were validated through document analysis,
surveys with six teachers and interviews with four teachers representing grades six, seven and
eight.
Validation of factual knowledge influences. To assess whether teachers knew what
factors caused gaps in learning, digital documents were analyzed in the math department files
and questions regarding factual knowledge were addressed in ta survey.
In the document analysis, a middle school math watch list exists to provide a watchful
eye over students of possible concern as they move through all grade levels in middle school.
The comprehensive list spans seven years and has been documented by all middle school math
teachers, including the six participants in this study. Qualitative teacher comments describe
demonstrated learning gaps, and patterns for each student are monitored from year to year to help
guide placement and to help plan interventions.
The 7
th
grade team has a separate document for students of concern in the current school
year cohort (see Appendix V). The spreadsheet includes data for all students but student names
are hidden, for the purpose of this study. A sample portion of the student concern data
spreadsheet is shown in Figure 13. Each row represents a different student. Color-coding boxes
in different areas of possible weakness highlight the students of concern. The spreadsheet lists
math concerns according to standards achievement data from the prior year in columns Z and
AA. In these two columns, the letter A references the performance level ‘Approaching’ and the
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letter B references the performance level ‘Below.’ Both levels are highlighted, indicating these
students did not meet grade level standards in the categories listed in Row 2, during the previous
school year. In addition, social emotional and behavior issues that occur most often in the
context of this grade level at this school are listed in columns AB to AL. Color coded boxes
indicates varying levels of concern in the identified categories, such as disruptive (Column AB),
difficult to self-start (Column AC), Low Math skills (Column AG), and so on. Specific
comments in Column AN address individual student gaps based on an analysis of the data
recorded in the spreadsheet and update with comments from the weekly student concern
meetings.

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Figure 13: A sample portion of the 17.18 Seventh grade students of concern, showing evidence
of learning gaps.
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Teachers’ confidence to identify learning gaps was assessed through a survey question,
which asked, “I know how to identify learning gaps.” Responses varied slightly with five of six
teachers agreeing they were confident, while one somewhat disagreed. Figure 7 highlights
teacher ability to collect data on all students in the grade level, to document learning gaps in
grade level standards (Columns Z and AA) or basic skills (Column AG), and to document
learning behavior difficulties that may result in a student’s poor performance.
Further comments on survey and interview results showed confidence in a system which
exists to prevent learning gaps and in the use of diagnostic assessments to initially identify
students that need support early in the year. One teacher expressed exceptional effort by all to
try to identify all gaps but questioned whether they are using proven tools to comprehensively
capture all of the gaps. Two teachers mentioned the need to work with students to be able to
understand individual learning gaps, rather than rely on data alone.
The student data collection shown in Figure 13 shows a comprehensive look at students’
performance and gaps in academic, social and behavioral categories. For example, the student in
Row 23 demonstrates a learning gap, as evidenced by the performance achievement level of B
(below expectations) which is highlighted in red, indicating a major concern. Looking at the
qualitative data from student concern meetings in Columns AB to AL, the student in Row 23
shows four red highlighted boxes for poor student learning behaviors, including difficulty to self
start (Column AB), difficulty to stay on task (Column AC), low reading skills (Column AE) and
difficulty in organization (Column AJ). In addition, the minor concerns that are highlighted in a
lighter shade of red indicate low writing skills (Column AF) and frequent reassessment in order
to meet grade-level standards (Column AH). However, the student in row 23 shows only minor
concern with low math skills (Column AG). Analysis of this data would show that the learning
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gap in mathematics is more likely resulting from learning behaviors and the low performance in
reading and writing, a challenge in a curriculum whose emphasis is reading and interpreting
word problems and writing descriptions explaining thinking and understanding of the problem.
Further analysis of the student’s academic background in Appendix V show that this student was
fortunate to be placed in an inclusion class the previous year but has not been placed in one this
year. However, the student has attended strategic learning classes. The next steps for this
student would be to contact the counselor and school psychologist to discuss strategies to address
the learning behaviors to find out what has worked in the past or to consider more structured
support and to contact the English Language Arts (ELA) teacher to discuss support for reading
and writing.
The student data collection document in Figure 13 is currently used only in 7
th
grade and
the student math data document in Figure 10 is currently only used in 8
th
grade. It is possible
that this document framework could be combined into one spreadsheet and shared with other
grade levels to support teachers that are looking for ways to comprehensively capture all of the
learning gaps and narrow down possible solution paths.
When asked, “ Is a deficiency the same as a learning gap?” all teachers responded with a
negative answer and proceeded to define both deficiency and learning gap. Within the two
concepts, however, teachers’ definitions sometimes overlapped. One teacher stated that a
learning gap is the amount someone actually learned versus what they should have learned. Two
teachers commented that a learning gap is the failure to acquire a specific skill or master a
learning standard. Two other teachers noted that a learning gap is content that was missed or
not encountered yet. On the other hand, three teachers described deficiency as missing skills or
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underdeveloped skills. The teachers who defined a learning gap as missing skills further defined
a deficiency as a defined learning challenge, such as dyscalculia, or a result of a cognitive issue.
Despite the differences in how teachers define gaps and deficiencies, the digital
documents in the middle school math department files, including the middle school watch list
spanning seven years of historical teacher comments, the sixth grade MAP data spreadsheet
(Figure 2), the seventh grade student concern list (Figure 7), the 8
th
grade’s Student Math Data
spreadsheet (Figure 5) and the Algebra 1 Lab Recommendations document (Figure 6) indicate
teachers are collecting and using information to identify missing skills, and they are considering
possible causes: 1) a student has been taught the content and has not yet mastered it (See
standards performance levels in Figure 7, columns Z and AA) and Figure 6, Column 2); 2) a
student has a learning challenge, diagnosed or undiagnosed; or 3) a student has never been taught
the content due to a transition from another school or from a transition from a lower to higher
ability-leveled course.
Validation of conceptual knowledge influences. To assess if teachers know grade level
expectations in mathematics and the learning progression from elementary through middle
school, digital documents were analyzed. The researcher reviewed the ‘MS Math Articulation of
Standards’ (See Appendix N) and the ‘Power Standards Math #SBG’ documents (See Appendix
O), created by the six stakeholders in this study and three other teachers who were in the middle
school math department during the 2016-2017 school year. A portion of the MS Math
Articulation of Standards can be seen in Figure 14.

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Figure 14. A sample portion from the middle school math department’s articulation of standards.
In Figure 14, a sequence of standards, based primarily on the sequence of the Common
Core State Standards, is displayed. Standards exist by grade level and are organized by
reporting domains (strands), such as number system, rational and proportional relationships,
geometry, statistics and probability, expressions and equations, linear relationships, and
functions. In the columns of Figure 14, on the right, the numbers represent the math courses in
middle school. The number with a plus, such as 6+, 7+ and 8+, are more advanced courses
where instruction is at a quicker pace and more extension work is provided. The color-coding
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indicates whether a standard is taught and assessed (green), whether there is exposure to content
but it is not assessed (yellow), whether a standard has been taught in the previous grade but
requires review (pink), whether a standard has been taught in the previous grade but requires
significant review (red), whether a standard has been unintentionally not been covered (gray) or
whether a grade level standard has been purposefully been deleted (black). Reflection on this
document was evident with comments from teachers indicating current practice along with
annual updates to the color-coding. The MS Math Articulation of Standards was the initial work
for the middle school math department to transition to standards based curriculum and to ensure
standards were being taught in a continuum rather than repeating the same content or missing
others when transitioning from one grade level to another or one course to another.
The Power Standards Math #SBG used the information and organized the standards in a way
where ‘power standards’ could be identified by grade level. Power standards are identified
within each strand (reporting domain) and are the most important content that needs to be
covered by each grade level for a student to be successful in the following mathematics course.
In Figure 15, each grade-level math PLC created a chart to represent the power standards, as
they deemed most important for teaching and assessing. All standards listed were pulled from
the MS Articulation of Standards and were the ones they taught throughout the year. Reporting
domains were found in the far left column and were specific areas of study. The power standard
was listed in the next column and summarized each important learning target that students
needed to know. The largest column contains ‘Descriptors’ which state the specific standards
that are taught and assessed. Within this column, specific standards or portions of standards
were highlighted in bold to focus on the primary learning that needs to occur for students to be
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successful. Narrowing course curriculum gives struggling students more time to focus on and to
practice the most important material they need to be successful at each grade level.

Figure 15: A sample portion of the ‘Power Standards Math #SBG’ document showing all
standards and highlighting most important standards.
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The ‘Power Standards Math #SBG’ document was created by the middle school
mathematics faculty and contained the standards for each grade level (See Appendix O). It was
organized by course, such as 6, 6+, 7, 7+ and 8, 8+. Progression from one skill level to another
was evident and the document clearly showed what was taught per grade level and per course.
Mathematics teachers focused their instruction with struggling students on the power standards
but they also articulated what specific content students still needed to learn and could project
next steps in students’ learning progression.
In addition to creating the continuum for standards of content knowledge, middle school
mathematics teachers created planning documents based on middle school mathematical
practices as written by the National Council for Teacher of Mathematics (NCTM, 2017). The
documents found in document analysis were ‘Unit Summative Assessment Itemized Math
Practices,’ (Appendix K) created by the Grade 6 Math PLC and “Unit Concept Checklists,”
(Appendix G) created by the Grade 7 Math PLC, which show evidence of NCTM math practice
standards alignment to content knowledge.
Figure 16 shows a sample of the alignment completed by the Grade 6 Math PLC in the
document ‘Unit Summative Assessment Itemized Math Practices.’ At the top of Figure 16, the
unit assessment is linked. In this case, the unit was called Prime Time and the test assessed
student ability in factorization, least common multiple, greatest common factor and using the
distributive property (Appendix P). The left column of Figure 10 lists the NCTM math practices
in linear order. The second column lists the question items that are associated with the math
practice. The last column of Figure 10 tallies the number of questions that are aligned with each
math practice. The number of items are recorded for each unit assessment and collected for all
assessments that are conducted throughout the school year.
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Figure 16: A sample portion of Grade 6 ‘Unit Summative Assessment Itemized Math Practices.
Figure 11 shows the itemized math practice standards alignment overview. An annual overview
provides a look at which practices are being assessed and valued more than others. In the pie
chart, in Figure 11, almost half of the questions on all unit assessments include MP6 (Attend to
precision) while only a negligible number of questions were related to MP 7/8 (Look for and
make use of structure and Look for and express regularity in repeated reasoning) and MP5 (Use
appropriate tools strategically). The analysis allowed teachers to see where they could add
further instruction, by noting the areas that were covered significantly less than others.
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Figure 17. Year overview of Grade 6 Math ‘Unit Summative Assessment Itemized Math
Practice Standards Alignment.’
In 7
th
grade, unit concept checklists were created for each unit to put learning targets into
student friendly language and to align content knowledge standards with math practices. In
Figure 18, the first unit of study during the year was called ‘Shapes and Designs’ and was a
geometry concept standard, as stated in the first row. In row two of Figure 12, the statement
relates what a student should be able to do, based on content knowledge standards from the
Power Standards Math #SBG document (Appendix O). The standard is aligned to specific math
practice standards as specified by the Grade 7 Math PLC, based on the NCTM math practice
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standards (NCTM, 2017). Vocabulary words are highlighted and listed at the bottom of the
concept checklist. Continual revision and improvement was evident through document analysis
of other concept checklists.

Figure 18. A sample portion of a 7
th
grade concept checklist aligning math practice standards to
content standards in student friendly language.
In Figure 19, a section was added to the start of the document to summarize what a
student should be able to do at the completion of this unit. The 7
th
grade concept checklists
aligned math practices to content standards in an accessible way for students’ use. Teacher
MS7.2 indicated during interviews that students add these documents to their math notebooks at
the start of the unit and revisit as they are progressing through the lessons.
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Figure 19: Sample portion of 7
th
grade unit concept checklist with added summary of student
objectives.
Four of six stakeholders in this study have contributed to this work to deliberately
incorporate math practice standards into curriculum. There are initial indications that math
practice standards are being documented formally although it is not consistent across grade
levels. ISSEA’s Grade 6 Math PLC modeled how to document alignment of math practices by
assessment question in a way that helps inform future instruction of math practices and ISSEA’s
Grade 7 Math PLC modeled how to align math practices to student-friendly learning targets.
The middle math department next steps could be to share expert models of math practice
alignment with each other.
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Comparison of ISSEA’s ‘Power Standards Math #SBG’ (Appendix O) to the
recommended clusters of study by the Partnership for Assessment of Readiness for College and
Careers (PARCC, 2014) -- Appendix Q -- revealed the matching of Common Core State
Standards (Mathematics Standards, 2018) to PARCC’s recommendations for major, supporting
and additional clusters for the purpose of ISSEA’s middle school mathematics course
continuum. In Figure 20, a sample of the clusters recommended by PARCC is shown with
color-coding to identify major (green), supporting (blue) and additional (yellow) clusters. The
color-coding is evident in the power standards in column two of the sample portion of ISSEA’s
middle school document titled ‘Power Standards Math #SBG (Appendix O),’ where a sample is
referenced in Figure 15.

Figure 20. A sample of PARCC’s recommendations for clusters of study for Grade 8
mathematics (PARCC, 2014).
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One specific example of a PARCC major cluster and CCSS power standard can be
tracked from ‘Power Standards Math #SBG’ Grade 7+ Math (See Figure 15) to the PARCC
recommended clusters, as defined in Figure 20. First, in the ‘Ratio Proportional Reasoning’
Domain of Figure 15, column one is followed by the power standard in column two, which
stated, “Understand the connection between proportions, lines, and linear equations.” The Grade
7+ Math power standard was correlated to the same statement in the PARCC recommended
major cluster in Figure 20, under Expressions and Equations Part B, highlighted in green.
Furthermore, when comparing the Geometry Domain of Grade 7+ Math from Figure 15 to the
PARCC recommended additional cluster for Geometry, the color-coding was consistently
yellow, indicating an area that is not most important for grade level content mastery.
Document analysis revealed ISSEA’s middle school mathematics faculty has used the
CCSS and PARCC recommendations to guide their alignment of mathematics standards and
learning progression. While the research determined the middle school mathematics teachers
fully understand the middle school grade level expectations and the learning progression from
sixth to eighth grades, there is no evidence that collaboration with elementary math teacher has
occurred to ensure there are no gaps in the transition from ISSEA’s elementary to ISSEA’s
middle school.
Validation of procedural knowledge influences. To validate whether teachers know
how to identify learning difficulties and choose the correct solution path, the researcher first
assessed teacher confidence level by asking the following questions in the survey:
a) What is your confidence to collect data on all of your students? (None, Some, Just the
right amount, Very high level)
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b) What is your confidence to use data to inform decisions with an intervention? (None,
Some, Just the right amount, Very high level)
c) What is your confidence to conduct an intervention? (None, Some, Just the right
amount, Very high level)
Overall, teachers all possessed confidence, ranging from feeling somewhat confident to very
confident in their ability to collect and use data to inform decisions for an intervention. Two
respondents felt just the right amount of confidence to conduct an intervention and four others
felt very confident to do the same. Comments included requests for organization of data for
quick and efficient accessibility, a researched diagnostic assessment that was manageable for
students, and specific strategies to form groupings and identify RTI needs.
For specific teacher involvement in the process to identify learning gaps and choose the
right path, teachers were asked the following survey questions:
a) I was involved in collecting data on students to create a database that is used in
identifying students in need of support (All six teachers agreed).
b) I received helpful information that contributed to my understanding of my students’ areas
of weakness (Three teachers agreed and three disagreed).
When asked to describe their understanding of the process to identify student learning gaps or
deficiencies and to create possible solution paths, teachers’ comments were similar. They
collectively stated that learning gaps could be identified throughout the year during lessons, pre-
assessments, standardized testing and information from prior school years. Further comments
stated that this information allows for teachers and students to communicate the importance of
goal-setting and personalized learning, which allows students to identify and celebrate their
progress throughout the year. The findings show middle school mathematics teachers know the
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process to identify learning gaps and they place emphasis on student self-efficacy when creating
a solution path. The findings also provide an indicator that middle school math teachers are
possibly not receiving support they need as part of a school wide intervention (RTI) team.
In document analysis, intervention solution paths are beginning to be recorded in eighth
grade unit plans and evidence showed that these plans are currently being updated as the school
year progresses. In Figure 21, the bottom left box of the Math 8 unit-planning template, includes
a section titled ‘Interventions’ and it includes specific interventions with designated teachers but
no further details are given which describe the students involved. Later units describe
interventions as the daily use of the three-tiered mathematics problems. In interview responses,
Teacher MS8.1 stated Eighth grade used the lowest level, the green level, as intervention.
Teachers MS6.1, MS6.2 and MS7.1 confirmed during the interview process that sixth and
seventh grades used the green level problems to provide standard, assessed content knowledge.

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Figure 21. A sample unit plan from the Grade 8 Math PLC, showing recorded interventions.
Interview data further specifies teachers’ collective understanding of the process to
identify learning gaps and create possible solution paths. All four teachers that were interviewed
commented on the use of standardized test data, in-house diagnostic testing, unit pre-assessment
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and daily formative assessments to re-sort and re-group students based on their needs. Teachers
also noted they created interventions based on the need presented by results from a formative
assessment. Sometimes they would create a solution path for small group interventions by
taking work from last year or sometimes they would design something new.
Each grade level had a pre-planned intervention based on the need of their core content.
For example, in sixth grade, Teacher MS6.1 and MS6.2 stated the use of daily skill drills and
online computer programs to support students with deficits in basic skills. Teachers MS6.1 and
MS6.2 explained during the interview that the daily skills review took place at the start of the
lesson as students came into the room and got settled and ready for class. Teacher MS6.1 further
described online skills programs for students who demonstrated a need in basic skills practice
based on previous grade level performance and initial diagnostic testing. Teacher MS6.1 noted:
If students don’t have basic skills, we put them on computer programs, such as ‘Reflex
Math’ and ‘Extra Math.’ ‘Extra Math’ takes a baseline of automatic skills and encourages
students while drilling problems. ‘Reflex Math’ is a game-oriented skill drill without a
timer.

Teacher MS6.1 stated that the online review was conducted at home with the assistance of
parents and parents were given support on how to help their child. Documentation of the
intervention results was evident in the Grade 6 Math PLC tracking document for spring and fall
MAP results in the Number Systems category for all students of concern (See Figure 7).
In seventh grade, digital files show an intervention before the unit on simplifying
expressions and solving equations using algebra tiles to support students who presented a deficit
in operations on standardized and diagnostic testing. The intervention was designed to give
students a visual way of displaying expressions and equations and was not assessed. This
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intervention was built into curriculum for all students during the last week before Winter Break
and during the first week in January after students returned to school. The choice of timing was
due to two factors: 1) the unit on simplifying expressions and solving equations was to start in
the New Year at the start of the second semester and 2) the intervention would give students
something meaningful to learn at a time when the year was ending and was filled with
celebrations, plays, concerts and the festivities of the holiday season.
The intervention was scheduled to continue into the New Year when students returned
from overseas holidays to give students a chance to readjust to the time zone. The Grade 7 Math
reflection document, as seen in Figure 9, noted that the seventh grade mathematics teachers were
not sure this was a good choice of timing. There were no further details on how the intervention
timing was revised or incorporated in the next school year lessons. Documentation for the
intervention was included in the reflection document and included numerical data only. It did
not list student performance and progress by name but revealed that MAP results from spring to
fall were considered as a measure for success. The next steps for the Grade 7 Math PLC could
be to use a format to record data for individual students, similar to grade six in Figure 7.
Eighth grade teacher, MS8.1 talked about pre-planned interventions for students who had
not yet mastered integer operations and the distributive property. There was evidence in Figure
21 that showed the Learning Support teacher conducted an intervention over a two-day period on
integer operations, but no further details were given. MS8.1 stated during the interview that this
intervention was conducted in a pullout model with small groups getting focused instruction
from the Learning Support teacher. MS8.1 further explained that the eighth grade teachers
created these interventions based on past experience showing that there are always students who
will need this support before their solving equations unit. The eighth grade teachers identified
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students who needed this support based on the previous year performance in grade level
standards and based on the students’ initial diagnostic results. Individual student progress was
tracked in the summary of eight grade student math data (See Figure 10).
Collectively, the middle school mathematics teachers have an extensive system to help
identify learning gaps; they modify instruction and conduct small group interventions daily
based on formative assessment; and they have created at least one pre-planned intervention to
meet the most pressing need for each grade level.
Validation of metacognitive knowledge influences. To validate whether teachers know
how to guide students in awareness of their own thinking and learning process and to help them
to self-regulate their learning, documents were analyzed, a survey question was asked and
interviews were conducted. First, the survey question asked teachers to choose the top for
variables to support students who have gaps in learning. While three variables were collectively
deemed more important overall than metacognition and goal-setting, three of six teachers
included metacognition and goal-setting in their top four choices. Further review through
document analysis and interviews revealed commonalities across grade levels: 1) Three-tiered
problem solving instruction method, giving choice to the student; 2) learning behavior self-
assessment rubrics; 3) reflection on learning targets or concept checklist followed by goal
setting, for each unit; 4) test corrections and error analysis templates; 5) reflection and
preparation for three way conferences in the fall with parents, teacher and student; and 6) explicit
teacher instruction guiding students on the process of self-reflection.
Three-tiered problem-solving instruction method. The three-tiered problem solving
instruction method is new this school year but has been adopted by all teachers in all grade levels
of middle school. The approach presents a topic of study then provides three levels of challenge
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for the problem solving practice. Students are guided in their choice of problems but ultimately
choose their own problems to work on. Class slides in grade six reveal teacher time to guide
students on how to identify the problems that best fit their level of knowledge. Grade seven PLC
agendas report that this method indicates that students are progressing faster to the more
challenging problems. All grades show evidence of student learning in cooperative groups. The
interview with Teacher MS6.2 was conducted in the math classroom and she walked around the
room, showing where students sit together to work on problems and how they accessed the
problems and solutions on the boards. Teacher MS8.1 noted he purposefully built collaboration
in the classroom by creating a ‘Positive Learning Zone’ so students would be comfortable to
share with their partners, their teams and with the whole class. He credited the high performance
of his students to their ability and comfort to ask questions during problem solving. Middle
school math teachers are guiding students to be able to identify where they are in their
understanding of the daily math concepts and then they give them the choice to select problems
that fit their individual needs.
Learning behavior self-assessment rubrics and reflection on learning targets.
Document analysis revealed that all grade levels had templates and rubrics for learning behaviors
and learning targets per unit. Templates for students follow similar formats and are used at the
end of each unit, as evidence on the pacing and planning calendars for grade six and seven.
Teacher MS8.1 indicated, during the interview, that the learning behavior self-assessment
occurred along with a reflection on learning targets for each unit. Grade eight had unit plans for
teacher planning purposes but also revised them in student friendly language, showing learning
targets per unit. Grades six and seven also had active documents in their digital files showing ‘I
can…’ statements (grade six) and ‘Concept Checklists’ (grade seven). Collective evidence
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validated that middle school math teachers are guiding students in their understanding of how
behaviors, such as responsibility, collaboration, organization and reflection, affected their ability
to learn and teachers are guiding students in their understanding of the learning target
progression.
Test correction and error analysis. All middle school math teachers employ a similar
system of test corrections and error analysis to help guide students to understand patterns in their
mistakes and wrong answers. Documents in each grade level’s digital files reveal active use of
test correction documents and templates. The method designed and improved upon by ISSEA
middle school math faculty is called TOFU, an acronym for Transfer, Operation, Finishing or
Understand type errors (See Figure 22).

Figure 22. TOFU test error analysis document.
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Transfer errors occur when students inadvertently change the number or operation sign
somewhere in their problem solving steps. Operation errors result from mistakes in adding,
subtracting, multiplying and dividing. Finishing errors happen when students are rushed and fail
to finish a problem. Understanding errors imply students do not understand the concept and need
more practice and time to fully understand the desired learning target.
The Grade 6 and 7 Math PLCs have modified the document for their classes. In Figure
23, the Grade 6 Math PLC includes a quantitative summary of test results with the error
correction analysis. In the far left column, the numbers of the questions on the test were listed.
In the middle column titled ‘Skill,’ the specific assessed task was identified and in the next
column, the number the student scored correct would be entered. The teacher adds a check in the
‘Developing’ column if the student has not yet mastered the content. A check in the ‘Mastered’
column indicates the student has successfully achieved the performance level for the grade level
standard. The second portion of the document provides space for students to correct their work.
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Figure 23. An example of a Grade 6 TOFU test correction document including error analysis.
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The test correction document for the Grade 7 Math PLC has also been modified slightly
from the original template and can be seen in Figure 18 on the following page. The categories of
‘Transfer’ errors and ‘Finishing’ errors have been replaced with more specific language in steps
1, 4, 5 and 6 in the modified description of error types.

Figure 24. The Grade 7 Math PLC template for test corrections and reflections.
In addition to using templates to guide students understanding the patterns of their errors,
planning calendars and class slides for daily lessons showed that class lesson time is dedicated to
students’ reflection and test correction.
An example from the Grade 7 Math PLC traced the timing allowed in class for test
correction from the math-planning calendar to a daily student update with links to the test
correction document instructions. In Figure 25, a practice test occurred before a unit summative
assessment on December 7 and December 11.
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Figure 25. A sample portion of the Grade 7 Math PLC planning and pacing calendar. (All grade
level PLCs use the same template for planning calendars, See Appendices C (Grade 7), R (Grade
6), and T (Grade 8)
A M7, Daily Student update, found on ISSEA’s PowerSchool calendar, showed on
December 11th and 12th that there were instructions for what students should do after the
practice test (See Figure 26).

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Figure 26. A sample portion of the M7, Student Daily Update found in the ISSEA PowerSchool
calendar for the M7 classes.
When the link on December 11/12 was opened, it showed specific instructions on how to
correct the test, using the test correction document, during the class period (See Figure 27).
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Figure 27. M7 math class instructions for how to complete the test correction document
following a practice test.
The example above was presented in this research through the analysis of the Grade 7
Math PLC planning documents. Other examples showing time for teachers to facilitate and
guide students in test corrections and error analysis for the Grade 6 and Grade 8 can also be
found in Math PLC documents (See Appendices P and Q). Document analysis across grade
levels, in planning calendars (Appendices R (G6), A (G7), and S (G8)), on agendas, and in
student instructions validates that teachers are collaboratively guiding students’ reflection on
their mistakes and why they occur to better understand how to improve their learning.
Three way conferences and explicit teacher instruction. At the International School,
South East Asia (ISSEA), three way conferences are conducted in the middle of October,
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between teacher, parent and student. Preparation for conferences is evident in each grade level
through PLC meeting agendas, planning and pacing calendars, class slide shows and conference
reflection templates. In grade six, class slides for a lesson show explicit instruction on how to
reflect on learning and share it with parents. Teachers guide students with prompts such as
“What are you going to focus on to help improve your learning?” Time is allowed for practice.
In grade seven, the conference template in digital files is similar but students prepare a math
reflection video based on prompts, as evidenced in class slides. In grade eight, a conference
form includes reflection on formative and summative assessments, homework completion, and
several reflection questions. Questions for struggling students include, “What do I do when I
don’t understand a concept or when I get stuck on a problem?” and “How can I better help
myself?” For students who need a challenge, the question was “What do I do to extend my
learning when I already understand the concept?” Students find their answers to these questions
based on prior reflection on learning behaviors, formative and summative assessment data, and
homework completion. Reflection templates for all grades end with a section for specific,
student created goals. With the help of learning behavior rubrics, error analysis, and reflection
templates, students possess and demonstrate self-efficacy by leading a conference with their
parents about their learning.
The middle school mathematics teachers’ documentation validates their ability to guide
students in the metacognitive realm of the knowledge dimension. They identify strategies in the
form of rubrics and templates. They give students tools and instruction so students can predict
their own responses to parents’ questions in conferences and so students better understand how
to choose problems based on their own knowledge level. They guide students to deconstruct
their learning process to find and analyze patterns in their mistakes to improve their overall
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performance. Finally, middle school math teachers purposefully guide student reflection so their
students can work towards personal goals. The last step in Anderson and Krathwohl’s (2001)
taxonomy is to create an innovative learning portfolio and it was not yet evident that students are
collecting and documenting their learning process in the form of a portfolio. There were no
portfolios found in any grade level in any of the digital files of the mathematics department nor
were they referenced in planning documents, agendas or class lessons. The next step for the
mathematics department could be to create a portfolio and start implementation slowly.
Motivation
Survey questions were asked and interviews were conducted to validate middle school
mathematics teachers’ motivation to implement interventions for struggling students. The four
assumed motivation influences, outlined in Chapter 3 were validated or validated in part, as
summarized in Table 12.
Table 12
Validated Assumed Motivation Influences
Motivation Validated
Category Assumed Influence Validated in Part
Active Teachers value the goal of every student
Choice achieving proficiency in grade level power
standards by the end of the school year. √

Persistent Teachers believe that engaging in a RTI
Effort program will contribute to students filling √
their learning gaps.

Confidence Teachers are confident in their ability to
execute the RTI program to reach 100% √
students achieving at grade level power
standards.

Confidence Teachers are confident in their PLC team and
with their collaboration with the LS teachers √
to be able to reach the goal.

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The findings and results were grouped according to the three indicators for motivation: active
choice, persistent effort and confidence. Following the table, there is a discussion of the
motivation influences on middle school mathematics teachers’ ability to create sustainable
interventions for students that present learning gaps.
Active choice. To assess motivation, data was examined for the presence of active
choice, persistence and effort. To measure the active choice of teachers, a question on the
teacher survey asked if they valued a goal stating that each student achieves 100 percent
efficiency in grade level power standards by the end of the school year. On a Likert scale of one
to four, fifty percent of respondents definitely agreed and sixty-eight percent of teachers at least
agreed to some degree. However, one-third of the teacher respondents disagreed. In order for an
intervention program to succeed, teachers need to collectively choose to implement interventions
in an energizing way. For every student to be able to learn at high levels and for ISSEA to
realize a strategic focus area, all teachers need to believe that they can help every student and
they need to be motivated to put a program in place. Further research will show some teachers’
confidence in their ability affects the value they place on the goal to support all students in their
understanding of every grade level critical standard.
Persistent Effort. An intervention program such as RTI depends on the collective
choice of teachers to implement in a stimulating way that reflects effort. Effort is evident by the
number of documents mentioned previously to collect qualitative and quantitative data on
students to identify learning gaps, by the professional documents which thoroughly identify and
outline the mathematics learning progression, by the creation of documents which further specify
learning targets by unit in both teacher and student friendly language, by the time spent, as
evidenced in agendas, in collaborative meetings to create common formative and summative
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assessments and by the number of templates, rubrics and lessons which are designed to guide
student metacognition and goal-setting. Persistent effort can be seen by the constant and recent
updates to all of these documents by multiple teachers, including the six stakeholders in this
study, indicating reflection and commitment to continual improvement.
In addition to demonstrating persistent effort in interventions that lead toward a goal,
teachers that are motivated to support the intervention program must demonstrate that they value
the process. To assess if teachers believed that engaging in a RTI program would contribute to
students filling their learning gap, they were asked on a survey question to choose one most
important factor, from a list of eleven factors proven effective during the literature review, that
provides the most significant support for students who exhibit learning gaps. Five of six teachers
chose the RTI intervention program as the most important factor leading a student to success and
one teacher chose personalized learning. All teachers believed that an individual approach to
interventions is the most valuable to supporting students with learning gaps and the
overwhelming majority believed that the RTI intervention program was the best method to
achieve this goal.
Confidence. Teachers were asked if they felt confident in their ability to execute the RTI
program to achieve 100 % student on grade level power standards.
All six respondents responded positively and with confidence that they could conduct an
intervention and they could track and monitor progress of students while also communicating
results to parents. Two teachers, however, felt they were only somewhat confident in their use of
data to inform their decisions regarding interventions. One teacher was not very confident in
collecting data. The area showing the least amount of confidence was garnered from the
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question regarding the knowledge gained from the Learning Support (LS) teacher assigned to the
grade level. Fifty percent of respondents showed low confidence in this area.
While teachers were asked about individual confidence in their ability, they were also
asked about their confidence in the ability of their PLC team to collaborate with the LS teachers
to reach the goal. Results were similar to the above question regarding individual confidence,
and indicated a lack of confidence in collaboration. Two of four teachers did not believe they
received helpful information on students’ weaknesses before the school year began, and they
didn’t feel early identification of students contributed positively to their work as math teachers.
Despite professional development training at the start of the year on a new teaching method,
another teacher still felt that they had not learned any new strategies to help create interventions.
Although two teachers appear to be less invested in the process than the other four survey
respondents, all six teachers agreed that their work on interventions and through RTI led to
stronger relationships with their PLC team members.
When asked to describe the intervention process, survey and interview responses
indicated respondents’ high motivation and confidence through lengthy descriptions and
animated responses on interviews. Five of six teachers showed clear understanding of the process
by describing it step-by-step. All four interviewees responded and discussed the process in detail
with a lift in the tone of their voice. They described the extra time needed to give students
during breaks, recesses, at lunch, and after school. They described this in a matter-of-fact way,
as it was necessary for student learning. While all four described after hours help and help
outside the classroom time, there were zero complaints about the use of their personal time.
Respondents demonstrated active choice in the execution of the RTI intervention program to
help struggling students through their descriptions of what they do for support. They
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demonstrated persistent effort through the documented work they have created and implemented
and through their dedicated time given to students. Overall, teachers seem to be mostly
confident in their ability to conduct interventions from start to finish, but their collective
responses also seem to indicate a need for some teachers to have more help with data collection
and analysis and for more assistance from the assigned Learning Support teacher.
Organization
The analysis for this study assessed the organization by looking into the structures,
policies and procedures, resources, stakeholder values, and the culture of the middle school
mathematics department. This study reviewed the roles of the administrative leadership team in
support of mathematics interventions for struggling students and reviewed professional
development in support of teachers’ ability to conduct interventions.
The four assumed organizational influences, outlined in Chapter 3 were validated or
validated in part, as summarized in Table 13. The findings and results were grouped in the
following three categories: clear vision, structures and resources and monitoring for
effectiveness. Following the table, there is a discussion of the organizational influences on
middle school mathematics teachers’ ability to create sustainable interventions for students that
present learning gaps.

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Table 13
Validated Assumed Organizational Influences

Organization Validated
Category Assumed Influence Validated in Part
Clear Teachers have a clear vision of what
Vision they are working toward. √

Structures Teachers have the resources needed to
and achieve the goal. √
Resources
Structures The organization has structures in place
and where roles and responsibilities have
Resources been delineated and an effective √

communication process exists to share
necessary information.
Structures The organization has policies and procedures
and for RTI to support teachers in achieving the √
Resources goal.
Monitor There is a process in place that ensures the
for teachers get timely, concrete feedback about √
Effectiveness their intervention effectiveness.

Intervention team and role of administrative leadership. A collective,
interdependent, collaborative community is comprised of three teams: a leadership team, teacher
teams and intervention teams (Buffum et al., 2012). The leadership team holds the greatest
responsibility for the RTI implementation process to be successful. The influence they have is
centered around several main factors: communicating a clear vision that is shared by the
community, creating structures and providing resources to both the LS and math teams to be
successful, and monitoring progress to ensure effectiveness. In the survey, when respondents
were asked to describe the process for interventions from start to finish, not one teacher indicated
assistance from any other person outside of the math department. The middle school math
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teachers’ perception of the lack of involvement by the leadership team indicates there are areas
for improvement in the organization.
Communicating a clear vision. The strategic focus of the International School, South
East Asia fosters a culture of excellence where every student learns at high levels. While
research shows that a commitment to students learning at high levels is paramount to success and
while the ISSEA vision states the importance of students learning at high levels, the leadership
team in the middle school is pulled in different directions by the different initiatives that take
place in support of the strategic plan. The strategic plan includes five priorities and within four
of the priorities, there are five subheadings, for a total of 26 goals. Under the priority for
Professional Learning Communities (PLCs), one of the 26 goals includes “Structures exist and
are utilized to support intervention and extension.”
Creating structures and providing resources. In order to be successful in any
intervention implementation, the organization needs to have policies, procedures and structures
to support the teachers in achieving the goal. In the middle school, the LS and math departments
appear to work separately to achieve the school-wide goal that every student achieves at high
levels. Leadership for the intervention team is split into two factions. A deputy principal has
been specifically assigned to lead the Support Services and a different deputy principal has been
designated as administrative leadership for the math department. A lack of collaboration is also
seen in digital documents where the two departments have different files and these documents
are not shared with each other. While math department documents are under Middle School
Curriculum oversight, it is not clear who owns the sharing rights for the files created by the LS
department.
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In addition to different leadership authority and different documentation, the 2016-2017
school year showed that LS teachers had different planning meeting times than math PLC
meeting times. In the current school year, there is some common planning time but it is
inconsistent across grade levels. Teacher MS6.2 stated that in sixth grade the LS teacher is split
between support for mathematics and English Language Arts (ELA). The sixth grade LS teacher
does not have common planning time with the sixth grade math team. Support mathematics
classes are scheduled but the sixth grade LS teacher is unable to come to those class sessions due
to scheduling conflicts. Teacher MS7.1, however, reported common planning time with the
seventh grade LS teacher who, unlike the sixth grade LS teacher, was designated as a math
support specialist only. While some common planning time exists, Teacher MS7.1 indicated
interest in more planning time for the LS teacher to be able to push-in to the classroom more
often for pre-planned small group interventions. Teacher MS7.1 indicated it was difficult to
have regular interventions because the seventh grade LS teacher was split between three
teachers, overseeing approximately 330 students. Teacher MS8.1 did not talk about planning
with the eighth grade LS teacher but noted working closely with the eighth grade LS teacher for
small group pullouts. While there is some collaboration between the math and LS departments,
it seems inefficient due to the lack of document sharing and the lack of common planning time.
Leadership team oversight and management of the two departments is necessary to achieve
greater success interventions. A more positive impact will result after roles and responsibilities
are discussed and shared accountability for all teams involved in the RTI effort agree upon
student learning.
Coburn’s (2009) study states the responsibility and accountability for struggling students
is shared among multiple units in order for decision-making to be stretched among multiple
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units. Currently, the LS department has two pages of duties and responsibilities that do not
coincide with the recommended responsibilities for RTI interventions, as described by Buffum
(2012). While every school contextualizes programs to fit their needs, there is no organization
chart at ISSEA that defines roles of accountability for each team supporting the RTI effort and
the responsibility by support team (Leadership Team, Intervention Team and Teacher Team) and
the associated tier of intervention (Buffum et al., 2012). In addition, student behavior
management is a component of the RTI intervention program, but currently, recess monitors,
who are primarily teacher aides and assistants, are not included in the planning meetings or
weekly student behavior concern meetings. They are tasked with accountability for behavior but
are not given tools to enact responsibility within the scope of their position. Teachers and staff
are not aware of the interdependence of their collaborative work and it is not evident that clear
expectations for different roles are given at the start of the year. The organization has a
structural gap where roles and responsibilities have not yet been delineated and an effective
communication process is not fully implemented to share necessary information between
stakeholders.
Monitoring progress to ensure effectiveness. In addition to communicating a clear
vision and providing structures and resources, the leadership team’s influence on teachers’
ability to conduct interventions comes from monitoring progress to ensure effectiveness. There
is currently a process where teachers get timely, concrete feedback based on classroom
observation. Administrators observe teaching in the classroom based on a template and with pre-
scripted targets of inquiry. Feedback is provided by email regarding what was noticed in the 20
to 30 minute observation. At this time, email reply or an office visit is the possible response for
teacher discussion of feedback. The focus of these observations is only on interventions, if
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requested by the teacher. There are Professional Growth Evaluation (PGE) meetings for
individual teachers and PLC leaders that allow structured time for discussion of goals and
progress. In addition to these regularly scheduled bi-annual observations and PGE meetings,
Teacher MS8.1 noted additional observations this school year based on communicating learning
targets with kids, providing consistent instructions across the three eighth grade math
classrooms, and small group interventions with students. According to this researcher, specific
progress monitoring for effectiveness of interventions is not yet formalized, but there are
structures in place that would support intervention monitoring, if specifically added to templates
that guide interventions and meetings.
Summary of Gaps Found
The research confirmed knowledge, motivation and organization influences identified in
the literature review. The stakeholders in the middle school math department know factors
causing learning gaps, know grade level expectations and know the learning progression of skills
for middle school. They use effective strategies to deliver appropriate instruction for all students
and can guide students in awareness of their own thinking and learning process. More efficient
use of data and a research-based, proven diagnostic tool will help to better capture all student
learning gaps and documentation of preplanned interventions in unit plans will help systematize
future instruction and allow for continual improvement. Collaboration with elementary math
teachers will help to ensure that the learning progression from elementary to middle school does
not create opportunities for gaps in learning. Teacher confidence will improve with additional
support with data collection and further advice on data analysis as well as consistent support
from grade-level Learning Support teachers. Finally, the organization needs to provide more
clarity in the structure of the intervention team both in leadership and in the individual roles and
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responsibilities of each person involved in the team. Policies and procedures need to be
improved to help provide clearer and more formalized coordination and accountability.
Structures in scheduling need to be improved to maximize efficiency. An in-depth literature
review will be conducted in Chapter 5, and research-based solutions will be developed for
addressing how the middle school intervention team can work together more efficiently to
support students exhibiting learning gaps.
Results of Research Question Three: Possible New Programs
What programs might ISSEA implement to help middle school math students fill learning gaps
and achieve proficiency for math grade level standards?
The interview responses were varied with each grade level offering different ideas on
possible new programs to help all students achieve at high levels.
Additional Problem Solving Time
Flipped classrooms. Teachers are always looking for ways to find more time to cover
required curriculum and to meet the needs of all students. Some of the suggestions or thoughts
from the mathematics teachers at ISSEA are flipped classrooms and flexible class time. With
flipped classrooms, students would cover basic content at home the night before the class. They
would read or watch videos on how to solve problems, look at examples and procedural steps,
and try a few simple problems. Class time with a mathematics teacher would be spent tackling
harder problems with the help of peers and an instructor and would also be spent on application
problems rather than basic problems. Flipped classrooms allow teachers to assist students in
making connections between the mathematics problem and practical application. Teacher MS8.1
noted that the flipped classroom model helps to develop deeper conversations in the classroom
due to the higher level questions that often accompany application problems. The flipped
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classroom also provides additional time for the instructor to work in smaller groups, allowing
time for interventions with students who are struggling with the basic content. Teacher MS8.1
stated that ISSEA provides the perfect opportunity for flipping the classroom because it is
already a one-to-one school, with each student required to have a laptop computer.
Flex time. Flex time describes a structured time that is built into the weekly academic
schedule. Teacher MS7.1 specified flex time as a time where kids could go where they need to
go to get extra help. He further explained that if the schedule would provide an hour per week
where teachers could pull-in students that were really struggling, the extra support could be
given during the school day rather than taking up time students could be spending on
extracurricular activities. Retests for students that require extra time to learn regular content
could also be taken during a flex time during the school day. While after school help is
available, flex time would allow for more structured support that would occur during the school
day.
Alternate Assessments. Different students express their understanding of content in
different ways. Teacher MS8.1 suggested performance tasks be used either in addition to regular
testing or sometimes in place of written tests. Through document analysis in the mathematics
department files, there is evidence by all grade levels that demonstrates planning for alternate
assessments in the form of periodic, ‘cornerstone’ tasks. A folder exists that has sample
‘cornerstone’ tasks by grade level, assessing content through performance tasks based on math
practices rather than assessing content alone on a written test. Some unit plans, assessments and
documents show planning and preparation for instruction with an emphasis on math practices.
Grade 7 has implemented a performance task for assessment of their geometry unit of study.
Performance tasks allow for a different way of demonstrating understanding and helps students
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to make authentic connections with practical applications of mathematics, an essential
component for some students to learn mathematics.
Teacher collaboration
While collaboration between the mathematics department and the Learning Support
department has been noted as needing improvement, it is clear that the middle school
mathematics faculty at ISSEA desire more collaboration as evident by teacher response through
interviews. Teachers specifically mentioned increasing collaboration with each other in the form
of co-teaching in ‘math hubs’ and increasing collaboration with LS teachers in regular math
classrooms on a daily basis.
Math hubs. Teachers MS6.1 and MS6.2 both described co-teaching in the library to
accommodate the needs of all students based on their performance on formative assessments.
Teacher MS6.2 stated, “Right now, we have kids moving back and forth all of the time and it’s
fine, but Teacher MS6.1 and I have to be focused to the minute.” Approximately 50 students are
moving from regular classrooms to the library for the sixth grade teachers to collaboratively
provide specific instruction to targeted groups. The sixth grade teachers have formally requested
to pilot a ‘math hub’ model during the next school year and the administrative leadership team
has approved the request. The next step will be to find the space to accommodate this large
number of students.
Co-teaching model. All four teachers who were interviewed commented that it would
be helpful to have the LS teacher in the standard mathematics class every day. Currently the
math classes are separated in two tracks, a standard instruction class and a more advanced class
that approaches the same topic at a faster speed and to a greater depth of knowledge. Within the
standard mathematics classes, some classes are designated as a support class. This designation
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allows a structure for students who struggle to have access to a LS teacher. However, given the
number of students at ISSEA, one LS teacher is responsible for overseeing intervention and
extension for approximately 330 students, sometime for both mathematics and English Language
Arts (ELA). The request for an every day, co-teaching model would require the current LS
teacher to attend a minimum of three to six mathematics classes and these classes would need to
be scheduled at different times, and not at the time of the support classes for ELA. Currently, the
schedule is not structured to accommodate this model. Once the roles and responsibilities for
the LS teachers are more specifically defined, the administration leadership team could look to
how they could support the mathematics teacher request for additional teacher support in the
designated support classroom.
Summary
When asked to describe new programs, all teachers indicated they would continue with
RTI as the intervention program, but they would modify and contextualize it meet the needs of
the middle school. They offered suggestions that would add value to the current intervention
program. However, to evaluate effectiveness of the RTI program, the faculty would need to
implement it with integrity. In order to have a sustainable program for intervention, however,
the program’s foundation should be formalized and documented before attempting to modify it.
Before attempting any new program, efforts should be made by the intervention team to address
the gaps identified in the structure of the existing program.
Conclusion
The qualitative data collected from the survey responses, interview responses and
document analysis revealed much about the middle school mathematics faculty’s knowledge and
skills, motivation and organization influences that support teachers’ efforts to implement the RTI
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intervention program to ensure all students achieve the major standards for their grade level by
the end of the school year. Four findings emerged from this study.
The first finding from the research study regards the knowledge and skills influences on
teachers’ ability to achieve the goal for all students to meet grade level standards in mathematics.
Middle school mathematics teachers have demonstrated a multitude of effective teaching
strategies and have the knowledge to identify learning gaps. When presented with a volume of
qualitative and quantitative student data, however, teachers requested advice on more efficient
use of this data. In addition, they requested a researched-based, proven diagnostic tool that
would capture all learning gaps for middle school students across the sixth, seventh and eighth
grade levels.
Second, while documentation in the math department files is extensive, this study did not
find any direct evidence of documents that show individual students benefitting directly from
interventions. The information may exist somewhere, but in the math department files, there
were no descriptions of interventions including pre and post test results and details about the
intervention. While no specific intervention details could be found, the sixth grade example of a
daily spiral review addressed a collective need for 10 percent of the class, with documentation of
tracking student progress using MAP fall and spring tests as the pre and post test analysis.
While teachers described preplanned unit interventions in the interview process, there was
minimal evidence of preplanned interventions in unit plans. In order to have a sustainable
system of intervention, documentation is necessary for reflection, continual improvement and
transition of teachers.
Overall, benefits of specific, individual middle school mathematics interventions were
hard to identify and there was no way to analyze whether the attempted intervention was the
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reason for the student’s eventual success. Consequently, teachers do not know whether or not to
continue on with current practice or to try something new. When given the opportunity to
suggest new instructional practices, however, all teachers had suggestions, indicating that the
middle school mathematics teachers felt that their interventions were not as successful as they
hoped they would be. Together, their suggestions included additional problem solving through a
structured ‘flex time’ that is built into the schedule or flipped classrooms, alternative
assessments, and more collaboration through math hubs and co-teaching classroom models.
Teachers would like to try performance tasks as an alternative assessment or may want to flip
classrooms but feel stifled by the need to be consistent across the three different teachers per
grade level. Teachers see the need for extra time for students to practice solving problems
because there is such high attendance in the after school help sessions and during lunch, break,
and recess time. Teachers are concerned about the time that students should be spent eating,
socializing and participating in after school activities. The math hubs and co-teaching model
classroom are ideas that have come as a reaction to the lack of availability of the learning support
teacher due to scheduling conflicts with English Language Arts commitments. Some of these
requests for new programs came as a result of problems associated with current practice.
The third finding also relates to documentation but is more focused on tracking the
progress of the individual learner by the learner himself or herself. At ISSEA, personalized
learning is an area of strategic focus and in general, personalized learning is paramount to the
success of an intervention for a struggling student. Personalized learning allows students to have
ownership of their own learning and the final stage of metacognition, according to Krathwohl
and Anderson (2001), would be for students to create an innovative, learner portfolio.
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The fourth finding relates to the structure of the intervention team. Currently, there is no
defined intervention team. Instead, there are several departments, led by different people on the
administrative leadership team, that do not efficiently collaborate or communicate due to a lack
of clarity in roles and responsibility and due to structures that do not allow for common planning
or consistent support in the classroom. The faculty perceived a lack of involvement by the
leadership team and the LS teachers in the process of intervention.
In Chapter 5, these findings will be discussed in light of the research on the influences on
mathematics teachers’ ability to support students that present learning gaps. Implications for
practice and future research will also be presented.

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CHAPTER 5: SOLUTIONS, IMPLEMENTATION AND EVALUATION
Every new school year begins with the arrival of students with a range of academic
abilities and differing levels of social and behavioral maturity. Most students will be able to
achieve proficiency in grade level mathematics standards through regular instruction, but
approximately 15 to 20 percent of students will need additional support (Buffum et al., 2012).
The purpose of this chapter is threefold. The first is to summarize the strategies that currently
exist at a high-performing middle school that correspond to strategies that have been proven
effective in review of the literature, and that help students improve their performance so they all
meet grade level standards for math proficiency/achievement. The second is to respond to the
second research question posed in this study, “What are the knowledge and skills, motivation,
and organizational (KMO) influences that exist to provide support to the middle school teachers’
ability to fill their students’ learning gaps while at the same time guide them to achieve grade
level standards by the end of the school year?” Solutions will be offered in this chapter to
address KMO barriers identified in the findings of this study. The third purpose of this chapter is
to present new, evidenced-based program recommendations that ISSEA or other middle schools
might implement to help middle school students fill learning gaps in mathematics so they can
achieve proficiency for math grade level standards. An evaluation plan will be presented to
provide guidance for implementing the proposed solutions and assessing progress towards
achieving positive outcomes.
Qualitative study methods were used to gather data from a purposeful sampling of middle
school mathematics teachers at ISSEA. Six teachers participated in surveys and four of those
teachers participated in follow-on interviews. Survey results helped to identify confidence levels
of the respondents and helped to assess motivation in terms of active choice, persistence and
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mental effort. Interviews were transcribed and coded for themes related to knowledge,
motivation and organizational categories. Document review of mathematics departmental files
was conducted to triangulate survey and interview results.
Validated Effective Strategies for Middle School Mathematics Instruction
To address the challenges that students sometimes face, the mathematics teachers at
ISSEA middle school utilized a variety of complex instruction strategies and considered other
factors such as social and behavioral issues, to best meet the needs of all students. First, they
have established a learning progression based on the Common Core State Standards and PARCC
recommendations, and they have created power standards to focus the need-to-know content for
each grade level. They have also aligned content standards with math practices developed by the
NCTM. Together, they have worked in PLCs to create common formative and summative
assessments. Because of this collaborative planning, middle school mathematics teachers were
able to guide their students in their understanding of the learning target progression for each unit
of study. Second, ISSEA has adopted the RTI program for intervention and thus acknowledged
students learn in different ways and data can be used to make decisions about what students need
to be successful. Every grade level in the middle school has tracked student performance with
qualitative and quantitative data to help identify learning gaps and was able to create and
implement data-based interventions. Students showed growth in learning.
Validated Influences
The data collected during this study ‘validated’ or ‘validated in part’ all of the
knowledge, motivation and organizational influences that support middle school mathematics
teachers’ ability to fill students’ learning gaps while guiding them to achieve grade level
standards by the end of the school year. Table 14 lists the KMO influences that were validated
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through survey and interview responses as well as document analysis and Table 15 outlines the
KMO influences that were only ‘validated in part’ because some small gap still existed.
Table 14
Summary of Validated Influences
Category Validated Influence
Knowledge - Factual Teachers know factors that could cause gaps in learning.

Knowledge - Procedural Teachers know how to use effective strategies to deliver the
appropriate instruction.

Motivation - Persistent Teachers believe that engaging in a RTI program will Effort
contribute to students filling their learning gaps.

Organization - Structures Teachers have the resources needed to achieve the goal. and
Resources

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Table 15
Summary of Influences Validated in Part
Category Influence Validated in Part
Knowledge - Procedural Teachers know how to identify learning difficulties and choose the
correct solution path.

Knowledge - Conceptual Teachers know grade level expectations in mathematics and the
learning progression from elementary to middle school.

Knowledge - Metacognitive Teachers know how to guide students in awareness of their own
thinking and learning process and help them to self-regulate their
learning.

Motivation – Active Teachers value the goal of every student achieving
Choice proficiency in grade level power standards by the end
of the school year.

Motivation - Confidence Teachers are confident in their ability to execute the RTI program
to reach 100% students achieving at grade level power standards.

Motivation - Confidence Teachers are confident in their PLC team and with their
collaboration with the LS teacher to be able to reach the goal.

Organization – Clear Teachers have a clear vision of what they are working Vision
toward.

Organization - Structures The organization has structures in place where roles and
Resources and responsibilities have been delineated and an
effective communication process exists to share
necessary information.

Organization - Structures The organization has policies and procedures for RTI to and
Resources support teachers in achieving the goal.

Organization - Monitor There is a process in place that ensures the teachers get
Effectiveness timely, concrete feedback about their intervention
effectiveness.

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Solutions
Research Question Two - Recommendations
This section describes the recommended KMO solutions to support middle school
teachers’ ability to fill learning gaps and achieve 100 percent of students reaching grade-level
standards in mathematics.
Knowledge and Skills
The solutions provided in this section will be based on three sources of research. The
first source of research is the Anderson and Krathwohl (2001) framework, a revised version of
the original Bloom’s Taxonomy. This framework categorizes the knowledge and cognitive
processes for learning, teaching and assessing. The four types of knowledge processes include
factual, procedural, conceptual and metacognitive. The cognitive process dimension represents a
continuum of increasing cognitive complexity from remember to create. While the middle
school mathematics faculty showed strong factual knowledge, they also demonstrated a need for
more procedural, conceptual and metacognitive knowledge to ensure that they would have the
ability to support all students achieving proficiency/achievement in grade level standards. The
second source of research is the Clark and Estes’ (2008) gap analysis framework. Enhancements
to an already strong middle school mathematics program will be recommended for the
knowledge and cognitive process dimensions, based on the Clark and Estes’ framework. Finally,
the third source of research is a review of literature to inform how the recommended solutions
can be adapted to fit in the context of mathematics instruction in the middle school at ISSEA.
Gap analysis. According to Clark and Estes (2008), one of the ‘big three’ causes of a
performance gap is people’s knowledge and skills. The purpose of the knowledge analysis is to
identify and discuss whether people know how (and when, what, why, where and who) to
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achieve their performance goals. Rueda (2011) states that solutions to problems fail because we
attempt to apply solutions before the problem is fully understood. To help identify and analyze
causes of performance gaps, Clark and Estes (2008) suggest assessment should include listening
openly and fully to stakeholders, reexamining learning, motivation and organization theory and
rethinking the literature of the subject matter.
To determine any possible gaps in knowledge and skills, an initial survey was conducted
with the middle school math teachers and document analysis was conducted on digital files in
the middle school math department folder. Initial results revealed some gaps in stakeholders’
declarative (conceptual), procedural and metacognitive knowledge but to varying degrees. To
further define the performance problem, interviews were conducted. Table 16 shows the
knowledge gaps that exist and summarizes the proposed solutions. There are four possible
knowledge and skill solutions (information, job aids, training or education) based on level and
use of relevant past experience. The proposed solutions will be described more fully in the
following section.

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Table 16
Knowledge Gaps and Solutions
Knowledge Gaps Proposed Solution
Procedural – Individual student progress Provide teachers with job aids, such as
during interventions was not consistently student progress templates, and peer
reported or documented across grade levels. models of student progress tracking
documentation. Provide instructional coach to help
PLCs create their own student data tracking
document, then provide training and monitoring/
feedback to ensure repeated practice through all
grade level PLCs.

Conceptual - There was no documented Provide middle school math and
evidence that middle school math teachers elementary department leaders the
have collaborated with elementary math training time to review current status and
teachers to ensure no gaps exist in the to share and compare standards
transition from elementary to middle sequence with each other.
school.

Metacognition – There were no student Provide teachers with job aids, such as
learning portfolios found in any grade sample student learning portfolios, and
level in any digital files or in records of the peer models of portfolios. Connect
mathematics department to document teachers with an instructional coach for
student progress. guidance during the training and
implementation phase. Individual PLCs should
break down the process into manageable parts and
set intermediate goals, comparing results with other
grade-level PLCs.

Procedural knowledge gap and solutions. The procedural knowledge gap demonstrated by
teachers was that middle school mathematics teachers did not track individual student learning
progress as students progressed through learning interventions.
Proposed Solution: Teachers will be provided with peer models of student tracking
documentation and an instructional coach will help guide teachers in creating their own
template for the middle school math department. A coach will provide training and guidance
during the implementation phase.
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Middle school mathematics teachers are asked to track individual student progress, as
recommended by the RTI intervention program (Buffum et al., 2012). While all middle school
math PLCs currently track student concerns as a group, they have not yet tracked individual
progress. In order for teachers to implement individual tracking for a middle school program
with over 1,000 students, a template needs to be created to streamline the process. Learning is
increased when learners are pre-trained using known examples (worked examples effect) and are
presented complex material in manageable parts (completion effect) (Kirschner et al., 2016). In
addition, learning is increased when learners are provided a demonstration of how to do things
by a credible and competent model (Clark et al., 2012). A sample of an Individual Learning
Plan (ILP) is presented as an option in Appendix U and was designed based on the work by
Buffum, Mattos and Weber (2012) and modified to fit the context of ISSEA. Teacher training
by an instructional coach and peer models will help scaffold and guide teachers in creation of an
individual student tracking document template to be used in the context of the middle school
mathematics program at ISSEA.
Conceptual knowledge gap and solutions. The conceptual knowledge gap demonstrated
by teachers was that teachers did not document or show evidence of collaboration with the
elementary school teachers to ensure there were no gaps in the transition from elementary to
middle school.
Proposed Solution: The administrative leadership team will provide training time to the
department leaders of the elementary and middle school mathematics programs to share and
compare the current standards sequence for the purpose of ensuring that no gaps exist in the
transition from elementary to middle school.
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The middle school mathematics faculty created a learning progression based on the
CCSS and on recommendations by PARCC, with some alignment to the math practice standards,
which were developed by NCTM. An articulation of standards across the three grade levels of
middle school, called ‘MS Articulation of Standards,’ was created in the school year 2015-2016
(Appendix N) and from this work, the continuum of power standards, called ‘Power Standards
#SBG,’ was completed in the school year 2016-2017 (Appendix O). Both of these documents
should be discussed regularly by all middle school math PLCs to ensure they continue to
effectively address benchmarks in their instruction. “By reviewing the learning progressions for
each standard, teachers can get better at sequencing learning and avoiding repeating material that
was taught in earlier grades” (Gregory et al., 2016). The elementary school teachers also
adopted CCSS and math practices by NCTM. However, as of the time of this study, there was
no evidence that the elementary and middle school mathematics teachers shared their work on
the learning progression and standards sequence with each other.
Because the CCSS sequence was modified to contextualize to the needs of the middle
school at ISSEA, the middle school mathematics teachers need to collaborate with the
elementary teachers to ensure there are no gaps for students as they transition from elementary to
middle school. The administrative leadership team could provide time for mathematics
department leaders from elementary and middle schools to meet to share and compare standards
sequence during a professional development day or during the regular school day with substitute
teacher coverage for classes missed. Prior to the meeting, middle school math teachers should
review the ‘MS Articulation of Standards’ (Appendix N) and the ‘Power Standards #SBG’
(Appendix O) to ensure they are teaching all standards to integrity. During the meeting between
elementary and middle school math departments, the department leaders should share the current
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reality of the grade level math PLCs effectiveness of their instruction of the power standards.
The math department leaders for elementary and middle schools should create a long-term goal
to define their outcome, such as the alignment of the mathematic standards from elementary to
middle school, and they should also define their first steps, the short term goals, to address any
gaps that are discovered that will need to be filled to achieve the longer term alignment goal.
Goals should be measureable and timely and ultimately reviewed by the administrative
leadership team for completion and effectiveness.
Metacognitive knowledge gap and solutions. Although middle school mathematics
teachers know how to guide students in awareness of their own thinking and learning process
and there is evidence showing that they help students to self-regulate their learning, the teachers
did not facilitate the use of student-learning portfolios, the final stage of growth in Krathwohl
and Anderson’s knowledge and cognitive dimensions.
Proposed Solution: Teachers will be provided with sample student learning portfolios and
peer models of portfolios. An instructional coach will guide grade-level math PLCs through
the process of developing a template for the digital portfolio and during the implementation
phase. Teachers will break down the process and set intermediate goals, comparing results
with other grade-level PLCs.
The addition of digital portfolios to document student-learning progression will enhance
an already strong mathematics program and the implementation of portfolios will support one of
ISSEA’s priorities to personalize learning for every student. Before creating an innovative
digital learning portfolio, the middle school mathematics department should address the
following questions (Niguidula, 2005):
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• Vision: What skills and content should students master and demonstrate in their
portfolios?
• Purpose: Why do we collect student work?
• Audience: Who are the audiences for portfolios?
• Assessment: How do the entries in portfolios reflect the school’s assessment vision, and
how can we assess the quality of those entries?
• Technology: What hardware, software, networking and technical support will our school
need to implement a digital portfolio assessment system?
• Logistics: How will students enter their work into digital portfolios?
• Culture: Is discussing student work already part of our school culture?
When considering the purpose of the digital portfolio, math teachers should think about
showcasing students’ best work, providing evidence of mastering grade-level expectations, and
using the portfolio to communicate with parents and other audiences about what students are
learning.
When including work from all subject areas, the portfolio could have additional benefits
such as demonstration of 21
st
century skills, use of reflection to inform individual growth in
learning, and data collection for teachers use to better guide their students. First, an e-portfolio
could be used to showcase high impact instructional practices with learner outcomes across
disciplines in 21
st
century skills, such as communication, collaboration, critical thinking,
creativity and innovation, character, and cultural competence. Second, digital portfolios could
also provide another avenue for student self-reflection, answering questions such as “How does
this entry fulfill the school’s expectations?” or “What skills did you use in this project?”
Learning is increased when learners set goals and monitor their own performance (Clark et al.,
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2008). Students who complete a digital portfolio every year will begin to recognize how their
skills have grown over time. Third, digital portfolios could be set up to collect data to help
inform teachers’ instruction. For example, one high school had a link for graduation
expectations. When a student clicked on this link, they could see what expectations they still
needed to provide evidence for and then a teacher could help plan how to fill in the gaps
(Niguidula, 2005). Student use of an e-portfolio, when incorporated into daily classroom
activities, can provide benefits for students and teachers alike.
Three examples of digital portfolios are presented in this section to give middle school
mathematics teachers an idea of how to get started using e-portfolios to enhance student learning
in their classrooms.
Option 1: Portfolio for purchase. The first example, ‘Richer Picture’ was recommended
by Niguidula (2005) and is a platform for purchase that provides a great way to exhibit growth
over time. A sample one minute and 29 second video walks through the portfolio platform and
demonstrates the ease in uploading files in text, audio, streaming media, animation video or any
software-specific format on any device (Richer Picture, 2018). This option is not connected to a
grade book but teachers can set up a checklist of requirements for graduation where students can
monitor their progress of learning targets and update as necessary. This feature allows for more
self-efficacy on the part of the students and a better understanding of their own path in learning.
Option 2: PowerSchool Learning Portfolio, an addition to existing platform. The
second example is called PowerSchool Learning and is used as an additional application to the
PowerSchool platform that is already utilized at ISSEA for attendance and grade book recording.
This option would provide an advantage to ISSEA middle school teachers because PowerSchool,
the main platform, is already in use and PowerSchool Learning would provide additional features
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that would further integrate with ISSEA’s existing Google App accounts. PowerSchool
Learning allows teachers to assign collaborative activities, to give assessments, to record grades
and create reports for teacher analysis, to move between Google docs and the class, to share
resources with the class in a few clicks, and to empower students to compile work across classes
in an e-portfolio. While the grade reports are not attached to the e-portfolio, this option allows
the students access to a comprehensive system where they can easily compare their grades and
assignments to what they are adding to the e-portfolio. The student can take the e-portfolio with
them in a single HTML file when they leave the school. Instructions for how to use the e-
portfolio function in PowerSchool Learning can be found in PowerSchool Learning How-To
(Teachers) (PowerSchool Learning, 2018).
Option 3: EPortfolios with Google Applications. While the second option requires
administrative intervention and approval for addition of a new application to an existing system,
the third option, an e-portfolio created through Google Applications can be utilized immediately
without any additional support. ISSEA has fully integrated Google applications into its
education support structure. Document analysis confirmed that ISSEA’s middle school used
Gmail for correspondence, Google calendar to organize the schedule, Google drive to store
documents, Google docs to share collaborative or independent work, Google sheets for data
analysis, Google slides for classroom lessons and school-wide presentations, and Google groups
to share specific information. Some teachers, such as the Grade 6 Math PLC used Google sites
to create an online classroom environment. Creating an interactive portfolio with Google Sites
would be a natural progression for teachers who are already familiar with the use of the Google
platform in education. ‘How-to’ instructions for creating an e-portfolio using Google sites can
be found at ePortfolios with Google Apps (Google Sites, 2018). The ePortfolio with Google
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Apps does not link to a grade book, as does Option Two’s PowerSchool Learning portfolio or a
graduation checklist, as in option one’s the ‘Richer Picture’ portfolio. The disadvantage of a
Google ePortfolio, compared to the other options, is the lack of data that the teacher would be
able to immediately collect and analyze to support their students’ learning progression.
Whether the middle school math teachers choose option one, two or three, the
implementation path is the same. Middle school mathematics teachers need to discuss the
questions listed above and decide on the vision, purpose, audience, assessment, technology
required, logistics, and status of school culture to determine the type of e-portfolio that will be
used. Training for teachers will be required, followed by planned training for students. An
instructional coach can plan for and conduct the teacher training in use of the e-portfolio and one
teacher can take the lead on how to best instruct students on the process of developing the e-
portfolio. Together, the middle school mathematics teachers should set intermediate goals on
how and when they should implement the portfolio into their calendar year of instruction.
Grade-level mathematics teachers could monitor their progress through frequent check-ins with
the other grade-level math PLC teams and they could all improve their process based on
feedback with each other and follow-on reflection.
Motivation
The three facets of motivational behavior are active choice, persistence and mental effort.
When the three facets are present, motivation is increased and combined with effective
knowledge, skills and work process results in goal achievement (Clark & Estes, 2008). At
ISSEA, the middle school mathematics teachers exhibited great persistence and mental effort to
support daily small group interventions for struggling students. “The starting point for
implementing RTI is a shared belief that we educate all children who come through our doors
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(Windram et al., 2012).” Through survey and interview responses, however, the math teachers
indicated a lower than desired level of motivation in response to being able to support 100
percent of students in filling learning gaps and achieving proficiency in grade level standards by
the end of the school year as seen in Table 17.
Table 17
Motivation Gaps and Solutions
Motivation Gaps Proposed Solution

Active Choice/Confidence - Not all Master teachers will share specific
teachers believe that they can help every examples of instruction and assessment
student achieve 100 percent proficiency in that demonstrate how the efforts of
grade level standards by the end of the math teachers made a difference to all
school year and they are unable to meet the learners, especially for the students
needs of students in their class that need struggling to fill a learning gap.
Tier 2 and Tier 3 support.

Confidence - Teachers are not confident Provide teachers with a research proven
that the start of the year diagnostic diagnostic tool that comprehensively
assessment of skills captures all learning captures 100 percent of learning gaps.
gaps and teachers are not confident in their Provide math teachers with a consistent
ability to collect and analyze data format to collect data and provide
efficiently. training for data analysis that leads to the most
efficient path to effective interventions.

Active choice motivation gap and solution. The gap in motivation through active
choice was that teachers do not believe that they can support 100 percent of students achieving
grade level proficiency by the end of the school year and they don’t feel they are able to provide
assistance for all students needing Tier 2 and Tier 3 support in their classes.
Proposed Solution: Master teachers or leaders will share specific examples of
instruction and assessment that demonstrate how mathematics teachers’ efforts made a
difference to all learners, especially those who struggled to fill learning gaps.
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Successes or failures, which are attributed to effort, are more adaptive and will more
likely lead to more positive expectancies for success (Pajares, 2010). The middle school
mathematics teachers at ISSEA have demonstrated persistence and effort in their attempts to help
all students to achieve a high level of learning. However, they have received little to no
feedback regarding their efforts because there is no monitoring system in place. The
mathematics teachers’ perception is that they are not successful because they are unable to
support 100 percent of the students. Rather than rely on perceptions or overall data based on
collective student results, there are several checklists that could be utilized by the administrative
leadership team to give specific feedback to teachers on their efforts to implement RTI, to use
research-based curriculum and instruction to support struggling students, to use ongoing
measurement and assessment techniques and tools, to collaborate in their PLC team, to their use
of a problem-solving process and to measure their effectiveness in Tier 2 interventions. Sample
checklists are provided as suggested rough drafts or starting points in Appendix U, and are based
on reproducible templates created by Windram, Bollman and Johnson (2012). Specific feedback
using suggested checklists allows teachers to evaluate their strengths and challenges and to
create an action plan with specific steps towards continual improvement. In addition to
collecting data on teachers’ effort and persistence, the administrative leadership team should
allocate two to five minutes at each faculty meeting to share examples of student success as a
result teacher effort.
Confidence motivation gap and solution. The gap in motivation due to a confidence
issues demonstrated by teachers was twofold. Teachers felt unable to comprehensively capture
all learning gaps at the start of the year because they felt the start of the year inventory was
inadequate, and current data collection and analysis methods were inefficient.
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Proposed Solution: Teachers will be provided a diagnostic tool that has been proven
successful through research. They will be given a framework to organize their data in a
consistent manner across grade levels that is easy to access and understand. Training will be
provided on analyzing data to find the best solution path for interventions.
Currently, the middle school mathematics teachers have developed their own diagnostics
assessment tool to be used at the start of the year. The skills inventories have been developed by
the math PLC leaders based on their experience and the skills necessary for each grade level.
While the assessments help to identify learning gaps, the other members of the mathematics team
feel the assessments do not comprehensively capture all learning gaps. A tool, proven in
research, would bring more confidence to teachers, helping them to understand all of the needs
of their students at the start of the school year.
Suggested screening tools, proven in research. Proactive screening at all grade levels is
necessary so schools do not wait for parent or teacher referral of students of concern, which has a
greater potential for bias and constitutes a “wait until the student is failing model” (Windram et
al., 2012). Universal screening methods need to be improved to ensure more students are
identified for Tier 3 support at the start of the year. Currently, the LS department reports there
are more students with more serious learning deficiencies than the students with Individual
Education Services Plans (IESPs) that are accepted into the Tier 3 pullout LS program (LS 17.18
Planning and Thinking). “We do lots of planning for kids with IESPs, but some students with
the biggest gaps or deficits don’t have IESPs.” Windram, Bollman and Johnson (2012)
recommend selecting screening instruments of basic skills that have established reliability and
validity coefficients in acceptable ranges.
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Some suggestions for research proven diagnostic testing start with a Screening Tool
associated with RTI4Success (Screening Tools Chart, 2018), followed by diagnostic interviews
with students on problem solving in mathematics and then a diagnostic interview with student
and parent, which is recorded (Eaker & Keating, 2015). Other possible diagnostic tools include
General Outcome Measures (GOMs) that are representative of the grade-level content but not
direct excerpts from local textbooks. GOM materials and data storage systems can be found with
the AIMSweb product (sold through Pearson) or Yearly Progress Pro (sold through McGraw
Hill). Pro-Ed publishes a program called Monitoring Basic Skills Progress (MBSP) and includes
assessments for math computation and math application (word problems) (Windram et al., 2012).
AIMSweb is currently used by the elementary school at ISSEA therefore, this is the
recommended program of choice due to onsite support from teachers already trained in use of the
program as well as familiarity of testing for students as they progress by grade level. MBSP is
recommended for more specific targeting of student support after the initial screening is
complete.
Suggested data collection tool for consistent monitoring throughout middle school.
Middle school mathematics teachers have shown collective strength in collecting data that helps
to understand students learning gaps and what may be causing the gap. The next step would be to
understand how to use this data most effectively when choosing a solution path for intervention.
First, the math department will need a consistent and organized method to collect and store data.
Currently there are a variety of strategies used by the different grade levels. A starting point for
all grade-level teams could be to copy the model used in the 7
th
grade for their student concerns.
In one spreadsheet, the Grade 7 Math PLC combines and stores quantitative and qualitative data
on one page. The 7
th
grade math teachers pulled information from the comprehensive ‘Watch
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List,’ from standardized test results, from in-class skills assessments, and from students’
PowerSchool standards performance level results. The data collection spreadsheet also includes
comments from weekly ‘Kids’ Chats,’ documenting ongoing academic, behavioral, and social
issues that students are experiencing. The document also includes strategies that have worked or
not worked in a comments section. See Appendix V for a sample portion of the Grade 7 Student
Concern document for 2017-2018. The Grade 7 Math PLC team’s framework for data collection
is the model and proposed solution for an efficient collection method for all grade levels in the
middle school.

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Organizational Barriers
Table 18
Organization Barriers and Solutions
Organization Barrier Proposed Solution
Clear Vision - The leadership team is The administrative leadership team will
pulled in different directions due to the prioritize initiatives to help guide the
large number of strategic initiatives in direction of teachers’ effort and will
the strategic plan for 2020, leaving an provide motivational support by being
unclear message on the priority present with teachers, listening to them
of each teachers’ work. and understanding demands on time.

Structure and Resources – File ownership should be updated to
Communication between math and include sharing rights between LS and
learning support departments is inefficient math teachers so there is equal access to
due to different leadership of each all information.
department and due to separate digital
files storage with a lack of sharing rights.

Structure and Resources – Collaborative Once roles and responsibilities are
planning time for LS and math teachers is defined, the administrative leadership
not consistent between grade levels, and is team should monitor for effectiveness
sometimes nonexistent, leaving teachers and adjust scheduling for planning time
a perception of creating and implementing, as well as reconsider the placement of
by themselves, interventions necessary for LS teachers compared to the number of
all students to achieve grade-level students needing support and the levels
standards. of support needed.

Structure and Resources – Roles and Provide an organizational chart at the
responsibility are not delineated on the start of the school year and explain the
intervention team so LS and math teachers expectations for each role at the same
are not aware of the interdependence of time. Ensure that both the LS and math
their collaborative work. departments are aware of the interdependent nature
of their work to support struggling students.

Monitor Effectiveness – There is no formal Add an intervention section to the
method in place to monitor effectiveness of current observation template and
interventions. discussion protocols that are already in place in the
Professional Growth and Evaluation (PGE) process.

Clear vision - organization gap and solution. The gap in organization is due to an
unclear vision where the administrative leadership team is pulled in different directions due to
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the large number of initiatives for the strategic plan for 2020, leaving teachers with an unclear
message on the priority for their work.
Proposed Solution: The administrative leadership team in the middle school will
prioritize initiatives to help guide the mathematics teachers’ efforts and will provide
motivational support by being present with the teachers, listening to them and providing
avenues for feedback.
The key elements for successful change are found in the connection between a
compelling vision, a sound business process to reach that goal, clear work goals accompanied by
effective work procedures, motivational support for everyone and assessment of results that
reflect both the achievement of the vision and connected business and work goals (Clark &
Estes, 2008). At ISSEA, there are 21 initiatives in the strategic plan for 2020 separated into five
all-encompassing priorities (ISSEA Strategic Vision, 2017). Support structures for interventions
and extensions are one of the 21 initiatives, imbedded under ‘Professional Learning
Communities, one of five priorities. Another related initiative focuses on practices and programs
allowing for personalization of learning, under the priority of ‘High Impact Instructional
Practices.’ In order for mathematics teachers and PLCs to plan their work for the year, they need
guidance from the administration team, early in the school year, on where their efforts should be
focused. During the teacher professional workdays before the school begins, the administrative
leadership team could describe the current state of the school in the Strategic Plan for 2020 and
provide guidance on the goals for the current school year. Another checkpoint where the
administrative leadership team can provide continued guidance for the priority of initiatives
would be during the mid-semester or the start of the second semester professional workdays.
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Structure and resource - organization gaps and solutions. There are two gaps in
organization due to structure issues.
1. Roles and responsibilities are not delineated on the intervention team so LS and
mathematics teachers are not aware of the interdependence of their collaborative
work.
2. Consistent, collaborative planning time does not exist with mathematics teachers and
LS teachers, leaving mathematics teachers with a perception that they are creating
and implementing interventions primarily on their own.
Proposed Solution 1: The administrative leadership team should define the roles of
accountability and define the responsibilities for each team supporting the RTI effort.
“A culture of collective responsibility is the shared belief that the primary responsibility
of each member of the organization is to ensure high levels of learning for every child” (Buffum
et al., 2012). A collective, interdependent collaborative community is comprised of three teams,
including a leadership team, an intervention team and a teacher team. ISSEA’s middle school
included these three teams but the mathematics and LS teachers perceived the roles of each team
differently, as evidenced through survey and interview responses from mathematics teachers, as
well as through documentation defining LS teacher roles.
Suggested solutions for accountability include defining roles and responsibilities for each
team supporting the RTI effort. First, an organizational chart should be created to show
interdependence of teams. A sample organization chart is included in Figure 28 as a rough draft
to get the work started within the teams at ISSEA and is based on the roles and responsibilities as
defined by Buffum, Mattos and Weber (2012).
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Figure 28. Suggested organization chart for RTI intervention team in the middle school at
ISSEA.
An additional document should be created to ensure roles and responsibilities of each team are
outlined and shared with all people involved in the intervention effort so that each team is aware
of each other’s work. A suggested rough draft is included as Table 19.

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Table 19

Roles and Responsibilities of Intervention Team

Leadership Team Intervention Team Teacher Team

Principal
Deputy Principal
(Intervention Team Leader)
Deputy Principal (Mathematics
Department Team Leader)
Grade Level Counselor,
School Psychologist,
Speech and Language,
Pathologist, Nurse
Reading Specialist
Math Specialist
Librarian
Grade Level Teaching
Assistants

Grade Level Math PLC Teams
(As experts in their field, they
know the content best, have
assessment data and know
students best.)
Primary Purpose: To provide
structures and resources that allow
teams to support all students learning
at high levels, and to monitor the
effectiveness of the intervention
program
Primary purpose: To lead
schools focused micro
view on the specific
students in need of Tier
3 intensive support
Primary purpose: To design Tier
1 core instruction and lead
school’s response when students
require additional instruction to
achieve these critical learning
outcomes.
• Build consensus that the mission
of RTI is one of collective
responsibility.
• Create a master schedule
providing time for team
collaboration, core instruction,
supplemental interventions and
intensive interventions.
• Coordinate human and fiscal
resources to best support
• Articulate essential learning
outcomes per grade level
• Lead school’s universal screening
efforts to identify students in
need of Tier 3 interventions
BEFORE they fail.
• Lead school efforts for Tier 1
behavioral expectations,
including awards and
recognitions.
• Ensure all students have access to
grade level core curriculum.
• Ensure sufficient effective
resources are available to provide
Tier 2 interventions for students
in need of supplemental support
in motivation, attendance and
behavior.
• Ensure sufficient resources are
available to provide Tier 3
interventions for students needing
intensive support.
• Monitor school wide evidence of
student learning.
• Determine specific
learning needs of
each student in
need of intensive
Tier 3 support
• Diagnose the cause
of the student’s
struggles in Tier 1
and Tier 2
• Determine the most
appropriate
intervention to
address the student
needs in Tier 3.
• Frequently monitor
the students’
progress to see if
interventions are
achieving the
desired outcomes
• Revise the
students’
interventions when
they are not
achieving the
desired outcomes

• Clearly define essential
student learning
outcomes
• Provide effective Tier 1
core instruction
• Assess student learning
and the effectiveness of
instruction
• Identify students in
need of additional time
and support
• Take primary
responsibility for Tier 2
supplemental
interventions for
students who have
failed to master the
teams identified
essential standards.

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In summary, the math PLC teams should lead Tier 1 and Tier 2 interventions and assist in
planning Tier 3 interventions. The Intervention Team, including the principal, counselor,
psychologist, speech and language pathologist, nurse, English language specialist, reading
specialist, and math specialist, should lead the Tier 3 interventions. Currently, the LS
department has two pages of duties and responsibilities, but that list should be condensed and the
primary role of LS teachers should be focused on Tier 3 interventions. Because behavior is a
large component of RTI, teacher assistants who monitor breaks and recess should join weekly
student concern meetings. Finally, the Leadership Team should provide structure and support to
the Math PLCs and the Intervention Team. Once roles and responsibilities are defined,
expectations for different roles should be clearly explained at the start of the school year and the
administrative leadership team should monitor for effectiveness and accountability by attending
PLC meetings in the LS and mathematics departments. The administrative leadership team
should conduct frequent observations, providing constructive and continuous feedback and
allowing time for discussion with teachers.
Proposed Solution 2: Collaborative planning time for Tier 2 and 3 interventions should
be worked into the schedule in a way that support is provided consistently across grade levels.
The administrative leadership team should adjust structures that are currently in place to
allow mathematics specialist support for Tier 3 interventions for any student that has
demonstrated the need after the start of the year diagnostic assessment or a pre-unit
assessment.
Survey and interview responses showed that mathematics teachers felt a lack of support
from the LS department. The lack of communication between the two departments led to feelings
of frustration at the inability to support the achievement of 100 percent students at grade-level
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standards by the end of the school year. Specific claims noted a lack of collaborative planning
time, inconsistent planning time or different amounts of planning time compared to other grade
levels or past experience. The current organization of the LS support is not consistent across
grade levels.
Grade six. In sixth grade, there is a LS math specialist for one third of the students due to
a pilot program for integrated curriculum. The LS teacher for the remaining two-thirds of the
class provides support for both mathematics and English Language Arts (ELA). Consequently,
the Grade 6 LS teacher’s time is split between the two disciplines. She is not able to attend all
mathematics support classes due to concurrent scheduling of ELA classes and there is little to no
common planning time with the Grade 6 Math PLC.
Grade seven. In seventh grade, there is a LS math specialist assigned to the entire grade
level, which includes three mathematics teachers and approximately 330 students. There is
common planning time and she is able to come into math support classes for Tier 2, small group
interventions or she is able to pull students out for more targeted Tier 3 interventions. Despite
common planning time and scheduling, the sheer number of students does not allow consistent
support with preplanned interventions for any one teacher.
Grade eight. Eighth grade is similar to seventh grade where a LS math specialist is
assigned to the entire grade level. Unlike seventh grade, however, Teacher MS8.1 noted during
interview responses that there was a lack of common planning time with the LS teacher. With
the addition of two new teachers this school year, the administrative team focused on
observations to ensure consistency across the eighth grade level rather than monitor effectiveness
of interventions for students needing support.
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Common planning time for math teams and LS teachers would alleviate the problems
resulting from a lack of communication regarding how support is given to students expressing
learning gaps. However, common planning time needs to be structured in a way that teachers are
sharing what work they are doing for support.
Math PLC agenda: add a line-item for intervention progress. The suggested solution for
all grade levels starts with a line item addition to every math PLC agenda to document the
progress of current interventions and the plans for future interventions. The line item could be
titled ‘Intervention Progress and Next Steps.’ The agenda for the math PLC meeting should be
sent several days before the meeting so that the LS teacher has the opportunity to add
information, if unable to attend the meeting in person. An additional step to ensure
communication between math and LS departments is to add interventions to unit planning
documents. While the planning and details of the intervention can be in the unit planner, the
mathematics teachers and the LS teacher should also prepare evidence for the PLC meetings that
document an educational problem in Tier 1 based on students results of the common pre-
assessments and/or common formative assessments for the unit of study, a discussion of the data
resulting from the implementation of the assessment tools, and research based instruction
practices and recommended actions for further support (Bender, 2012).
Interventions recorded in unit planners. Unit planning documents need to be an active
part of the planning process and available at all meetings. They should be linked to the agendas
for the math grade level PLC meetings for easy access to necessary information. Document
analysis showed that the Grade 8 Math PLC started adding interventions into their unit planners
and these documents could be used as the framework model for all other grade levels. For an
example of a Grade 8 Unit plan, see Appendix W. For more specific organization of the
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intervention section in the unit planner, the following intervention plan in Table 20 is suggested
as a more detailed template (Buffum et al., 2009). The intervention plan could be saved each
year in the reflection section of the unit planner. A new intervention plan template would be
used each new school year but would include the history of assessment tools and research-based
instructional practices that were found to be most effective in the previous year.
Table 20
Intervention Plan
Prior Skills Needed
(Prevention)
Taught By Taught When Assessment Tool Research-Based
Instructional
Practices to be Used

Tier 1 Core
Instruction
(Goal 75+% Proficient)
Taught By Taught When
Assessment Tool for
Progress Monitoring
Research-Based
Instructional
Practices to be Used

Tier 2
Supplemental
Interventions
Taught By Taught When
Assessment Tool for
Progress Monitoring
Research-Based
Instructional
Practices to be Used
Failed Learners:

Tier 3 Intensive
Interventions
Taught By Taught When
Assessment Tool for
Progress Monitoring
Research-Based
Instructional
Practices to be Used
Failed Learners:

Once interventions are added to unit planners, the documentation provides the opportunity for
reflection and improvement, leading to a more sustainable intervention program. Unit planners
should be continually revised and updated to reflect the most successful practices.
A perception exists with MS mathematics teachers that there is a lack of support and that
more planning time is needed. However, once roles and responsibilities are identified and
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shared, awareness for accountability will be improved and work will be streamlined. In addition,
Tier 2 and Tier 3 interventions can be planned based on expected areas of struggle and these
interventions can be documented in unit planners for continual improvement. Common formative
assessments guide the understanding of which students need support, which instructional
practices proved to be most effective, what patterns are identified from student mistakes and
what interventions are needed to provide failed students additional time and support (Buffum et
al., 2009). Planning for interventions should come before the common formative assessment and
should be documented in the unit planner, with minor variations based on individual student’s
results. Because mathematics teachers are responsible for Tier 2 interventions and LS support
teachers are primarily responsible for Tier 3 interventions, communication during planning time
can occur on the agenda, if it is not possible to be present in person, and in the unit planner.
With more prior planning, the support of the LS teacher can be more efficient and better utilized
and the time needed for meeting in person can be reduced. Because the concern has been raised,
however, the effectiveness of this collaborative support needs to be monitored from someone
outside of the math PLC and LS teams to best see where changes can be made and/or
improvements can be added.
Monitor effectiveness – organization gap and solution. A gap exists in the
organization due to the lack of a formal method for monitoring interventions for effectiveness.
Proposed Solution: The administrative leadership team will add a section to the
observation template and discussion tool used during the Professional Growth and Evaluation
process.
Currently, a system is in place at ISSEA for professional growth and evaluation once
each semester. The process of evaluation begins with a goal setting session at the start of the
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year between a teacher and their primary administrative leader, and the session is followed by an
observation. A leader from the administrative team informs the mathematics teacher of a date
and time for an observation. The administrative leaders give guidance on what will be observed
and also make observations based on the teacher’s request. Observations take place in the
classroom for about 20 minutes and a written report follows by email within two days of the
observation. There is a minimum of one observation each semester. Towards the end of the
year, another session with the administrative leader and the mathematics teacher is conducted to
assess effectiveness of teaching strategies and instruction as well as progress towards the
personal and PLC goals. Data is recorded in an online system called TeachBoost. The process in
place provides a framework for monitoring effectiveness of teaching and there is an opportunity
for feedback with subsequent in-person sessions. At this time, the process of observation and
follow-on discussion follows a template and protocol, which does not explicitly include
interventions. A simple fix for this gap would be to include a section in the observation template
and discussion protocol for interventions and expectations for interventions.
The ‘Supervisor Look-For’ observation template includes three sections covering student
focus, teacher focus and additional notes. See Appendix X for an example of ‘Supervisor Look-
For’ documentation of an observation of Teacher MS7.3. The ‘Student Focus’ section includes
type of grouping, what students are doing, evidence of engagement and observations of 3
different students or student groups. The ‘Teacher Focus’ section includes connection to
learning targets and what the teacher is doing. The additional notes section includes a highlight
to share and a question. To evaluate an intervention, there are several characteristics to consider,
such as a sense of urgency in implementation, a requirement for struggling students to
participate, timeliness of intervention, a targeted intended outcome, necessary training and
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215
resources for the intervention teacher, and a seamless movement in and out of the intervention
(Buffum et al., 2009). A suggested addition for the current observation template and discussion
protocol for administrators would include a bullet point in the additional note section for a
description of the intervention that takes place during the classroom time, considering the
characteristics listed above.
Structure and resources gap due to inefficient communication. Communication
between mathematics and learning support departments is inefficient due to different leadership
of each department and due to separate digital file storage with a lack of sharing rights.
Proposed Solution: File ownership should be updated to include sharing rights
between LS and mathematics teachers so there is equal access to all information.
Collaboration involves sharing information between all teams. Files and digital
information should be shared between the LS and mathematics departments and this digital
sharing should be monitored. The owners of each department’s digital folder should share the
rights of their documents with the other department. In addition, language usage throughout
student data files should be consistent so topics that may be difficult to discuss with parents and
students are easier for them to understand.
Integrated Implementation and Evaluation Plan
Implementation and Evaluation Framework
The model that informed this implementation and evaluation plan is the New World
Kirkpatrick Model (Kirkpatrick & Kirkpatrick, 2016) based on the original Kirkpatrick four level
model of evaluation (Kirkpatrick & Kirkpatrick, 2006).
This model suggests that evaluation plans return to the goals of the organization and work
backwards, identifying the “leading indicators” that bridge from the completion of one’s
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216
recommended solutions to the organization’s goals. The “leading indicators” are the external
and internal measures of success along the way. Furthermore, outcomes of the solutions that
focus on assessing behavior on the job are identified, then indicators of learning during the
implementation are identified, and finally, indicators of satisfaction with the implementation are
identified. By designing the implementation and evaluation plan in this manner, the researcher is
forced to consider the connections between the immediate solutions and the larger goal, and may
be better able to solicit “sign on” (Kirkpatrick and Kirkpatrick, 2016) rather than buy in from the
larger leadership to ensure success.
Organizational Purpose, Need and Expectations
The strategic focus of ISSEA is that every student is known and advocated for, learns at
high levels and personalizes their own learning (ISSEA Strategic Vision, 2017). The goal of the
Middle School Mathematics Department is to create systemic interventions for common deficits
in middle school mathematics so that students with identified mathematics content deficiencies
will be able to co-construct a learning plan to correct their deficit and achieve grade level power
standards in mathematics by the end of the school year. With a thoughtfully planned, systemic
intervention program, students will get immediate support for their learning gaps and will be able
to create their own learning plan for success. They will learn to persevere with a challenge and
know that they have a support network to lean on from the mathematics teacher, parent and
learning support teacher at school. The will gain confidence and participate more in
class. Consequently, they will fill their learning gap at the same time as gaining new knowledge
at their grade level. The future impact will become evident when more students voluntarily
choose mathematics courses in high school and more students will choose to persevere in the
more challenging mathematics tracks. With experience working through a challenge and
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learning how to tap into their support network, students will have a better chance of completing
university in their chosen field and more students will have the tools they need to persist in more
challenging (pre-med, STEM) majors or more demanding top-tier universities.
Level 4: Results and Leading Indicators
If the middle school mathematics department achieves their goal of every student
learning at high levels, ISSEA will demonstrate increased performance on internationally
recognized measures of achievement in mathematics, such as the Program for International
School Assessment (PISA) and Trends in International mathematics and Science Study (TIMSS)
assessments. With recognition of high scores, other educators from international, local and US
schools will visit ISSEA for advice on how to improve their own mathematics
programs. Professional education journals would report ISSEA achievement levels and how
they were able to achieve increased growth in learning in mathematics. ISSEA’s annual report
would include improved mathematics scores on MAP testing and parents would be confident
their child was in a rigorous program. They would continue to enroll and possibly increase
donations to the endowment fund. With an increased focus on data-driven instruction and a
commitment to collaboration, the middle school mathematics department itself would be more
motivated and confident about their work. The specific metrics and methods appear in Table 21,
below.

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Table 21

Expected Outcomes, Metrics and Methods
Outcome Metric(s) Method(s)
External Outcomes
Recognition
in
international
education
publications
and media
regarding the
achievements
of ISSEA in
mathematics
Higher numbers of ISSEA students scoring
in international measures of mathematics
(for example PISA and TIMSS)
Results from PISA and TIMSS
Visits by other professional educators to
observe teaching practices
Journal reports (Professional Education)
reporting ISSEA achievement levels and
growth in learning in mathematics
Internal Outcomes

International
School South
East Asia
Higher levels of student achievement in
middle school mathematics
All students will achieve greater than 50% on
the mathematics section of the Spring MAP
test.
Parents will express their feelings that their
child is known and advocated for. They
will feel their child has been provided
extraordinary care and they may refer
potential students to the school.
Survey or interview parents at the end of the
school year.
MAP test results posted in ISSEA annual
report will reflect growth in learning and
parents will continue to enroll their
students in ISSEA. Donors will contribute
more money to the endowment fund.
Compare donations from one year to another
to see if increase funds relates to increased
MAP scores.

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Level 3: Behavior
Critical behaviors. The three teams that support the RTI effort are the Administrative
Leadership Team, the Intervention Team and the Mathematics Teacher Team. Each team has
critical outcome behaviors that lead the overall effort to support all students at ISSEA achieving
Internal Outcomes

Middle School
Mathematics
Department
More students are identified for Tier 3
support at the start of the year due to a
universal screening tool, proven in
research.
Compare the numbers of students from the
start of each school year to see if more
students are identified each year. Also
compare the initial number of students
identified as needing to support to actual
number of students served over the course of
the school year.

Improved communication between teacher
and intervention teams.

Document analysis to verify that both
departments share digital documents and files.
Higher levels of engagement for students in
mathematics class
Administrative leadership team observations
of student interventions (Tier 2 small groups
in classrooms and Tier 3 pullout support) and
pre/post surveys with students on their
perception of support and how it affects their
success.
More effective teacher lessons due to
focused work effort and focused roles and
responsibilities for teacher and intervention
teams.
Pre/post surveys with teachers based on
administrative review of teacher effort using
checklists and administrative leadership team
observations of classroom lessons.
No gaps in the sequence of standards
during the transition from elementary to
middle school mathematics.
Administrative leadership team reviews
standard documentation for elementary and
middle school math departments and
interviews department leaders for continual
review of effective, current practice.
Grade Level
PLC teams
Improved collaboration in the teacher and
intervention teams, focused on student data
collected in a standard spreadsheet across
grade levels.
Students learning mathematics in classrooms
that use the RTI model of flexible grouping,
based on students’ academic ability level for
each unit and/or behavior and social
emotional needs.
Teachers positively and flexibly move
students to different classrooms and track
individual progress through interventions
on individual learning plans (ILP).
Classroom observations and document
analysis of student ILPs.
Teachers guide students in understanding
of their own learning progression through
the use of digital portfolios.
PLC and classroom observations as well as
document analysis of student portfolios.
Teachers use data to purposefully plan,
document and discuss interventions,
reflecting on effectiveness for continual
improvement.
PLC observations and document analysis to
verify preplanned interventions are used and
documented in unit planners and are discussed
in PLC meetings.
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220
grade-level standards by the end of the school year. The administrative leadership team
supervision and reflection, as well as revised structures and processes can occur at the same time
as the grade level Professional Learning Community (PLC) teams and the Learning Support (LS)
teacher collaborate and plan for improved interventions. For example, the administrative
leadership team can clarify roles and responsibilities of all team members, can update sharing
rights on department digital folders, and can observe and document teachers’ current efforts
towards interventions. At the same time, math PLC teams can create a consistent student data
spreadsheet across grade-levels, choose a universal screening tool and choose a digital student
portfolio platform. Elementary and middle school mathematics department leads can collaborate
to ensure sequence and articulation of standards leaves no gaps in the transition from elementary
to middle school. The administrative leadership team can work together with the math PLC and
intervention teams to provide data to be used to create interventions that are documented in unit
planners for continual reflection and improvement. Math teachers can document individual
progress with Individual Learning Plans (ILPs) to verify students success over the course of the
school year. The specific metrics, methods and timing for each of these outcome behaviors
appear in Table 22.

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Table 22
Critical Behaviors and Metrics and Methods for Assessing Progress.
Outcome Metric(s) Method(s) Timing
Critical
Behaviors

1. Administrative
leadership team
(deputy principals
leading intervention
team and math teacher
PLC teams) shares
digital folder for
department with each
other, allowing for
visibility of
information between
departments.

Digital folders are
shared between
intervention team and
math department team.
PLC team leads will
conduct a document
analysis to verify all
members have access to
view shared information.
Document sharing rights can be
updated immediately but should
be updated no later than the end
of the current school year.
2. Administrative
leadership team uses
checklists to document
teacher efforts towards
supporting all students
learning at high levels,
provides feedback and
publicly acknowledges
persistent effort at
faculty meeting.

PLC team knows the
effect of their effort
and has concrete
feedback to improve.
PLC team members
feel confident that they
can support all
students filling
learning gaps and
achieving grade level
standards by the end of
the year.
Administrative leadership
team will complete
checklists and share with
PLC team members,
providing feedback and
allowing for discussion.
Checklists should be completed
by the end of the school year and
public acknowledgement should
occur at one of the final faculty
meetings of the school year.
3. Math grade-level
PLC teams,
collaborating with
their designated LS
teacher, create a
comprehensive student
data spreadsheet
following the model of
Grade 7 (Appendix V).

PLC team evaluates
and updates
‘Articulation of
Standards’ and ‘Power
Standards #SBG’ to
reflect instruction that
took place during the
current school year.
Administrative leadership
team will observe PLC
meetings and conduct
document analysis to
ensure consistency between
grade levels.
First draft should be completed
before the end of the school year
and should be shared with the
following grade-level math PLC
team.
4. PLC teams evaluate
current reality of
standards sequence as
taught in the classroom
during the current
school year. Teams
compare articulation of
standards at a
department meeting.

PLC team evaluates
and updates
‘Articulation of
Standards’ and ‘Power
Standards #SBG’ to
reflect instruction that
took place during the
current school year.
Administrative leadership
team will observe PLC
meetings and conduct
document analysis to
ensure gaps are addressed
and solutions are proposed.
Update should be completed by
the end of the current school
year, allowing time for the
department leads to also meet
before the end of the school year
(proposed due date would be
May 1).

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Outcome Metric(s) Method(s) Timing
Critical
Behaviors

5. Middle School Math
Department Lead
meets with Elementary
Math Department Lead
and compares
articulation and
sequence of math
standards, noting any
gaps.

Department Lead
teachers share
standards sequence
and note any gaps.
Administrative leadership
team will observe PLC
meetings and will conduct
a document analysis of
revised ‘Articulation of
Standards’ and ‘Power
Standards #SBG’.
During the month of May, or
before leaving for summer break
so information can be shared for
PLC team’s use in planning the
next school year’s instruction.
6. PLC teams decide
on universal screening
tool and gain approval
from administrative
leadership team.
Administration
provides support and
resources, as needed.

PLC team studies
different options then
discuss benefits and
disadvantages of each,
ultimately choosing
the one that best fits
the needs of the
middle school at
ISSEA.
Administrative leadership
team observes PLC
meetings and provide
resources as necessary.

The choice of a universal
screening tool should be made by
the end of the current school
year, or as soon as possible.
Training on how to use this tool
should be conducted as soon as
possible and before the start of a
new school year. Coaching
should occur concurrently with
first use with students.
7. PLC teams will
choose a digital
portfolio option and
will learn how to use
the digital portfolio.
PLC team studies
different options then
discuss benefits and
disadvantages of each,
ultimately choosing
the one that best fits
the needs of the
middle school at
ISSEA. PLC teams
will create a template
for the digital portfolio
that will be presented
to students.
Administrative leadership
team observes PLC
meetings and provide
resources as necessary.

Choice of digital portfolio should
be made and template should be
created by the end of the current
school year, or as soon as
possible.
8. PLC teams will
design a training
program to guide
students in creation of
a digital portfolio to
document learning
progression and to
help students
understand how to use
the portfolio to gain
understanding of their
own learning strengths
and areas to work on.
PLC team will conduct
training with students
during class lessons on
using digital portfolios
and all students will
create a personal
portfolio, beginning
with one documented
entry of individual
work and reflection
Student digital portfolio
will be shared at the Oct
parent conference, students
will talk through their
learning plan with their
parents and at the March
parent conference, students
will reflect on progress and
articulate to parents next
steps.

Digital portfolios should be
created after Fall MAP testing
during the first week of
September.

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Outcome Metric(s) Method(s) Timing
Critical
Behaviors

9. Administrative
leadership team
provides PowerSchool
data in a format that
shows students who
have not met power
standards for previous
grade level.

10. Administrative
leadership team
provides intervention
and teacher teams the
consolidated
mathematics MAP test
data after the Spring
MAP test.
PLC teams and LS
teacher will request
and apply data
informing them of
incoming students who
have not yet power
standards and also data
for students who are
below the 50% on the
mathematics MAP test
as well as reading
scores.
Observations of PLC using
data to inform intervention
planning.
This could start in May of the
current school year or may take
the first few weeks of the
academic calendar year (August)
to establish during scheduled
meeting times
11. PLC teams will
continue to collect data
for students who don’t
don’t achieve higher
than 50% on the Fall
mathematics MAP test
and for students who
exhibit gaps in
academic, behavioral
and social emotional
issues.
PLC team and
Learning Support (LS)
teacher will input data
into the student data
spreadsheet that
includes grade reports
for student indicating
if they met standards
in each of the six to
eight possible
mathematics power
standards in the
previous school year
and their Spring MAP
test results in
mathematics and
reading.
Analysis of collected data
in the student data
spreadsheet.
This could start in May or may
take the first few weeks of the
year (August) to establish during
scheduled meeting times.
12. PLC teams identify
students needing
support and create a
learning plan to be
used to document
interventions for that
school year.

PLC team will use
data to help students to
identify learning
deficiencies and
document
interventions on an
Individual Learning
Plan (ILP).
Document analysis of ILPs
for each grade-level math
PLC.
ILPs will be created in the first
week of school, based on
previous year PowerSchool
record of standards achievement
and will be updated after Fall
MAP testing during the first
week of September.

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Outcome Metric(s) Method(s) Timing
Critical
Behaviors

13. Administrative
leadership team
reflects on the
assignment of learning
support teachers and
the structures
supporting
collaborative planning
time, making
improvements for the
next school year as
necessary.
PLC team will
perceive full support
from administration
and from the
intervention team and
will be confident in
their ability to support
all students with
learning needs.
Survey or interview PLC
math team and intervention
team members regarding
collaborative effort to
support all students
achieving at high levels.
Survey conducted now and
compared to survey results
conducted after improvements
are made in the new school year.
14. Administrative
leadership team creates
and shares the
organization chart of
the RTI team for
ISSEA’s middle
school.

15. Administrative
leadership team revises
duties and
responsibilities of
intervention team
members and shares
with all members.
PLC team and
intervention teams
acknowledge
understanding of the
organization chart and
duties and
responsibilities of
other team members.
Survey or interview PLC
math team and intervention
team members regarding
knowledge of their place in
the RTI effort and their
confidence to achieve
success in 100% of
students achieving grade-
level standards at the end of
the school year.

Survey conducted now and
compared to survey results
conducted after improvements
are made in the new school year.
16. PLC teams
collaborate to
complete creation of
common formative
pre-assessments for
every unit.
PLC teams will use
scheduled meeting
time to create pre-
assessments for every
unit.
PLC observations and
conversations
Pre-assessments will be created
and executed at least one week
before the unit begins.
17. PLC teams and LS
teacher will identify
common errors for
students who don’t
have the prior
knowledge for the unit.
PLC team will use
data to help students to
identify learning
deficiencies and to set
a goal to achieve
success by the end of
the unit.
Classroom observation
check of learning targets on
‘I can’ or ‘Concept
Checklists’ in student
notebooks.
During the week prior to the
beginning of a new unit.

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Outcome Metric(s) Method(s) Timing
Critical
Behaviors

18. PLC team and LS
teacher will co-create
intervention programs
for those who have
deficiencies in what
they need to know to
be successful in this
unit as well as
extension for those
who already know the
material. Interventions
are documented in unit
planners.
PLC teams and
Learning Support
Teachers group
students flexibly
between mathematics
classes on the different
7th grade sides that
meet concurrently.

Record the sessions in the
intervention section on the
unit planner located in the
shared mathematics
department folder (the date,
which students moved,
where the session was held,
who was the facilitator for
the session, what was
accomplished).
Every intervention and extension
19. PLC teams will use
interventions based on
identified deficiencies
on skills necessary for
the unit and will
document individual
student progress on
ILPs.
PLC teams and LS
teacher will plan
interventions based on
identified common
errors.
PLC observations and
conversations
After the pre-assessment and at
the start of the new unit.

Required drivers. Success at Level 3 is the key to Level 4 results. Required drivers are
the process and systems that reinforce, monitor, encourage, and reward performance of critical
behaviors on the job. The required drivers, which support the critical behaviors, are tools that
reinforce, encourage and reward. Required drivers that support the critical behaviors can include
follow-up modules, work review checklists, on the job training, refreshers, job aids, reminders,
modeling, self-directed learning, coaching, mentoring and recognition. To enhance the RTI
effort in the middle school at ISSEA, all teams should follow and regularly update the Critical
Behaviors checklist to ensure certain steps have taken place. Team leaders should provide
mentorship and coaching to encourage teachers that are implementing new strategies and tools
for the first time or for those who wish to have more practice and additional strategies. Team
leaders, peers in the PLC team or administrators should look for evidence of student learning
from interventions and provide job-aids or modeling, as necessary. Recognition should take
place in a mathematics department meeting so teachers can learn from those who found
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successful strategies and processes and recognition should also take place in all middle school
faculty meetings so effort and results are recognized. See Table 23 for proposed suggestions for
required drivers that support critical behaviors.
Table 23
Required Drivers, which support Critical Behaviors
Method(s) Timing
Critical
Behaviors
Supported
1, 2, 3 Etc.
Reinforcing
Math department team leader and intervention team leader will check for
Tier 2 and Tier 3 intervention documentation, such as student data
spreadsheets, student individual learning plans (ILPs), digital portfolios,
common pre-assessments for each unit, learning target goal sheets, and unit
planner intervention sections for recording student deficiencies and for
record of data for intervention sessions.

Prior to
intervention
3, 6, 7, 8, 12,
16, 17,and 18
An administrator will check to make sure there are no gaps in the
articulation of standards from elementary to middle school.
At the end of the
school year.
4, 5
Math department team leader and intervention team leader will check for
Tier 2 and Tier 3 intervention documentation, such as student data
spreadsheets, ILPs, digital portfolios, learning target goal sheets, and unit
planners for student progress updates and revisions.
After
interventions
(mid-quarter
timeframe)
3, 6, 7, 8, 12,
16, 17, and 18
A coach and/or an administrator will review learning interventions, flexible
grouping in classrooms, and documentation of interventions and provide
feedback to PLC teams.
Every week for
the first six
weeks.
2, 3, 8, 9, 10,
11, 13
Encouraging
Department team leaders and/or administrator will observe and provide
positive feedback to PLC team and LS teacher on persistent effort towards
interventions.

Ongoing 2
Rewarding
Acknowledge successful strategies during department meeting so teachers
from other grade levels can learn from others and acknowledge persistent
effort at a middle school faculty meeting.

Every few
months
2, 3, 5, 8, 11,
12, 16, 17, 18,
19
Success is recognized and acknowledged by an administrator in PGE
meeting documentation of performance.

Semi-annual 2, 3, 5, 8, 11,
12, 16, 17, 18,
19

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Monitor. Monitoring is a required driver that helps to ensure alignment and compliance
with critical behaviors and progress towards desired results. Types of monitoring for
accountability include action learning, interviews, observation, self-monitoring, document
analysis, survey and touching base in meetings. Document analysis on student data documents,
common pre-assessments and formative assessments, and individual student learning plans with
positive feedback can encourage continual reflection and improvement. Through interviews,
observations and touching base in meetings, the administrative leadership team and peer teachers
in PLC teams should make note of the resources, tools and professional development or coaching
that mathematics teachers require to be successful. Together, all teams involved in the RTI
effort should liaison with the administration leadership team for resources and needs that are
identified along the way.
Level 2: Learning
Learning goals. There are five components of Level 2 Learning: knowledge, skills,
attitude, confidence and commitment. It is easy to get caught up by adding ‘bling’ to lessons and
by adding more structure to different types of levels of learning objectives. However, it is
important to remember that learning is a means to and an end and to focus only on whether or
not learning is taking place in the mathematics classroom. Some common methods to evaluate
Level 2 learning are mathematics knowledge checks through tests, discussion, individual and
group activities, and simulations (Kirkpatrick & Kirkpatrick, 2016).
Program. The Response to Intervention RTI program explores strategies to support
students who need to correct mathematics content deficiencies and achieve grade level
mathematics power standards. At the same time, teachers are guiding students to understand and
own their learning and to believe they can be successful in mathematics. The Middle School
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228
administration team will provide support for the interventions by providing clarity of roles and
responsibilities in the RTI organization, by creating structures in schedules, and by providing
necessary proficiency and assessment data to middle school mathematics grade level teams and
the intervention team. In addition, the Middle School mathematics teachers will collaborate with
Learning Support (LS) teachers to identify from the data the students who have common
mathematics content deficiencies and will work together to create pre-assessments for unit
deficiencies and then create interventions to support students with learning gaps. Teachers will
consistently document student progress and analyze data for effectiveness of interventions.
Teachers will guide students to creating innovative digital portfolios to document their learning
progression. Math PLCs will annually review articulation of standards to ensure integrity of the
standard sequence and to ensure there are no gaps in content. This intervention program is
designed to provide students with a supportive network to help them to persevere with a growth
mindset to achieve proficiency in grade level mathematics power standards. Following
completion of the proposed solutions, the stakeholders, the PLC grade level teams will be able
to:
1. Capture all students needing Tier 3 interventions at the start of the year, rather than
waiting for students to fail to provide necessary support.
2. Identify common misconceptions that students make on common pre-assessments,
create and document interventions in unit planners, and document student progress
during interventions in ILPs throughout the year.
3. Reflect on documented intervention plans and student progress results to improve
future interventions.
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4. Group students flexibly between mathematics classes within the different grade-level
classes that meet concurrently with mathematics teachers leading Tier 2 interventions
with assistance from a LS teacher and with LS teachers leading Tier 3 interventions
with advice from mathematics teacher.
5. Guide students’ self-efficacy in their learning progression through the use of digital
portfolios.
Components of learning. Kirkpatrick and Kirkpatrick (2016) state the five components
of learning as “the degree to which participants acquire the intended knowledge, skills, attitude,
confidence and commitment based on their participation in the training” (Kirkpatrick &
Kirkpatrick, 2016, p. 15). From a teacher’s voice, knowledge indicates “I know it” and skill
means “I can do it right now.” A positive attitude suggests, “I believe this will be worthwhile”
and confidence shows “I think I can do this.” Finally, a firm commitment by a teacher states, “I
will do this.” Merrill (1983) further elaborates on the topic of learning by defining the principles
of instruction, beginning with the emphasis on problem-centered instruction and followed by
four phases for effective instruction, as listed below.
1. Learning is promoted when learners are engaged in solving real-world problems.
2. Learning is promoted when existing knowledge is activated as a foundation for
new knowledge.
3. Learning is promoted when new knowledge is demonstrated to the learner.
4. Learning is promoted when the learner applies new knowledge.
5. Learning is promoted when new knowledge is integrated into the learner’s world
(Merrill, 1983).
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The Learning Technology Center at Vanderbilt (Merrill, 1983) describes a learning cycle that
involves the implicit and explicit principles of effective instruction. The cycle begins with an
activator activity to generate ideas and continues through feedback and revision checkpoints as it
completes its journey. The learning cycle is depicted in Figure 29.

Figure 29. Learning Cycle, a framework for instruction. (Merrill, 1983)
The learning cycle, as well as the five principles of instruction, can be applied to the
implementation of an RTI program in the middle school mathematics department. For the
implementation of RTI, the learning components include factual, conceptual, procedural and
metacognitive knowledge for teachers’ understanding of what is expected and steps on how to
proceed. Teachers will demonstrate their skill when they actively begin to identify all students
with deficiencies and when they begin to plan, conduct and document interventions. They will
exhibit a positive attitude when they see the value in the work they are doing to help students
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231
create their own learning plan with their personal, data-driven goal based on work collected in a
digital portfolio. Finally teachers will feel confident to continue to persist in their efforts for
successful interventions, supporting all students in the goal to achieve grade-level standards by
the end of the school year. See Table 24 for an organized view of the components of learning for
the RTI implementation.
Table 24
Components of Learning for the RTI Implementation Program.
Method/Tool/Technique Timing
Before, during or
after training
Factual, conceptual, procedural or metacognitive Knowledge: “I know it”

Factual: Teachers know the common misconceptions made by students on unit pre-
assessments and can evaluate progress during an intervention. (check for knowledge, pre
and post test, record progress in ILP)
Every unit
Conceptual: Teachers know that the sequence of standards from elementary to middle
school is continuous with no gaps and they can identify which standards they taught and
assessed during the school year.
At the end of every
school year
Conceptual: How is RTI different or the same as the interventions and differentiation that
we have used in the past? (open-ended question used to analyze effectiveness of
interventions and to suggest improvements for future interventions)
End of each unit.
Procedural: Teachers will know how to use data consistently across grade levels and will
be able to efficiently to identify all students with needs.
Identify data: Before
the start of the school
year and before each
unit.
Procedural: Teachers will design different interventions based on deficiencies noted upon
assessment results and will track progress on student individual learning plans
(ILPs). (Proof of knowledge will be recorded).
End of each
assessment.
Meta-cognitive: Teachers will use a rubric to assess their own progress in executing
interventions for each unit, reflect on practice to improve procedures for the next unit.
End of each unit.
Meta-cognitive: Teachers will compare digital portfolios with peers from another grade-
level and will improve their own practice to better guide students in their understanding of
their own learning progression.
Once every quarter
for a total of 4 times
in one school year.
Skills: “I can do it right now”

Teachers can assess a student’s active demonstration of knowledge using a rubric. End of each unit
assessment.
Teachers will apply interventions and will group students efficiently and maximize
instruction time for all students of different ability levels (teacher rubric to assess PLC
effectiveness).
During intervention.

Attitude: “I believe this is worthwhile to do on the job”

Survey or interview questions for PLC team and LS teacher: I can see that my students are
more engaged in lessons and I feel I can support all students in their understanding of
mathematics (rating scale). The importance of applying what I learned on this intervention
is… (open-ended question).
After the intervention
at the end of the unit.
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232
Confidence: “I think I can do this on the job”

Survey for PLC team and LS teacher: Rate on a scale: I have confidence in my ability to
pre-assess and group students, providing FAST (Frequent, Accurate, Specific, Timely)
feedback. Additional questions could include: what support will you need to implement an
intervention? What barriers do you anticipate that could limit your success in executing the
intervention? How confident do you feel about starting the process of RTI
implementation?

Other questions to assess a lack of confidence could also be included in a ‘check all that
apply’ list on a survey. My confidence is not high because:
• I do not have the necessary knowledge and skills to conduct this particular
intervention.
• I do not have a clear picture of what is expected of me.
• I have other, higher priorities.
• I do not have the necessary resources to begin an intervention.
• I do not have the support to begin an intervention.
• I don’t think what I have learned thus far will work in this case.
• There is not an adequate system of accountability to ensure application of an
effective intervention.
• Other (please explain):

Start of the
intervention
Commitment: “I will do it on the job”

Survey or discussion with PLC team and LS teacher:
• Rating scale or ‘thumbs up’: I am committed to collaboratively creating and
conducting pre-assessments and planning intervention programs. I am committed
to executing the plan and reflecting on what worked and what didn’t work.
• Discussion: How do you plan to begin your implementation? What will an
intervention look like for you?
Start of the
intervention.

Level 1: Reaction
Level 1 is “the customer satisfaction requirement of training” (Kirkpatrick & Kirkpatrick,
2016, p. 17). It is the degree to which the participants define their engagement in the activity and
whether or not the activity is relevant in their role as a math teacher. A critical component of the
reaction is to monitor and adjust as needed to help removing barriers and to help keep the
program on track. In Table 25, several methods describe how engagement, relevance and
monitoring are connected to the implementation of RTI in the middle school mathematics
department. A survey will be conducted prior to the first phase of implementation to see whether
the math teachers and Learning Support teacher received supportive information, were involved
in the data collection and creation of intervention programs. Another survey will be used at the
LEARNING GAPS IN MIDDLE SCHOOL MATH

233
end of each quarter to ensure mathematics teachers felt that they had the help they needed to
create and document interventions and that by collaborating with colleagues, their bonds with
each other became stronger. Teachers would know their work was relevant if the students were
actively participating in the interventions and achieving their goals. Finally, teachers will express
confidence in March of the school year, when the time comes to start over planning all over
again with a new set of incoming students on the horizon.
Table 25
Components to Measure Reactions to the RTI Implementation Program.
Method/Tool/Technique
Survey
Timing

Engagement: the degree to which the PLC team and the LS teacher are actively involved in the
intervention
The degree to which the team received helpful information before the start of the
school year (student concern data, PowerSchool achievement levels for standards
from previous year, previous year MAP test results, screening data)
Before the school year
begins.
The degree to which the team was involved in the data collection and planning the
Tier 2 interventions and likewise, the degree to which the LS teacher planned the
Tier 3 interventions.
After the unit pre-assessment
and before the unit begins.
The way the people involved in the RTI effort collaborated to support all students
learning mathematics contributed positively to my learning experience in using
interventions.
End of the first semester
Relevance: the degree to which the intervention process directly relates to intervention team
members’ responsibilities
I learned new strategies that will be helpful to document interventions and reflect
for further improvement.
End of each unit
I believe I will see an impact in the following areas: stronger relationships with
team members in my PLC and in the overall intervention team; students actively
participating in intervention process; students’ positive outcomes after receiving an
intervention.
End of each semester
Customer satisfaction: the degree to which participants react favorably to the initiative/course
Rate the following on a sliding scale from 1 to 5, with 1 being the lowest measure.
1. Knowledge gained from a coach (or mentor teacher). If you have 3 or less, what
needs to happen to increase your knowledge?
2. Confidence to collect data in a consistent spreadsheet, efficiently use data to inform
decisions with intervention, and conduct interventions so that all students are
successful in achieving grade-level standards. What needs to happen to increase
your confidence?
3. Confidence to document, track and monitor intervention progress with students and
communicate results to parents. What needs to happen to increase you confidence?
4. Commitment to the RTI process? What needs to happen to increase your
commitment?
5. Please share any additional comments.
Survey in March when the
year is starting to wrap up
and planning for the new year
begins.
(Likert scale 1-5)

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234
Evaluation Tools
Mid-quarter survey. A survey will be conducted to evaluate the perception of
mathematics teachers as they get started with interventions during the first quarter of the school
year. It is worthwhile to capture the thoughts of the mathematics teachers and close any gap that
may exist between what the data shows and their perceptions. It is also valuable to see how well
the participants feel at the start of the implementation and to address initial concerns. Too often,
potential barriers are not carefully thought out and planned for and crisis results. A survey can
help prevent crisis and shift teams to more pro-active prevention of trouble spots. See Appendix
A for “Getting Started with the RTI Effort in MS Mathematics.”
Second quarter survey. This second survey collects information about the successes
experienced by the second quarter of the school year. The data can be used for celebrations of the
process along the way. In addition, challenges can be noted and time can be set aside to address
possible solutions. This survey is also a pulse check on the commitment level of the
mathematics teachers to see if they need support or if they are inspired to continue with the
intervention process. To see the suggested survey for “Second Quarter Pulse Check in the RTI
Effort for MS Mathematics”, see Appendix Y.
Data Analysis and Reporting
During a time of change when there are many initiatives that teachers are working on, it
will be difficult to adjust to more than one requirement at a time. Therefore, the use of a Wordle
to capture the survey respondents’ words that are shared more often will visually and quickly
highlight their main successes and their biggest concerns (See Table 26).

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235
Table 26

A Wordle showing early signs of success and challenge in the first quarter

Further analysis of the data from each survey will help establish a solution path and get the team
focused on improvement in one area. If this survey were to be given twice a year, there could be
two improvement cycles in one academic calendar year and focused effort would yield
quantifiable improvement. See Table 27 for measurement indicators and target goal status.

Success Challenges

LEARNING GAPS IN MIDDLE SCHOOL MATH

236
Table 27

A Sample RTI Effort Dashboard based on Survey Results
Key Measurement Target Percent of
Successful RTI
interventions
Actual Status
Teachers know the common misconceptions
made by students on unit pre-assessments and
can evaluate progress during an intervention
100% 100%
→
Teachers know the sequence of standards from
elementary to middle school and know that
there are no gaps in the continuum.
100% 80%
↓
Teachers confidently reflect and revise
interventions based on effectiveness and
record their thoughts in the unit planner.
100% 30%
↓
Teachers confidently create pre-assessment
documents and flexibly group for interventions
according to roles and responsibilities of
teachers (Tier 2 led by Math PLC and Tier 3
led by LS teacher).
100% 30%
↓
Teachers confidently track student progress
after interventions using Individual Learning
Plans (ILPs).
100% 0%
↓
Percentage of teachers using digital portofolios
to document students’ learning progression.
100% 10%
↓
Percentage of barriers exist that will impede
student progression in RTI intervention.
0% 30%
↑
Teachers who received helpful information
prior to their first intervention.
100% 60%
↓
Percentage of teachers involved in collecting
data for a consistent, student data spreadsheet
and planning interventions.
22% 22%
→
Percentage of teachers that think it is
worthwhile to co-create, track and monitor
student progress.
100% 88%
↓

LEARNING GAPS IN MIDDLE SCHOOL MATH

237
Summary
In this chapter, the New World Kirkpatrick Model was used to evaluate and make
recommendations to all members of the RTI effort. The purpose of evaluating the intervention
process is to improve the program, to maximize the transfer the learning of RTI strategies to
actions by the mathematics teachers that result in positive outcomes for student learning. In
addition, the evaluation and documentation of the RTI implementation in the middle school
mathematics department is critical to demonstrating the value of RTI to the math department and
to the school community. The implementation and evaluation of RTI will help to achieving the
organizational goal of 100 percent of students achieving grade level power standards and
achieving at high levels by the end of the school year.
Future Research
Additional time for students and more collaborative work with other teachers was a
common request by the middle school mathematics teachers at ISSEA. While the program that
is currently in place is highly successful at capturing learning gaps and supporting the majority
of students to achieve at high levels, the teachers are always looking for ways to improve even
more. The change to ‘Challenge by Choice’ leveled problem solving in all middle school classes
is new this year and is still in the pilot phase. The shift allowed more time for individual and
small group problem solving in the classroom, with teachers as facilitators. There is little
research on this specific approach and additional time, analysis and evaluation is necessary to
understand the effects of this approach on all students’ learning.
Teachers proposed additional methods to gain more time to work on problems in the
classroom setting and during the school day. These ideas included flipped classrooms, a
structured flex time rather than unstructured help sessions and office hours after school, more
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238
structured co-teaching with LS teachers and math hubs, which would provide a common space
for teachers to teach together. Using a flipped classroom is an alternate pedagogy that has not
yet been used in the middle school mathematics department and would need to be researched to
see how to best implement it given the existing program. In addition, while math hubs are
currently in the pilot phase in one-third of math classes in the sixth grade, careful attention to
students’ growth in learning should be measured and analyzed to ensure this is a successful
approach, before rolling this out throughout the entire middle school. Some analysis of other
exemplar schools using flipped classrooms and math hubs would be beneficial to inform future
practice.
Conclusion
In 2016, a Measures of Academic Progress Assessment (MAP) revealed that 25 percent
of middle school students achieved less than the 70
th
percentile at the start of the year and 11
percent of students started below grade level standards in mathematics. At ISSEA, a high-
performing school, the goal is to have 100 percent of students achieving grade-level standards
and performing at high levels. In order to achieve the benchmark of all students learning at high
levels, middle school mathematics teachers differentiate instruction and level problems in the
classroom with the varied amounts of assistance from a Learning Support teacher.
Approximately four percent of students are enrolled in a Tier 3 support program allowing
additional time and instruction. However, an achievement gap of 21 percent exists, where
students are still struggling to achieve grade level standards with regular classroom instruction.
While the joint efforts of all stakeholders is necessary to achieve the goal of all students
achieving grade level standards by the end of the school year, middle school mathematics
teachers have been selected as the key stakeholders. Teachers at ISSEA have created inclusive,
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239
collaborative classroom environments that support comfortable yet challenging learning. They
engage in high impact instructional strategies, showing great teaching, every day for every
student. They demonstrate evidence of nurturing the growth mindset in students, and they
promote the use of metacognitive skills. All teachers fully participate in Professional Learning
Communities (PLCs) and can clearly articulate content progressions with prioritized standards
for each grade level. Collaboratively, they collect data and identify students that exhibit learning
gaps. While the middle school mathematics teachers have created a model mathematics
program, document analysis, interviews and surveys reveal that not all teachers are confident in
their ability to support all students in achieving desired performance levels.
Gap Analysis Framework
In order to identify the root causes of teacher inability to achieve 100 percent of students
reaching grade level standards at the end of each school year, a gap analysis framework was
applied, which focused on the knowledge, motivation and organizational influences that affected
the performance gap. Effective strategies that are currently in use by middle school
mathematics teachers were documented in this project and the key question that was studied was,
“ What are the knowledge, motivation and organizational causes that exist to provide support to
the middle school teachers’ ability to fill their students’ learning gaps while at the same time
guide them to achieve grade level standards by the end of the school year?
Through an extensive literature review of learning theories of knowledge acquisition and
the most effective strategies to bridge learning gaps for struggling students, document analysis of
middle school mathematics department digital files, online surveys with all stakeholders and four
Skype interviews with teachers representing all grade-levels, five assumed knowledge
influences, four assumed motivational influences, and five assumed organizational influences
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240
were identified. There were a total of 14 assumed influences. Of the 14 assumed influences,
four were validated, and ten were validated-in-part. The ten influences that were only partly
validated were analyzed for areas that could be improved. The findings were triangulated and
validated with document analysis of 6
th
, 7
th
and 8
th
grade digital files and with teacher interviews
for all three grade-levels.
Assumed knowledge influences. The final results exhibited validation of one factual
and one procedural knowledge influence. Teachers know the factors that cause gaps in learning
and teachers know how to use effective strategies to deliver the appropriate instruction.
However, there was exactly one influence in procedural, conceptual and metacognitive
knowledge categories, where the influence was only validated in part. Teachers need more
assistance to be able to identify all learning gaps and choose the correct solution path for all
students who struggle to learn. While teachers know grade-level expectations in mathematics,
they still need to articulate the progression from elementary to middle school to ensure no gaps
exist as a result of the transition. Teachers know how to guide their students in awareness of
their own thinking and learning process, but they still need to guide students to creating their
own digital portfolio to better personalize their own learning and to help them self-regulate their
learning. Proposed solutions to enhance an already strong program and to help teachers support
all students who struggle were presented in Chapter 5.
Assumed motivational influences. The final results demonstrated validation of
persistent effort from all middle school mathematics teachers. Teachers believe that engaging in
a RTI program will contribute to students bridging their learning gaps. Three other motivational
influences were validated in part only. For the category of ‘active choice,’ not all teachers
valued the goal of every student achieving grade-level, power standards by the end of the school
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241
year. For the ‘confidence’ category, not all teachers were confident in their ability to support
100 percent of students achieving the goal. In addition, not all teachers were confident that their
collaboration with their learning support teacher would lead them to the end goal of all students
achieving at high levels. Proposed solutions were recommended in Chapter 5.
Assumed organizational influences. The final results showed validation of one
organizational influence. Teachers have the resources needed to achieve the goal. However,
there were four organizational influences that were only validated in part. Teachers need a
clearer vision of what they are working towards and need intermittent direction for the priority of
their work effort as the school year progresses. Roles and responsibilities for collaboration
between the math and intervention teams need to be delineated and the communication process
between the mathematics department and the intervention support team needs improvement so
all parties have access to necessary information. Current scheduling structures and learning
support placement needs annual review to ensure consistent support throughout all grade levels
in the middle school. Minor adjustments to the current performance monitoring system need to
be made to ensure teachers get timely, concrete feedback about their intervention effectiveness.
Proposed solutions were recommended in Chapter 5.
Proposed Solutions
To address the teacher knowledge, motivational and organizational influences that were
only validated in part, the proposed solution is to provide middle school mathematic teachers
with several suggestions to systematize and streamline their processes to become more efficient
and to maximize their collaborative efforts with the other members of the intervention team.
Knowledge influences. To systematize and streamline processes, Professional Learning
Communities (PLCs) for mathematics in each grade level will work together to accomplish three
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242
goals: 1) to create a common framework for data collection throughout the middle school; 2) to
ensure the integrity of grade-level power standards each year and to articulate the content
knowledge sequence with the elementary math department; and 3) to facilitate digital student
learning portfolios. The student data collection document for middle school math PLCs will be
based on the framework already utilized by the Grade 7 Math PLC. Grade-level student data
will be shared with the learning support department and other members of the intervention team.
Using the same framework in each grade-level will save time, as the framework will not have to
be recreated each year and sharing it with the intervention team will ensure that information is
available for all who need it. Annually, the grade-level math PLCs will evaluate the integrity of
their instruction of standards and the math department leaders for elementary and middle schools
will meet to articulate the sequence so that no gaps exist for students transitioning from 5
th
to 6
th

grade. The middle school mathematics department will choose a platform for digital student
portfolios and will facilitate the rollout of their use for students to begin to be able to
demonstrate and understand their learning progression. A focus for the next school year on
improved data collection, on articulation of mathematics standards between elementary and
middle schools, and on the implementation of a digital portfolio will improve efficiency in the
middle school math department and will increase the ability for teachers to capture all students’
learning gaps, thus improving confidence of middle school mathematics teachers as they persist
in their goal of all students achieving grade-level standards.
Motivational influences. To increase motivation, middle school mathematics teachers
will employ a research-proven, screening tool to capture all learning gaps at the start of the year
and their efforts to support struggling learners will be documented. A universal, diagnostic tool
will more likely capture more students with learning gaps, allowing teachers to plan ahead for
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243
interventions that these students will need. Master teachers will document the efforts of
mathematics teachers’ instruction and assessment of students struggling to fill a learning gap,
using templates provided. Documentation will pinpoint areas of strengths, unequivocally
showing teachers that their efforts are productive, and will isolate areas where more work can be
done, focusing teachers on a specific achievable task.
Organizational influences. To increase sustainability of the intervention program and to
increase effectiveness in the collaboration within the intervention team in the middle school
mathematics department, the administrative leadership team will continue to prioritize initiatives
to maximize teachers’ efforts and will change sharing rights of documents to ensure all members
have easy access to the information available. They will create an organizational chart for the
intervention team and will define the roles and responsibilities of all members. They will
continue to monitor teachers’ ability to improve all students’ growth in learning and will
specifically look for intervention progress. The improvements suggested for the administrative
leadership team for better communication and clarity in accountability could be completed
immediately.
Proposed Implementation
To increase the sustainability and full integration of the proposed solutions, 14 external
and internal outcomes and 19 critical behaviors have been defined with specific short-term goals,
which range over the course of a school year. The middle school mathematics department will
work more closely with the members of the intervention team and department leaders and
administrators will reinforce goals by checking on intervention documentation, such as student
data spreadsheets, student individual learning plans, digital portfolios, common pre-assessments,
learning target goal sheets, unit planner intervention sections and record of data for interventions.
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244
An administrator will also check the articulation of standards, ensuring no gaps exist in the
transition from elementary to middle school. Mathematics department team leaders or
administrators will observe and review learning interventions, flexible grouping, and
documentation and provide feedback to the grade-level PLC teams. Persistent effort and
successful strategies will be recognized and acknowledged.
Proposed Evaluation Plan
As part of the proposed evaluation plan, surveys will be conducted with middle school
mathematics teachers to determine if the integration of suggested solutions has been successful
and to check on the commitment level of the mathematics teachers to see if they need support or
if they are inspired to continue with the intervention process. Applying the New World
Kirkpatrick’s evaluation framework, the impact of improved collaboration between the middle
school mathematics faculty and the intervention team along with more efficient processes in the
grade-level math PLCs will be assessed. The administrative leadership team will continue with
ongoing monitoring of the implementation of proposed solutions and will provide feedback for
corrections, as needed.
Implications for Wider Mathematics Community
While delimitations in the design of this study preclude the results from being
generalizable, the discovery of the most effective practices in student learning, the knowledge,
motivational and organizational influences on teachers’ ability to support struggling students,
and the proposed solutions to enhance effectiveness of instruction and interventions have value
for education programs in all school communities, whether in the international school system or
in the United States. The literature review found the eight most effective strategies to improve
student learning: 1) teacher effectiveness through cooperative learning; 2) teaching practices,
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245
such as mathematics teaching practices; 3) explicit instruction of mathematical practices in the
21
st
century instruction; 4) math practice at home and tutoring (peers or hired); 5) parent
involvement; 6) personalized learning; 7) metacognition and goal setting; and 8) response to
intervention. Furthermore, the literature review consolidated and documented the scientific
evidence as to why these strategies worked. The review of the middle school mathematics
department at ISSEA confirmed the most effective strategies found in literature review were also
effective in practice at this high-performing, international school. Considering the extensive
literature review, the detailed gap analysis and the explicit, step-by-step proposed
implementation and evaluation plan, the documentation of the high-performing middle school
mathematics program at ISSEA and the suggested strategies to improve effectiveness of
interventions that were presented in this study are widely applicable to all education programs
aiming to create a sustainable intervention program for learning gaps in middle school
mathematics.

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Appendix A
“Getting Started with RTI in MS Mathematics” (A survey to evaluate the perception of middle
school mathematics teachers at the start of the school year)

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Appendix A (continued)
“Getting Started with RTI in MS Mathematics” (A survey to evaluate the perception of middle
school mathematics teachers at the start of the school year).

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Appendix B
“First Quarter Pulse Check in RTI for MS Mathematics” (A survey determine success and
challenges experienced by middle school mathematics teachers).

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Appendix C
Grade 7 Mathematics Planning and Pacing Calendar for September 2016

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Appendix D
Grade 7 Math PLC Agendas from September 2017

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Appendix D (continued)
Grade 7 Math PLC Agendas from September 2017

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Appendix E
Grade 8 Class Lesson Demonstrating Differentiation

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Appendix F
Grade 8 Learning Target Sheet Sample (page 1 of 2)

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Appendix F (continued)
Grade 8 Learning Target Sheet Sample (page 2 of 2)

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Appendix G
Grade 7 Concept Checklist Sample

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Appendix H
Grade 6 “I Can” Statements Sample

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Appendix I
Samples of Tiered Problems for MS Mathematics (page 1 of 2)

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Appendix I (continued)
Samples of Tiered Problems for MS Mathematics (page 2 of 2)

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Appendix J
Examples of MS Mathematics PLC agendas showing student thinking (page 1 of 2)

GRADE 6

GRADE 7

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Appendix J (Continued)
Examples of MS Mathematics PLC agendas showing student thinking (page 2 of 2)

GRADE 8

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Appendix K
Grade 6 Math PLC’s ‘Unit Summative Assessment Itemized Math Practice Standard Alignment’
(page 1 of 8)

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Appendix K (Continued)
Grade 6 Math PLC’s ‘Unit Summative Assessment Itemized Math Practice Standard
Alignment’ (page 2 of 8)

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Appendix K (Continued)
Grade 6 Math PLC’s ‘Unit Summative Assessment Itemized Math Practice Standard
Alignment’ (page 3 of 8)

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Appendix K (Continued)
Grade 6 Math PLC’s ‘Unit Summative Assessment Itemized Math Practice Standard Alignment’
(page 4 of 8)

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Appendix K (Continued)
Grade 6 Math PLC’s ‘Unit Summative Assessment Itemized Math Practice Standard Alignment’
(page 5 of 8)

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Appendix K (Continued)
Grade 6 Math PLC’s ‘Unit Summative Assessment Itemized Math Practice Standard Alignment’
(page 6 of 8)

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Appendix K (Continued)
Grade 6 Math PLC’s ‘Unit Summative Assessment Itemized Math Practice Standard Alignment’
(page 7 of 8)

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Appendix K (Continued)
Grade 6 Math PLC’s ‘Unit Summative Assessment Itemized Math Practice Standard Alignment’
(page 8 of 8)

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Appendix L
Summary of Existing, Effective Strategies Currently in Use in MS Math at ISSEA (page 1 of 4)

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Appendix L (continued)
Summary of Existing, Effective Strategies Currently in Use in MS Math at ISSEA (page 2 of 4)

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Appendix L (continued)
Summary of Existing, Effective Strategies Currently in Use in MS Math at ISSEA (page 3 of 4)

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Appendix L (continued)
Summary of Existing, Effective Strategies Currently in Use in MS Math at ISSEA (page 4 of 4)

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Appendix M
Eight Effective Mathematics Teaching Practices

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Appendix N
MS Math Articulation of Standards – Sample (page 1 of 4)

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Appendix N (continued)
MS Math Articulation of Standards – Sample (page 2 of 4)

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Appendix N (continued)
MS Math Articulation of Standards – Sample (page 3 of 4)

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Appendix N (continued)
MS Math Articulation of Standards – Sample (page 4 of 4)

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Appendix O
Power Standards Math #SBG – Grade 6

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Appendix O (continued)
Power Standards Math #SBG – Grade 6+

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Appendix O (continued)
Power Standards Math #SBG – Grade 7

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295
Appendix O (continued)
Power Standards Math #SBG – Grade 7+

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296
Appendix O (continued)
Power Standards Math #SBG – Grade 7+

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297

Appendix O (continued)
Power Standards Math #SBG – Grade 8

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Appendix O (continued)
Power Standards Math #SBG – Grade 8

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Appendix O (continued)
Power Standards Math #SBG – Grade 8

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300
Appendix O (continued)
Power Standards Math #SBG – Grade 8+

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301
Appendix O (continued)
Power Standards Math #SBG – Grade 8+

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Appendix O (continued)
Power Standards Math #SBG – Grade 8+

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Appendix O (continued)
Power Standards Math #SBG – Grade 8+

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Appendix P
Prime Time Unit Assessment (Grade 6 Math PLC) – Pages 1, 2 and 6

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305
Appendix P (continued)
Prime Time Unit Assessment (Grade 6 Math PLC) – Pages 1, 2 and 6

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Appendix P (continued)
Prime Time Unit Assessment (Grade 6 Math PLC) – Pages 1, 2 and 6

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Appendix Q
Recommended Clusters of Study by PARCC (Partnership for Assessment of Readiness for
College and Careers)

GRADE 6

GRADE 7

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308
Appendix Q (continued)
Recommended Clusters of Study by PARCC (Partnership for Assessment of Readiness for
College and Careers)

GRADE 8

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309
Appendix Q (continued)
Recommended Clusters of Study by PARCC (Partnership for Assessment of Readiness for
College and Careers)

ALGEBRA 1

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Appendix R
Grade 6 Mathematics Planning and Pacing Calendar – Sample (page 1 of 2)

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311
Appendix R (continued)
Grade 6 Mathematics Planning and Pacing Calendar – Sample (page 2 of 2)

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312
Appendix S
Grade 8 Mathematics Planning and Pacing Calendar – Sample

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Appendix T
Suggested Individual Learning Plan (ILP)

Individual Learning Plan (ILP)
Name:
Grade: Birth:
ESL Level: Entry Date:
MAP Test Scores
Grade 6 Grade 7 Grade 8
Reading:
Math:
Power Standards Achieved
English Language Arts

Mathematics
o Understand ordering and absolute
value of rational numbers
(6.NS.C.7)
o Understand ratio concepts and use
ratio reasoning to solve problems
(6.RP.A1, 2, 3)
o Represent, analyze and solve one
and two-variable problems using
models, expressions, equations,
and inequalities (6.EE.A.3,
6.EE.B.7, 8, 6.EE.C.9)

o Compute fluently with rational
numbers (integers, fractions, and
decimals – 7.NS.1, 2, 3)
o Use properties of operations to
generate equivalent expressions
(7.EE.1, 3)
o Represent and solve problems
using expressions and equations
(7.EE.4)
o Analyze and use proportions and
percentages to solve problems
(7.RP.2, 3)
o Work with radicals and
integer exponents (8.NS.1,
8.EE.1, 2, 3, 4)
o Analyze and solve linear
equations and pairs of
simultaneous linear equations
(8.EE.7, 8)
o Define, evaluate and compare
functions (8.F.1, 2, 3)
o Understand the connection
between proportional
relationships, lines and linear
equations (8.EE.5, 6)
o Understand and apply the
Pythagorean Theorem
(8.G.6, 7, 8)
Interventions

1.
Begun on:

2.
Begun on:

3.
Begun on:

4.
Begun on:

5.
Begun on:

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Appendix U
Suggested Checklists for Evaluation of MS Math Teachers’ Efforts (page 1 of 6)

Parent Involvement
Not in place
Limited Practice
Partial
Implementation
Well established
All parents are provided information regarding the RTI
framework and what it means for them and their child.

Communication with families exists in a language or mode that
is meaningful to them.

There is meaningful communication between families and staff
about all students’ strengths and needs and additional
collaboration when concerns are identified.

Parents are notified when their child begins a supplemental (Tier
2 or 3) intervention.

Parents are provided with materials and training in the provision
of supports in the home setting when appropriate.

Parents of children who receive interventions at any tier are
provided reports on their child’s intervention, goals, and
progress toward their goals.

Comments:

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Appendix U (continued)
Sample Checklists for Evaluation of MS Math Teachers Efforts (page 2 of 6)

School Climate and Culture
Not in place
Limited Practice
Partial
Implementation
Well established
All educators have attended an overview presentation of the
RTI framework that included information on implications for
curriculum and instruction, assessment practices, and school
wide organization and problem solving.

All educators understand how the RTI framework is
represented in their school (including curriculum, assessment
and organization).

All educators understand that RTI is a school wide framework
designed to benefit all students.

Educators feel shared responsibility and play meaningful roles
in ongoing activities to sustain the RTI framework.

Research-based practices are understood and accepted by
educators and are consistently incorporated in classroom
instruction.

Educators are committed to ongoing professional development
regarding research –based practices and instruction of diverse
learners.

Consultation, feedback, and collegial exchange about
curriculum, instruction and behavioral expectations are
supported by administration and valued by educators.

Shared responsibility for all children is evident among
educators.

Expectations for academic performance and positive behavior
have been agreed on and shared with all stakeholders.

Educators believe that communication with families and
community is an integral part of their jobs

Growth and learning is celebrated with members of the school
and community and with families.

Comments:

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316

Appendix U (continued)
Sample Checklists for Evaluation of MS Math Teachers Efforts (page 3 of 6)

Curriculum and Instruction
Not in place
Limited Practice
Partial
Implementation
Well established
There are clear, high-quality core curricula in academic and
social behavior areas implemented with well-defined scope and
sequence plans across grades.

Ongoing work to align the core curricula with other grade-levels
and from elementary to middle school is evident.

Universal screening results are linked to ongoing discussions
about high-quality core curriculum for academics and social
behavior.

Teachers are knowledgeable about and implement principles of
effective instruction (high rates of engagement, opportunities to
respond, immediate error corrections, and so on).

Teachers understand how to embed basic skills instruction
within content areas classes and do this regularly.

Evidence-based curricula, instruction, or strategies are
identified for tiered intervention supports of increasing intensity.

Criteria and procedures for moving between tiers of
intervention are set.

By combining high-quality core instruction with intensive tiered
supports, the school has a plan to accelerate learning for all at-
risk students so they meet grade-level standards by the end of the
school year.

Comments:

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317
Appendix U (continued)
Sample Checklists for Evaluation of MS Math Teachers Efforts (page 4 of 6)

Measurement and Assessment
Not in place
Limited Practice
Partial
Implementation
Well established
An assessment plan exists that includes screening procedures
for all students at least three times per year; diagnostic
assessment as needed; a plan for progress monitoring those at
risk; and outcomes evaluation at least annually.

Data is stored in a database that is easily accessible by all
teachers and administrators in a timely manner.

Educators communicate about the purpose and value of the
assessments used, as well as their limitations.

Educators are skilled at interpreting assessment results and
making decisions based on these results.

School wide assessment data are used to identify students who
may be at risk in academic or social-behavioral areas.

Diagnostic assessments occur as needed to better understand
specific needs of identified at-risk students.

Schedules for progress monitoring are set based on the
intensity of students’ needs, and assessment occurs at least
monthly for all identified students.

Teachers regularly use data from progress monitoring to drive
instructional decisions throughout the continuum of supports.

Educators conduct an outcomes evaluation at least once per
year to identify areas of strength and need for continuous
program improvement.

Comments:

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318
Appendix U (continued)
Sample Checklists for Evaluation of MS Math Teachers Efforts (page 5 of 6)

Collaborative Teams
Not in place
Limited Practice
Partial
Implementation
Well established
Grade-level and school-level teams follow a consistent,
problem-solving process to make data-based educational
decisions that promote improvement in academic and social-
behavioral outcomes for students.

There is common understanding of the purpose and unique roles
of each team within the school and the way they interrelate.

Team meetings are regularly scheduled, of sufficient duration,
and frequent enough (monthly grade-level team meetings,
weekly problem-solving team meetings) to complete necessary
tasks. All members regularly attend meetings.

Grade-level teams exist in grades 5-9 and include general and
special educators who serve students at each grade level.

Meeting agendas are clearly communicated and include goals
and tasks. There is effective facilitation/leadership at each
meeting.

All teams maintain records on students they have served.
Effective communication exists between teams.
Comments:

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319
Appendix U (continued)
Sample Checklists for Evaluation of MS Math Teachers Efforts (page 6 of 6)

Problem-Solving Process
Not in place
Limited Practice
Partial
Implementation
Well established
Universal screening results for academic and behavior are used
to identify students for problem solving by grade-level teams.

Team members effectively and efficiently identify and
prioritize problems for every student or group of students
served through intervention services.

The prioritized problem for each student or group of students is
observable, measurable, and described as a discrepancy between
what is expected and what is occurring as measured on one
assessment tool, with additional converging evidence from
other sources.

An individual, specific and measurable goal is set for each
student.

Interventions selected by the problem-solving team are
supported by research.

Interventions selected by the problem-solving team address the
student need identified in the discrepancy and hypothesis
statements.

Intervention plans are implemented in a timely manner.
A plan to gather regular progress-monitoring data toward the
student goal is a part of each intervention plan.

Student responsiveness is evaluated based on progress-
monitoring data.

Intervention plans are evaluated in a timely manner, and
resulting decisions are documented.

The team cycles through the problem-solving process again and
again when students’ performance is not sufficiently responsive
to the current intervention.

Comments:

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320
Appendix V
Grade 7 Student Data Collection Spreadsheet – Frame 1
(Presented in five frames that would be combined in one spreadsheet from left to right.)

LEARNING GAPS IN MIDDLE SCHOOL MATH

321
Appendix V (continued)
Grade 7 Student Data Collection Spreadsheet – Frame 2
(Presented in five frames that would be combined in one spreadsheet)

LEARNING GAPS IN MIDDLE SCHOOL MATH

322
Appendix V (continued)
Grade 7 Student Data Collection Spreadsheet – Frame 3
(Presented in five frames that would be combined in one spreadsheet)

LEARNING GAPS IN MIDDLE SCHOOL MATH

323
Appendix V (continued)
Grade 7 Student Data Collection Spreadsheet – Frame 4
(Presented in five frames that would be combined in one spreadsheet)

LEARNING GAPS IN MIDDLE SCHOOL MATH

324
Appendix V (continued)
Grade 7 Student Data Collection Spreadsheet – Frame 5
(Presented in five frames that would be combined in one spreadsheet)

LEARNING GAPS IN MIDDLE SCHOOL MATH

325
Appendix W
Grade 8 Unit Plan Sample

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326
Appendix X
Supervisor ‘Look For’ Observation Protocol Example
Supervisor ‘Look For’
PRINCIPAL has just completed ISSEA: PGE Look-For. Results are listed below.
Message from PRINCIPAL:
Thanks MS7.3 Teacher. I enjoyed my time in your class yesterday.
INTRO
Teacher (s) MS7.3 TEACHER
Type Supervisor Look-For
Date of Visit 04/04/2017
Observer (s) PRINCIPAL
Time of Visit 12:30 pm
End Time 12:50 pm
Subject Math
Grade 7
DEMOGRAPHICS
What part of the lesson is being taught? Middle
STUDENT FOCUS
What grouping is occurring? Small group
What are the students doing? Practicing a skill
Evidence of engagement All students on task
Observation of Student 1 Knows why they are learning and/or how it
applies; seems to be challenged appropriately
Comments: The first group of students I spoke with said that trying to figure out area was hard. They
said they had to think “out of the box” and that it was fun, but a little stressful. When asked why you
don’t just tell them how to do it, they responded, “If she told us, it wouldn’t make us think.” One thing
that you do that helps them learn is re-teach when they don’t understand.
Observation of Student 2 Can articulate learning targets, seems to be
challenged appropriately
Comments: When asked on a 1-5 scale, was the task easy or difficult, this group said “In the middle.”
They shared that you help them by giving hints when they are stuck.
Observation of Student 3 Can articulate learning targets, seems to be
challenged appropriately.
Comments: This group seemed the most confident. They were initially confused by the diagonal
square area, but seemed to figure it out. They were doing a lot of estimating and trial and error. One
student said her brain was hurting, but “like when you exercise and your muscles hurt, it means it’s
getting stronger.”
TEACHER FOCUS
Connection to learning targets Task reflects the learning target
What is the teacher doing? Helping individual students, facilitating small
groups
HIGHLIGHT TO SHARE
When I arrived, you were circulating among the groups, offering hints and re-teaching. The students
were trying to derive how many squares they could find on a grid of dots. The students were all
engaged in the practice and seemed committed to figuring it out. About halfway through my time, you
had the students refocus on you and gave a “hint” to help move some of the students from being
“stuck.”
Boosters to student agency: Challenge: Teacher presses student to think deeply and rigorously.
Teacher expects students to persist when the fast a difficult task.
QUESTION TO ASK
How might this lesson be framed to help students connect to the “real world” relevance more
readily?
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327
Appendix Y
Second Quarter Pulse Check in the RTI Effort for MS Mathematics (page 1 of 2)

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328
Appendix Y (continued)
Second Quarter Pulse Check in the RTI Effort for MS Mathematics (page 2 of 2)

Abstract (if available)

Abstract At the start of every new school year, there are always middle school students who arrive with learning gaps in mathematics. Middle school mathematics teachers want to support these students in their collective goal of achieving grade-level standards in mathematics by the end of the school year. However, sometimes teachers do not possess the confidence or ability to support these struggling students. The intention of this study was to understand what teaching strategies were effective in teachers’ability to support students learning mathematics. Through a gap analysis, this study also looked at the knowledge, motivational and organizational influences that affected at teachers ability to guide students in closing their learning gaps and understanding mathematics at a high level. This qualitative case study explored the middle school mathematics department at a high-performing international school in South East Asia to find what strategies they effectively used that were proven in research. Data was collected through document analysis, survey and semi-structured interviews. Data was triangulated by data collection methods and by interviews with teachers from each grade level in the middle school. Data was then coded and checked against a conceptual framework generated from an in-depth literature review to validate the influences on middle school mathematics teachers’ ability to guide 100 percent of students to achieve grade-level standards by the end of the school year. The findings revealed the middle school mathematics teachers knew the factors causing gaps in learning, knew how to use effective strategies to deliver instruction, believed that the Response to Intervention contributed to students filling learning gaps and had the resources to achieve the goal of 100 percent of students achieving grade-level proficiency. The findings also revealed there was more work that could be done to help close the gap that currently existed in students who were unable to achieve grade-level standards and those who were able to achieve at high levels by the end of the school year. This study documents the strategies that work to support teachers’ability in supporting all students goals to reach grade-level standards, suggests several improvements to an already high-performing middle school mathematics department and offers some new possibilities to consider for any middle school mathematics program interested in creating a sustainable intervention program.

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Asset Metadata

Creator San Jose, Monica Mary (author)

Core Title Sustainable intervention for learning gaps in middle school mathematics: a gap analysis

School Rossier School of Education

Degree Doctor of Education

Degree Program Education (Leadership)

Publication Date 03/27/2018

Defense Date 02/15/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag education,effective strategies,intervention,learning progression,mathematics, response to intervention,middle school,OAI-PMH Harvest,RTI,support

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Picus, Lawrence O. (committee chair), Chung, Ruth H. (committee member), Darrough, Rebecca L. (committee member)

Creator Email monicamsanjose@gmail.com,msanjose@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-488733

Unique identifier UC11268051

Identifier etd-SanJoseMon-6128.pdf (filename),usctheses-c40-488733 (legacy record id)

Legacy Identifier etd-SanJoseMon-6128.pdf

Dmrecord 488733

Document Type Dissertation

Rights San Jose, Monica Mary

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA