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Bridging theory and practice: developing children’s mathematical thinking through cognitively guided instruction
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Bridging theory and practice: developing children’s mathematical thinking through cognitively guided instruction
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Content
Bridging Theory and Practice: Developing Children’s Mathematical Thinking through
Cognitively Guided Instruction
By
Lydia M. Song
Rossier School of Education
University of Southern California
A dissertation submitted to the faculty
in partial fulfillment of the requirements for the degree of
Doctor of Education
August 2022
© Copyright by Lydia Myungjin Song 2022
All Rights Reserved
The Committee for Lydia Myungjin Song certifies the approval of this Dissertation
Angela “Laila” Hasan
Kimberly Hirabayashi
Kenneth Yates, Committee Chair
Rossier School of Education
University of Southern California
August 2022
iv
Abstract
Students need to learn mathematics in a way that prepares them for their future. Standardized
achievement data shows a downward trend in mathematics scores from elementary to high school.
The purpose of this curriculum has a dual focus for teachers: to develop mathematical thinking in
their students through instruction that is centered on cognitively guided instruction and to provide
a process for planning robust instruction through the 5 practices for orchestrating productive
mathematics discussions. Two theories that support the approach to this curriculum are funds of
knowledge and culturally responsive mathematics education. The curriculum consists of eight
modules which occur over four days. Teachers will learn about the structure of problems, the
trajectory of student strategies, and intentional acts that facilitate instruction and dialogue. After
completing the modules, teachers will have lessons that they created for their classrooms. There is
a process for ongoing collaboration and follow-up at their site to ensure success. This curriculum
bridges the tenets of cognitively guided instruction with the 5 practices for orchestrating productive
mathematics discussions.
v
Dedication
To my mom and dad, for sacrificing so much, for inspiring me with your integrity and care for
others, and for the legacy that you imprinted on me.
To my loving husband, for believing in me and supporting me in countless ways so that I could
flourish.
To my children: Josiah, Audrey, and Claire for being patient, understanding, and encouraging.
vi
Acknowledgements
I would like to thank my committee members for your support and guidance in designing
this curriculum for teachers. Thank you for your time, attention, and willingness to share your
expertise to make this curriculum more poignant. Your comments and feedback encouraged me
to make this curriculum better. Thank you for sharing in this cause of supporting teachers who in
turn, teach our children. Math is so much more than a set of skills to learn.
I am so grateful for my Trojan colleagues who encouraged me to complete this
dissertation. Marisela, you supported me to be a better writer and your words of wisdom nudged
me forward one step at a time to the finish line. Ofelia, you are the biggest cheerleader and you
always encouraged me at the right moments. Jade, thank you for being on the other end of my
texts and keeping it real. In addition, I am thankful for my critical friends who always reminded
me of their faith in me and supported me in countless ways. Finally, I am thankful to the students
and teachers who inspired me to pursue this. I learned how to teach and think about math through
the eyes of the children who revealed their amazing thinking to me, and I was challenged by
teachers who invited me into their classrooms and pushed me to consider how teachers learn and
what they needed to be successful as learners too.
vii
Table of Contents
Abstract …………………………………………………………………………………………...iv
Dedication …………………………………………………………………………………………v
Acknowledgements ………………………………………………………………………………vi
List of Tables ……………………………………………………………………………………...x
List of Figures ……………………………………………………………………………………xii
Chapter One: Overview of the Project and Needs Assessment …………………………………...1
Problem of Practice ................................................................................................................ 1
Evidence for the Problem of Practice .................................................................................... 4
Importance of Solving the Problem ....................................................................................... 8
Instructional Needs Assessment ............................................................................................ 9
Implications for Achieving Goals ........................................................................................ 10
Curriculum Purpose, Goal, Assessment, and Outcomes ...................................................... 12
Learning Environment ......................................................................................................... 15
Potential Issues with Power, Equity, and Inclusion ............................................................. 16
Definition of Terms .............................................................................................................. 16
Organization of the Design Blueprint .................................................................................. 17
About the Author ................................................................................................................. 18
Chapter Two: Review of the Literature …………………………………………………………23
Prior Attempts ...................................................................................................................... 23
The Content of the Curriculum ............................................................................................ 25
Chapter Three: The Learning Environment and the Learners …………………………………...41
Description of the Learning Environment ........................................................................... 41
viii
Learner Characteristics ........................................................................................................ 44
Implications of the Learning Environment and Learner Characteristics for Design ........... 50
Chapter Four: The Curriculum…………………………………………………………………...51
Cognitive Task Analysis (Information Processing Analysis) .............................................. 51
Overview of the Modules ..................................................................................................... 55
Delivery Media Selection .................................................................................................... 68
General Instructional Platform Selection in Terms of Affordances .................................... 68
Specific Instructional Platform Selection in Terms of Restrictions ..................................... 73
Client Preferences or Specific Conditions of the Learning Environment ............................ 74
Specific Media Choices ....................................................................................................... 75
General Instructional Methods Approach ............................................................................ 78
Chapter Five: Implementation and Evaluation Plan ……………………………………………..81
Implementation Plan ............................................................................................................ 81
Evaluation Plan .................................................................................................................... 82
Data Analysis and Reporting ............................................................................................... 96
References ………………………………………………………………………………………100
Appendix A: Lesson Plan and Instructor’s Guide ……………………………………………...109
Module 1a, Overview ......................................................................................................... 109
Module 1b, Designing the Classroom Space ..................................................................... 119
Module 2, Structure of the Problems ................................................................................. 129
Module 3, Noticing Strategies (Join) ................................................................................. 139
Module 4, Noticing the Strategies (Separate) .................................................................... 153
Module 5, Structure of Multiplication and Division Problems .......................................... 167
ix
Module 6, Structure of Multi-Digit Addition Problems .................................................... 179
Module 7, Multi-Digit Subtraction Problems .................................................................... 191
Module 8, Base-Ten Number Concepts ............................................................................. 203
Appendix B: Materials for Module 3 …………………………………………………………...214
Appendix C: Keynote Slides for Module 3……………………………………………………..221
Appendix D: Evaluation Administered at the End of Each Day of the Course ………………...240
Appendix E: Evaluation Administered Delayed for a Period After the Program ………………241
x
List of Tables
Table 1: Percentage of All California Students Who Took the Smarter
Balanced Assessment System in Math…………………………………………………. 6
Table 2: Comparison of All Students Who Took the Smarter Balanced Assessment
System in Math to Economically Disadvantaged (ED) Students Listed by Ethnicity….. 7
Table 3: Declarative and Procedural Knowledge Required for Learning Goals……………… 39
Table 4: Scope and Sequence…………………………………………………………………. 67
Table 5: Dimensions of Media………………………………………………………………… 69
Table 6: Key Considerations for Media Selection…………………………………………….. 74
Table 7: Media Choices in Developing Mathematical Thinking……………………………… 76
Table 8: Indicators, Metrics, and Methods for External and Internal Outcomes……………… 84
Table 9: Critical Behaviors, Metrics, Methods, and Timing for Evaluation………………….. 86
Table 10: Required Drivers to Support Critical Behaviors……………………………………. 88
Table 11: Evaluation of the Components of Learning for the Program………………………..92
Table 12: Components to Measure Reactions to the Program …………………………………94
Table A1: Instructional Activities, Module 1a………………………………………………..112
Table A2: Instructional Activities, Module 1b………………………………………………..122
Table A3: Instructional Activities, Module 2…..……………………………………………..132
Table A4: Instructional Activities, Module 3…..……………………………………………..142
Table A5: Instructional Activities, Module 4…..……………………………………………..156
Table A6: Instructional Activities, Module 5…..……………………………………………..170
Table A7: Instructional Activities, Module 6…..……………………………………………..182
Table A8: Instructional Activities, Module 7…..……………………………………………..194
xi
Table A9: Instructional Activities, Module 8…..……………………………………………..206
xii
List of Figures
Figure 1: Major Steps to Develop Mathematical Thinking through Problem-Solving ………..58
Figure 2: Visual Representation of the Modules in the Curriculum …………………………...65
Figure 3: Third Grade CAASPP Student Achievement Data...………………………………...97
Figure 4: Discussions About Lessons and Student Thinking During Collaboration
Meetings………………………………………………………………………………. 98
Figure 5: Use Observations of Student Strategies to Plan Lessons ……………………………99
Figure A1: Module 1a, Overview…………………………………….……………………….109
Figure A2: Module 1b, Designing the Classroom Space……………………………………...119
Figure A3: Module 2, Structure of the Problems……………………………………………...129
Figure A4: Module 3, Noticing Strategies (Join)……….……………………………………..139
Figure A5: Module 4, Noticing the Strategies (Separate)……………………………………..153
Figure A6: Module 5, Structure of Multiplication and Division Problems…………………....167
Figure A7: Module 6, Structure of Multi-Digit Addition Problems…….……………………..179
Figure A8: Module 7, Multi-Digit Subtraction Problems………………………………….. ….191
Figure A9: Module 8, Base-Ten Concepts……………………………………………………..203
Figure A10: Module 3 First page Slide Handout……………………………………………….214
Figure A11: Module 3 Second Page Slide Handout...…………..……………………………...215
Figure A12: Problem Type Chart in CCSS Appendix…………………...……………………..216
Figure A13: Student Strategies……………………..…………………………………………..217
Figure A14: Lesson Planning Template………………………………………………………..218
Figure A15: Planning Reflection…………………...…………………………………………..219
Figure A16: Lesson Planning: Allison and Cookies…..………………………………………..220
1
Chapter One: Overview of the Project and Needs Assessment
The vision of the California Department of Education (CDE) is to prepare students to be
contributing members of society as they pursue personally fulfilling lives and careers. According
to the US News and World Report ranking of the 100 Best Jobs, the top 14 of 15 jobs require a
strong background in mathematics (U.S. News Best Jobs Rankings, 2021). All of these jobs
require post-secondary degrees. One job requires an associate's degree, five require a bachelor's
degree, three require a master’s degree, and the remaining six require a doctorate degree. More
importantly, according to the draft of the newly revised Mathematics Framework, a student’s
eligibility to attend a top university may be determined by the end of the fifth grade (CA
Department of Education, 2021). In short, how will students in California who are struggling in
mathematics compete for admission to higher education in order to qualify them for these
jobs? This curriculum attempts to respond to this question by teaching elementary educators
how to develop students’ mathematical thinking through problem solving. This chapter will
present the problem of practice, how my curriculum addresses the problem of practice, an
introduction to my curriculum, and the experiences and theories that influenced my positionality
in developing my curriculum.
Problem of Practice
Internationally, students from various countries participate in the Trends in International
Mathematical and Science Study (TIMSS) and the Program for International Student Assessment
(PISA). TIMSS assesses fourth and eighth graders on math content knowledge while the PISA
assesses how 15-year-old students apply their mathematical understanding to solve real-world
problems. The United States examines the data on these assessments to compare progress with
other countries and to determine which countries may provide insight to improve achievement
2
amongst US students. According to the American Institutes for Research report, a lack of
national curriculum, let alone national standards, may be a contributing factor to mediocre
performance on the 2003 assessments, the data that was used to inform the Common Core State
Standards (Ginsburg et al., 2005). Hence, the National Governors Association Center for Best
Practices and the Council of Chief State School Officers led a process to develop the Common
Core State Standards. In 2010, California adopted the Common Core State Standards for
Mathematics (CCSS-M). These standards were created to improve achievement in mathematics
by providing focused and coherent content standards that describe what students are expected to
understand and do. The CCSS-M emphasizes what learners require to develop mathematical
conceptual understandings, procedural skills, and Habits of Mind (Cuoco et al., 1996), all of
which will ultimately prepare students for college, career, and civic life.
To assist educators, parents, and curriculum publishers with implementing the California
content standards, the State Board of Education (SBE) appoints members of the Mathematics
Curriculum Framework and Evaluation Criteria Committee (CFECC) to revise the Mathematics
Framework every 7 years. For this current revision cycle, the members of the CFECC are
teachers, and personnel from district offices, county offices, and universities. The CDE and SBE
direct the work of the CFECC, which is to describe effective standards-based instruction, and
provide explanations, examples, and research-based teaching principles and practices. The draft
of the newly revised Mathematics Framework delineates a deeper innovation change in
pedagogy. In fact, it calls for significant reform in mathematics instruction in the state of
California.
The draft of the revised Framework describes the five components of equitable and
engaging teaching as:
3
1. Plan teaching around big ideas
2. Use open, engaging tasks
3. Teach toward social justice
4. Invite student questions and conjectures
5. Center reasoning and justification
The expectation of the draft of the revised Framework is that mathematical content is developed
through investigations which align with at least one of the Drivers of Investigation (DI). The
purpose of the DI is twofold. First, it piques students' curiosity which motivates and gives
purpose to mathematical endeavors. Second, it grants students opportunities to understand and
explain, predict, and/or affect the world through a mathematical lens. In short, the DI provide a
reason to learn mathematics. According to the 2021 draft of the revised Mathematical
Framework (CA Department of Education, 2021), the DIs are:
● DI1: Making Sense of the World (Understand and Explain)
● DI2: Predicting What Could Happen (Predict)
● DI3: Impacting the Future (Affect)
The experiences of historically marginalized students of color may point to another
reason for the need for reform. Historically marginalized students of color are restricted from
experiencing mathematics through robust and meaningful learning opportunities, which in turn
leads to lower status within hierarchies of mathematical ability and to disadvantaged social
hierarchies (Louie, 2017). Math educators in California need to address inequities for historically
marginalized students of color in terms of access to higher mathematics courses and post-
secondary educational opportunities. Culturally and linguistically diverse students have unique
needs, and their diversity should be seen as an asset rather than a deficit (NCTM, 2020). The
4
context and cultural background of students matter. Students, who were unsuccessful with
“school mathematics” problems, are in fact, able to accurately solve similar problems when
problems are connected to a familiar context (Louie, 2017; Nasir, et al., 2008). Equitable
instruction includes culturally relevant and engaging tasks. This means reframing mathematics
instruction by replacing “unidimensional” (traditional) classrooms that focus on memorizing
facts and fluency with basic skills, with “multidimensional” classrooms that develop meaningful
and meaning-making mathematics (Boaler & Staples, 2008; Gutierrez, 2002; Louie, 2017).
The work that students engage in should be relevant to them and over time, the
investigations should build mathematical conceptual understandings, procedural skills, and
Habits of Mind (Cuoco et al., 1996). Changing instructional practice will not be accomplished in
one year. In order to capture students’ attention and prepare them for the future, teachers need to
engage students with tasks that are relevant and through pedagogical practices that garner
interest and equip students with essential 21st century skills. Mathematics teachers in California
need to understand and implement shifts in instructional practices to motivate and support
students by addressing their needs, especially historically marginalized students, by
implementing the instruction as described in the new Mathematics Framework. Thus, the
problem of practice addressed by this curriculum is that students are not learning mathematics in
a way that prepares them for their future.
Evidence for the Problem of Practice
Ball et al. (2005) found that a teacher’s Mathematical Knowledge for Teaching (MKT) is
a predictor of student achievement. A higher MKT score correlated with higher student gains
over the year. Conversely, a lower MKT score was tied to lower levels of student achievement.
Furthermore, Hill and Lubienski (2007) conducted a study with 438 teachers in California and
5
found a mild to moderate correlation that teachers in schools with higher numbers of
socioeconomically disadvantaged and Hispanic students, had a lower MKT score. Thus, these
students inevitably fell further behind. This finding mirrors the overall state data.
Many students are not meeting or exceeding standards on the state assessment. In grades
three through twelve, the state measures progress towards academic goals in mathematics
through the California Assessment of Student Performance and Progress (CAASPP) System.
There is a downward trend in scores on the Smarter Balanced Assessment System in
mathematics from third grade through eleventh grade. As shown in Table 1, in 2018–2019, there
is a downward trend of the percentage of students who have “Met or Exceeded Standard,”
starting with about half of all third graders to only about one third of all 11th graders. The scores
indicate that students have not developed robust mathematical thinking. Thus, this trend in scores
demonstrates the urgency to change the trajectory of scores before middle school for three
reasons. First, the biggest drop in scores is from third grade to fifth grade. Second, the revised
California Mathematics Framework states that eligibility for a top university and a Science
Technology Engineering Art Mathematics pathway are determined by the end of fifth grade (CA
Department of Education, 2021). Lastly, low test scores are an indication that students may be
less prepared for post-secondary work.
6
Table 1
Percentage of All California Students Who Took the Smarter Balanced Assessment System for
Math in 2018-2019
Met or exceeded standard Did not meet or exceed standard
Grade 11 32.24% 67.76%
Grade 8 36.63% 63.37%
Grade 5 37.99% 62%
Grade 3 50.22% 49.78%
Note. Adapted from 2019 State Smarter Balanced Detailed Test Results: CAASPP Reporting
(CA Dept of Education), 2019.
Table 2 shows that student achievement scores on the Smarter Balanced State
Assessment System for economically disadvantaged (ED) Hispanic, Latino, and Black African
American students are consistently lower than both All students and ED White. The percentage
of ED Hispanic Latino students and ED Black African American students are consistently lower
than ED White students, at times less than half or even less than one third. Current instruction
and pedagogical practices are not supporting ED students overall, and even less our ED students
of color. The newly Revised Mathematics Framework proposes that teachers recognize and
respond to California’s diverse population by implementing lessons that are culturally relevant
and draw on students’ assets (CA Department of Education, 2021).
7
Table 2
Comparison of Math Scores on Smarter Balanced Assessment System: All Students to
Economically Disadvantaged (ED) Students Listed by Ethnicity
All students ED Hispanic Latino ED Black African American ED White
Grade 3 50.22% 36.21% 28.46% 44.74%
Grade 4 44.94% 29.75% 20.58% 45.95%
Grade 5 37.99% 22.78% 14.62% 48.80%
Note. Adapted from 2019 State Smarter Balanced Detailed Test Results: CAASPP Reporting
(CA Dept of Education), 2019.
The state test scores show that students are not making progress. Researchers found that
teachers with a lower MKT score are correlated with lower student achievement levels, so
working to improve the teacher’s MKT is warranted especially in schools with disadvantaged
students (Ball et al., 2005, Hill & Lubienski, 2007). MKT is specialized knowledge for teaching
in addition to content specific knowledge of mathematics. Hill et al. (2005) define MKT as “the
mathematical knowledge used to carry out the work of teaching mathematics” (p. 373). Thus,
helping teachers better understand and interpret student thinking processes is a significant step in
improving mathematics instruction and learning (Fenema et al., 1996; McDonough et al., 2002).
An intentional synthesis of the knowledge about student thinking and developmental processes
8
supports teachers to transform instruction in their classroom and hence develop robust
mathematical thinking in students (Fenema et al., 1996).
Importance of Solving the Problem
Reports of low-test scores and insufficient instruction fuel fears that students who are
inadequately proficient in mathematics will be unprepared for the global market and workforce
(Swafford et al., 2001). According to US News and World Report, many “top jobs” require a
strong foundation in mathematics (U.S. News Best Jobs Rankings, 2021). One measure of a “top
ten job” is income level which in turn is connected to level of education. According to the
Bureau of Labor and Statistics, a higher level of education translates into a higher weekly income
and a lower unemployment rate (Education Pays: U.S. Bureau of Labor Statistics, 2021). Thus,
access to college is key to future earning potential and financial stability. Students who are
struggling in mathematics before entering high school may find college entrance inaccessible.
Moses and Cobb (2002) state that algebra is a gatekeeper course because students who cannot
pass algebra are undeniably excluded from further educational opportunities. In California,
algebra content is mostly found in eighth grade. Unfortunately, the concern begins even before
eighth grade. According to the 2021 draft of the California Mathematics Framework, a student’s
progress at the end of fifth grade may determine their chance at attending a top university and
often, before entering sixth grade, many educational systems have decided eligibility criteria for
which students will be on a Science Technology Engineering Art Mathematics pathway (CA
Department of Education, 2021).
In 2001, the U.S. Congress passed the No Child Left Behind act to reform education by
improving student achievement and addressing the achievement gap. The magnified lens on
student achievement, indicated by higher test scores, resulted in an over-emphasis on test scores
9
themselves rather than robust instruction (Smith et al., 2020; Neill et al., 2004; Smyth, 2008).
The focus was on raising test scores and narrow skills that did not translate into higher order
thinking skills (Darling-Hammond, 2004). This disparity was especially apparent for students of
certain socioeconomic status and race because instead of measuring achievement, the tests
became the purpose of instruction (Smyth, 2008). The No Child Left Behind Act sunsetted.
Afterwards, the US adopted the CCSS-M. The new standards call for a shift in pedagogical
practices that focus on learning through problem solving and dialogue which ultimately leads to
a more substantive understanding of mathematics (Myers et al., 2020).
Many historically marginalized students of color come from homes that face
generational poverty. It is imperative that students learn mathematics so that they can be better
prepared for the future. This is an opportunity to shift math instruction. A student’s status at the
end of fifth grade may greatly impact their post-secondary opportunities (CA Department of
Education, 2021). Elementary students need to develop robust mathematical thinking to improve
their future outlook and their job prospects. As such, elementary teachers need to revamp their
instructional practices by focusing on sense making strategies for children and by teaching
through problem solving.
Instructional Needs Assessment
The state assessment data shows that students are not achieving at high levels. The SBE
is preparing to adopt a new framework. It illuminates “new” areas of emphasis and attention such
as equity for various marginalized student populations (CA Department of Education, 2021).
Smith and Ragan (2005) state that before designing new curriculum, it is important to determine
whether there is an actual need for new materials. This framework does not propose new
standards. However, it does call for an intentional focus to ameliorate the outlook for students
10
who have not been successful in the past. It requires educators to scrutinize their practice to
make adjustments to meet the needs of all students, especially historically marginalized students
of color. Shifting pedagogy requires innovation that is grounded in research.
Implications for Achieving Goals
The California SBE will adopt a new Mathematics framework in July 2022 that
prioritizes equity in the classroom. In order to deliver equitable teaching, teachers need to
understand the big ideas of mathematics so that they can instruct through meaningful and
purposeful tasks and activities. These tasks build on students’ assets, are interesting, real-world
based, and culturally relevant. In a classroom that promotes equitable teaching, teachers
orchestrate discussions where students and student thinking are at the center of conversations. As
students learn to communicate their thinking and mathematical reasoning, they will solidify their
understanding and connections for how mathematics is integral to their world.
Effective and equitable teaching includes being able to facilitate and guide students
toward understanding. Schulman (1986) and Hill et al. (2008) speak about pedagogical content
knowledge which is essential for being an effective math teacher because it requires knowing
more than only the content of the mathematics discipline. Strong math teachers understand the
content and how to present it and engage students in lessons that are relevant and interesting.
They are familiar with the landscape of concepts, student errors, and how to reroute student
thinking. Furthermore, teachers who serve historically marginalized populations in urban settings
need to know how to identify and develop lessons that speak to the needs of the community, and
hence are pertinent to their students.
The adoption of the new math framework is an impetus for change. The proposed
curriculum follows the Innovation Model which acknowledges a change in the instructional
11
program that may in turn call for revised goals within the organization (Smith & Ragan, 2005).
Teachers need to learn strategies based on what the new mathematics framework says. The new
framework specifically talks about big ideas in each grade level (not a focus in the previous
framework), equity and culturally relevant pedagogy (specifically named in the new framework)
and learning through experiences and tasks.
As teachers have a better understanding of what is required at each grade level, they can
identify, determine, and create tasks and lessons that appropriately support the learning goals of
their grade level. This will support a depth of knowledge rather than a sporadic, isolated
knowledge of a list of disconnected skills. Students may have a chance to learn and understand
mathematics beyond a specific test or a particular course.
Culturally responsive mathematics education is an undercurrent in the curriculum
because of the way problems are written, the choice of problems that teachers pose, and the
driving force behind how learning unfolds in the classroom. First, when writing problems,
teachers are directed to consider topics and contexts that make sense to the children in the room.
The contexts should not be a barrier for them. Rather, the context should help students make
sense of the problem and it should build on their lived experiences. Next, this curriculum builds
on what students know. The choice of subsequent problems depends on the strategies that
students are using or the challenges that students are facing. Teachers will decide what problem
is appropriate to pose next. It is not predetermined by a textbook. Students and students’ thinking
drive the instruction. Teachers will examine their students’ current strategies to gauge
appropriate numbers (or number ranges) for problems. It is not predetermined by a textbook
sequence. The teacher orchestrates learning opportunities and discussions to capitalize on what
students know, to stretch and develop robust mathematical thinking.
12
Curriculum Purpose, Goal, Assessment, and Outcomes
The purpose and goal of this curriculum is to support teachers to develop mathematical
reasoning in their students through instruction that is centered on problem solving. To
accomplish this, there is a three-pronged approach to this professional development. First,
professional development will focus on the content knowledge teachers need to implement
effective math instruction. Second, this course will provide opportunities for teachers to rehearse
pedagogical strategies that mirror classroom scenarios. Finally, teachers will examine problem
solving situations as a means to acknowledge students’ assets and cultural background as a key
to presenting meaningful and culturally relevant tasks.
The multifaceted approach frames the overarching considerations to the design of this
curriculum. First, the research on cognitively guided instruction (CGI) is the foundation of the
design of pedagogical math content (Carpenter et al., 1996; Carpenter et al., 2015). Three basic
tenets of CGI research are: students come to school with mathematical ideas, the problem type
framework is a classification system of word problems, and the identification and progression of
student strategies helps teachers determine instructional moves centered on those strategies
(Carpenter et al., 1996; Carpenter et al., 2015). The problem type framework helps teachers
focus on the structure and components of problems and how those features affect student
thinking and strategies. Furthermore, understanding the structural features of problems helps
teachers analyze student thinking and generate new problems. In turn, teachers will use the data
from students’ thinking to craft subsequent problems and lessons. By doing this, the teacher is
able to purposefully connect learning for students and make progress towards students’ content
goals. The second design consideration is how teachers will process and then implement the
content of the curriculum. The five practices for orchestrating mathematical discussions provide
13
the structure for helping teachers with instruction from the planning phase through
implementation of lessons (Stein et al., 2008; Stein & Smith, 2011; Smith & Stein, 2018). Stein
et al. (2008) describe the five practices:
● Anticipating: Teachers solve the problem they are presenting to students so that they
can engage in the mathematics first, anticipate student responses, and formulate some
thoughts about potentially salient discussion points.
● Monitoring: During the lesson, teachers circulate through the room and observe
students solving the problem. Teachers may engage in conversations to ask about
student strategies, help clarify questions or ask questions to extend student thinking.
● Selecting: One purpose of the previous phase (monitoring) is to notice the strategies
students are using to intentionally select certain students to share their strategy.
Teachers select strategies that highlight a point or contribute to the overall
mathematical discussion and storyline.
● Sequencing: The order in which students share strategies is not random. A teacher
selects specific student strategies and determines the order to present them. The
teacher considers what order would make sense and support the goal of the problem.
● Connecting: Teachers discuss and deconstruct strategies that students share. In order
to avoid a show and tell situation, the teacher may compare and contrast different
student strategies. This would help students see similarities between strategies and
they may also get a sense of the different levels of sophistication between strategies.
Through the conversations, students may have a new strategy to try next time.
Finally, the third design consideration is focused on incorporating students’ strengths and needs
into problem solving situations. One assumption of CGI is that students enter school with
14
knowledge of mathematical ideas. To start, teachers need to create opportunities for students to
show their thinking. Then, to build on that knowledge, teachers need to consider topics that are
relevant, interesting, and engaging for students. Acknowledging and naming topics that are
relevant to students makes learning more connected and purposeful for them.
The intent of layering the five practices with CGI is to learn, practice and apply the
content within the course so that teachers may feel more prepared when implementing new
strategies with their students in their classrooms. Within the professional development session,
teachers are learning the problem type framework and the strategies that students are likely to
use when solving word problems. Teachers will be keenly aware of planning problems that will
capture students' attention. As teachers learn to connect specific strategies to specific word
problems, teachers will become more comfortable in anticipating student responses. They will
also become more proficient in noticing, identifying, and classifying the strategies that students
use as they are monitoring progress, first within the professional development session and then in
a live classroom. Next, depending on the goals or pre-determined talking points, the teacher will
select strategies that best contribute to the goals. It can be challenging to determine the goals for
discussion. Sometimes teachers may think there is one right storyline, but there are many
plausible possibilities. The exact story to pursue will depend on the needs of the class and
purpose set by the teacher. Then, the teacher will consider the benefits of selecting the order to
present students’ strategies. The sequence of strategies needs to be intentionally determined to
best support the mathematical goals of that lesson.
Four salient outcomes of this curriculum are: (a) teachers understand how to teach
mathematics through problem-solving, (b) teachers understand and interpret student thinking to
build on it during classroom discussions, (c) students learn mathematics in a way that makes
15
sense to them and is culturally relevant, (d) both teachers and students see that mathematics is
connected to the real world. As a result, teachers will confidently and effectively plan lessons to
implement what they have learned. Furthermore, as students engage in problem-solving, they
will wrestle with mathematical concepts which will strengthen their mathematical thinking.
There will be one more measure to assess learning. Teachers will take a pre and posttest (sample
questions) from the CGI teacher knowledge assessment which was designed to determine the
impact of CGI on mathematics instruction (Fuentes, 2019). The change in pre and post tests will
demonstrate growth if the teacher’s understanding of CGI concepts has increased.
Learning Environment
This curriculum may be used in a professional learning series for in-service teachers. As
such, it falls within the category of non-formal learning (Smith & Ragan, 2005). It is intended to
support an organization’s overarching goals as well as site specific goals and objectives.
However, the curriculum is voluntary. There is no formal certification and grades are not
awarded for participating. California is adopting a new Mathematics Framework which
emphasizes culturally relevant pedagogy, asset-based models, and meaningful tasks. This might
encourage districts and sites to consider professional learning activities to help teachers align
their teaching methods and styles to the tenets of the new framework. Also, as some districts may
be in the process of adopting new instructional materials, this training series may provide an
additional lens to examine materials and act as a catalyst for shifting practice. Those who are
facilitating learning opportunities need to understand what the framework is proposing. In some
instances, the task will be to usher in new strategies for teaching. In other situations, it will be to
solidify sound practices to develop mathematical thinking.
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Potential Issues with Power, Equity, and Inclusion
Historically marginalized students of color come to school with strengths and experiences
that can serve as a basis for further learning. However, the American Institutes for Research
Report (2005) shows that historically marginalized students of color and female students’
performance on the TIMSS and PISA are not favorable. Historically marginalized students of
color and female students are not achieving as well as White male students. Rather than blaming
students and expecting them to conform to a long-standing curriculum, it is beneficial to consider
how to adjust curriculum to meet the needs of our historically marginalized students. By
acknowledging and honoring their backgrounds and strengths, these students can develop their
best self. CGI teachers choose problems using contexts that are culturally relevant to their
students. They draw on their students’ backgrounds and the assets within the community to teach
math in a meaningful and culturally relevant manner. By starting with their students’ thinking
and background, CGI teachers build lessons that are student-centered. All students are capable of
achieving, given conditions that foster understanding, focus on strengthening self-efficacy, and
connect learning to real life in meaningful ways.
Definition of Terms
● Five practices model is a systematic method for planning the facilitation of rich and
robust mathematical discussions. Stein et al., (2008) name the five steps for
orchestrating mathematical discussions: 1. Anticipating 2. Monitoring 3. Selecting 4.
Sequencing 5. Connecting. It is cumbersome to facilitate meaningful mathematical
discussions impromptu. By pre-planning, teachers will be able to craft discussions
that will propel the mathematical understanding of the group as a whole.
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● Learning trajectory is the natural path that children follow as they learn and develop.
Clements and Sarama (2014) denote the three parts of learning trajectories:
o A math goal connected to the big ideas of math
o A developmental path that children typically progress on as they formulate
understanding and skills
o Teaching practices, including educational environments, interactions, and
activities (or tasks), that are linked to each of the levels of thinking in that path
which support children construct higher levels of thinking.
When teachers understand these learning trajectories, they are better equipped to plan
and deliver effective and responsive instruction.
● Pedagogical content knowledge is subject matter knowledge that a teacher has
concerning teaching (Shulman, 1986).
● Teacher/professional noticing encompasses two processes according to Sherin, et al.
(2011):
o Attending to particular events in an instructional setting.
o Making sense of events in an instructional setting.
Teachers attend to different occurrences during instruction to make sense of events
and interact with the classroom. They need to consider and analyze many different
components during instruction.
Organization of the Design Blueprint
This curriculum design blueprint is divided into five chapters. Chapter 1 provides the
context and the need for this curriculum that addresses a problem of practice. Chapter 2 is a
review of the literature that informs the theoretical approach to and the content for the design of
18
the curriculum. Chapter 3 details the profile of the learner, the learner’s environment, and the
context for learning. Chapter 4 presents the curriculum goals, analysis of the learning tasks, and
outcomes for the curriculum. Finally, Chapter 5 describes the implementation and evaluation
process in determining the effectiveness of the curriculum.
About the Author
I am the youngest daughter of parents who lived through the Japanese occupation. From
an early age, I knew education was something my parents cherished perhaps because they had to
fight for every educational opportunity. Neither parent graduated high school because school
buildings were taken over by soldiers. Yet somehow my mother became a nurse. She sought
after the chance to learn, study, and make a difference helping others. More than two decades
later, my mother’s nursing career gave my family access to immigrate to the United States.
Education was a gateway for new prospects.
Coming to the United States, a country of promise and endless opportunities, was not
easy. Growing up, I rarely found another student who looked like me and I was painfully aware
that I had to try harder than those around me to fit in and to achieve my goals. Reading and
writing were especially hard for me, given that I was an unidentified language learner with no
support at school and limited English support at home. Math was a little easier because I was
able to make sense of it.
All of the prior struggles help inform my priorities as an educator. Seeing firsthand how
educational achievement provides access for individuals, fuels my social justice lens on
education. In fact, it fuels my passion for making sure students learn math. Moses and Cobb
(2002) say that algebra is the gatekeeper. Students who do not pass algebra have little chance to
advance through college. My students have to learn math and learn it well in order to have a
19
chance to achieve their future goals. I have noticed that sometimes there is a disconnect for
students of color. School learning may not make sense. The only reason math was easier for me
was because I could replicate the example problems and follow along with the arithmetic.
However now, mathematics is taught through context and situations. Students are expected to
learn more than just arithmetic. They need to develop an understanding of mathematics as a
discipline. Teachers have to ground lessons and concepts by making explicit connections to their
students’ lives. Problems need to be culturally relevant. Finally, it is important to teach from the
mindset that students come to school with knowledge and resources. It is up to the teacher to
recognize this and mine for their students’ assets. My teachers were not aware of my rich history.
If given the opportunity, perhaps my family could have contributed to the educational space
more. Unfortunately, school was not the most welcoming place for my limited English, and timid
parents.
I hope to make a love of or at least a fondness for mathematics a priority. The purpose of
this curriculum is to help teachers know how to develop mathematical thinking in students
through problem solving. I began my CGI journey as a third-grade teacher. Eventually, I left the
classroom to work for the county office to support CGI professional development. In this role, I
became curious about what it would take to support children to learn, how to inspire and
encourage teachers to create rich mathematical environments, and how to sustain these efforts.
Facilitating learning so that learners create meaning for themselves and feel confident became a
personal charge. Making sure that children learn math well, in a way that makes sense to them
and empowers them for their future is my passion. Students need to make sense of mathematics
so that they can use it in their life. The importance of real-world connections and solid
conceptual understanding became even more clear during my tenure as a middle school math
20
teacher. The teacher’s potential impact is paramount because they can help students make
connections through the way they pose problems and engage with students’ strategies to
reinforce mathematical content goals. Every student has the right to learn mathematics well and
to have access to opportunities.
Theories That Support the Author’s Positionality
Teaching and learning mathematics involve social interactions that draw on the
background and strengths of both educators and students. Developing mathematical thinking in
students, needs to connect to their experiences and make sense to them. Unfortunately,
sometimes there is a disconnect between school mathematics and that which students experience
in the world (Civil, 2002). Two theories that support the approach to this curriculum are Funds of
Knowledge (Moll, 1992; Moll & Gonzalez, 2004) and Culturally Responsive Mathematics
Education (Abdulrahim & Orosco, 2020; Aguirre, 2009; Gay, 2009).
Asset Based Funds of Knowledge
One of the assumptions in CGI is that young children enter school with mathematical
knowledge (Carpenter et al., 1996; Carpenter et al., 2015). So, one might wonder how children
develop mathematical thinking outside of a formal school setting. Children’s homes, families,
and communities are the source of potentially unorthodox, and untapped resources (Aguirre,
2009; Aguirre et al., 2013; Gonzalez et al., 1995). Examining local community practices,
businesses, markets, and libraries may help a teacher contextualize mathematics that already
makes sense to children (Aguirre et al., 2013). This would allow a teacher to build on and
connect to the knowledge that students bring to school. Authentic problems are motivating,
increase engagement, and are a catalyst for student agency and a sense of ownership in their
learning (Turner & Strawhun, 2007). In developing my curriculum, I had to counteract the deficit
21
mindset concerning historically marginalized students of color, the misperception that they are
ill-prepared or ill-equipped for school. The quandary of what counts as knowledge stems from
the disconnect between school mathematics and mathematics that is practiced in the community
(Civil, 2002). Students who were labeled as “more successful” were inclined to use traditional,
formulaic algorithms as opposed to the “less successful” students who were likely to make sense
of the mathematics and connect it to real life unless the school mathematics lacked relevance or
if students were not allowed to draw on their real-life experiences (Civil, 2002).
Culturally Responsive Mathematics Education
Mathematics instruction that considers the learners and their identities, is powerful and
empowering. Teaching students in an equitable manner means recognizing and responding to the
fact that every student has different needs and thus, instruction needs to adapt to provide access
to all students (Aguirre, 2009). Aguirre (2009) presents a conceptual frame that includes two
spectrums that emphasize equity and mathematics: the pedagogy of access and transformation on
one side and the pedagogy of problem solving and problem posing on the other side. Culturally
responsive mathematics education (CRME) practices support students from linguistically diverse
backgrounds (Abdulrahim & Orosco, 2020). Cultivating an appreciation and understanding for
cultural identity helps teachers recognize that families and communities are co-constructors of
knowledge that is a valuable asset in the classroom (Abdulrahim & Orosco, 2020). In addition,
teachers can increase instructional motivation by designing a space that attends to a variety of
students’ needs and background (Abdulrahim & Orosco, 2020). Teachers who reflect on their
practice and critically scrutinize their beliefs, are more readily able to construct mathematical
learning environments that truly support students (Abdulrahim & Orosco, 2020). Teaching
mathematics and developing robust mathematical thinkers is more than a job. It is a passion that
22
motivated the development of this curriculum. Most of my career has been working with
historically marginalized students of color and their families. Because I see mathematics
achievement as a key that provides access, it is imperative that we do it well. Students deserve to
be empowered and will benefit from understanding that they are not starting from ground zero,
but rather, that they have many resources and strengths on which to build.
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Chapter Two: Review of the Literature
This literature review is presented in two parts. The first part includes a description of
prior attempts and an analysis of those attempts by discussing some of the benefits and
drawbacks of the different configurations for training. The second part contains the content of
the curriculum. This section elaborates upon developing mathematical thinking through CGI.
Prior Attempts
There are many models and providers that deliver professional development in CGI.
Educators may find training through a provider that is geographically close (through the county
office, a local university, or district), a conference (regional math conferences or the national
biennial conference), or through math organizations like Teacher’s Development Group or the
Teacher Learning Center. Professional development offered through the county office tends to be
a multiple day training with follow-up support. Teachers may attend additional levels of training
in subsequent years. The content is mostly CGI, but it may include topics that are not part of the
CGI research base. Another source of training is through conferences. The speakers may be local
educators or researchers depending on their availability. These sessions tend to be shorter and
focus on a specific topic. Finally, math organizations like the Teacher Learning Center also
provide training. The training course is very intense, and it requires a 3-year commitment with
initial training in the summer (4 days) and additional follow-up days during the school year (two
in the fall and two after January). Currently, there is a 3-year research study in Florida involving
first and second grade teachers from 22 schools (Schoen et al., 2020). The Teachers
Development Group is providing the training for this study. In short, there are different models
for training. Educators need to evaluate their resources and goals, and then determine which
provider best suits their needs.
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Each of the professional development models have their advantages and disadvantages.
The depth and thoroughness of a 3-year program is an advantage. Training that is easily
accessible is also advantageous. My curriculum has a strong focus on implementation. The
theory behind CGI and student thinking is examined with the lens of praxis. Teachers need to see
how new knowledge is directly related to their everyday practice. So, one of the focal points of
my curriculum is lesson planning. Whenever teachers learn about student thinking, strategies, or
problem types, they have to consider how to use that knowledge to develop effective problem-
solving lessons. The content always goes back to how to implement it in the classroom through
lesson planning.
Some of the prior attempts discussed above require a 3-year commitment. The time
commitment may be a major hurdle. School districts and individual sites may not be ready or
willing to commit to 3 years of ongoing professional development. Site administrators must
consider staff needs, budget, and district agendas each year as they plan for professional
development. So, a 3-year commitment upfront may not be feasible. In addition, some training
sessions attempt to build the case for the CGI philosophy by connecting it to other related topics.
Thus, teachers may leave training with a loose understanding of why they should develop
student-centered lessons but they are left wondering how to start teaching this way. My
curriculum is shorter and focused on connecting theory to classroom instruction. Teachers are
expected to plan lessons within the course so that they are ready to implement immediately.
Teachers may be more successful in implementing this approach to teaching mathematics
if they learn within a context that is tightly tied to their practice of teaching. This curriculum will
make the connection explicit between examining and understanding student strategies and
learning to make instructional decisions based on student thinking. The purpose for noticing and
25
analyzing student thinking is to gather evidence to inform instructional decisions. Teachers will
engage in this work during the training so that when they return to their classrooms, they will
have confidence in trying this new approach.
Student achievement is another lens through which to scrutinize training models. The
initial year one results of the Florida study showed a large positive effect in first grade and a
large negative effect in second grade (Schoen et al., 2020). In this case, the researchers
determined that the computational expectations were considerably different for the two grade
levels and so it will be necessary to make adjustments in the training to attend to those
differences (Schoen et al., 2020). It is important for the content of the training proposed herein to
account for local standards. This current curriculum will address local needs by supporting and
guiding teachers in their decision processes as they design instruction for their classrooms.
The Content of the Curriculum
This curriculum was designed to teach teachers how to teach mathematics through
problem-solving. The first section is an introduction and importance of problem solving. The
next section is how to teach problem solving which includes a discussion about learning
trajectories and orchestrating mathematical discussions. Finally, the last section describes
cognitively guided instruction and how to use it to teach problem-solving.
Introduction and Importance of Problem Solving
The importance of mathematics education extends beyond K–12 education. Math
educators need to examine what competencies students need to become successful adults. The
National Research Council declares that balanced instruction which develops mathematical
proficiency in students attends to all five strands of mathematical proficiency: conceptual
understanding, procedural fluency, strategic competence, adaptive reasoning, and productive
26
disposition (Swafford et al., 2001). Furthermore, Swafford et al. (2001) state that one strand in
particular, strategic competence, highlights the importance of problem solving as it is the ability
to translate and represent situations mathematically to solve a problem. Ensuring that all students
learn to think mathematically was an impetus for a shift in mathematics pedagogy from an
emphasis on rote memorization and procedures to learning in context with understanding
(Swafford et al., 2001). Thus, problem solving is at the heart of teaching and developing
mathematical thinking in students, which in turn supports students in becoming successful adults.
The National Governors Association and Council of Chief State School Officers (2010)
examined other high performing countries, long-standing research, and reform efforts to
formulate a new set of standards for students in America, namely, the Common Core State
Standards. There are two types of standards: the Standards for Mathematical Practice and the
Standards for Mathematical Content. The Standards for Mathematical Practices hinge on the
process standards of the National Council for Teachers of Mathematics for problem solving and
the five strands of mathematical proficiency as defined by the National Research Council’s
report (NGA & CCSSO, 2010).
The Common Core Standards Writing Team (2011) contextualizes the meaning of
addition, subtraction, multiplication, and division through problem situations that students
investigate. One way students learn about the four major operations is through problem solving
situations, namely word problems. Furthermore, students grow in sophistication by how they
attend to the structure of problem situations, how they represent the problems, and the strategies
they use to solve those problems (The Common Core Standards Writing Team, 2011).
Mathematics instruction involves more than developing computational fluency, at the center is
27
problem-solving. The shift in mathematics instruction from traditional methods to one that
focuses on problem-solving requires different approaches and emphasis.
How to Teach Problem Solving?
Teaching problem solving in mathematics is quite different from other disciplines. First
problem solving is not an isolated topic nor a separate unit of study in mathematics (Lester &
Cai, 2016). Second, problem solving hinges on problems that are connected to real-life and are
applicable beyond the classroom (Hiebert et al., 1996). Finally, problem solving is pervasive in
that it involves designing a space that supports thinking and struggling with problems on a
regular basis.
Lester and Cai (2016) propose that it is important to teach mathematics through problem
solving rather than teach problem solving alone. Thus, there is a shift in the conceptualization of
problem solving. It is an approach to teaching mathematics where teachers pose a variety of
problems and students solve problems in a way that makes sense to them. Furthermore, students
collectively negotiate meaning through their social interactions as they share their thinking and
justify their thoughts (Lester & Cai, 2016). Teachers are not prescriptively teaching a list of
specific strategies for specific problems. Rather, teachers are teaching through problem-solving
which leads to a deep conceptual understanding and improved performance in mathematics
(Lester & Cai, 2016).
Historically, problem solving means acquiring knowledge and then being able to apply it
(Hiebert et al., 1996). However, problem solving in mathematics goes beyond application in the
classroom space because it utilizes tasks that are connected to real-life situations (Hiebert et al.,
1996). When students solve a variety of problems connected to real-life, they will be able to
transfer their knowledge to other real-life situations (Hiebert et al., 1996). Developing a palette
28
for wanting to solve problems takes intentional effort. Students need multiple opportunities to
solve tasks. Tasks need to be worthwhile (Lester & Cai, 2016). Furthermore, tasks that may be
difficult or perplexing are more engaging for students to solve (Hiebert et al., 1996).
A classroom that supports problem solving is a dynamic place with key dimensions that
work together to create a space that is conducive to learning. One dimension is the teacher’s role,
which is a facilitator rather than a dispenser of knowledge (Hiebert et al., 1997). As a facilitator,
the teacher poses a problem and guides the discussion about approaches and solution strategies.
Another dimension involves the nature of tasks. Students regularly solve cognitively demanding
tasks that warrant different strategies, tools, or representations (NCTM, 2014). The opportunity
to select tools and the choice of tools used in solving problems, purposefully supports students’
reasoning and understanding. Next, the social culture of the classroom is another dimension that
describes how members of this community expect to present ideas, examine thinking, and
negotiate meaning (Hiebert et al., 1997). Conceptual understanding of mathematics is
constructed in this space. Attention to each previous dimension ensures students have access to
mathematics. The last dimension is equity and access. Each student has the right to learn and is
expected to achieve at high levels (Hiebert et al., 1997).
Learning Trajectories
Teaching problem solving means teaching mathematics that reaches beyond the
classroom. The approach to teaching, the selection of worthwhile tasks, and the design of the
classroom space, support teaching mathematics through problem solving. In addition, knowledge
of how students develop mathematical thinking influences teachers who teach mathematics
through problem solving. Knowledge of learning trajectories illuminates a lens on student
thinking and strengthens praxis.
29
First, knowledge of learning trajectories helps teachers see that students develop
understanding of math in a predictable manner (Clements & Sarama, 2014). Clements and
Sarama (2014) further describe the learning trajectories as more like benchmarks on a landscape
than ones that follow a strict, linear sequence. For example, the strategies students use for adding
two numbers is predictable. Students do not necessarily start with the most concrete model, then
progress through each level of abstraction. In fact, students may “skip” certain levels or “go
back” to more concrete levels depending on a variety of factors. This knowledge of learning
trajectories helps teachers develop a schema for student strategies (Wilson et al., 2014). It may
even mean revamping their understanding of how students reason mathematically (Wilson et al.,
2013). Visualizing the schema and knowing the landscape of strategies that students use will
help the teacher gauge where a student is thinking and possibly predict their strategies for other
problems.
Next, knowledge of learning trajectories has an impact on praxis. Teachers grow this
knowledge of learning trajectories by examining student strategies more and thus, they get better
at recognizing and interpreting student thinking. In turn, this knowledge of learning trajectories
bolsters teachers’ mathematical content knowledge by giving teachers insight into the cognitive
steps that students take when tackling a task (Wilson et al., 2014). It also supports the teacher’s
view of the schema for how students think (Wilson et al., 2014). This may mean adjusting the
teacher’s own understanding of mathematics as well as revamping how they interpret student
thinking (Wilson et al., 2013). Teachers then base instructional moves on what their students are
doing and determine the goals that are appropriate for those students. Thus, the instructional
decisions and design of the learning environment are rooted in the trajectory along which
children naturally progress (Clements & Sarama, 2014).
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Orchestrating Mathematical Discussions
Effective discussions are carefully developed by implementing an intentional planning
process that supports classrooms centered on discourse. Discourse is one of three classroom
features that fosters an environment that allows students to collectively develop robust
mathematical thinking (Franke et al., 2007). It sets the stage for this productive exchange
between the members of the math community in the classroom. Students learn from negotiating
meaning and discussing their ideas. Thus, designing lessons centered on discourse requires
forethought, planning, and are at the center of enacting effective instruction.
To support intentional planning of robust lessons, the five practices for orchestrating
mathematical discussions are a pedagogical model, much like a roadmap, that help teachers
effectively facilitate discussions centered on student responses (Stein et al., 2008). The five
practices are: anticipating, monitoring, selecting, sequencing, and connecting (Stein et al., 2008).
This recursive planning process allows teachers to devote their attention to student thinking in
the moment and to the work of building the mathematical story for the task and lesson.
The first two steps, anticipating and monitoring, are connected. Anticipating strategies
beforehand gives teachers time to think through and predict what they might see during a
problem-solving lesson, so that they are not trying to interpret everything in the moment (Stein et
al., 2008). This helps teachers easily recognize and identify student strategies when they see it
during the lesson. Furthermore, it lowers the cognitive load during the lesson when the teacher is
monitoring student thinking. Teachers monitor by observing and conferring with students to
understand the strategies that students are using. Monitoring is like gathering data. Hence,
careful listening to students’ mathematical thinking, especially to the details of their strategies
supports mathematical discourse in the classroom (Franke et al., 2007). If students happen to
31
solve in a way that does not match or is puzzling to the teacher, then the teacher can probe for
clarification or focus-clarifying questions that will redirect students. It gives teachers a place to
start.
Next, in terms of selecting and sequencing strategies, there is no one right or prescribed
way. Teachers look for strategies that support the goal of the lesson. However, if no one uses that
particular strategy, the teacher will select different ones and perhaps adjust the goal of the lesson
or save it for next time. In that case, the teacher may also consider what intermediary experiences
students need to explore to encourage students to try different or novel strategies next time. Once
the strategies for discussion are selected, the teacher decides how to sequence them. A teacher
may start with the most concrete or popular strategy, or the teacher may select a common
misconception. The important part of sequencing is the process of deciding how to develop the
mathematical story in a way that best supports the goal of the lesson (Stein et al., 2008).
The final step is connecting. This is the time for discussion. The teacher will facilitate the
discussion about strategies students used to solve the problem. Both the teacher and students may
ask clarifying questions. It is important to compare and contrast the different strategies. One way
to do this is to trace the numbers in the problem. For example, when adding two numbers, it may
be interesting to note how students represented the numbers (both sets, one set, or none), who
counted on, how they counted, or how they combined the numbers to get the total. The numbers
in the problem and how students use them, tell a story that helps unveil the mathematical story in
the problem (Stein et al., 2008).
All in all, this recursive planning process allows the teacher to focus energy on the
mathematical content of the lesson. Practicing and rehearsing an instructional activity helps
teachers focus and tighten the facilitation of the task (Kazemi et al., 2009). Furthermore, it
32
lowers cognitive load so that the teacher can focus attention on what students are doing and
judiciously make decisions for the lesson at hand.
The first two steps of orchestrating discussions, anticipating, and monitoring, require
closer examination. Providing a spotlight on professional noticing helps the teacher pinpoint
areas which will strengthen anticipating and monitoring. Professional noticing entails three
components which are: focusing on student strategies, being able to accurately interpret them,
and judiciously making sound instructional decisions based on the evidence from those strategies
(Jacobs et al., 2010). As such, attentive listening to and analysis of the details of student
strategies provides valuable data that is the backbone of instructional practice. In understanding
the three components of professional noticing, recognizing the benefits of professional noticing,
and identifying the evidence that indicates improvement in professional noticing, the teacher has
a tangible way to enhance the planning process and support the goals of the lesson.
The first component of professional noticing is the ability to observe strategies that
students use. When teachers observe students as they solve a problem, they need to be able to
recall the strategies with a detailed level of specificity. Second, the teacher synthesizes the
specifics of student strategies and analyzes them. Those details piece together as evidence for
interpreting children’s thinking (Jacobs et al., 2010). Then after observing students solve
multiple problems, the teacher connects this evidence with what mathematical research on
children’s thinking says, to formulate a picture of what each student understands (Jacobs et al.,
2010). Then, the teacher determines how to best respond to students and make evidenced-based
decisions.
Evidence-based decisions are neither unilateral nor singular. Teaching is a complex
activity and there is no one prescription for teachers to make decisions during a lesson. For
33
example, there are a variety of instructional moves that teachers may utilize. Teachers may select
one of eight categories of instructional moves based on the purpose or target (Jacobs & Ambrose,
2008). Depending on whether the student has arrived at a solution or not, the instructional move
may be different. Instructional moves are intentional, and it is beneficial to base instructional
decisions on evidence.
Implementing evidence-based instructional decisions requires a level of craftsmanship in
professional noticing. There are identifiable signals that indicate skill. These indicators are
signals of skill in professional noticing. A teacher’s ability to: notice pertinent and specific
details of student strategies, determine the root cause of misconceptions, predict how students
would solve a given problem, and write problems that are tailored to students’ needs, are
evidence of teacher’s growth in craftsmanship (Jacobs et al., 2010). If a teacher is struggling with
professional noticing, then it may be beneficial to consider working on one of the indicators as a
place to start. Overall, orchestrating robust mathematics instruction is a complex endeavor. A
teacher needs to be able to take careful note of student thinking and analyze that thinking to
inform next steps of instruction. As such, cognitively guided instruction (CGI) provides a frame
for understanding student thinking within the context of word problems and according to a
trajectory of solution strategies.
What Is Cognitively Guided Instruction?
Planning for and teaching mathematics through problem solving can be a messy process.
Cognitively guided instruction provides order through the conceptual framework for teaching
mathematics through problem solving. First, CGI outlines a categorization of problem types and
the trajectory of student strategies to support teaching mathematics through problem solving.
Next, it is important to note that the framework and this approach to teaching mathematics is
34
developed from studying how students naturally think about and solve problems. Finally, the
CGI framework supports teachers to deepen their conceptual understanding of mathematics.
The categories in the problem type chart are organized according to what is happening in
the problem. Some problems are action problems, and they represent a joining or a separating
situation (Carpenter et al., 1988). Other problems are non-action problems, and they consist of
sets and subsets, or comparisons of sets (Carpenter et al., 1988). The solution strategies children
use are consistent with the structure of the problem and their solution strategies follow a
predictable trajectory. Children intuitively attend to the action and relationships within a word
problem to solve it (Carpenter et al., 1996). This may contradict how adults or older children
solve problems. The strategies that adults use to solve problems do not necessarily match how
students conceive of and solve problems. Teachers who teach mathematics through problem
solving need to see the problems and solution strategies the way that children inherently see
them.
The problem type chart and trajectory of solution strategies were developed after
observing how children solved problems. Allowing children to solve in a way that makes sense
to them supports the development of their mathematical thinking (Carpenter & Fennema, 1992).
Furthermore, it is important that mathematics learned in school makes sense and is connected to
real-world experiences of children (Carpenter & Fennema, 1992). Children need to be able to
solve problems in context, that is problems connected to real-life situations, using strategies that
make sense to them. Then, knowing this helps a teacher honor how children naturally solve
problems, and it prevents them from teaching strategies that are counterintuitive to students.
Children come to school with an informal understanding of mathematics (Carpenter et al., 1996).
They develop mathematical thinking by solving problems in ways that make sense to them and
35
by using strategies that they choose (Carpenter & Fennema, 1992). This informal understanding
of mathematics, with which children enter school, is foundational for developing formal
mathematical knowledge (Carpenter et al., 1996).
Posing problems, giving students space to solve however they want, and observing
students, informs the teacher’s schema of mathematics. As teachers pose problems and facilitate
teaching through problem solving, their conceptual understanding of mathematics deepens. The
subject matter of teaching mathematics to children includes more than the mathematics content
itself. Knowledgeable and effective teachers understand how children generally think about
mathematics, how they develop their thinking, what common misconceptions and errors they
have, and the strategies that their students utilize. Shulman (1986) identifies this specific
knowledge as pedagogical content knowledge. CGI teachers recognize semantic distinctions in
the problems and how those distinctions affect a student’s ability to solve the problem (Carpenter
et al., 1988).
The CGI framework of problem types and trajectory of solution strategies informs CGI
teachers as they base their instructional decisions on their students’ thinking. First, the
framework illuminates the content of what CGI teachers teach. Second, there are principal
components of CGI and CGI lessons that support instruction.
Primary grade teachers need to teach concepts and skills for adding and subtracting. CGI
teachers refer to the problem type framework when selecting problems to pose. They understand
which problems are easier or more difficult based on the problem type framework. The problem
type framework is a tool that supports teachers when determining the problem to pose, what
questions to ask, and what to focus on when listening to students (Carpenter & Fennema, 1992).
So, CGI teachers align their goals for lessons based on the problems they choose. Furthermore,
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when designing lessons, teachers reflect on their students’ strategies so that they can pose
problems that appropriately challenge them (Carpenter & Fennema, 1992). CGI teachers
formatively assess their students as they listen to students and take note of the strategies they
employ (Carpenter & Fennema, 1992).
The teacher needs to consider some principles for interacting with students during a
lesson as well as some key components of CGI lessons. Carpenter et al. (2015) point out four
“steps” to getting started: engage children in problem solving, select and pose problems, elicit
student thinking, and share strategies. In terms of eliciting student thinking, certain principles are
helpful when interacting with students. Eliciting student thinking may entail asking questions to
illuminate the details of a student’s strategy or clarify what a student did (Carpenter & Fennema,
1992). It is critical not to superimpose the teacher’s interpretation or method on a student’s
strategy (Carpenter & Fennema, 1992). The focus should be on student generated strategies and
explanations.
Next, regarding the step of sharing strategies, this occurs within a culminating discussion.
This time of discussion is not a time for merely reporting strategies. Rather, the teacher is
facilitating a discussion and needs to consider the purpose for that exchange. The purpose of
sharing strategies is to compare ideas, focus on details of different student’s ideas, or to
amplify/extend ideas (Carpenter & Fennema, 1992). Discussions are messy and it will take
repeated implementation attempts to feel more comfortable. Teachers report that in the
beginning, this approach to teaching mathematics may engender uneasiness, yet the initial
struggle is temporary and insignificant compared to the rewards (Carpenter et al., 2015). Both
teachers and students benefit from this approach to teaching mathematics. Carpenter and
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Fennema (1992) have found that student achievement is correlated with the teacher’s ability to
predict the strategies that their students will use to solve problems.
Summary of the Curriculum Content
The learning goals of the curriculum can be understood through the lens of the two types
of knowledge: declarative and procedural knowledge. Smith and Ragan (2005) describe
declarative knowledge as knowing what, that, or why something is, and procedural knowledge as
knowing how, that is how to apply knowledge in instruction. Both types of knowledge are
important for teaching problem solving.
Declarative Knowledge
Smith and Ragan (2005) list three types of declarative knowledge: labels and names
(linking or connecting the label and the name), facts and lists (how concepts are related), and
organized discourse (synthesized comprehension of complex material). The first and second
learning goals of this curriculum pertain to labels and names. Being able to identify and
distinguish the problem types and strategies is at the heart of those two goals. Next, the
expectation for learning goal three is to be able to take the knowledge gained from the previous
learning goals and connect them to each other. Learning the names of the problem types is
insignificant if it is not connected to the types of strategies one would expect to see for that
problem type. Finally, the last learning goal involves the implications of the previous learning
goals for instruction and how that knowledge is then applied to instructional practice. The
teacher needs to synthesize what they see their students doing with the types of problems and
strategies to design effective instruction.
Procedural Knowledge
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Procedural knowledge is knowing how to apply knowledge (Smith & Ragan, 2005). The
procedural knowledge explicates the learning goal with concrete steps and essential learning.
This clarifies the expectations for the learning goals. The focus of procedural knowledge is how
to accomplish the objectives of the learning goals. In Table 3, the declarative knowledge and
procedural knowledge are listed for each learning goal.
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Table 3
Declarative and Procedural Knowledge Required for Learning Goals
Learning goal Declarative knowledge Procedural knowledge
Learn how children think
and solve problems
intuitively and
conceptually.
Know the structural
components of the 14
problem types.
Identify and distinguish the
problem type according to
the taxonomy of problem
types.
Distinguish between action and
non-action problems.
Represent problems with an
appropriate graphic
organizer.
Determine strategy for note
taking.
Recognize the trajectory
of strategies that
students employ to
solve problems.
Acknowledge the level of
strategies for solving
problems: direct modeling,
counting, invented
algorithms/derived facts.
Notice and notate student
strategies in a detailed
manner.
Use evidence to identify the
level of the strategy used by
the student.
Describe the next plausible
level of sophistication that a
student might use based on
the current strategy.
Analyze student strategies
and use of tools to
gauge the level of
thinking based on the
problem.
Identify and connect the
continuum of strategies with
representations.
Analyze the representation and
the way a student utilizes it
to determine the utility of
that representation.
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Learning goal Declarative knowledge Procedural knowledge
Identify when a student uses a
less sophisticated
representation with a more
efficient explanation.
Ask appropriate questions to
highlight discrepancies
between representation and
strategy explanation.
Design instruction that is
responsive to students’
needs.
Understand the structural and
contextual components of
problems that make problems
easier or more difficult.
Manipulate the context of the
problem to make it easier or
make it more difficult.
Select number sizes and
combinations to
appropriately challenge
students.
Utilize the order of the
numbers presented to
highlight certain
mathematical properties or
strategy use.
Know how to adjust the
complexity of the text to
support students.
Next, in Chapter 3, the analysis of the learning context and the learners is addressed.
Chapter 4 offers a description and rationale for the formulation of the curriculum. The content
and knowledge types discussed in the curriculum will be presented and configured as units of
instruction for the detailed design.
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Chapter Three: The Learning Environment and the Learners
This chapter will focus on the learning environment and the learners. Smith and Ragan
(2005) define the learning system as all aspects that are affected by and influence the learning.
Hence, the first section of this chapter will examine the learning context by describing
teacher/trainer/facilitator characteristics, existing curricula/programs, available equipment and
technology, and classroom facilities and learning climate. The second section will attend to the
learners. It is important to customize instruction to a particular target audience so that instruction
can be effective for those learners (Smith & Ragan, 2005). Thus, this section will focus on the
cognitive, physiological, affective, and social characteristics of the intended target audience.
Description of the Learning Environment
In this section, the characteristics of the facilitators will be described as a component of
the learning environment. The location of the proposed curriculum will be described in terms of
any existing curricula or programs. And finally, the physical environment consisting of
technology and facilities available to the learners are addressed.
Teacher/Trainers/Facilitator Characteristics
Facilitators for this course should exhibit expertise in pedagogical content knowledge for
teaching mathematics and approaches to andragogy. The facilitator needs to be trustworthy and
credible on all accounts. First, the pedagogical content knowledge for teaching mathematics is
derived from the research on CGI. Facilitators need to have a robust conceptual understanding of
the problem type framework in terms of the hierarchy of problems and the corresponding student
strategies. This understanding should be rooted in direct classroom experience with CGI. The
facilitator will be to address implementation questions and issues based on experience from the
field. In addition to working with students in the classroom, the facilitator needs to exhibit the
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ability to work effectively with adult learners. It is important that the facilitator is proficient with
facilitation and presenter skills such as listening, paraphrasing, asking questions, and managing
participant engagement. At every step, the facilitator is modeling strategies that are applicable to
the classroom as well. The facilitator models questioning techniques, recording, and
documenting participant thinking, in short, the components of orchestrating mathematical
dialogue. A skillful facilitator is able to lead the group and help the group unpack the
instructional moves underlying the lesson. It is making the invisible moves of a polished lesson,
visible and thus reproducible.
Existing Curricula/Programs
There are many models and providers for CGI training. Sessions range from one hour to
40 hours per week over three years. Currently, Teachers Development Group is one of the most
prominent providers of training on CGI (Schoen et al., 2020). The Institute of Educational
Sciences is sponsoring a three-year study in Florida, and they released a report in 2020 on their
findings from the first year. In this report, they found a large positive impact on first graders’
mathematics abilities but a large negative effect on second graders’ mathematics abilities,
specifically with computational ability (Schoen et al., 2020). Thus, the program developers were
encouraged to revisit the content focus and expectations for second graders versus first graders.
Smith and Ragan (2005) note the importance of considering the learning environment in
designing curriculum. It is necessary to be aware of state standards, expectations on state or local
assessments, as well as local curriculum. The proposed curriculum would avoid a potential
mismatch between content within the curriculum and expectations for students. Local educators,
who know their state standards, assessments, and curriculum, will have the opportunity to
articulate goals for their students and then plan problems that work toward those goals. The
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problem type framework and the trajectory of strategies are a staple component of most CGI
training. However, supporting educators to know how to recognize strategies and utilize that
knowledge to plan effective lessons is essential. Smith and Stein (2018) propose a template for
planning lessons for students through five steps: anticipating, monitoring, selecting, sequencing,
and connecting. The curriculum will bridge the content knowledge of CGI and practice in
classrooms through the five-step planning process. By simulating classroom discussions within
the training and providing opportunities to plan and reflect on lessons taught, participants may
have more confidence. Educators will discuss their experiences, struggles, and successes with the
group. Then, they will determine how to use that data to plan subsequent lessons.
Available Equipment and Technology
This professional development session will be delivered over the course of four
days. Ideally, some of the days should occur during the school year. However, it is acceptable if
the training is completed during the summer before school starts in the fall. In terms of supplies,
the facilitator will need access to pads of sticky chart paper, chart markers, speakers, computer,
projector that will project onto a screen, wall or SmartBoard, and a document camera. The
facilitator will record information on chart paper and post them so that participants can access
content from earlier conversations throughout the session. Participants will view the slides and
videos during the presentation, and they will conduct discussions in their small groups. At times,
the facilitator will circulate in the room to observe and interact with the table groups as they are
working on problems. There should be enough room for the facilitator to circulate the room
inconspicuously and on the walls to display posters throughout the training session.
Classroom Facilities and Learning Climate
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The training room will be designed to model an optimal learning space. The session can
accommodate from four to 36 participants. Tables or workstations should be set up to
accommodate groups of four to six participants. The tables would be arranged so that everyone
has a clear, unobstructed view of the front board or wall. Each table will have a box of supplies
and a box containing math manipulatives. The supply box should contain post-its, blank paper,
pens, sharpened pencils, markers (a minimum of four to eight colors), erasers, highlighters,
scissors, glue sticks, and scotch tape. The math manipulative box should contain: unifix cubes,
red and yellow counters, a small hundreds chart (six small ones fit on a page), a full-page
hundreds chart (1-100), a full page 99 chart (0-99), coins (pennies, nickels, dimes, quarters),
number lines (0-20), blank number lines, and base ten blocks (unit cubes, ten rods, and hundred
flats). Participants will receive a binder or folder to store their handouts and notes from the
training.
Learner Characteristics
This section will focus on the learner characteristics. Smith and Ragan (2005) attend to
learner’s cognitive characteristics which includes general characteristics and specific knowledge,
physiological characteristics, affective characteristics, and social characteristics. In designing an
effective curriculum, it is important to consider the audience in terms of these learner
characteristics so that the curriculum may account for and better support individual learners.
Cognitive Characteristics
Participants in this curriculum are expected to be educators who are currently employed
in an elementary school. This curriculum is most effective for those who have prior experience in
an elementary classroom and who currently have access to students for whom they plan and
execute lessons. Some may be newer to the profession while others may be more experienced. It
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is important that participants have an opportunity to transfer what they have learned in training
into a classroom whether it is the participant’s own classroom or one that someone is borrowing.
If one is borrowing a classroom, it is most beneficial to have access to this classroom on multiple
occasions so that the participant can implement lessons in that same classroom. Having multiple
opportunities to observe one group of students over time will help the educator notice patterns
and tendencies in solution strategies amongst those students.
Educators will access and engage with this curriculum using visual, auditory, and
kinesthetic senses. They will view video excerpts, listen to conversations between students and
teachers and amongst students, and utilize math manipulatives to model solutions to problems.
Participants will be expected to analyze student thinking, role play classroom interactions, and
discuss their ideas with other participating educators. These activities and experiences will
support participants as they develop upcoming lessons.
In order to ground the learning to their work, participants will determine how state
standards, pre-existing curricula in their classrooms, and the cultural background of their
students need to be considered in planning future lessons. The state standards delineate learning
goals by grade level and grade band. It will be important to make sure that lessons support the
work at that grade level. Textbooks and other forms of curricula contain examples of grade level
content and academic language. Sometimes the content does match the state standards at the
grade level, so it is important to check the content with the exact expectations at each grade
level. Educators need to consider the academic language found in textbooks. It is an example of
formal language and register which is beneficial for language learners. They need to be exposed
to academic language on a regular basis. Furthermore, problems that are written with students’
background and experiences in mind, will be more tangible and connected for students. A
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familiar context allows students to attend to the mathematics without being confused or
distracted by the situation of the problem and thus, will make the mathematics more accessible.
General Characteristics
Educators need to acknowledge and consider the way that students develop their
mathematical thinking. Recognizing the categories of the problem types and solution strategies,
helps educators understand the systematic process of developing mathematical thinking.
Developmental Level
Teachers attending this training may differ in terms of the number of years of
experience, grade levels, and backgrounds. The notion that students employ different strategies
depending on their experiences and developmental levels may be unfamiliar. Thus, it will be
important to support teachers as they become familiar with the types and structures of problems,
in conjunction with the trajectories of strategies that children use. This curriculum is based on the
framework of problem types as identified by CGI. The classification of problem types helps
teachers know which problems are easier or more difficult. It allows teachers to predict strategies
that students use. Usually, students develop their strategies in a predictable manner based on the
problem type. Student strategies develop through phases of direct modeling, counting strategies,
and derived facts or invented strategies. Not every student will progress through every strategy
or stage. The strategies and levels of stages serve as markers to interpret how students develop
mathematical thinking. Once teachers understand this multidimensional framework, they will be
able to capitalize on it to present problems that are tailored to their students’ needs. The training
will prepare teachers to know what to look for and how to support struggling students.
Prior Knowledge
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Participants bring both general and specific knowledge to the training. Their prior
knowledge is rooted in their experiences as a student in school and as a teacher. It is helpful to
draw on those experiences to personally ground the material in the curriculum with their own
framework of learning and understanding mathematics. Hopefully, teachers will then build on
and expand their framework.
General World Knowledge
The common denominator for all participants of this curriculum is that they are all
elementary educators which means they have earned their multiple subject credential.
Furthermore, everyone has had experiences learning math as a student in school. Some may have
had particularly bad or good experiences with mathematics. These experiences may motivate
them to create mathematics environments that support learning in a way that they were supported
or differently because they did not have a good experience.
Specific Prior Knowledge
Participants need to have taught mathematics to elementary students so that they have
familiarity with how students approach and solve problems. This approach to teaching
mathematics presented in the curriculum may be challenging for some especially if they are more
comfortable with direct instruction. It is helpful if educators come with an open mind to learning
about children’s thinking and how it can be used to design effective mathematics instruction. The
instructor will scaffold learning opportunities to highlight and orchestrate conversations about
various solution strategies. This curriculum is designed to bridge theory to practice. Participants
will learn how to develop mathematical thinking through problem solving and how to design
problems that are relevant and appropriate for their students, that is, ones that are based on or
48
connected to their students’ interests. Knowing this background information will facilitate the
planning process and make it easier.
Physiological Characteristics
The educators participating in this curriculum need to be in relatively good health to
participate in this professional development. They will need to be able to sit at a table with other
participants and at times walk around the room to view posters on the wall or move to new
groups to collaborate. Furthermore, in the event that the state has imposed mandates such as
wearing masks, participants need to be able to tolerate wearing masks for the duration of the
professional development as it is occurring indoors. In addition, a variety of accommodations
will be made available for participants with special needs per the Americans with Disabilities
Act.
Affective Characteristics
Some elementary teachers have math anxiety. In fact, some teachers choose to teach a
lower grade such as kindergarten or first grade because they feel confident with the mathematics
at those grade levels but not at the higher grades. While it is also likely that teachers have
attended professional development sessions before with the best intent to implement what they
have learned, they may have been unsuccessful at putting their new learning into practice.
Sometimes there is a disconnect between what is presented and the actual transfer to their
classroom situation. Another possibility is that educators have a negative attitude towards
professional development in general. If they have faced numerous mandates to attend training,
involuntarily signed up for sessions, or find themselves in sessions that are totally irrelevant to
their current situation, educators are highly unmotivated. With the introduction of the CCSS-M,
some educators do not understand the different instructional models and approaches for teaching.
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They also may not believe the standards are appropriate for their students because they are too
rigorous or not rigorous enough. All of these underlying beliefs will impact how the training is
received.
Self-efficacy is the perception that one holds about their own capabilities (Schunk, 2020).
Educators attending this training will grapple with their perceived ability to deliver effective
mathematics instruction. After completing the course and learning about the strategies that
children use to solve problems, educators need to believe that they can orchestrate effective
instruction that develops mathematical thinking. Confidence in their ability to facilitate
discussions, to understand and teach mathematics, and to foster an environment that supports
learning amongst students, contributes to their level of self-efficacy. Low self-efficacy has
negative effects. Even when educators know that certain approaches produce desirable outcomes,
if they do not believe they can do it, it is futile (Zee & Koomen, 2016). Thus, it is important that
educators believe that they and their students are capable of learning.
Social Characteristics
Learning mathematics is a social activity. The instructor sets an example by creating this
environment in the synchronous session, as the teacher needs to create it in their classroom.
Effective mathematics instruction depends on the interaction between the teacher, student, and
mathematical content (Swafford et al., 2001). Students learn to communicate and develop their
thinking through discussions. The instructor models and facilitates conversations between
participants that mirror classroom practice to help bridge learning from the sessions to the
classroom. In addition, participants have opportunities to reflect on their thinking and practice
with others so that they can process what they have learned and how they might apply it. This
helps foster learning mathematics in community.
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Implications of the Learning Environment and Learner Characteristics for Design
This curriculum was designed with the learners, elementary math educators, in mind.
There is a delicate balance between presenting theory and research-based practices with
strategies to implement it in the classroom. The training has to be relevant and practical. In other
words, it has to be applicable to the individual’s classroom. Participants will learn about the
research-based problem type framework and levels of strategies that students use as they work on
developing their knowledge base to recognize their student strategies and how to orchestrate
productive discussions based on their students’ thinking. It is important to simultaneously
develop the participant’s mathematical content understanding as a teacher with the practical
aspects of what to look for, what questions to ask and when, and next steps for instruction.
Teachers may experience the training as though they are in a lab, where they examine student
thinking and then, they determine how they would manage the learning environments and
opportunities. The research-based theory, CGI, is presented in tandem with the process for
planning effective instruction, the five practices for orchestrating productive mathematics
discussions. The marriage of the theory, grounded in a process for implementing it, will support
teachers to enact their new learning.
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Chapter Four: The Curriculum
Chapter four introduces the curriculum and explains the rationale for it. There are two
components in this chapter. The first part is a curriculum analysis, and the second part is a lesson
analysis. The goal of this curriculum is to train teachers how to develop mathematical thinking
through problem solving. In particular, teachers will be able to identify and distinguish the
problem type according to the taxonomy of problem types, recognize the trajectory of strategies
that students employ to solve problems, analyze student strategies and use of tools to gauge the
level of thinking based on the problem, and design instruction that is responsive to students'
needs. The starting point for the curriculum analysis is the conduct of a cognitive task analysis
(CTA) to capture expertise in the domain.
Cognitive Task Analysis (Information Processing Analysis)
An information processing analysis (Smith & Ragan, 2005), also known as a cognitive
task analysis (CTA; Clark et al., 2010), was carried out to determine the necessary steps to
execute a task (Smith & Ragan, 2005). Studies show that when a CTA is conducted to create a
training, it is more effective than training that is not based on a CTA (Clark et al., 2010; Hattie,
2012). The purpose of conducting a CTA is to ascertain the declarative knowledge (the discrete
steps), procedural process (performing a skill and being able to practice it), and associative
processes (paying attention to the cues that signal cognitive or action steps needed to achieve a
goal) that are automatic for an expert (Clark et al., 2010).
Bootstrapping the literature (Clark et al., 2010) is recommended to capture preliminary
information about a task and was used in the first part of establishing the main steps for
developing mathematical thinking through problem solving. I conducted a search in Google and
Google Scholar using the following phrases: how to teach problem solving in mathematics and
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how to develop mathematical thinking in children through problem solving. Then, after a review
of the literature, I was able to distill the major steps which I confirmed in the CTA interviews.
I conducted two CTA interviews with teachers who have been using CGI in their
classrooms. One teacher has been using CGI in their classroom for 13 years and the other for
seven years. I selected these teachers because they were able to trace their journey, reflect on
significant catalysts that changed their practice or made a difference in their journey, and
identified moments that were difficult and how those were resolved. Their reflections and voices
were especially helpful because one underlying goal for my curriculum is to address potential
barriers to implementation. Just as I believe every child has the potential to learn and develop
strong mathematical thinking, I believe that every teacher can strengthen their craftsmanship as a
math educator. Both teachers offered insights to critical moments and challenges that were
important for continuing to grow on this journey. They are also positive role models for teachers
who may be skeptical. Students in their classroom demonstrate their mathematical prowess by
their confidence and state test scores. They are successful CGI teachers.
It is important to make sure that the training provides support in the content as well as for
the implementation process. When considerable time is spent on teaching the content knowledge
without sufficient time to practice and apply it, teachers may leave without feeling confident that
they can enact the new approaches. The experts affirmed the need to learn how students develop
mathematical thinking and how to facilitate classroom spaces that are conducive to orchestrating
learning through problem solving. Teachers need practice that mirrors what they do in their
classroom within the scope of the training. The synchronous sessions should balance components
of learning and applying practice with feedback. Conversations about how to record student
thinking during a live lesson, proven strategies to help students better comprehend the problems,
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and techniques to plan a culminating discussion, are important to process before attempting the
first lesson with students. With that in mind, teachers will have an opportunity to write problems
and immerse themselves in the planning process. As teachers implement the problems that they
have planned, they will return to the next session with student work samples. During the next
session, teachers will have the chance to examine student work and analyze student strategies to
further solidify the learning goals of my curriculum. This practical connection with their
students and their classrooms will reinforce the implementation process. This repeated
connection between the training and classroom practice points to the cyclical nature of the steps
for my curriculum.
Both CTA interviews in conjunction with the literature review substantiated the steps and
goals of the curriculum. The content of this curriculum will be offered synchronously and in-
person with optional additional asynchronous activities. Synchronous in-person sessions are
preferred. However, in the event that synchronous in-person sessions are not a viable option, the
sessions may be offered as a synchronous, virtual session.
The following are the relevant results of the CTA:
● Objective: Instruct teachers to develop children’s mathematical thinking through
problem solving.
● Cue: Teachers will need to know how to effectively teach mathematics when they
sign a contract with a district to teach. This professional development may be
voluntary, or attendance may be required by a site or district administrator.
● Condition: Elementary teachers are responsible for students in their classroom. They
will provide instruction in mathematics per their grade level standards and
requirements.
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● Standards: Teachers will complete a professional development cycle within the time
agreed upon between the presenter(s) and site or district administrators. The course is
delivered in two parts, days one and two then days three and four after several weeks.
In between meeting dates, teachers are expected to implement lessons. Teachers will
discuss challenges and successes of implementation with the presenter(s), grade level
teams, administration, or other site support personnel.
● Equipment:
o Meeting space: The meeting space needs to be designed to support the learning
goals of the sessions. Optimally teachers should sit at a table of four to six people
to encourage collaboration and conversations. Throughout the session, the
facilitator(s) and teachers will post charts on the wall. The room will need ample
wall space and a clear line of vision for those charts.
o Supplies: Each table should have a box that contains pens, pencils, chart markers,
highlighters, and post-its. In addition, each teacher will receive a folder to
organize their handouts and materials, along with two books: Children’s
Mathematics: Cognitively Guided Instruction (2015) by Carpenter et al. and
Principles to Actions: Ensuring Mathematical Success for All (2014) by NCTM.
o Technology: The facilitator will need a computer, projector, clicker (to advance
slides), and speakers to amplify sound for video clips. Participants may choose to
bring a device to take notes during the session. Table groups will need access to a
computer or other device with internet access.
● Steps: As a result of the literature bootstrapping and interviews with the experts, the
following major steps are required to instruct teachers to develop mathematical
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thinking through problem solving. The notation following the step indicates whether
it was supported in the literature (LIT) and/or the cognitive task analysis (CTA).
o Learn how children think and solve problems intuitively and conceptually
(LIT, CTA)
o Listen to students and take note of the strategies they use (LIT, CTA)
o Plan lessons that engage students in problem solving activities (LIT, CTA)
o Facilitate dialogue around mathematical ideas (LIT, CTA)
o Reflect on student strategies and grade level expectations to determine next
instructional steps (LIT, CTA)
These major steps, which were informed by the literature review and the CTA, formed the basis
of the curriculum. The curriculum will attend to the content knowledge and application educators
need to develop mathematical thinking in children through problem-solving.
Overview of the Modules
The modules in my curriculum are the result of a 21-year journey. Initially, I learned
about CGI as a participant in various trainings and through my classroom experience. Then, I
began providing professional development in my role as coordinator, presenter, and coach. I also
benefited from extensive interactions with two integral members of the original research team:
first, one of the original researchers and author of many CGI publications and second, one of the
20 CGI teachers in the original study (Carpenter et al., 1989). Both were extensively involved in
the CGI research from the beginning and are still active within the CGI community.
After implementing this approach to teaching mathematics in my own classroom, I began
to study this approach from the vantage point of professional development, through training
others. I became curious about what content piqued curiosity amongst the teachers, what nudged
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teachers to take risks and try new strategies, the role of reflection in sustaining or sparking
growth, and finally, how to pack those “ingredients” into an effective curriculum. The literature
and my experiences convinced me that the focus of my curriculum had to be on bridging theory
to practice. I have encountered many teachers who appreciate the training content but do not
implement it in their classroom. I wonder what would support teachers who are not
implementing need. I am seeking to develop a curriculum that intentionally instigates or grows
generative change in teachers. Generative change, or self-sustaining change, signals a deep-
rooted shift in one’s perspective, practical inquiry that leads to a fresh understanding of learning,
and how it applies to classroom praxis (Franke et al., 1998).
Thus, my curriculum is rooted in the literature, input received from the CTA interviews,
prior attempts of CGI training, my experiences with teachers and researchers, and the children I
have met over the years. One of the main outcomes of this curriculum is to connect what is
learned in training to classroom practice. Priority is given to ensuring that teachers would learn
the content and have ample opportunity to practice scenarios that they would face in the
classroom, so that they would be ready to implement the new strategies in their classrooms. As
such, the following statements are major steps to ensure that all students have access to equitable
teaching practice:
• Learn how children think and solve problems intuitively and conceptually.
• Listen to students and take note of the strategies they use.
• Plan lessons that engage students in problem solving activities.
• Facilitate dialogue around mathematical ideas.
• Reflect on student strategies and grade level expectations to determine next
instructional steps.
57
The major steps of the curriculum are not linear. It is a cyclical process that teachers
continue to engage in as they grow in knowledge and refine their practice. This process is
essential because it is iterative, and this repeating process is what helps bind the instructional
approach to classroom practice. Theoretically (in one’s classroom), a teacher may join the cycle
at any point because any “step” may be the trigger for a new learning cycle. A teacher may
become curious or seek training after unsuccessfully trying to bring closure to a lesson or the
teacher may be puzzled by a student’s strategy that seemingly works but the teacher does not
know why it works. Whatever the reason behind seeking training, or engaging in this cycle, the
point of entry can involve any of the steps. For the purposes of training, we begin with step A.
The decision to label the steps with letters was intentional. In a five-component pattern, it is the
repetition of the parts that is important, not the genesis of the pattern. The pattern does not have
to begin with A. In fact, it may begin with C (for example: CDEABCDEABCDEAB). So,
outside of the training sessions, a teacher may begin their journey at C or E or another step.
Labeling the steps with numbers might subtly send the message that it is proper to start at
number one because “1” is the beginning. Teachers will engage with or observe every step in
each of the modules. They may engage in the step by participating in an activity or they may
observe the step in a video clip or an article. Hence, this reinforces the fact that the steps are
cyclical. The steps are represented in Figure 1.
58
Figure 1
Major Steps to Develop Mathematical Thinking Through Problem-Solving
List of Modules (6-hour day plan for 1/2 day per module)
There are eight modules within this training. To accommodate district and site needs,
there is some flexibility to the delivery timeline of the modules. First, the best scenario is to
conduct synchronous, in-person training over 4 days, 2 days of training at the beginning, with the
A. Learn how
children think and
solve problems
intuitively and
conceptually.
B. Listen to
students and take
note of the
strategies they
use.
C. Plan lessons
that engage
students in
problem solving
activities.
D. Facilitate
dialogue around
mathematical
ideas.
E. Reflect on
student strategies
and grade level
expectations to
determine next
instructional
steps.
59
remaining 2 days spaced out (ideally 3–5 weeks apart). Since each module is approximately
three hours, two modules will be covered each day. Although the teacher workday is typically
7.5 hours, planning for a 6-hour day is more realistic given lunch and breaks within the day. The
second option is to offer the training over eight synchronous, in-person sessions that are three
hours long. Some adjustments may be made to support teachers depending on the timing. For
example, at the end of each training day, teachers will have a problem or problems to try in their
classroom. They are expected to return with student work to analyze during the next session.
These assignments may be different depending on the delivery format (full day sessions or
separate sessions by module). In the event that in-person training is not possible, the training
could be delivered through a video-conferencing platform like Zoom or Google Meets.
The curriculum begins with an overview in Module 1 that sets the stage for teaching
mathematics through problem-solving. Then, five steps of the curriculum are represented in each
of the modules except Module 2. This is because the focus of Module 2 is the taxonomy of
problem types which is one of the major concepts of CGI. Teachers will learn the structure of the
problems and how it is tied to how students think of mathematics within the problems. Thus, this
module focuses on learning (Step A) the specific concept of the taxonomy of the problem types.
Ultimately, the repetition and reinforcement of the process through the cycle will lower the
cognitive load of implementing new strategies.
Module 1a: Overview
● Research base (Steps A, B)
o History of research
o 6 Research questions (statistics)
● Connection to Common Core State Standards by grade level (Step E)
60
o What are the grade level expectations for problems?
Module 1b: Designing the Classroom Space
● Video clip 5.1 (What do you notice? What do you wonder?) (Steps B, C, D, E)
● Lizzie Beetle problem (Steps B, C, D)
o Multiple solution strategies
o Classroom environment/expectations
Module 2: Structure of Problems (the problem type framework)
● Join and separate problems: pay attention to structural differences (Step A)
o “Build” this section of the problem type chart
o Write problems
o Attach number sentences to the problems
o Notice the problems have action (there is a sequence of events)
● Part-part-whole problems and compare problems: pay attention to structural
differences (Step A)
o “Build” this section of the problem type chart
o Write problems
o Attach number sentences to the problems
o Notice these problems do not have action (part-part-whole problems are about
categories of items - the whole set and part of the set; Compare problems are
about the comparison of two sets. What distinguishes them is the information that
is given and what needs to be determined.)
Module 3: Noticing Strategies (Join problems)
61
● Examine the different levels of solving strategies (direct modeling, counting, flexible
strategies, derived facts) as shown on the “Children's Solution Strategies”. (Steps A,
B, C)
● Learn purpose and determine strategy for note taking. (Steps B, D, E)
● Unpack the closing discussion (Steps D, E)
o What is the purpose?
o What are possible storylines?
▪ Comparing an idea to another student’s idea
▪ Attending to the details of another student’s ideas
▪ Building on or adding to another student’s ideas
o What are possible lesson goals?
● Use the planning template to plan a problem from scratch. (Steps B, C, D, E)
o Anticipate strategies that students might employ
o What is the purpose of posing this problem (goal)?
o Prepare for a storyline for the closing discussion.
● Use the planning template to plan a problem, given student work samples. (Steps B,
C, D, E)
Module 4: Noticing Strategies (Separate problems)
● Examine the different levels of solving strategies (direct modeling, counting, flexible
strategies, derived facts) as shown on the “Children's Solution Strategies”. (Steps A,
B, C)
● Practice note taking strategies. (Steps B, D, E)
● Use the planning template to plan a problem from scratch. (Steps B, C, D, E)
62
o Anticipate strategies that students might employ
o What is the purpose of posing this problem (goal)?
o Prepare for a storyline for the closing discussion.
● Use the planning template to plan a problem, given student work samples. (Steps B,
C, D, E)
Module 5: Structure of Multiplication and Division Problems (the problem type framework)
● Multiplication, Measurement Division, Partitive Division: pay attention to structural
differences (Step A)
o “Build” this section of the problem type chart
o Write problems
o Attach number sentences to the problems
o Connect to the graphic organizer
● Examine the different levels of solving strategies (direct modeling, counting, flexible
strategies, derived facts) (Steps A, B, C)
● Practice note taking strategies. (Steps B, D, E)
● Use the planning template to plan a problem from scratch. (Steps B, C, D, E)
o Anticipate strategies that students might employ
o What is the purpose (goal) of posing this problem?
o Prepare for a storyline for the closing discussion.
● Use the planning template to plan a problem, given student work samples. (Steps B,
C, D, E)
Module 6: Multi-digit Addition Problems
63
● Examine the different levels of solving strategies (direct modeling, counting, flexible
strategies, derived facts) as shown on the “Children's Solution Strategies”. (Steps A,
B, C)
● Practice note taking strategies. (Steps B, D, E)
● Use the planning template to plan a problem from scratch. (Steps B, C, D, E)
o Anticipate strategies that students might employ
o What is the purpose of posing this problem (goal)?
o Prepare for a storyline for the closing discussion.
● Use the planning template to plan a problem, given student work samples. (Steps B,
C, D, E)
Module 7: Multi-digit Subtraction Problems
● Examine the different levels of solving strategies (direct modeling, counting, flexible
strategies, derived facts) as shown on the “Children's Solution Strategies”. (Steps A,
B, C)
● Practice note taking strategies. (Steps B, D, E)
● Use the planning template to plan a problem from scratch. (Steps B, C, D, E)
o Anticipate strategies that students might employ
o What is the purpose of posing this problem (goal)?
o Prepare for a storyline for the closing discussion.
● Use the planning template to plan a problem, given student work samples. (Steps B,
C, D, E)
Module 8: Base-Ten Number Concepts
64
● Examine how multiplication and measurement division problems to support base-ten
concepts (Step A)
● Examine the different levels of solving strategies (direct modeling, counting, flexible
strategies, derived facts). (Steps A, B, C)
● Practice note taking strategies. (Steps B, D, E)
● Scrutinize tools and representations to notice the level of strategy and the utility of the
tools (Steps B, C, D, E)
● Use the planning template to plan a problem from scratch. (Steps B, C, D, E)
o Anticipate strategies that students might employ
o What is the purpose of posing this problem (goal)?
o Prepare for a storyline for the closing discussion.
● Use the planning template to plan a problem, given student work samples. (Steps B,
C, D, E)
Visual Overview of the Modules
Figure 2 represents the eight modules in this curriculum. Modules 1 and 2 must be
presented first. However, because all the major steps of the curriculum are represented in each
module, the order of Modules 3 through Module 8 may be adjusted to better meet the needs of
the teachers in the training.
Figure 2
Visual Representation of the Modules in the Curriculum
While Figure 2 depicts a visual representation of the modules, Table 4 is a scope and sequence chart of the modules that will further
describe the content of the curriculum.
Scope and Sequence Table
The scope and sequence table (see Table 4) is a visual representation of how the curriculum was designed (Smith & Ragan,
2005). In Table 4 the scope denotes the learning goals, and the sequence shows the modules as they are presented. There are four
symbols in the table that signal the level of mastery of the learners for each learning goal within a given module. The “P” symbol
Module 1a:
Overview
Module 1b:
Designing the
Classroom
Space
Module 2:
Structure of
the Problems
Module 3:
Noticing
Strategies
(Join)
Module 4:
Noticing
Strategies
(Separate)
Module 5:
Structure of
Multiplication
& Division
Problems
Module 6:
Multi-Digit
Addition
Problems
Module 7:
Multi-Digit
Subtraction
Problems
Module 8:
Base-Ten
Number
Concepts
65
66
indicates that the learning goal is given a brief introduction with no expectation of mastery. Next,
the symbol, “I,” indicates a more in-depth introduction. It will lay the groundwork for a deeper
understanding as the concepts are later reinforced (R) in subsequent exposures. At the point of
“R,” learners will have a better grasp of, practice, and receive feedback on their understanding of
the concepts. Finally, the “M” symbol indicates a level of “Mastery.” By this point, the learners
will have had multiple opportunities and scenarios to demonstrate, practice, give and receive
feedback from the instructor and other learners. The scope and sequence table reinforces the
cyclical nature of the steps. From module 3, every learning goal is reinforced until mastery in
module 8.
Table 4
Scope and Sequence
Learning goal Module
1
Module
2
Module
3
Module
4
Module
5
Module
6
Module
7
Module
8
Learn how children think and solve
problems intuitively and
conceptually.
P I R R R R R R, M
Recognize the trajectory of the
strategies students employ to solve
problems.
I, R R R R R R, M
Analyze student strategies and use of
tools to gauge the level of thinking
based on the problem.
I, R R R R R R, M
Design instruction that is responsive to
students’ needs.
P P I, R R R R R R, M
Note: P = preview; I = introduce; R= reinforce; M = master.
67
68
Delivery Media Selection
The choice of media impacts the learner and the learning process. Media is a vehicle for
delivering content. Thus, the choice of media needs to align with the goals of the curriculum.
Clark et al. (2010) delineate a two-pronged process for selecting media. It is important to
consider instructional methods and the particular media that best supports the cognitive processes
of the learner. Next, it is essential to consider the amount of cognitive load imposed on the
learner. Mayer’s Principles of Multimedia Learning give insight to reduce extraneous processing,
manage essential processing, and foster generative processing (Mayer & Alexander, 2016). The
final matter concerning media selection is instruction design. Merrill’s First Principles of
Instruction and the Guided Experiential Learning (GEL) model will serve as guidelines for
instructional design of the curriculum (Clark et al., 2010; Merrill, 2002). Careful deliberation
over choices for media will benefit the learner and learner process.
General Instructional Platform Selection in Terms of Affordances
When considering media selection, it is important to consider the benefits of different
dimensions (Clark et al., 2010). The first dimension is whether the training is synchronous or
asynchronous. It is important to consider the goals of the curriculum to determine whether
synchronous activities are necessary or if and when asynchronous activities are feasible. The
second dimension to consider is whether sessions will occur in-person or online. Given the
recent pandemic, it is difficult to ignore the benefits of online programs and the possibility that
in-person training may be prohibited. Table 5 outlines the benefits of the different combinations
of the two dimensions.
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Table 5
Dimensions of Media
In-person Online
Synchronous There are benefits of having people
in the same space and engaged in
the same learning activities
together. Certain dynamics are
difficult to replicate online or
asynchronously.
The instructor is able to read cues
from participants such as body
language. For example, after
giving participants time to discuss
a topic, the instructor listens for a
lull. This naturally happens as
groups slow their conversations.
Participants could not engage in
conversations in small groups
while being aware of the rest of
the group in another format.
Participants are able to engage in
small group and whole group
conversations fluidly.
Synchronous, online sessions allow
participants to join from wherever
they are. Some may be in remote
locations or are unable to travel to
a physical training location.
The instructor can design instruction
to maximize the use of a platform
such as Zoom.
There are many effective
instructional methods that
maximize engagement amongst
participants. One is the use of
breakout rooms that allows
participants to engage in
conversations in a small group.
Another is the use of a whiteboard
where the instructor may
demonstrate concepts there or
participants may post and share
their ideas.
Asynchronous The needs of individuals within a
group may be varied. So,
asynchronous, in-person may best
differentiate and tailor instruction
for each member of a group.
One way is to offer a menu of
activities or sessions with which to
engage. This may be like a mini
conference where participants
choose the session that best
addresses their needs for learning.
Asynchronous, online sessions
maximize opportunities for
participants to customize learning
time around their own schedules.
Teachers are able to choose the times
that work best for their schedule.
Some may prefer to engage late at
night or early in the morning. This
accommodates different schedules.
70
In-person Online
Groups attend the learning event
together while receiving
customized instruction. There will
be opportunities and times where
groups may come together, reflect,
and share what they have learned.
Some participants may feel pressured
to respond in-person. Here, they
may take as much time as they
need to respond to a prompt or a
colleague’s post. It can be
challenging to formulate an idea
and wait for just the right
opportunity to insert a thought into
a synchronous conversation. This
format allows participants to
respond at their own rate.
Another advantage is the ability to
watch and rewatch sessions. A
teacher can rewind and watch a
session multiple times.
Access
Access examines ideas such as who and where and how many learners (Clark et al.,
2010). The content of this curriculum is beneficial for all teachers working with elementary
students who teach math. In its current state of development, it is designed for in-service
elementary teachers. However, there is room to modify and adapt it to accommodate pre-service
teachers in elementary credential programs as a module within their math methods course. If the
training is in-person, the number of participants can range from four to 36 with an ideal range
between 20 and 28. In the case of smaller groups, like four to eight participants, a school may
request training at their site for a specific cluster of teachers. In this case, a smaller cohort may
be more advantageous because the participants work together and are able to collaborate with
each other. Although individual teachers (those who attend by themselves, without colleagues
71
from the same grade level or school) may attend, it is not as beneficial. Teachers are expected to
collaborate with their colleagues about implementation after the training session is over. If the
training is online, the capacity is much higher. It will be important to check the bandwidth
capacity of the platform for both the instructor and participants.
Consistency
Consistency examines how important it is to have the same content and pedagogy
delivered to all the learners (Clark et al., 2010). Consistency of the program in terms of content
and pedagogy speaks to the integrity of the program. One way to ensure reliable content is to
have set slides, handouts, manipulatives, and posters (for room environment) predetermined for
each of the sessions. The baseline presentation will be on slides so that each training will have
consistent content. In addition, the structure of the activities and the handouts are carefully
developed to build proficiency in the content. While it is important to ensure stable content, it is
equally important that the training is relevant and meets the needs of the participants. The trainer
will be prepared to address specific questions and concerns that participants raise. While the
prompts for participant input may be the same, comments will vary from one session to another.
Depending on the needs demonstrated by the participants, the trainer may expound on the certain
areas more.
Whether the training is online or in-person, the instructor will utilize methods that are
best suited for the platform of choice. For example, instructors may have a “Parking Lot” poster
to collect questions from participants attending in-person. Online, participants may post
questions in the chat box, or the instructor may have a separate place to pose questions such as
on a Jamboard. Furthermore, activities may have to be modified to adapt to the online or in-
person format.
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Cost
Cost of delivery may include one-time costs and recurring costs (Clark et al., 2010).
There are different types of costs associated with the training. The development costs are the
same no matter how it is implemented. It is important to determine whether the training sessions
will be online or in-person because the materials needed for each format are different. With in-
person training, the first type of expenditures are one-time costs to develop the curriculum,
including time, software, and hardware needed to create and maintain the program, and materials
(like manipulatives and participant boxes) for the training sessions. It may be both beneficial and
necessary to have presentation supplies and technology ready to go for every training: items such
as a portable projector, speakers, chart stand, presenter with a laser pointer, and a laptop
computer. The second type of expense for in-person training will depend on how it is delivered.
If the training is broken up into shorter chunks, the cost increases because of the number of times
a trainer is sent out to the site. One of the baseline costs is the per diem fee of the trainer. First of
all, it is more cost effective to have the trainer work with a larger group than a single grade level
group at a site because the per diem costs can be shared by multiple participants. More
participants translate into a smaller base fee for each individual. One way to avert a high per
diem burden on a small grade level group, may be to opt for online modules.
Online sessions are an alternative choice for training. The one-time costs for developing
the program are the same as in-person training. If the training is online, special attention to the
video conferencing platforms and subscriptions for accessory programs such as Google Suites
(for Google Docs, Jamboard, Google Slides) or Nearpod and video recording support software
may be essential. Per diem costs will depend on whether sessions are asynchronous or
synchronous. If sessions are asynchronous, the base fee is at the discretion of the program
73
designer. It may be a flat, pre-determined charge for all participants. If the training is
synchronous, per diem cost structures are similar to in-person sessions.
Another consideration for cost is the amount of time available for the training. Districts
and sites face many competing agendas for professional development. Thus, they may need to
find a way to accommodate different demands. Currently, there are eight, 3-hour modules in the
course. A site may not be able to devote 24 hours to training in one year. Certain modifications
may need to take place. One option is to stretch the course over two years. A second option is to
offer the course one module at a time and instead of utilizing a full day, present modules during
after school sessions. A final option is to shorten each module. There are multiple opportunities
for planning, creating, and designing lessons. Some of this work can be done at the site with
colleagues, perhaps during a grade level collaboration time. Ideally, it is better to keep the
training intact and as is. However, it may be necessary to make accommodations to make the
course more accessible for sites and districts.
Specific Instructional Platform Selection in Terms of Restrictions
According to Clark et al. (2010), there are three factors to consider when selecting
media. The first factor, conceptual authenticity, ensures that the media creates the appropriate
conditions for learning so that learners may apply their learning. The second consideration is
immediate feedback. Given the need for immediate feedback, participants need to have
opportunities to observe modeling and receive correct feedback as they engage in practice.
Finally, the third factor is sensory requirements. These sensory requirements are those beyond
visual or aural inputs. Table 6 shows the key considerations for media selection.
74
Table 6
Key Considerations for Media Selection
Key
consideration
Media considerations
Conceptual
authenticity
It is important that teachers have the opportunity to practice interpreting
student strategies and to select talking points to highlight during a
debriefing session and thus, bring closure to a lesson. This takes learning
the content to an application level.
Immediate
feedback
Participants will interpret strategies, write problems to pose in their own
classroom, and design a debriefing conversation. Whether they work
independently or in small groups, participants will receive feedback from
the instructor and other participants. The different voices will broaden
their perspective.
Special sensory
requirements
The in-person training will mimic what happens in the classroom. Some of
the learning is through tactile means from using physical manipulatives.
There are digital versions of manipulatives. However, as students are
learning new strategies or more complex mathematics, it may be
unnecessary to increase cognitive load by introducing multiple platforms.
It is important to attend to the purpose which is to focus on the strategies
for learning and representing mathematics. Multiple platforms could
potentially raise one’s affective filter or cognitive load.
Client Preferences or Specific Conditions of the Learning Environment
Another reason for in-person, synchronous training is that this mode matches the current
culture of professional development. Time is a limited resource for teachers, and it is unfeasible
to expect teachers to view recordings prior to synchronous training. In fact, teachers are not
normally asked to watch asynchronous training sessions. It is different from a university course
where flipped instruction is often expected. There are a few occasions when teachers may be
required to engage in asynchronous professional development such as: risk management training
75
on sexual harassment or calibrating videos from the CDE for state testing preparations and
accommodations. These training sessions are mandated and informational. Even then, schools
may provide a block of time after or during a staff meeting where teachers will watch these
sessions in groups. The purpose of this professional development course is more than
informational. It is to instruct new strategies for teaching math and ultimately shift instructional
pedagogy to better match the objectives laid out in the state framework.
Specific Media Choices
After contemplating all of the factors, in-person training is deemed the preferred platform
for this course. In-person sessions have certain benefits that are difficult to replicate with
asynchronous sessions. Participants have opportunities to engage with slides, video clips, and
copies of student work collaboratively with others. They may pose questions to the presenter
and receive responses immediately. Another advantage of a live session is that presenters ensure
that the training is relevant to participants by responding to individual and group needs by
clarifying ideas and offering additional scenarios and examples to solidify understanding as
needed. Although asynchronous instruction may be more convenient because participants can
access the training when it best suits them, it does not allow participants to engage with the
content with others at the same time in the same way. Thus, if the curriculum is delivered
asynchronously, participants may forfeit the opportunity to learn from other participant’s
questions, comments, or thinking process.
The media selected for the curriculum include digital resources and media that will be
utilized during the synchronous, in-person sessions. Table 7 shows the media selections for the
curriculum.
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Table 7
Media Choices in Developing Mathematical Thinking
Media Purpose Benefits
Instructor-led The instructor will facilitate
conversations and activities to
guide participants through the
content.
The instructor will clarify content,
provide extra examples, and help
learners connect new content to
prior experiences
Student work
samples
Student work is presented so that
teachers can analyze the
work. First, teachers need to
examine and identify the
strategies used. Next, they will
generate questions to ask the
student based on their work.
As teachers practice examining,
identifying and interpreting student
work, they will be more prepared to
do so during a lesson. Facility in
quickly identifying and interpreting
student work may allow the teacher
to focus on appropriate questions or
next steps.
Math
manipulatives
Manipulatives are used to represent
thinking and solve problems.
The variety of manipulatives allows
teachers to explore different
strategies using different
manipulatives. Then, teachers may
have conversations comparing and
contrasting the strategies used for
solving. It helps teachers develop a
more nuanced understanding of
strategies and sophistication that
the different tools support.
Video clips Videos of students demonstrate
strategies that students
use. Classroom clips show
teachers using strategies in an
authentic setting.
Learners will be about to view and
review strategies during and after
the sessions. It will allow teachers
to practice their own note taking
techniques. Furthermore, it allows
teachers to assess the current
situation and consider next steps
for instruction.
PowerPoint
Slides
Slides anchor the presentation as
visual records of the content.
Participants can refer back to the
slides later.
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Media Purpose Benefits
Handouts Handouts are provided to help take
notes, organize work and engage
in activities to work on learning
the content. They will provide
scaffolds for learning and
applying the content.
Participants will have material to read
and refer back to later. They will
have notes on student strategies and
problems that they can use in their
own classroom.
Book:
Children’s
Mathematics:
Cognitively
Guided
Instruction
This book was written by the
original CGI authors. It explains
the framework of problems and
the strategies that students
use. Furthermore, it has QR
codes for videos of students that
teachers can review outside of the
session. It is the foundational
content.
Teachers may need a refresher or
clarification when they are
implementing in their classroom.
School cohorts may refer back to
the book to do further study.
Book:
Principles to
Actions:
Ensuring
Mathematical
Success for
All
This book incorporates ideas from
the National Research Council’s
book, Adding It Up (2001) and it
builds on the work of Stein and
Smith, 5 Practices for
Orchestrating Productive
Mathematics Discussions (2011).
It begins with data to make a case
for shifting practice. Then, it
provides more details about
effective teaching and learning,
essential elements of a
mathematics classroom, and
actions educators may take to
improve mathematics
instruction.
This is an anchor text for
mathematics teachers. Participants
will learn the overarching
principles in the book during the
sessions. The text will provide
another medium to reinforce the
ideas learned in the training.
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General Instructional Methods Approach
This curriculum was designed to support teachers in developing mathematical thinking
through problem solving that will initiate generative, self-sustaining practices for instruction.
The modules balanced supplantive and generative strategies to maximize learning. Three main
theories informed the design of the curriculum: cognitive load theory (Schunk, 2020; Smith &
Ragan, 2005), GEL (Clark et al., 2010), and sociocultural theory (Schunk, 2020).
The tenets of cognitive load theory influenced how information within the modules is
balanced. Knowing that teaching is a complex activity which requires many in-the-moment
decisions, it is important to consider how to manage the amount of information and the way new
approaches for teaching were presented. One technique that is used is segmenting (Smith &
Ragan, 2005). The new content within the module is centered around a different operation and
the corresponding solution strategies, which in turn, allow teachers to focus on one thing at a
time.
Next, the GEL principles shape how activities knit together in each module. In particular,
modules three through eight follow a similar cyclical pattern through the elements of GEL that
include (a) Objectives; (b) Reasons for learning; (c) Overview; (d) Conceptual knowledge; (e)
Demonstration of procedure; (f) Part and whole-task practice of procedures with corrective
feedback; and (g) Challenging, competency-based tests that include reactions and learning
performance (Clark et al., 2010). The structure in each of those modules is the same. First,
teachers learn about the problem type(s), their structure, and the natural, predictable strategies
that children use to solve those problems. Then, teachers determine what they should take note of
and practice it while watching or interacting with students as they solve. Finally, they will
engage in a planning process to prepare for a live lesson with children. Teachers will gather
79
evidence to formulate an underlying storyline for the closing discussion. They will discuss and
refer to videoclips as well as student work samples they analyzed within the module. At each
step of the way, teachers will have opportunities to give and receive feedback, reflect on
different viewpoints, practice small segments of what they will do in their classroom before
having to piece together an entire lesson in front of live children.
Sociocultural theory was another theory that informed the design of this curriculum in
two ways. It is important to create a space where learning was through communicating and
negotiating ideas and second, to attend to familiar cultural contexts to frame mathematics
lessons. According to sociocultural theory, humans learn through social activities and within a
context (Schunk, 2020). Communication of ideas is important in developing meaning in
mathematics. It is beneficial for children as well. Children learn mathematics as they negotiate
meaning and communicate ideas with each other (NCTM, 2014). Thus, teachers need time to
discuss ideas and work out the details of complex acts of teaching. They will have ample
opportunities to deliberate over ideas and create meaning within this social context. In fact, this
process mirrors the classroom experiences that the teachers will replicate in their classrooms.
Furthermore, Schunk (2020) notes that the connections between social, cultural-historical, and
individual factors influence the developmental process and future growth. A learner’s cultural
background and experiences shape their mental structures (Schunk, 2020). Hence, teachers will
consider their students’ contexts to write problems and orchestrate meaningful discussions.
This curriculum is designed to teach teachers how to develop students' mathematical
thinking through problem-solving. Teachers will be trained in the tenets of CGI, and they will
learn how to plan effective lessons by using the five-step planning template. The first and second
modules lay the foundation and background for CGI. Then, Modules 3 through 8 cover the
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mathematical content of the different operations (adding, subtracting, multiplying, and dividing).
It may be the first training teachers attend on CGI or it may be a refresher course.
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Chapter Five: Implementation and Evaluation Plan
This chapter presents the purpose for, learning objectives of, and evaluation plan for the
curriculum. The evaluation plan describes a process to monitor and measure learning and
implementation. Kirkpatrick and Kirkpatrick (2016) propose that evaluation measures effective
training (provides relevant knowledge, teaches skills, and bolsters confidence to apply learning),
determines training effectiveness (the training and accountability to support that produces
positive organizational results), and underscores the value of training for an organization
(demonstrates organizational goals and mission).
Implementation Plan
Smith and Ragan (2005) address key concepts regarding implementation: diffusion,
dissemination, adoption, and stakeholders. The stakeholders, teachers and those supporting
implementation (administration and other site support staff), are invested in ensuring the transfer
of knowledge to the classroom. Diffusion happens when groups of teachers apply training to
their practice and to this, dissemination adds a layer of intent for implementation. Adoption
occurs when implementation becomes more apparent and intentional. A thorough evaluation
process is essential to ensure implementation. The more teachers practice and apply what they
learned, the greater the impact on student learning and math achievement.
The first pilot course will be offered to primary teachers (kindergarten to third grade) in
two schools within a given district and enrollment will be capped at 24 participants
(administrators, instructional coaches, and math specialists may attend without impacting the
enrollment cap). Formative and summative data will be used to evaluate the course to make
improvements after the first pilot. During the training, the instructor will collect formative data
from observations, charts (notes from group discussions), and surveys to serve as evidence for
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future decisions. When appropriate and feasible, the instructor will address needs that surface on
surveys at the end of each training day. Surveys immediately following the course and the
delayed end of course surveys will be scrutinized to determine necessary revisions or
improvements to the course. In addition, a separate focus group will be conducted with site
leadership, including administrators, instructional coaches, and math specialists. They will
provide insights from the perspective of those who are supporting accountability and monitoring
implementation.
Evaluation Plan
Curriculum Purpose, Need and Outcomes
The purpose of this curriculum is to teach teachers how to develop mathematical thinking
in children through problem solving. Many educators attend training and do not transfer that
knowledge to their classrooms. This curriculum supports teachers' bridge theory to practice in
the classroom. After completing the eight modules, teachers will be able to identify and analyze
student strategies, plan problem-solving lessons that help students enhance their mathematical
thinking and collaborate within a community to foster robust discussions.
Evaluation Framework
The evaluation framework for this curriculum is the new world Kirkpatrick model
(2016). The new world Kirkpatrick model differs from the previous model because planning
occurs in reverse order of how it is executed (Kirkpatrick & Kirkpatrick, 2016). Beginning with
the end in mind helps make sure every component of the course is designed to support and
achieve the outcomes. The descriptors of the four levels are: Reaction, Learning, Behavior, and
Results. Level 4 (Results) looks at the big picture from measurable external and internal
outcomes. From the beginning, these will serve as a constant reminder of the course outcomes so
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that every activity supports and builds toward those goals. Teachers will observe and practice
new skills during the sessions to support them in their classrooms. This curriculum has activities
built in to provide formative data about whether teachers are learning the content (Level 2 –
Learning) and to help bridge theory to practice, ultimately to change behavior (Level 3 –
Behavior). Finally, Level 1 addresses the teacher’s reaction by measuring engagement,
relevance, and satisfaction. The Kirkpatrick model for evaluation starts with the end first to make
sure every step supports the intended outcomes.
Level 4: Results and Leading Indicators
Level 4 of the Kirkpatrick model looks at the connection between the intended training
outcomes with the results of training, support, and accountability (Kirkpatrick & Kirkpatrick,
2016). Leading indicators specify the desired results through tangible observations and
measurements (Kirkpatrick & Kirkpatrick, 2016). These short-term observations and
measurements are sorted as external (they are publicly visible) or internal (they are visible to
members within the organization). Both types of indicators serve as evidence that the curriculum
outcomes were met. This curriculum was designed to teach teachers how to develop
mathematical thinking through problem-solving. State and district assessments, along with other
measures of positive acknowledgements (surveys and communication) are examples of external
data to check outcomes. Internally, records of lesson plans, data from local assessments and
surveys will demonstrate achievement of outcomes. Table 8 specifies the external and internal
indicators to show whether course outcomes are met or not.
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Table 8
Indicators, Metrics, and Methods for External and Internal Outcomes
Outcome Metric(s) Method(s)
External outcomes
Increased student
achievement for
mathematics on state
achievement
assessments
Student’s overall mathematics
score on CAASPP (California
Assessment of Student
Performance and Progress)
Summative assessment data
from the CAASPP website
Increased student
achievement for
mathematics on district
level assessments
Student’s mathematics score on
district assessments
Data collected from district
administered tests
Increased community
perception of
mathematics
achievement
Data from survey Survey administered to
families at the end of the
year
Increased School Board
approval and
recognition of progress
in mathematics
Number of positive comments
from the School Board (email
communication, district
highlights, verbal comments
during board meetings)
School administration team
tracks comments
Internal outcomes
Increased number of
problem-solving
lessons for math.
Number of problem-solving
lessons
Daily lesson plans check by
the administrative team
Increased self-efficacy in
teaching problem-
solving math
Data from Likert scale survey Survey administered at the
beginning and end of the
course by the course
instructor
Increased educator’s
ability to predict
student strategies
Number, variety, and details in
descriptions of anticipated
strategies
Lesson planning template by
the administrative team
Increased educator’s
ability to plan problem-
solving lessons
Number of lesson plans per
week
Weekly/daily lesson plans by
the administrative team
Increased student
achievement in school
assessments
Data from common assessments Record of data (google sheets
for all local assessments)
reviewed by grade levels
and administrative team
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Level 3: Behavior
Level 3 is perhaps the most important level of the Kirkpatrick model because it is
concerned with the extent that learners will apply what they have learned back in their workplace
(Kirkpatrick & Kirkpatrick, 2016). No training can possibly cover every scenario that a learner
will face. Thus, it is important that learners are able to transfer knowledge from the training
situation to new ones, whether they are similar situations (near transfer) or other, future tasks (far
transfer) (Smith & Ragan, 2005). Such learning results in behavioral changes. According to the
cognitive view, a change in behavior is initiated by changes in the learner’s knowledge and their
environment (Mayer, 2011). One element of this course that supports behavioral changes is the
multiple opportunities to engage in planning. Learners experience the planning process as a
collaborative endeavor (learner’s environment) and they learn to plan problem-solving lessons
using a five-step planning template (new knowledge).
Critical Behaviors Required to Perform the Course Outcomes
Within Level 3, Kirkpatrick and Kirkpatrick (2016) call out critical behaviors which
learners need to consistently perform in order to accomplish the targeted outcomes. These are
most important for organizational success (Kirkpatrick & Kirkpatrick, 2016). For this
curriculum, teachers need to identify, analyze, and predict student strategies to effectively plan
problem-solving lessons that are responsive to students’ needs. As teachers understand the
strategies that children develop, they are able to plan discussions that reinforce, clarify, and
stretch students’ that ultimately develops mathematical thinking. Table 9 outlines the critical
behaviors that support the course outcomes along with the methods, metrics, and timing
associated with measuring those behaviors.
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Table 9
Critical Behaviors, Metrics, Methods, and Timing for Evaluation
Critical behavior for
course outcomes
Metric(s)
(Unit of measure)
Method(s)
(How measured)
Timing
(How often)
Teachers write daily
problem-solving
lesson plans using
the lesson
planning template.
Lesson plans using
the lesson plan
template with the
five practices
Lesson plans are shared
during collaboration
and grade level
meetings as well as
during observations
by site support
personnel
Daily/weekly/monthly
Teachers reflect on
and adjust
instruction to
ensure student
growth in
problem-solving
skills after each
daily lesson.
Adjustments to
lessons based on
observed student
strategies
Measurable/observable
growth on student
strategies chart
Daily
Teachers engage in
conversations
with colleagues
about problems,
student work, and
observed
strategies during
weekly or bi-
monthly staff
meetings.
Data from common
problems posed
as a grade level
Student work and
anecdotes/notes from
lessons
Weekly/bimonthly
Required Drivers
In addition to the critical behaviors to support course outcomes, Kirkpatrick and
Kirkpatrick (2016) identify four “drivers” that support the critical behaviors: reinforcing,
monitoring, encouraging, and rewarding. These required drivers encompass systems and
processes that will ensure accountability for implementing critical behaviors up to 85%
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(Kirkpatrick & Kirkpatrick, 2016). This support and accountability provide structures that ensure
transfer of knowledge to practice. The required drivers within this course, outline expectations
for regular follow up and monitoring of teachers that is built on a system of layered supports.
Teachers will be accountable to and supported by their colleagues as well as administration.
They will share lesson plans and problems, engage in coaching cycles, report successes, and
collaborate with their grade level team. All in all, the processes and systems are proactive and
supportive. Table 10 identifies the method, timing, and associated critical behavior for each
required driver.
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Table 10
Required Drivers to Support Critical Behaviors
Method(s) Timing
Critical
behaviors
supported
Reinforcing
Teachers meet with site support staff
(administration, instructional
coach, math specialist) and report
progress in planning and
implementing lessons.
During data chat meeting and
professional development
sessions
1, 2, 3
Grade level teams will post the
problems for each unit of study on
a shared google document.
Weekly 1
Encouraging
Teachers meet with grade level
teams to discuss upcoming
lessons/topics.
Bimonthly during grade level
meeting time
1, 2, 3
Rewarding
Administration and colleagues
acknowledge teachers
Monthly staff meetings 1, 3
Staff members will acknowledge
successes from teaching problem-
solving lessons.
Monthly staff meetings 1, 2, 3
Monitoring
Teachers will bring lesson plans to
share with the grade level team.
Bimonthly grade level meetings 1, 2, 3
Teacher will schedule coaching
cycle with site support personnel
(administrator, instructional coach,
math specialist)
One cycle per trimester 1, 2, 3
Teachers will present grade level
focus areas for the trimester during
grade level release
day/professional development.
Once per trimester 1, 2, 3
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Organizational Support
Kirkpatrick and Kirkpatrick (2016) state that the four dimensions of required drivers are a
package that need to be implemented as such. The success of this curriculum depends on support
and follow-up by administration, the instructional coach, math specialists, and grade level teams.
They represent layers of both accountability, assistance, and encouragement. Beginning with
grade level teams, they will collaborate with each other to plan problems and lessons. They will
be the first line of defense in case one member has a question or needs a second opinion about a
strategy or approach. Next, the math specialist and instructional coach will work with individuals
and grade level teams for consulting and coaching cycles as content expert support.
Administration will check on the teacher's lesson plans and progress in implementation. It will
take all members to work together to support planning and implementing robust problem-solving
lessons.
Level 2: Learning
Level 2 of the Kirkpatrick model delineates the five components (commitment,
confidence, attitude, skills, and knowledge) of what participants have learned after the course
(Kirkpatrick & Kirkpatrick, 2016). In the New World Kirkpatrick model, confidence and
commitment are additions to Level 2 for the purpose of connecting learning and behavior
(Kirkpatrick & Kirkpatrick, 2016). Level 2 addresses the dimensions that participants need to
learn and acknowledge before changing behavior and putting it to practice (Level 3). Hence, the
goals of Level 2 are learning goals, not outcomes. Learning goals are statements of what
participants should be able to do after completing the course (Smith & Ragan, 2005).
Terminal Learning Objectives
After successfully completing the course, teachers will be able to demonstrate the critical
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behaviors listed in Table 9 and achieve the learning goals listed below:
● Identify grade level expectations regarding problem types and levels of strategies.
● Identify characteristics and practices of a CGI classroom.
● Identify and distinguish the structural features of join, separate, part-part whole
problems and compare problems.
● Identify and analyze strategies that children use when solving
○ Join problems
○ Separate problems
○ Multiplication, measurement division, and partitive division problems
○ Multi-digit addition problems
○ Multi-digit subtraction problems
● Identify and analyze the types of problems that support base-ten concepts and the
strategies that children use when solving multi-digit subtraction problems
● Synthesize strategies that students use to solve problems to create (write) problems
and lessons that respond to students’ needs (are culturally responsive)
● Plan problem-solving lessons to implement in the classroom
Components of Learning Evaluation
During the course, the facilitator engages participants in various activities so that they can
attend to the learning goals. Throughout the course, the facilitator observes participants to assess
whether they are learning what was intended. Level 2 of the Kirkpatrick model focuses on five
components (Kirkpatrick & Kirkpatrick, 2016). These components, shown in Table 11, represent
different dimensions to evaluate learning. The first component is knowledge, and this course
utilizes Kahoot and sorting activities to check for declarative knowledge. Next, there are many
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opportunities and occasions for discussions and collaborative work to provide evidence for three
other components of learning: procedural, attitude, and confidence. The facilitator observes,
asks, and answers questions that may arise during those activities. Data from these observations
will serve as formative feedback on what teachers are learning along all five components of
learning. The fifth component of learning is commitment. Lesson plans that the teachers create
are evidence of ongoing commitment to apply what was learned in the course. During the course,
teachers work on writing lesson plans using the template and continue to do so after the course is
completed. Bridging this practice from the course to the classroom underscores a commitment to
apply the learning.
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Table 11
Evaluation of the Components of Learning for the Program
Method(s) or activities Timing
Declarative knowledge “I know it.”
Check for understanding through Kahoot or
sorting activities
During
Elbow conversations about the content During
Procedural skills “I can do it right now.”
Observations of collaborative planning
activities
During
Individual lesson plans for a problem-solving
lesson
During, after
Attitude “I believe this is worthwhile.”
Observation of discussions During
Observation of closing discussions about
what left an impression and what they are
wondering about.
During
Confidence “I think I can do it on the job.”
Observation of discussions During
Survey about self-efficacy regarding ability to
identify student strategies and adjust
lessons accordingly
After
Commitment “I will do it on the job.”
Daily/weekly lesson plans During and after
Level 1: Reaction
Level 1 of the Kirkpatrick model is an evaluation of participants’ reactions to the course
in terms of whether the course was engaging, favorable, relevant to their work (Kirkpatrick &
Kirkpatrick, 2016). Formative and summative evaluations provide different types of information
and serve different purposes. First, formative evaluation is ongoing and often happens in real
93
time. The instructor is observing participants’ reactions to discussions and activities and can then
make adjustments right away. This may garner positive reactions from participants because the
responsiveness of the instructor. All throughout this course, the instructor is observing partner
conversations and group work to measure engagement and check for a perception of relevance.
The instructor is also constantly checking in with table groups as they are working to make sure
participants have clarity about tasks and to answer questions. In doing this, the instructor is also
monitoring relevance, as many of the activities are geared toward planning and practicing what
they will do after the course. If they are struggling during the course, they will struggle with
implementation after. Thus, the instructor will try to make the connection to the work in the
teacher’s classroom as seamless as possible. The second type of evaluation is summative. In this
course, it will be collected through an end of course survey to capture satisfaction data. Table 12
details the methods and timing of measures.
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Table 12
Components to Measure Reactions to the Program
Method(s) or tool(s) Timing
Engagement
Observation of teacher’s engagement during
learning activities
During
Observation of discussions with elbow
partners or table group
During
Relevance
Observation of teacher comments about what
left an impression on them
During
Check-in conversation with teachers During, after
Customer satisfaction
Course evaluation End of course
Personal reflections During
Evaluation Tools
Kirkpatrick and Kirkpatrick (2016) note that it is important to decrease the gap between
participants’ perceptions and how those perceptions are interpreted, by being mindful of six steps
to ensure successful evaluation. Planning for evaluation begins with contemplating what data
would be informative and how data would be analyzed. In terms of data, it is also important to
consider the participants. Thus, this curriculum utilizes a blended approach to evaluation to
prioritize the learner’s needs. First, to avoid survey fatigue, the evaluation instrument will
measure all four levels of evaluation instead of having multiple surveys to measure each level
separately. Second, the tools are learner-centered in that the questions in the instrument will be
95
phrased from the learner’s perspective and jargon free (Kirkpatrick & Kirkpatrick, 2016). Third,
surveys will be administered both immediately following the course and after a delay. The focus
of evaluation administered immediately after the course is on Levels 1 (Reaction) and 2
(Behavior). Later, another survey will be given to gather data on all four levels. It is important to
give teachers time to implement their new knowledge before examining Level 3 (Behavior) and
Level 4 (Results).
Immediately Following the Program Implementation
The course will occur over four days. At the end of each day, a brief survey will be given
to teachers to gather formative data. Teachers will reflect and answer open-ended questions
describing what they learned, what they are committed to trying, and what they need to
implement their learning. In addition, teachers will rate how engaging, relevant, and satisfying
the course has been thus far on a Likert scale. This formative data will allow the instructor to be
responsive to teachers’ needs by adjusting or addressing issues immediately. The focus of this
instrument, found in Appendix D, will be on Levels 1 (Reactions) 2 (Learning).
Delayed For a Period After the Program Implementation
Three months after the completion of the course, the evaluation instrument predominantly
focused on Levels 3 (Behavior) and 4 (Results) will be administered. It is important to wait to
evaluate Levels 3 and 4 because teachers need time to implement their new learning and the
required drivers need to be activated (Kirkpatrick & Kirkpatrick, 2016). Teachers will rate items
and have an opportunity to provide comments for each item to further explain or describe their
rating. Instrument questions will also address Levels 1 (Reaction) and 2 (Behavior). This
instrument appears in Appendix E.
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Data Analysis and Reporting
After course completion, a data story will be shared with site and district stakeholders.
Data will be presented first to administration so that they can in turn distribute the findings with
key site stakeholders such as instructional coaches, math specialists, teachers, and community
members. The results will inform future decisions to continue professional development and to
monitor progress. Although the data included here is fictitious, it is representative of the type of
data that will be provided to stakeholders. Figure 3 presents state assessment data for third
graders (kindergarten, first, and second graders do not take the annual state assessment) to show
improvement in student achievement data over time. Figures 4 and 5 are visual representations of
responses to Likert scale items from the end of course survey. These questions target critical
behaviors (Level 3) and intended results (Level 4) from the training. In addition, all survey data,
quantitative and qualitative, will be analyzed to identify areas of strength and areas that need to
be improved. Evaluation and data analysis provide the means to examine the effectiveness of a
program and to pinpoint areas to make improvements (Kirkpatrick & Kirkpatrick, 2016).
97
Figure 3
Third Grade CAASPP Student Achievement Data
98
Figure 4
Discussions About Lessons and Student Thinking During Collaboration Meetings
99
Figure 5
Use Observations of Student Strategies to Plan Lessons
100
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Appendix A: Lesson Plan and Instructor’s Guide
Module 1a, Overview
Figure A1
Module 1a, Overview
Module Duration: approximately 90 minutes synchronous engagement
Introduction:
This is part one of the first module of an 8-module course to teach teachers how to implement cognitively guided instruction (CGI) in
their classrooms. Throughout the course, teachers will learn about the different problem types and strategies that students use. They
will also learn about a process for strategically planning lessons and robust mathematical discussions. The purpose of this first module
is twofold. First, it will provide background knowledge of the foundational research behind CGI. Second, teachers will examine the
Common Core State Standards and see that the problem types and strategies are embedded in the standards.
Module 1a:
Overview
Module 1b:
Designing the
Classroom
Space
Module 2:
Structure of
the Problems
Module 3:
Noticing
Strategies
(Join)
Module 4:
Noticing
Strategies
(Separate)
Module 5:
Structure of
Multiplication
& Division
Problems
Module 6:
Multi-Digit
Addition
Problems
Module 7:
Multi-Digit
Subtraction
Problems
Module 8:
Base-Ten
Number
Concepts
109
110
Learning Objective(s)
Terminal Objective:
● Given the background information of CGI and the Common Core State Standards,
teachers will identify grade level expectations regarding problem types and levels of
strategies.
Enabling Objective(s):
● For a given grade level, teachers will:
○ Know the meaning of problem types
○ Know the meaning of student strategies
○ Know the meaning of level of student strategies
○ Identify the word problems that students are expected to learn at each grade level
Learning Activities
● After introductions and attention activities and learning objectives, assess prior
knowledge of the meaning of problem types, student strategies, and levels of student
strategies.
● Teach any necessary prerequisite knowledge by providing definitions and examples and
nonexamples.
● Provide opportunities for learners to generate their own examples and nonexamples.
● Model the procedure for identifying the word problems that students are expected to learn
at each grade level.
● Provide practice and feedback on identifying the word problems that students are
expected to learn at each grade level.
● Provide opportunities to transfer knowledge to the teachers’ own classroom practice.
Summative Assessment
● Given the problem type chart from the appendix of the standards document, teachers will
identify grade level expectations for the different problem types and they will identify the
appropriate range of strategies for each grade level with 100% accuracy.
Lesson Materials
● Written resource: Math Common Core State Standards, grades K-5
● Written resource: Appendix for Math Common Core State Standards
● Presentation slides
● Handout with key slides
● Handout with reflection questions
● Sorting Activity (definitions)
● Blank chart paper
● Charts/Posters:
○ Agenda: Day 1
○ Learning Objectives for Module 1a
○ Six Research Questions (for the table rankings)
○ Burning Questions
○ “Gots” and “Needs”
111
Learner Characteristic Accommodations
Learners will complete this module in person. Teachers will sit in table groups to support small
group conversations and collaboration. Slides will be projected so that all may access the
materials. Teachers will receive handouts with key slides on them to facilitate listening and note
taking. If needed, teachers may request further accommodations.
Facilitator’s Notes
This opening session of the course will present the background information about CGI. It is often
reassuring to know that this way of teaching math is not a fad. In fact, it is grounded in research
over the last 40 years and the research is continuing today. Teachers will also examine the Math
Common Core State Standards and find that problem types are weaved into the standards. Thus,
a change is warranted. This first part of module 1 begins to examine the differences between the
way adults perceive and solve problems versus the way children think of them. Throughout the
course, teachers will have opportunities to observe children, classrooms, and discuss ideas with
colleagues. All sessions will be synchronous and face-to-face. Teachers will be asked to present
problems to their class and return to subsequent sessions with student work.
Table A1 provides the instructional activities and details the instructional sequence, duration,
descriptions of the learning activities, instructor actions/decisions (supplantive events of
instruction) and learner actions/decisions (generative events of instruction).
112
Table A1
Instructional Activities, Module 1a
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Gain attention 15
Introduce teachers in
the room. Ask
teachers to reflect
on their
expectations for the
course.
Instructor gives
teachers time to
personally think
about their
responses, then
within a table
group and then
with the whole
group.
Instructor synthesizes
and captures main
points from each
table on a chart.
Teachers reflect on
their reasons and
expectations for
attending the
course.
Teachers share out
with the whole
group.
Learning
objectives
5 The learning
objectives for the
module are posted
at the beginning of
the course content.
In order to establish
relevance and to
facilitate learning,
ask learners to read
the terminal and
enabling learning
objectives.
Ask teachers to write
their personal
learning objectives
on a post-it and add
it to the chart.
Teachers read the
terminal and
enabling learning
objectives for
themselves.
Teachers will post
their personal
learning objectives
on the chart.
Purpose for
learning
• Benefits
• Risks
3 Benefits:
• It is important to
understand that
children pay
attention to the
structure of word
problems.
• The CCSS-M
acknowledge this
Ask teachers to
discuss challenges
they have faced
when posing word
problems to
students.
Invite teachers to
record any burning
Teachers participate
in conversations at
their table and
reflect on their
practice.
Teachers may record
burning questions
113
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
structure and state
expectations for
them by grade
level.
Risks:
A teacher who does
not understand the
structure of
problems and its
nuances may
inadvertently
instruct students in
a way that confuses
them.
questions on a post-
it and add it to the
“Burning
Questions” chart.
The instructor will
peruse the
questions and make
notes to address
these throughout
the sessions.
on a post-it and add
it to the chart.
Overview
• Prior
knowledge
• New
knowledge
• Learning
strategies
(What you
already
know, what
you are
going to
learn, and
how you are
going to
learn it.)
17 Activate prior
knowledge by
examining
teacher’s
perceptions and
misperceptions of
how
kindergarteners
responded to and
solved the “Six
Research
Questions.”
Overview of the
agenda for the
session:
• Examine where
word problems
appear in the
kindergarten, first,
and second grade
standards.
• Describe the
progression of
word problem
types and range of
Instructor prompts
teachers to discuss
the “Six Research
Questions” with
their table and rank
problems from
easiest to most
difficult.
Instructor records
how each table
ranked the
problems.
Instructor leads a
discussion about
the similarities and
differences in how
the group ranked
the problems.
Instructor presents
data on how the
kindergarteners
solved: percentage
accurate solution
Teachers discuss their
thoughts about the
problems.
Table groups present
their ranking with
the whole group.
Teachers engage in a
discussion about
the different
rankings.
Teachers reflect on
the data presented.
Teachers review the
agenda and have an
opportunity to ask
for clarification or
elaboration on any
items.
Teachers revisit their
personal learning
objectives.
114
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
expected strategies
from kindergarten,
first, and second
grades.
and percentage
viable strategy.
Instructor provides an
overview of the
module and
reminds teachers
that the key slides
are available in the
handout.
Prerequisite
knowledge
5 Define problem
types, student
strategies, and
levels of student
strategies.
Check knowledge of
definitions through
a sorting activity.
Instructor explains
definitions.
Instructor facilitates
sorting activity to
identify gaps in
knowledge.
Instructor provides
feedback during the
activity.
Instructor prompts
teachers to ask
clarification
questions.
Instructor addresses
gaps in knowledge.
Teachers may record
notes on the
definitions
presented by the
instructor.
Teachers engage in
the sorting activity
by sorting cards
with examples and
non-examples.
Teachers receive
feedback on
performance.
Teachers reflect on
how these terms
might inform their
practice.
Teachers work with
their table group to
generate one
additional example
and non-example to
share with the
group.
Teachers have an
opportunity to ask
115
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
clarifying
questions.
Learning
guidance:
learning
context
• Lecture
• Demo
10 Examine the CCSS-
M documents: the
standards by grade
level and the
appendix that
contains the
problem type
framework starting
with kindergarten.
Identify standards
that reference word
problems.
Demonstrate the
process of
identifying
standards that
reference word
problems.
Model with the
standards in
kindergarten.
Highlight those
standards that
mention or refer to
word problems.
Model the color-
coding process by
highlighting the
problem types
kindergarteners are
expected to learn in
the appendix with
one color.
As the demonstration
proceeds, teachers
review the
standards with
references to word
problems in
kindergarten.
Teachers make notes
on the words or
phrases that
signaled a
reference to word
problems.
Teachers highlight
the kindergarten
standards in their
copy of the
standards.
Teachers reflect on
the problem types
kindergarteners are
expected to learn.
Teachers have an
opportunity to ask
clarifying questions
as instructor
monitors and
checks for
understanding.
Practice and
feedback:
The
learning
context
17 Proceed to examine
first-grade
standards in the
same way as was
demonstrated with
Instructor asks
teachers to examine
first-grade
standards.
Teachers engage in
the process for
examining first-
grade standards.
116
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
the kindergarten
standards.
Repeat the process
for the second-
grade standards.
Prompt teachers to
highlight
appropriate
standards that
mention or refer to
word problems.
Ask teachers to use a
different color
highlighter to mark
the problem type
chart in the
appendix that
matches first-grade
expectations.
Provide feedback
about progress and
give teachers
opportunities to ask
clarifying
questions.
Repeat the process
for the second-
grade standards.
Teachers highlight
the appropriate
standards in first
grade using a
different color than
they did for
kindergarten.
Then, repeat the
process for the
second-grade
standards.
Teachers reflect on
surprises or
realizations they
have from
examining grade
level expectations
about word
problems.
Teachers will discuss
reflections at their
table.
Authentic
assessment
10
Describe the
progression of the
word problems that
students are
expected to learn.
Instructor asks
teachers to
categorize the
problem types by
the grade level
expectations.
Instructor prompts
teachers to ask
clarifying questions
about ambiguous
concepts.
Teachers demonstrate
their knowledge of
the problem type
expectations at
each grade level.
Teachers may ask
questions or seek
assistance as
necessary.
Retention and
transfer
3 Learners reflect on
the objectives for
the lesson and
Ask teachers to
consider the
implications of this
Teachers reflect on
the objectives for
this module and the
117
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
identify and report
on what they have
learned.
knowledge on math
instruction in their
classrooms.
Ask teachers to state
the problem types
that they need to
focus on in their
classrooms.
Tell teachers that they
will examine
specific problems
in their curriculum
after they have
learned about
specific problem
types.
implications for
their classroom.
Teachers identify
problem types they
will focus on in
their classrooms
Big ideas
3 The expectations
about word
problems are
delineated in the
CCSS-M at each
grade level.
Ask teachers to write
what they have
learned about the
grade level
expectations about
word problems on a
post it.
Prompt teachers to
record questions
they may have on a
post it so that the
instructor may
address them.
Teachers identify one
thing they learned
and any questions
or concerns they
have.
Teachers may post
what they learned
(“Gots”) and their
questions
(“Needs”) on the
chart.
Advance
organizer
for the next
module
2 Refer to visual
representation of
the modules to
show the module
that was completed
and the topic of the
module that will
follow.
Ask teachers to voice
any questions that
are lingering.
Given the topic of the
next module,
“Designing the
Classroom Space,”
Teachers ask
lingering questions.
Teachers consider
questions they
might have
concerning the
118
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
ask teachers what
they are wondering
and what questions
might have been
triggered by the
topic.
Instructor will record
questions on a
chart.
design of the
classroom.
Total time 90
Module 1b, Designing the Classroom Space
Figure A2
Module 1b, Designing the Classroom Space
Module Duration: approximately 90 minutes synchronous engagement
Introduction:
This is the second part of the first module of an 8-module course to teach teachers how to implement cognitively guided instruction
(CGI) in their classrooms. Throughout the course, teachers will learn about the different problem types and strategies that students use.
They will also learn about a process for strategically planning lessons and robust mathematical discussions. The purpose of the second
part of the first module is to familiarize teachers with the characteristics and practices of a CGI classroom.
Module 1a:
Overview
Module 1b:
Designing the
Classroom
Space
Module 2:
Structure of
the Problems
Module 3:
Noticing
Strategies
(Join)
Module 4:
Noticing
Strategies
(Separate)
Module 5:
Structure of
Multiplication
& Division
Problems
Module 6:
Multi-Digit
Addition
Problems
Module 7:
Multi-Digit
Subtraction
Problems
Module 8:
Base-Ten
Number
Concepts
119
120
Learning Objective(s)
Terminal Objective:
● Given the background information of CGI and the Common Core State Standards,
teachers will identify characteristics and practices of a CGI classroom.
Enabling Objective(s):
● For a given grade level, teachers will:
○ Know the meaning of unpacking problems
○ Know the meaning of eliciting student thinking
○ Identify characteristics and practices of a CGI classroom
Learning Activities
● After introductions and attention activities and learning objectives, assess prior
knowledge of the characteristics and practices of a CGI classroom.
● Teach any necessary prerequisite knowledge by providing definitions and examples and
nonexamples.
● Provide opportunities for learners to generate their own examples and nonexamples.
● Model the procedure for identifying the characteristics and practices of a CGI classroom.
● Provide practice and feedback on identifying characteristics and practices of a CGI
classroom.
● Provide opportunities to transfer knowledge to the teachers’ own classroom practice.
Summative Assessment
● Given the various examples of CGI classrooms, teachers will identify the characteristics
and practices of CGI classrooms.
Lesson Materials
● Video clip from YouTube: “Hillbilly Math!”
https://www.youtube.com/watch?v=MfgX0fyNeLc
● Video clips from Children’s Mathematics: Cognitively Guided Instruction
● Presentation slides
● Handout with key slides
● Sorting activity (Unpacking Problems and Eliciting Student Thinking)
● Handout with “Lizzie Beetle Problem”
● Handout with a signal light on it (reflection activity)
● Blank chart paper
● Charts/Posters
○ Agenda: Day 1
○ Learning Objectives for Module 1b
○ Burning Questions
○ Principles for Unpacking a Problem
○ Principles for Eliciting Student Thinking
○ Signal Light Poster
121
Learner Characteristic Accommodations
Learners will complete this module in person. Teachers will sit in table groups to support small
group conversations and collaboration. Slides will be projected so that all may access the
materials. Teachers will also receive handouts with key slides on them to facilitate listening and
note taking. If needed, teachers may request further accommodations.
Facilitator’s Notes
This opening session of the course will present the background information about CGI. It is often
reassuring to know that this way of teaching math is not a fad. In fact, it is grounded in research
over the last 40 years and the research is continuing today. Teachers will also examine the Math
Common Core State Standards and find that problem types are weaved into the standards. Thus,
a change is warranted. This second part of module 1 begins to examine the characteristics of a
CGI classroom focusing specifically on unpacking problems and eliciting student thinking.
Throughout the course, teachers will have opportunities to observe children, classrooms, and
discuss ideas with colleagues. All sessions will be synchronous and face-to-face. Teachers will
be asked to present problems to their class and return to subsequent sessions with student work.
Table A2 provides the instructional activities and details the instructional sequence, duration,
descriptions of the learning activities, instructor actions/decisions (supplantive events of
instruction) and learner actions/decisions (generative events of instruction).
122
Table A2
Instructional Activities, Module 1b
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner
action/decision
(generative)
Gain attention 10
Watch “Hillbilly
Math!” clip and
discuss reactions.
https://www.youtube.
com/watch?v=Mfg
X0fyNeLc
Instructor asks
teachers to discuss
their thoughts and
reactions to the
video.
Instructor facilitates a
brief discussion
about what may
have been
surprising (or not)
about the video
clip.
Teachers discuss their
thoughts and
reactions with their
table group.
Learning
objectives
5 The learning
objectives for the
module are posted
at the beginning of
the course content.
In order to establish
relevance and to
facilitate learning,
ask learners to read
the terminal and
enabling learning
objectives.
Ask teachers to write
their personal
learning objectives
on a post-it and add
it to the chart.
Teachers read the
terminal and
enabling learning
objectives for
themselves.
Teachers will post
their personal
learning objectives
on the chart.
Purpose for
learning
• Benefits
• Risks
3 Benefits:
• It is important to
explicitly focus on
certain
characteristics and
practices of a CGI
classroom.
• This will support
teachers who are
attempting to
Ask teachers to
discuss challenges
they have
witnessed in
traditional math
classrooms.
Ask teachers to
discuss the areas
they would like to
Teachers participate
in conversations at
their table and
reflect on their
practice and areas
for
growth/learning.
Teachers may record
burning questions
123
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner
action/decision
(generative)
transfer what they
have learned to
their classroom
practice.
Risks:
A teacher who does
not understand the
characteristics and
practices in a CGI
classroom may
flounder and
struggle to know
how to implement
this new learning
and may, in fact,
not implement at
all.
learn more about
with regards to
teaching with word
problems.
Invite teachers to
record any burning
questions on a
post-it and add it to
the “Burning
Questions” chart.
The instructor will
peruse the
questions and make
notes to address
these throughout
the sessions.
on a post-it and add
it to the chart.
Overview
• Prior
knowledge
• New
knowledge
• Learning
strategies
(What you
already
know, what
you are
going to
learn, and
how you are
going to
learn it.)
12
Activate prior
knowledge by
asking teachers to
describe a
traditional math
classroom.
Overview of the
agenda for the
session:
• Identify and
examine
characteristics and
practices of a CGI
classroom:
“Principles for
Unpacking
Problems” and
“Principles for
Eliciting Student
Thinking”
Instructor asks
teachers to
generate a
description of a
traditional math
classroom with
other teachers at
their table.
Instructor synthesizes
and captures main
points from each
table on a chart.
Instructor provides an
overview of the
module and
reminds teachers
that the key slides
are available in the
handout.
Teachers reflect on
their impressions
and descriptions of
a traditional math
classroom.
Teachers share ideas
with their table
group and prepare
to share out with
the whole group.
Teachers review the
agenda items and
have an opportunity
to ask for
clarification or
elaboration on any
items.
124
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner
action/decision
(generative)
Prerequisite
knowledge
14 Present and explain
“Principles for
Unpacking
Problems” and
“Principles for
“Eliciting Student
Thinking.”
Check knowledge of
the principles
through a sorting
activity.
Instructor presents
and explains the
“Principles for
Unpacking
Problems” and
“Principles for
“Eliciting Student
Thinking.”
Instructor facilitates
sorting activity.
Instructor provides
feedback during
the activity.
Instructor prompts
teachers to ask
clarifying
questions.
Instructor identifies
gaps in knowledge
through the sorting
activity.
Teachers may record
notes about the two
charts: “Principles
for Unpacking a
Problem” and
“Principles for
Eliciting Student
Thinking.”
Teachers engage in a
sorting activity
with examples and
non-examples.
Teachers will receive
feedback on
performance.
Teachers reflect on
how these
principles might
inform their
practice.
Table groups will
have an opportunity
to share something
that they learned
from this activity.
Teachers have an
opportunity to ask
clarifying
questions.
Learning
guidance:
learning
context
• Lecture
• Demo
18 Solve a non-
traditional problem
to experience a
lesson and
discussion.
Demonstrate the
process of posing a
word problem.
As the demonstration
lesson proceeds,
the instructor will
Teachers review the
“Principles for
Unpacking a
Problem” and the
“Principles for
Eliciting Student
Thinking.”
125
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner
action/decision
(generative)
highlight actions
that exemplify the
“Principles for
Unpacking a
Problem” and the
“Principles for
Eliciting Student
Thinking” (in
which case, the
teachers are the
“students”).
Facilitate a
discussion about
the problem and
the different
strategies that
teachers used.
Ask teachers to
discuss the benefits
of this lesson
format.
Prompt teachers to
discuss what they
learned with their
table group.
Instructor addresses
questions teachers
may have.
Teachers solve the
given problem.
Teachers participate
in the conversation
around the variety
of strategies that
different teachers
used.
Teachers reflect on
this lesson.
Teachers discuss
what they learned
about problem
solving lessons
with their table
group. They may
consider the
benefits of this
lesson format and
make note of
questions that
might have arisen.
Teachers have an
opportunity to ask
clarifying
questions.
Practice and
feedback:
The
learning
context
10 Watch a video clip of
a CGI teacher who
poses a problem to
her class to identify
“Principles for
Unpacking a
Problem” and
“Principles for
Eliciting Student
Show a video of a
CGI teacher who
poses a problem to
her students.
Instructor facilitates a
conversation about
the teacher’s
actions.
Teachers watch the
video clip to
extrapolate
evidence of the
“Principles for
Unpacking a
Problem” and
“Principles for
126
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner
action/decision
(generative)
Thinking” in the
video clip.
Prompt teachers to
identify the
“Principles for
Unpacking a
Problem” and
“Principles for
Eliciting Student
Thinking.”
Instructor facilitates a
debriefing
conversation about
what teachers
noticed in the
video.
Provide feedback
about progress and
give teachers
opportunities to ask
clarifying
questions.
Eliciting Student
Thinking.”
Teachers discuss
what they noticed
in the video with
their table group.
Table groups report
what they noticed
during the whole
group conversation.
Teachers have the
opportunity to ask
clarifying
questions.
Authentic
assessment
10
Distinguish
traditional and CGI
classroom practice
by calling out the
“Principles for
Unpacking a
Problem” and
“Principles for
Eliciting Student
Thinking.”
Instructor asks
teachers to make a
chart listing
practices in a
traditional
classroom and a
CGI classroom.
Instructor asks
teachers to focus
on the “Principles
for Unpacking a
Problem” and
“Principles for
Eliciting Student
Thinking.”
Teachers will
demonstrate their
knowledge of the
characteristics and
practices in a CGI
classroom by
identifying the
“Principles for
Unpacking a
Problem” and
“Principles for
Eliciting Student
Thinking.”
Teachers may ask
questions or seek
assistance as
necessary.
127
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner
action/decision
(generative)
Teachers may ask
clarifying
questions about
ambiguous
concepts.
Retention and
transfer
3 Learners reflect on
the objectives for
the lesson and
identify and report
on what they have
learned.
Ask teachers to
consider the
implications of this
knowledge on math
instruction in their
classrooms.
Prompt teachers to
reflect on their
practice and
implications of this
learning on their
practice.
Teachers reflect on
the objectives for
this module and the
implications for
their classroom.
Big ideas
3 The characteristics
and practices in a
CGI classroom are
presented.
Ask teachers to state
what they have
learned about the
characteristics and
practices in a CGI
classroom.
Ask teachers to
reflect on their
practice using the
signal light
reflection sheet.
Ask teachers to state
an area that they
would like to focus
on shifting.
Prompt teachers to
ask any questions
they may have so
the instructor may
address them.
Teachers state what
they have learned
about
characteristics and
practices in a CGI
classroom.
Teachers reflect on
their practice and
record their
reflections on the
reflection sheet
(with the signal
light):
- What will I stop
doing? (RED)
- What do I need to
consider and think
about more?
(YELLOW)
- What will I start
doing when I return
128
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner
action/decision
(generative)
to the classroom?
(GREEN)
Teachers share an
area they would
like to shift.
Teachers may ask
lingering questions.
Advance
organizer
for the next
module
2 Refer to visual
representation of
the modules to
show the module
that was completed
and the topic of the
module that will
follow.
Given the topic of the
next module,
“Structure of the
Problems,” ask
teachers what they
are wondering and
what questions
might have been
triggered by the
topic.
Instructor will record
questions on a
chart.
Teachers consider
questions they
might have
concerning the
structure of word
problems.
Total time 90
Module 2, Structure of the Problems
Figure A3
Module 2, Structure of the Problems
Module Duration: approximately 180 minutes synchronous engagement
Introduction:
This is the second module of an 8-module course to teach teachers how to implement cognitively guided instruction (CGI) in their
classrooms. Throughout the course, teachers will learn about the different problem types and strategies that students use. They will
also learn about a process for strategically planning lessons and robust mathematical discussions. The purpose of the second module is
to teach teachers the structural and distinguishing features of the problem types.
Learning Objective(s)
Terminal Objective:
● Given the background information of CGI and the Common Core State Standards, teachers will need to identify and
distinguish the structural features of three types of join problems, three types of separate problems, three types of part-part
Module 1a:
Overview
Module 1b:
Designing the
Classroom
Space
Module 2:
Structure of
the Problems
Module 3:
Noticing
Strategies
(Join)
Module 4:
Noticing
Strategies
(Separate)
Module 5:
Structure of
Multiplication
& Division
Problems
Module 6:
Multi-Digit
Addition
Problems
Module 7:
Multi-Digit
Subtraction
Problems
Module 8:
Base-Ten
Number
Concepts
129
130
whole problems and three types of compare problems and write problems that are
culturally responsive to their students’ lived experiences.
Enabling Objective(s):
● For a given grade level, teachers will:
○ Know the meaning of join problems (add to result unknown, add to change
unknown, add to start unknown)
○ Know the meaning of separate problems (take from result unknown, take from
change unknown, take from start unknown)
○ Know the meaning of part-part whole problems (put together/take apart - total
unknown, put together/take apart - addend unknown, put together/take apart -
both addends unknown)
○ Know the meaning of compare problems (compare difference unknown, compare
bigger unknown, compare smaller unknown)
○ Know the meaning of action problems
○ Know the meaning of non-action problems
○ Identify and distinguish structural features of the different problem types
○ Create (write) problems that respond to students’ needs (are culturally responsive)
Learning Activities
● After introductions and attention activities and learning objectives, assess prior
knowledge of the structural features and categories of the different problem types.
● Teach any necessary prerequisite knowledge by providing definitions and examples and
nonexamples.
● Provide opportunities for learners to generate their own examples and nonexamples.
● Model the procedure for identifying the structural features and categories of the different
problem types.
● Provide practice and feedback on identifying structural features and categories of the
different problem types.
● Provide opportunities to transfer knowledge to the teachers’ own classroom practice.
Summative Assessment
● Given the example problems on the problem type chart, teachers will identify and
distinguish the structural features and categories of the different problem types.
Lesson Materials
● Presentation slides
● Sorting activity (definitions of terms)
● Handout with key slides
● Written resource: Appendix for Math Common Core State Standards (from Module 1a)
● Written resource: Problem Type Chart from the Appendix (CCSS-M) (from Module 1a)
● Handout with blank problem type chart
● Handout with examples of problem types (to cut out)
● Scissors and glue (from participant boxes)
● Handout with reflection prompts (square, circle, exclamation point)
131
● One computer with internet access per table (link to google doc with the blank problem
type template on it)
● Blank chart paper
● Charts/Posters
○ Agenda: Day 1
○ Learning Goals for Module 2
○ Six Research Problems (chart with data from last module)
Learner Characteristic Accommodations
Learners will complete this module in person. Teachers will sit in table groups to support small
group conversations and collaboration. Slides will be projected so that all may access the
materials. Teachers will also receive handouts with key slides on them to facilitate listening and
note taking. If needed, teachers may request further accommodations.
Facilitator’s Notes
This opening session of the course will present the background information about CGI. It is often
reassuring to know that this way of teaching math is not a fad. In fact, it is grounded in research
over the last 40 years and the research is continuing today. Teachers will also examine the Math
Common Core State Standards and find that problem types are weaved into the standards. Thus,
a change is warranted. This second part of module 1 begins to examine the characteristics of a
CGI classroom focusing specifically on unpacking problems and eliciting student thinking.
Throughout the course, teachers will have opportunities to observe children, classrooms, and
discuss ideas with colleagues. All sessions will be synchronous and face-to-face. Teachers will
be asked to present problems to their class and return to subsequent sessions with student work.
Table A3 provides the instructional activities and details the instructional sequence, duration,
descriptions of the learning activities, instructor actions/decisions (supplantive events of
instruction) and learner actions/decisions (generative events of instruction).
132
Table A3
Instructional Activities, Module 2
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Gain
attention
10
Reflect on the data
from the “Six
Research
Questions.”
Instructor asks
teachers to consider
the problems from the
“Six Research
Questions.”
Instructor prompts
teachers to discuss
what might have made
these problems
accessible to
kindergarteners.
Instructor asks
teachers to name
features of problems
that might have
facilitated access.
Instructor asks table
groups to report their
findings.
Instructor records
findings on a chart.
Teachers discuss
features of the
problems that might
have facilitated
access.
Teachers report their
findings to the large
group.
Learning
objectives
5 The learning
objectives for the
module are
posted at the
beginning of the
course content.
In order to establish
relevance and to
facilitate learning, ask
learners to read the
terminal and enabling
learning objectives.
Ask teachers to
discuss their learning
objectives with a
partner at their table.
Teachers read the
terminal and
enabling learning
objectives for
themselves.
Teachers discuss their
personal learning
objectives with a
partner at their table.
133
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Instructor invites
teachers to comment
on or ask clarifying
questions about the
learning objectives for
this module.
Teachers may
comment on or ask
clarifying questions
about the learning
objectives.
Purpose for
learning
• Benefits
• Risks
3 Benefits:
• It is important to
note the structural
features of and
categories of
word problems.
• Knowing which
features and
categories make
problems easier
or difficult for
children will help
the teacher meet
the students’
needs.
Risks:
• A teacher who
does not
understand the
structural features
of word problems
may focus on
cues that
undermine
comprehension.
• A teacher who
does not
understand the
structural features
of word problems
may inadvertently
change the type
of problem when
intending to
Ask teachers to
discuss challenges
they have faced when
teaching word
problems.
Ask table groups to
generate and share
their top three
challenges when
teaching word
problems.
Instructor will record
challenges on a chart
paper to make sure to
address those
challenges throughout
the sessions.
Teachers participate in
conversations at the
tables.
Teachers discuss
challenges they have
encountered when
teaching word
problems.
Teachers distill their
list of challenges to
their top three to
share with the group.
134
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
simplify
linguistic
demands of the
problem.
Overview
• Prior
knowledge
• New
knowledge
• Learning
strategies
(What you
already
know,
what you
are going
to learn,
and how
you are
going to
learn it.)
12
Activate prior
knowledge by
asking teachers to
describe
problems that
children struggle
with and find
challenging.
Overview of the
agenda for the
module:
• Examine the
structural features
of four categories
of problem types:
join, separate,
part-part whole,
and compare.
• Demonstrate how
to categorize
problems as
action or non-
action.
• Demonstrate how
to write any of
the 12 problem
types presented
thus far.
Instructor asks
teachers to describe
problems that children
find challenging with
other teachers at their
table.
Instructor synthesizes
and captures main
points from each table
on a chart.
Instructor provides an
overview of the
module and reminds
teachers that the key
slides are available in
the handout.
Teachers reflect on
their impressions
and descriptions of
problems that are
challenging for
children.
Table groups share out
with the whole
group.
Teachers review the
agenda items and
have an opportunity
to ask for
clarification or
elaboration on any
items.
Prerequisite
knowledge
20 Define join,
separate, part-part
whole, and
compare problem
types.
Instructor explains
definitions.
Instructor facilitates
sorting activity to
identify gaps in
knowledge.
Teachers may take
notes of definitions
presented by the
instructor.
Teachers engage in the
sorting activity by
sorting cards with
135
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Define action
versus non-action
problems.
Check knowledge
of definitions
through a sorting
activity.
Instructor provides
feedback during the
activity.
Instructor prompts
teachers to ask
clarification questions.
Instructor addresses
gaps in knowledge.
examples and non-
examples.
Teachers receive
feedback on
performance.
Teachers reflect on
how these categories
might inform how
they approach
teaching them.
Teachers will generate
an additional
example and non-
example to share
with the group.
Learning
guidance:
learning
context
• Lecture
• Demo
45 “Construct” the
problem type
chart by
comparing and
contrasting the
structural features
of different
problems:
• Action versus
non-action
• Equation that
matches the
problem
Demonstrate the
process of examining
the structural features
of a problem by
comparing and
contrasting three
problems at a time.
Instructor models by
placing the problems
on a blank problem
type chart.
Instructor
distinguishes action
versus non-action
problems on the chart.
Instructor
demonstrates how to
generate the equation
that matches each
Teachers refer to
problem type chart
from the Appendix
in the CCSS-M.
Teachers cut out
examples of problem
types and glue onto
the blank problem
type chart as the
instructor models it.
Teachers record notes
on their copy of the
problem type chart.
Teachers engage in the
discussion to
compare and
contrast the given
problems.
136
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
problem and records it
under the problems.
Instructor asks
teachers to voice
questions and make
comments about the
structure of the
problems or the
categorization process.
Teachers make notes
of structural features
of the problems.
Teachers participate in
discussions about
problems that are
presented.
Teachers share what
they learned about
the structural
features of problems
with the group.
Teachers have an
opportunity to ask
clarifying questions
as instructor
monitors and checks
for understanding.
Practice and
feedback:
The
learning
context
45 Write new problems
for each problem
type.
Instructor reminds
teachers of the
problem type chart
from the Appendix
and the one they
constructed in the last
module (both are
worked examples).
Instructor asks table
groups to write their
own problems on the
blank problem type
chart in the google
doc.
Instructor will monitor
and provide feedback
on group work through
the google doc and
Teachers work
collaboratively to
write problems on
the blank problem
type chart in the
shared google doc.
Table groups report
what they noticed
during the whole
group conversation.
Teachers have the
opportunity to ask
clarifying questions.
137
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
while walking around
the room.
Authentic
assessment
13
Describe and
distinguish the
structural features
and categories of
problem types.
Instructor asks
teachers to make a
chart of action versus
non-action problems.
Instructor asks
teachers to describe
the category (action
versus non-action), list
the problem types for
each category, and one
example problem
under each category.
Instructor encourages
teachers to ask
clarifying questions
about ambiguous
concepts.
Teachers demonstrate
their knowledge by
creating a chart of
the problem types,
sorted by action
versus non-action
problems.
Teachers may ask
questions or seek
assistance as
necessary.
Retention
and
transfer
10 Learners reflect on
the objectives for
the lesson and
identify and
report on what
they have
learned.
Ask teachers to
consider the structural
features of the
different problem
types and the
expectations for their
grade level.
Prompt teachers to
reflect on the
implications of
knowing the structural
features of problems
on their practice.
Ask teachers to
determine problems
that they will try with
their students.
Teachers reflect on the
objectives for this
module and the
implications for their
classroom.
Teachers determine
problems that they
would like to try
with their students.
138
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Big ideas
5 The structural
features of
problems support
students when
solving word
problems.
Ask teachers to state
what they have learned
about structural
features of problems
and to ask any
questions they may
have so the instructor
may address them.
Ask teachers to reflect
on their practice using
the prompts on the
graphic organizer
(square, circle, and
exclamation point.
Teachers state what
they have learned
about structural
features of problems.
Teachers reflect on
their practice and
record their
reflections on the
reflection sheet (with
the square, circle,
triangle):
- This idea fits my
thinking (It resonates
with me).
(SQUARE)
- These ideas are
circling in my mind
(I’m still thinking
about it). (CIRCLE)
- This idea important
(EXCLAMATION
POINT)
Advance
organizer
for the
next
module
2 Refer to visual
representation of
the modules to
show the module
that was
completed and
the topic of the
module that will
follow.
Ask teachers to voice
any questions that are
lingering.
Given the topic of the
next module,
“Structure of the
Problems,” ask
teachers what they are
wondering and what
questions might have
been triggered by the
topic.
Instructor will record
questions on a chart.
Teachers ask lingering
questions.
Teachers consider
questions they might
have concerning the
structure of word
problems.
Total time 180
Module 3, Noticing Strategies (Join)
Figure A4
Module 3, Noticing Strategies (Join)
Module Duration: approximately 180 minutes synchronous engagement
Introduction:
This is the third module of an 8-module course to teach teachers how to implement cognitively guided instruction (CGI) in their
classrooms. Throughout the course, teachers will learn about the different problem types and strategies that students use. They will
also learn about a process for strategically planning lessons and robust mathematical discussions. The purpose of the third module is to
teach teachers the strategies children use when solving Join problems and a planning process to facilitate implementation.
Learning Objective(s)
Terminal Objective:
● Given the background information of CGI and the Common Core State Standards, teachers will identify and analyze strategies
that children use when solving Join problems, write problems and create culturally responsive lesson plans to implement in
their classroom per the rubric in the lesson template.
Module 1a:
Overview
Module 1b:
Designing the
Classroom
Space
Module 2:
Structure of
the Problems
Module 3:
Noticing
Strategies
(Join)
Module 4:
Noticing
Strategies
(Separate)
Module 5:
Structure of
Multiplication
& Division
Problems
Module 6:
Multi-Digit
Addition
Problems
Module 7:
Multi-Digit
Subtraction
Problems
Module 8:
Base-Ten
Number
Concepts
139
140
Enabling Objective(s):
● For a given grade level, teachers will:
○ Know the meaning of direct modeling strategies for Join problems
○ Know the meaning of counting strategies for Join problems
○ Know the meaning of number facts (derived fact) strategies for Join problems
○ Know the meaning of flexible choice of strategies
○ Know the meaning of anticipating
○ Know the meaning of monitoring
○ Know the meaning of selecting
○ Know the meaning of sequencing
○ Know the meaning of connecting
○ Identify and analyze strategies that children use to solve Join problems
○ Synthesize strategies that students are using to create (write) problems and lessons
that respond to students’ lived experiences
○ Design lessons using the 5-step planning template for a Join problem
Learning Activities
● After introductions and attention activities and learning objectives, assess prior
knowledge of the strategies that children use to solve Join problems.
● Teach any necessary prerequisite knowledge by providing definitions and examples and
nonexamples.
● Provide opportunities for learners to generate their own examples and nonexamples.
● Model the procedure for identifying the strategies that children use to solve Join
problems.
● Provide practice and feedback on identifying strategies that children use to solve Join
problems.
● Model the procedure for planning a lesson.
● Provide practice and feedback for planning a lesson.
● Provide opportunities to transfer knowledge to the teachers’ own classroom practice.
Summative Assessment
● Given the various videos and examples of student work, teachers will identify and
analyze the strategies that children use to solve Join problems and will write problems
and create lesson plans to implement in their classroom per the rubric in the lesson
template.
Lesson Materials
● Written resource: Appendix for Math Common Core State Standards (from Module 1a)
● Video clips from Children’s Mathematics: Cognitively guided instruction
● Presentation slides
● Handout with key slides
● Access to Kahoot site on internet (via phone or a computer)
● Handout with “Student Strategies”
● Handout with “Planning Reflection”
● Handout with “Children’s Solution Strategies”
141
● Handout with “Lesson Planning Template”
● Blank chart paper
● Charts/Posters
○ Agenda: Day 2
○ Learning Objectives for Module 3
○ Burning Questions
Learner Characteristic Accommodations
Learners will complete this module in person. Teachers will sit in table groups to support small
group conversations and collaboration. Slides will be projected so that all may access the
materials. Teachers will also receive handouts with key slides on them to facilitate listening and
note taking. If needed, teachers may request further accommodations.
Facilitator’s Notes
This opening session of the course will present the background information about CGI. It is often
reassuring to know that this way of teaching math is not a fad. In fact, it is grounded in research
over the last 40 years and the research is continuing today. Teachers will also examine the Math
Common Core State Standards and find that problem types are weaved into the standards. Thus,
a change is warranted. This second part of module 1 begins to examine the characteristics of a
CGI classroom focusing specifically on unpacking problems and eliciting student thinking.
Throughout the course, teachers will have opportunities to observe children, classrooms, and
discuss ideas with colleagues. All sessions will be synchronous and face-to-face. Teachers will
be asked to present problems to their class and return to subsequent sessions with student work.
Table A4 provides the instructional activities and details the instructional sequence, duration,
descriptions of the learning activities, instructor actions/decisions (supplantive events of
instruction) and learner actions/decisions (generative events of instruction).
142
Table A4
Instructional Activities, Module 3
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Gain
attention
7
Discuss the
differences
between how
children are
typically taught to
solve this problem
and how the
student in the video
clip (3.1) solves it.
Instructor asks
teachers to state
how students are
typically taught to
solve this problem.
Instructor shows
video clip (3.1).
http://fast.wistia.net
/embed/iframe/fces
d96yww?popover=t
rue&ls=1
Instructor facilitates a
brief discussion
about how the
student in the video
clip solves the
problem and how it
may differ from
how children are
typically taught to
solve this problem.
Teachers discuss how
students are
typically taught to
solve this problem:
Camilla has 7 dollars.
She wants to buy a
book that costs 11
dollars. How much
more does she need
to save so that she
can buy the book?
Teachers watch the
video clip.
Teachers discuss their
thoughts and
reactions to the
video clip with
their table group.
Teachers engage in a
discussion about
the differences
between how this
problem is typically
taught and how this
student solves it.
Learning
objectives
4 The learning
objectives for the
module are posted
at the beginning of
the course content.
In order to establish
relevance and to
facilitate learning,
ask teachers to read
the terminal and
enabling learning
objectives.
Teachers read the
terminal and
enabling learning
objectives for
themselves.
Teachers will share
their personal
143
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Ask teachers to share
their personal
learning objectives
with a partner.
learning objectives
with a partner.
Purpose for
learning
• Benefits
• Risks
3 Benefits:
• It is important to
know the strategies
children use to
solve Join
problems.
• This will support
teachers to
understand the
progression of
strategies that
children use as
they teach word
problems in their
classroom.
• Learning a
systematic method
for planning
lessons supports
teachers to be
prepared to teach
problem solving
lessons.
Risks:
A teacher who does
not understand how
children solve Join
problems may
inadvertently
confuse students
and be confused by
students when
teaching word
problems.
Instructor asks
teachers to discuss
challenges they
have witnessed
while teaching
word problems.
Instructor asks
teachers to discuss
the areas they
would like to learn
more about with
regards to teaching
with word
problems.
Instructor invites
teachers to record
any burning
questions on a post-
it and add it to the
“Burning
Questions” chart.
The instructor will
peruse the
questions and make
notes to address
these throughout
the sessions.
Teachers participate
in conversations at
their table and
reflect on their
experiences,
practice, and areas
for
growth/learning.
Teachers may record
burning questions
on a post-it and add
it to the chart.
144
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Overview
• Prior
knowledge
• New
Knowledge
• learning
strategies
(What you
already
know,
what you
are going
to learn,
and how
you are
going to
learn it.)
9 Activate prior
knowledge by
asking teachers to
review the “Add
To” (Join)
problems on the
problem type chart
from the Appendix
in the CCSS-M.
Provide brief
overview of the
module:
• Examine strategies
that students use by
watching videos of
students who use a
variety of strategies
to solve Join
problems.
• Demonstrate some
important features
when taking notes
on student
strategies.
• Examine a five-
step planning
template to use
when planning
lessons.
• Demonstrate how
to plan a lesson for
a Join problem that
starts with
strategies students
currently use and
helps them
progress to more
sophisticated
strategies.
Instructor prompts
teachers to review
the “Add To”
(Join) problems on
the problem type
chart from the
Appendix in the
CCSS-M.
Instructor provides an
overview of the
module and
reminds teachers
that the key slides
are available in the
handout
Instructor prompts
teachers to
articulate what they
will focus on
during this module.
Teachers review the
“Add To” (Join)
problems from the
problem type chart.
Teachers consider the
components of the
module and discuss
their learning focus
for this module at
their table.
Teachers have an
opportunity to ask
for clarification or
elaboration on any
items.
145
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Prerequisite
knowledge
13 Present the definition
of the different
levels of student
strategies for Join
problems.
Check knowledge of
examples and non-
examples through a
Kahoot activity.
Instructor provides
definitions of levels
of student
strategies.
Instructor identifies
gaps in knowledge
through the Kahoot
activity which
checks the level of
the student strategy
and asks teachers to
distinguish between
an example and a
non-example.
Instructor gives
teachers an
opportunity to ask
clarifying
questions.
Instructor addresses
any misconceptions
or questions.
Teachers engage in
the Kahoot activity.
Teachers answer
questions asking
them to identify the
level of the student
strategy. Additional
questions will focus
on whether the
strategy is an
example or non-
example of a
particular level of
student strategy.
Teachers will receive
feedback from
Kahoot and the
instructor.
Teachers reflect on
how these terms
might inform their
practice.
Table groups will
have an opportunity
to share something
that they learned or
questions they
might have from
this activity.
Learning
guidance:
learning
context
• Lecture
• Demo
40 Observe students
solving various
Join problems
(video clips and
student work
samples).
Instructor
demonstrates what
is important to pay
attention to when
watching a student
solve a problem.
Teachers watch video
clips and examine
student work
samples of students
solving various
Join problems.
146
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Focus on notetaking
strategies while
watching video
clips and
examining student
work.
Instructor focuses
notetaking
strategies to record
what students are
doing, how they
solve the problem,
what tools they use
(physical or in their
mind), how they
count, what number
they start on to
count, and how to
ask clarifying (not
leading) questions.
Instructor prompts
teachers to record
notes on the
“Student
Strategies”
handout.
Instructor asks
teachers to discuss
the benefits and
importance of
notetaking.
Instructor asks table
groups to share
what they learned
with the whole
group.
Teachers record notes
on “Student
Strategies”
handout.
Teachers discuss the
purpose of and the
techniques for
taking notes while
observing students.
Teachers reflect on
what they learned
about the range of
strategies and
discuss how taking
notes may provide
evidence/data that
will inform
instruction.
Teachers have an
opportunity to ask
clarifying questions
as instructor
monitors and
checks for
understanding.
Practice and
feedback:
The
learning
context
20 Apply notetaking
skills while
watching
additional video
clips.
Instructor shows two
more video clips of
a student solving a
Join problem.
Instructor prompts
teachers to take
Just as with the
preceding
demonstration,
teachers watch the
video and take
notes.
147
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Display notes on a
poster for the
different problems.
Discuss similarities
and differences and
the connection to
classroom practice.
notes individually
and then discuss
their notes with
their table group.
Instructor asks table
groups to record
their notes on a
poster to display to
the group.
Facilitate a gallery
walk.
Instructor leads a
discussion about
the similarities and
differences
between the
posters.
Instructor leads a
discussion about
the implications
(benefits) and
connections of
notetaking to
classroom practice.
Instructor provides
feedback and gives
teachers
opportunities to ask
clarifying
questions.
Teachers discuss
what they noticed
in the video and
share their notes
with other table
members.
Teachers make a
poster with their
notes to share with
the whole group.
Teachers walk around
the room with their
table group and
discuss what they
see on the different
posters.
Teachers engage in a
discussion about
what they saw and
noticed.
Teachers have the
opportunity to ask
clarifying
questions.
Prerequisite
knowledge
13 Provide definitions
for each of the five
practices.
Instructor provides
definitions for each
of the five practices
for orchestrating
Teachers engage in
the Kahoot activity.
Teachers answer
questions asking
148
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Check knowledge of
examples and non-
examples through a
Kahoot activity.
mathematics
discussions.
Instructor identifies
gaps in knowledge
through the Kahoot
activity which will
check the level of
the planning
process and asks
teachers to
distinguish between
an example and a
non-example.
Instructor gives
teachers an
opportunity to ask
clarifying
questions.
them to identify the
step of the planning
process.
Teachers will receive
feedback from
Kahoot and the
instructor.
Teachers reflect on
how this planning
process might
support their
practice.
Table groups will
have an opportunity
to share something
that they learned or
questions they
might have from
this activity.
Learning
context
• Lecture
• Demo
25 Present and complete
the planning
template organizer
with the five
practices.
Instructor presents the
planning template
with the five
practices.
Instructor
demonstrates the
process of
completing the
planning template
organizer by
modeling with a
Join problem.
Teachers listen to the
descriptions of each
step of the planning
process.
Teachers will record
notes on their own
organizer so that
they have a worked
example.
Teachers have an
opportunity to ask
clarifying questions
as instructor
monitors and
checks for
understanding.
149
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Practice and
feedback:
The
learning
context
24 Continue to plan
another problem
using the planning
template.
Instructor asks table
groups to select one
of the problems
that they wrote
during Module 2.
Instructor asks table
groups to discuss
the problem and
complete the
planning organizer.
Instructor asks table
groups to post their
plan on a chart.
Instructor facilitates a
gallery walk
activity to examine
the planning
posters.
Instructor prompts
teachers to record
their reflections on
the graphic
organizer,
“Planning
Reflection,” under
the categories:
Notice and
Wonder.
Instructor facilitates a
discussion about
what the groups
noticed and records
themes on a chart.
Instructor provides
feedback about
Teachers select a
problem from one
of those they wrote
in Module 2.
Teachers collaborate
with their table
groups to complete
a planning template
for the problem
they selected.
Table groups record
their planning
process of a chart
to share with the
whole group.
Teachers circulate
through the room
with their graphic
organizer.
Teachers record their
reflections on the
graphic organizer.
Teachers engage in a
discussion about
what they saw and
noticed.
Teachers have an
opportunity to ask
clarifying
questions.
150
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
progress and give
teachers
opportunities to ask
clarifying
questions.
Authentic
assessment
14
Plan a lesson to
implement in the
classroom using
the planning
template.
Instructor asks
teachers to describe
the five steps of the
planning process.
Instructor asks
teachers to plan a
lesson using the
planning template
given a problem
and samples of
student work.
Teachers will
demonstrate their
knowledge of the
levels of strategies
for a given Join
problem.
Teachers will plan a
lesson given 5
student work
samples (to mirror
the monitoring
phase in the
classroom).
Teachers may ask
questions or seek
assistance as
necessary.
Retention
and
transfer
3 Learners reflect on
the objectives for
the lesson and
identify and report
on what they have
learned.
Instructor asks
teachers to consider
the implications of
this knowledge on
math instruction in
their classrooms.
Instructor prompts
teachers to reflect
on their practice
and the
implications of this
learning on their
practice.
Teachers reflect on
the objectives for
this module and the
implications for
their classroom.
151
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Big ideas
3 The level of
strategies for
solving Join
problems and the
five-step planning
process are
presented.
Instructor asks
teachers to state
what they have
learned about the
strategies students
use to solve Join
problems and the
five-step planning
process.
Instructor prompts
teachers to ask
questions they may
have so the
instructor may
address them.
Teachers reflect on
and state what they
have learned about
student strategies
for solving Join
problems and the
five steps for
planning lessons
with an elbow
partner.
Teachers finish the
sentence prompts:
“Something that left
an impression on
me during this
module was…” and
“One thing I am
wondering about
is…”
Teachers (volunteers)
share their
impressions and
wonders with the
whole group.
Advance
organizer
for the next
module
2 Refer to visual
representation of
the modules to
show the module
that was completed
and the topic of the
module that will
follow.
Instructor asks
teachers to voice
any questions that
are lingering.
Given the topic of the
next module,
“Noticing the
Strategies
(Separate),” ask
teachers what they
are wondering and
what questions
might have been
Teachers consider
questions they
might have
concerning the
strategies that
children use with
Join problems or
the planning
template.
152
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
triggered by the
topic.
Instructor will record
questions on a
chart.
Total time 180
Module 4, Noticing the Strategies (Separate)
Figure A5
Module 4, Noticing the Strategies (Separate)
Module Duration: approximately 180 minutes synchronous engagement
Introduction:
This is the fourth module of an 8-module course to teach teachers how to implement cognitively guided instruction (CGI) in their
classrooms. Throughout the course, teachers will learn about the different problem types and strategies that students use. They will
also learn about a process for strategically planning lessons and robust mathematical discussions. The purpose of the fourth module is
to teach teachers the strategies children use when solving Separate problems and a planning process to facilitate implementation.
Learning Objective(s)
Terminal Objective:
● Given the background information of CGI and the Common Core State Standards, teachers will identify and analyze strategies
that children use when solving Separate problems, write problems and create culturally responsive lesson plans to implement
in their classroom per the rubric in the lesson template.
Module 1a:
Overview
Module 1b:
Designing the
Classroom
Space
Module 2:
Structure of
the Problems
Module 3:
Noticing
Strategies
(Join)
Module 4:
Noticing
Strategies
(Separate)
Module 5:
Structure of
Multiplication
& Division
Problems
Module 6:
Multi-Digit
Addition
Problems
Module 7:
Multi-Digit
Subtraction
Problems
Module 8:
Base-Ten
Number
Concepts
153
154
Enabling Objective(s):
● For a given grade level, teachers will:
○ Know the meaning of direct modeling strategies for Separate problems
○ Know the meaning of counting strategies for Separate problems
○ Know the meaning of number facts (derived fact) strategies for Separate problems
○ Identify and analyze strategies that children use to solve Separate problems
○ Synthesize strategies that students are using to create (write) problems and lessons
that respond to students’ lived experiences
○ Design lessons using the 5-step planning template for a Separate problem
Learning Activities
● After introductions and attention activities and learning objectives, assess prior
knowledge of the strategies that children use to solve Separate problems.
● Teach any necessary prerequisite knowledge by providing definitions and examples and
nonexamples.
● Provide opportunities for learners to generate their own examples and nonexamples.
● Model the procedure for identifying the strategies that children use to solve Separate
problems.
● Provide practice and feedback on identifying strategies that children use to solve Separate
problems.
● Model the procedure for planning a lesson for a Separate problem.
● Provide practice and feedback for planning a lesson for a Separate problem.
● Provide opportunities to transfer knowledge to the teachers’ own classroom practice.
Summative Assessment
● Given the various videos and examples of student work, teachers will identify and
analyze the strategies that children use to solve Separate problems and will write
problems and create lesson plans to implement in their classroom per the rubric in the
lesson template.
Lesson Materials
● Written resource: Appendix for Math Common Core State Standards (from Module 1a)
● Video clips from Children’s Mathematics: Cognitively Guided Instruction
● Presentation slides
● Handout with key slides
● Sorting activity (five step lesson planning)
● Access to Kahoot site on internet (via phone or computer)
● Handout with “Student Strategies”
● Handout with “Planning Reflection”
● Handout with “Children’s Solution Strategies”
● Handout with “Lesson Planning Template”
● Blank chart paper
● Charts/Posters
○ Agenda: Day 2
○ Learning Goals for Module 4
○ Burning Questions
155
Learner Characteristic Accommodations
Learners will complete this module in person. Teachers will sit in table groups to support small
group conversations and collaboration. Slides will be projected so that all may access the
materials. Teachers will also receive handouts with key slides on them to facilitate listening and
note taking. If needed, teachers may request further accommodations.
Facilitator’s Notes
This opening session of the course will present the background information about CGI. It is often
reassuring to know that this way of teaching math is not a fad. In fact, it is grounded in research
over the last 40 years and the research is continuing today. Teachers will also examine the Math
Common Core State Standards and find that problem types are weaved into the standards. Thus,
a change is warranted. Module 4 examines the strategies children use when solving Separate
problems and scaffolds a process for planning a lesson. Throughout the course, teachers will
have opportunities to observe children, classrooms, and discuss ideas with colleagues. All
sessions will be synchronous and face-to-face. Teachers will be asked to present problems to
their class and return to subsequent sessions with student work.
Table A5 provides the instructional activities and details the instructional sequence, duration,
descriptions of the learning activities, instructor actions/decisions (supplantive events of
instruction) and learner actions/decisions (generative events of instruction).
156
Table A5
Instructional Activities, Module 4
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Gain
attention
10
Watch clips 3.11 and
3.12 from
Children’s
Mathematics:
Cognitively Guided
Instruction.
Discuss the
differences
between the two
strategies.
Instructor asks
teachers to state
how children seem
to be confused
when they use a
counting strategy
for separate
problems.
Instructor shows two
video clips:
3.11
http://fast.wistia.net
/embed/iframe/u0tn
rhav89?popover=tr
ue&ls=1
3.12
http://fast.wistia.net
/embed/iframe/u0tn
rhav89?popover=tr
ue&ls=1http://smar
turl.it/CM3.12
Instructor encourages
teachers to take
notes of the
strategies so that
they can discuss
them.
Instructor facilitates a
brief discussion
about the strategies
the students used
and how the
counting strategies
for separate
Teachers watch the
video clips and take
notes on the
strategies that each
student uses.
Teachers discuss their
thoughts and
reactions with their
table group.
157
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
problems can be
tricky.
Learning
objectives
4 The learning
objectives for the
module are posted
at the beginning of
the course content.
In order to establish
relevance and to
facilitate learning,
ask learners to read
the terminal and
enabling learning
objectives.
Ask teachers to share
their personal
learning objectives
with a partner.
Teachers read the
terminal and
enabling learning
objectives for
themselves.
Teachers will share
their personal
learning objectives
with a partner.
Purpose for
learning
• Benefits
• Risks
3 Benefits:
• It is important to
know the strategies
children use to
solve Separate
problems.
• This will support
teachers to
understand the
progression of
strategies that
children use as they
teach word
problems in their
classroom.
• Learning a
systematic method
for planning
lessons supports
teachers to be
prepared to teach
problem solving
lessons.
Risks:
Instructor asks
teachers to discuss
challenges they
have witnessed
while teaching
Separate problems.
Instructor asks
teachers to discuss
the areas they
would like to learn
more about with
regards to teaching
with Separate
problems.
Instructor invites
teachers to record
any burning
questions on a post-
it and add it to the
“Burning
Questions” chart.
The instructor will
peruse the
Teachers participate
in conversations at
their table and
reflect on their
experiences,
practice and areas
for
growth/learning.
Teachers may record
burning questions
on a post-it and add
it to the chart.
158
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
A teacher who does
not understand how
children solve
Separate problems
may inadvertently
confuse students
and be confused by
students when
teaching word
problems.
questions and make
notes to address
these throughout
the sessions.
Overview
• Prior
knowledge
• New
knowledge
• Learning
strategies
(What you
already
know,
what you
are going
to learn,
and how
you are
going to
learn it.)
9 Activate prior
knowledge by
asking teachers to
read the word
problems on the
problem type chart
from the Appendix
in the CCSS-M.
Overview of the
agenda for the
session:
• Examine strategies
that students use by
watching videos of
students who use a
variety of strategies
Separate problems.
• Demonstrate some
important features
when taking notes
on student
strategies.
• Review five-step
planning template
to use when
planning lessons.
Demonstrate how to
plan a lesson for a
Separate problem
that starts with
Instructor asks
teachers to generate
a list of problems
that are considered
Separate problems
(“Take From”)
from the problem
type chart.
Instructor asks table
groups to discuss
any surprises,
problems, that do
not seem to belong.
Instructor synthesizes
and captures main
points from each
table on a chart.
Instructor provides an
overview of the
module and
reminds teachers
that the key slides
are available in the
handout.
Instructor prompts
teachers to
articulate what they
Teachers discuss their
thoughts about the
“Separate From”
problems on the
problem chart.
Teachers share ideas
with their table
group and prepare
to share out with
the whole group.
Teachers review the
agenda items and
have an opportunity
to ask for
clarification or
elaboration on any
items.
159
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
strategies students
currently use and
helps them progress
to more
sophisticated
strategies.
will focus on
during this module.
Prerequisite
knowledge
13 Present the definition
of the different
levels of student
strategies for
Separate problems.
Check knowledge of
examples and non-
examples through a
Kahoot activity.
Instructor provides
definitions of levels
of student
strategies.
Instructor identifies
gaps in knowledge
through the Kahoot
activity which will
check the level of
the student strategy
and ask teachers to
distinguish between
an example and a
non-example.
Instructor gives
teachers an
opportunity to ask
clarifying
questions.
Teachers engage in
the Kahoot activity.
Teachers answer
questions asking
them to identify the
level of the student
strategy. Additional
questions will focus
on whether the
strategy is an
example or non-
example of a
particular level of
student strategy.
Teachers will receive
feedback from
Kahoot and the
instructor.
Teachers reflect on
how these terms
might inform their
practice.
Table groups will
have an opportunity
to share something
that they learned or
questions they
might have from
this activity.
160
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Learning
guidance:
Learning
context
• Lecture
• Demo
40 Observe students
solving various
Separate problems.
Focus on their
notetaking
strategies while
watching video
clips.
Instructor
demonstrates what
is important to pay
attention to when
watching a student
solve a problem.
Instructor focuses
notetaking
strategies to record
what students are
doing, how they
solve the problem,
what tools they use
(physical or in their
mind), how they
count, what number
they start and stop
on, and how to ask
clarifying (not
leading) questions.
Instructor prompts
teachers to record
notes on the
“Student
Strategies”
handout.
Instructor asks table
groups to discuss
the benefits and
importance of
notetaking.
Instructor prompts
table groups to
share what they
learned with the
whole group.
Teachers watch video
clips of students
solving various
Separate problems.
Teachers record notes
on “Student
Strategies”
handout.
Teachers discuss the
purpose of and
techniques for
taking notes while
observing students.
Teachers reflect on
what they learned
about the range of
strategies and
discuss how taking
notes may provide
evidence/data that
will inform
instruction.
Teachers have an
opportunity to ask
clarifying questions
as instructor
monitors and
checks for
understanding.
161
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Practice and
feedback:
The
learning
context
20 Apply notetaking
skills while
watching additional
video clips.
Display notes on a
poster for the
different problems.
Discuss similarities
and differences and
the connection to
classroom practice.
Instructor shows
video clips of a
student solving a
Separate problem.
Instructor prompts
teachers to take
notes individually
and then discuss
their notes with
their table group.
Instructor asks table
groups to record
their notes on a
poster to display to
the group.
Facilitate a gallery
walk.
Instructor leads a
discussion about
the similarities and
differences
between the
posters.
Instructor leads a
discussion about
the implications
(benefits) and
connections of
notetaking to
classroom practice.
Instructor provides
feedback and gives
teachers
opportunities to ask
clarifying
questions.
Just as with the
preceding
demonstration,
teachers watch the
video and take
notes.
Teachers discuss
what they noticed
in the video and
share their notes
with other table
members.
Teachers make a
poster with their
notes to share with
the whole group.
Teachers walk around
the room with their
table group and
discuss what they
see on the different
posters.
Teachers engage in a
discussion about
what they saw and
noticed.
Teachers have the
opportunity to ask
clarifying
questions.
162
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Prerequisite
knowledge
12 Review the
definitions for the
five steps of the
planning process.
Check knowledge of
examples and non-
examples through a
sorting activity.
Instructor reviews the
definitions for the
five-step planning
template.
Instructor identifies
gaps in knowledge
through the sorting
activity which
checks the level of
the planning
process and asks
teachers to
distinguish between
an example and a
non-example.
Instructor gives
teachers an
opportunity to ask
clarifying
questions.
Teachers engage in
the sorting activity.
Teachers will match
the name and the
step of the planning
process.
Additional questions
will focus on
whether the
description matches
an example or non-
example of the
planning process.
Teachers will receive
feedback from the
instructor.
Teachers reflect on
how this planning
process might
support their
practice.
Table groups will
have an opportunity
to share something
that they learned or
questions they
might have from
this activity.
Learning
context
• Lecture
• Demo
23 Refer to the five steps
for the planning
template organizer.
Complete the
planning template
organizer for a
Instructor reviews the
five-step planning
template.
Instructor
demonstrates the
process of
completing the
Teachers review the
descriptions of each
step of the planning
process.
As the demonstration
proceeds, teachers
record notes on
163
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Separate problem
lesson.
planning template
organizer with a
Separate problem.
their own organizer
so that they have a
worked example
for a Separate
problem.
Teachers have an
opportunity to ask
clarifying questions
as instructor
monitors and
checks for
understanding.
Practice and
feedback:
The
learning
context
24 Continue to plan
another problem
using the planning
template.
Instructor asks table
groups to select one
of the problems
that they wrote
during Module 2.
Instructor asks table
groups to discuss
the problem and
complete the
planning organizer.
Instructor asks table
groups to post their
plan on a chart.
Instructor facilitates a
gallery walk
activity to examine
the planning
posters.
Instructor prompts
teachers to record
their reflections on
the graphic
organizer,
“Planning
Teachers select a
problem from one
of those they wrote
in Module 2.
Teachers collaborate
with their table
groups to complete
a planning template
for the problem
they selected.
Table groups record
their planning
process of a chart
to share with the
whole group.
Teachers circulate
through the room
with their graphic
organizer.
Teachers record their
reflections on the
graphic organizer.
164
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Reflection,” under
the categories:
Notice, and
Wonder.
Instructor facilitates a
discussion about
what the groups
noticed and records
themes on a chart.
Instructor provides
feedback about
progress and gives
teachers
opportunities to ask
clarifying
questions.
Teachers engage in a
discussion about
what they saw and
noticed.
Teachers have an
opportunity to ask
clarifying
questions.
Authentic
assessment
14
Plan a lesson to
implement in the
classroom using the
planning template.
Instructor asks
teachers to make a
chart listing
examples of the
levels of strategies
for the different
Separate problems.
Instructor asks
teachers to describe
the five steps of the
planning process.
Teachers may ask
clarifying questions
about ambiguous
concepts.
Teachers will
demonstrate their
knowledge of the
different levels of
strategies for
Separate problems.
Teachers will plan a
lesson given
student work
samples (to mirror
the result of the
monitoring phase in
the classroom).
Teachers may ask
questions or seek
assistance as
necessary.
Retention
and
transfer
3 Learners reflect on
the objectives for
the lesson and
identify and report
Instructor asks
teachers to consider
the implications of
this knowledge on
Teachers reflect on
the objectives for
this module and the
165
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
on what they have
learned.
math instruction in
their classrooms.
Instructor prompts
teachers to reflect
on their practice
and the
implications of this
learning on their
practice.
implications for
their classroom.
Big ideas
3 The level of strategies
for solving
Separate problems
are presented and
the five-step
planning process is
practiced.
Instructor asks
teachers to state
what they have
learned about the
strategies students
use to solve
Separate problems
and the five-step
planning process.
Instructor prompts
teachers to ask
questions they may
have so the
instructor may
address them.
Teachers reflect on
and state what they
have learned about
student strategies
for solving
Separate problems
and the five steps
for planning
lessons with an
elbow partner.
Teachers finish the
sentence prompts:
“Something that left
an impression on
me during this
module was…” and
“One thing I am
wondering about
is…”
Teachers (volunteers)
share their
impressions and
wonders with the
whole group.
Advance
organizer
for the
next
module
2 Refer to visual
representation of
the modules to
show the module
that was completed
Instructor asks
teachers to voice
any questions that
are lingering.
Teachers consider
questions they
might have
concerning the
strategies that
166
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
and the topic of the
module that will
follow.
Given the topic of the
next module,
“Structure of
Multiplication and
Division
Problems,” ask
teachers what they
are wondering and
what questions
might have been
triggered by the
topic.
Instructor will record
questions on a
chart.
children use with
multiplication and
division problems
or the planning
template.
Total time 180
Module 5, Structure of Multiplication and Division Problems
Figure A6
Module 5, Structure of Multiplication and Division Problems
Module Duration: approximately 180 minutes synchronous engagement
Introduction:
This is the fifth module of an 8-module course to teach teachers how to implement cognitively guided instruction (CGI) in their
classrooms. Throughout the course, teachers will learn about the different problem types and strategies that students use. They will
also learn about a process for strategically planning lessons and robust mathematical discussions. The purpose of the fifth module is to
teach teachers the structure of, and strategies children use when solving multiplication, measurement division, and partitive division
problems, and to plan a lesson.
Module 1a:
Overview
Module 1b:
Designing the
Classroom
Space
Module 2:
Structure of
the Problems
Module 3:
Noticing
Strategies
(Join)
Module 4:
Noticing
Strategies
(Separate)
Module 5:
Structure of
Multiplication
& Division
Problems
Module 6:
Multi-Digit
Addition
Problems
Module 7:
Multi-Digit
Subtraction
Problems
Module 8:
Base-Ten
Number
Concepts
167
168
Learning Objective(s)
Terminal Objective:
● Given the background information of CGI and the Common Core State Standards,
teachers will identify and analyze strategies that children use when solving
multiplication, measurement division problems, write problems and create culturally
responsive lesson plans to implement in their classroom per the rubric in the lesson
template.
Enabling Objective(s):
● For a given grade level, teachers will:
○ Know the meaning of multiplication, measurement division, and partitive division
problems (equal groups and array types)
○ Know the meaning of direct modeling strategies for multiplication, measurement
division, and partitive division problems
○ Know the meaning of counting strategies for multiplication, measurement
division, and partitive division problems
○ Know the meaning of number facts (derived fact) strategies for multiplication,
measurement division, and partitive division problems
○ Identify and distinguish structural features of multiplication, measurement
division, and partitive division problems
○ Identify and analyze strategies that children use to solve multiplication,
measurement division, and partitive division problems
○ Synthesize strategies that students are using to create (write) problems and lessons
that respond to students’ lived experiences
○ Design lessons using the 5-step planning template for a multiplication,
measurement division, and partitive division problem
Learning Activities
● After introductions and attention activities and learning objectives, assess prior
knowledge of the structure of and strategies that children use to solve multiplication,
measurement division, and partitive division problems.
● Teach any necessary prerequisite knowledge by providing definitions and examples and
nonexamples.
● Provide opportunities for learners to generate their own examples and nonexamples.
● Model the procedure for identifying the structure of and strategies that children use to
solve multiplication and division problems.
● Provide practice and feedback on identifying the structure of and strategies that children
use to solve multiplication and division problems.
● Model the procedure for planning a lesson for multiplication, measurement division, and
partitive division problems.
● Provide practice and feedback for planning a lesson for multiplication, measurement
division, and partitive division problems.
● Provide opportunities to transfer knowledge to the teachers’ own classroom practice.
169
Summative Assessment
● Given the various examples of CGI classrooms, teachers will identify the structure of and
strategies that children use to solve multiplication, measurement division, and partitive
division problems and teachers will plan a lesson to implement in their classroom.
Lesson Materials
● Written resource: Appendix for Math Common Core State Standards (Table 2)
● Video clips from Children’s Mathematics: Cognitively Guided Instruction
● Presentation slides
● Handout with key slides
● Access to Kahoot site on internet (via phone or a computer)
● Handout with “Lesson Planning Template”
● Handout with graphic organizer: “Multiplication, Measurement Division, and Partitive
Division Problems”
● Blank chart paper
● Charts/Posters
○ Agenda: Day 3
○ Learning Goals for Module 5
○ Burning Questions
○ Principles for Unpacking the Problem
○ Principles for Eliciting Student Thinking
Learner Characteristic Accommodations
Learners will complete this module in person. Teachers will sit in table groups to support small
group conversations and collaboration. Slides will be projected so that all may access the
materials. Teachers will also receive handouts with key slides on them to facilitate listening and
note taking. If needed, teachers may request further accommodations.
Facilitator’s Notes
This opening session of the course will present the background information about CGI. It is often
reassuring to know that this way of teaching math is not a fad. In fact, it is grounded in research
over the last 40 years and the research is continuing today. Teachers will also examine the Math
Common Core State Standards and find that problem types are weaved into the standards. Thus,
a change is warranted. Module 5 begins to examine the structure of multiplication, measurement
division, and partitive division word problems as well as the strategies that children use to solve
them. Throughout the course, teachers will have opportunities to observe children, classrooms,
and discuss ideas with colleagues. All sessions will be synchronous and face-to-face. Teachers
will be asked to present problems to their class and return to subsequent sessions with student
work.
Table A6 provides the instructional activities and details the instructional sequence, duration,
descriptions of the learning activities, instructor actions/decisions (supplantive events of
instruction) and learner actions/decisions (generative events of instruction).
170
Table A6
Instructional Activities, Module 5
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Gain
attention
10
Discuss strategies
used to model a
division problem
(without a context).
Discuss how students
may be more
comfortable with
one strategy over
another.
Emphasize the point
that without a
context, students
may not understand
one strategy or
another.
Instructor asks
teachers to show
how they would
teach children to
model a division
problem such as
15÷3.
Instructor facilitates a
brief discussion
about there are
different models to
represent 15÷3
and how children
decide which
strategy (model) to
use.
Teachers write down
the strategy that
they would use to
teach children to
solve a division
problem.
Teachers display their
strategy.
Teachers discuss the
different methods
used to teach how
to solve division
problems.
Teachers reflect on
the different
models to represent
a division problem.
Learning
objectives
4 The learning
objectives for the
module are posted
at the beginning of
the course content.
In order to establish
relevance and to
facilitate learning,
ask learners to read
the terminal and
enabling learning
objectives.
Ask teachers to share
their personal
learning objectives
with a partner.
Teachers read the
terminal and
enabling learning
objectives for
themselves.
Teachers will share
their personal
learning objectives
with a partner.
Purpose for
learning
• Benefits
• Risks
3 Benefits:
• It is important to
know the structure
of and the
Instructor asks
teachers to discuss
challenges they
have witnessed
Teachers participate
in conversations at
their table and
reflect on their
171
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
strategies children
use to solve
multiplication and
division problems.
• This will support
teachers to
understand the
progression of
strategies that
children use as they
teach word
problems in their
classroom.
• Practicing a
systematic method
for planning
lessons supports
teachers to be
prepared to teach
problem solving
lessons.
Risks:
A teacher who does
not understand how
children solve
multiplication and
division problems
may inadvertently
confuse students
and be confused by
students when
teaching word
problems.
while teaching
multiplication and
division problems.
Instructor asks
teachers to discuss
the areas they
would like to learn
more about with
regards to teaching
with multiplication
and division
problems.
Instructor invites
teachers to record
any burning
questions on a post-
it and add it to the
“Burning
Questions” chart.
The instructor will
peruse the
questions and make
notes to address
these throughout
the sessions.
experiences,
practice and areas
for
growth/learning.
Teachers may record
burning questions
on a post-it and add
it to the chart.
Overview
• Prior
knowledge
• New
knowledge
• Learning
strategies
10 Activate prior
knowledge about
teaching
multiplication or
division by
examining word
problems on the
Instructor asks
teachers to refer to
and read the
multiplication and
division problems
from the Appendix.
Teachers discuss their
thoughts about the
“multiplication and
division problems
on the problem
type chart.
172
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
(What you
already
know,
what you
are going
to learn,
and how
you are
going to
learn it.)
problem type chart
from the Appendix
in the CCSS-M
(Table 2), focusing
on the equal groups
and array types of
problems.
Provide brief
overview of the
module:
• Examine strategies
that students use by
watching videos of
students who use a
variety of strategies
to solve
multiplication,
measurement
division, and
partitive division
problems.
• Review five-step
planning template
to use when
planning lessons.
• Demonstrate how
to plan a lesson for
multiplication,
measurement
division, and
partitive division
problems that starts
with strategies
students currently
use and helps them
progress to more
sophisticated
strategies.
Instructor asks table
groups to discuss
any surprises,
problems, that do
not seem to belong.
Instructor synthesizes
and captures main
points from each
table on a chart.
Instructor provides an
overview of the
module and
reminds teachers
that the key slides
are available in the
handout.
Instructor prompts
teachers to
articulate what they
will focus on
during this module.
Teachers share ideas
with their table
group and prepare
to share out with
the whole group.
Teachers have an
opportunity to ask
for clarification or
elaboration on any
items.
Prerequisite
knowledge
21 Present the definition
of the different
Instructor provides
definitions of
Teachers take notes
on the graphic
173
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
types of
multiplication and
division problems
(multiplication,
measurement
division, and
partitive division)
using the graphic
organizer.
Check knowledge of
examples and non-
examples through a
Kahoot activity.
multiplication,
measurement
division, and
partitive division
problems.
Instructor identifies
gaps in knowledge
through the Kahoot
activity which
checks the type of
problems
(multiplication,
measurement
division, or
partitive division)
and asks teachers to
distinguish between
an example and a
non-example.
Instructor gives
teachers an
opportunity to ask
clarifying
questions.
Instructor addresses
misconceptions or
questions.
organizer for
multiplication,
measurement
division, and
partitive division
problems,
especially noting
the structural
differences of the
problems.
Teachers engage in
the Kahoot activity.
Teachers answer
questions asking
them to identify
type of problems
(multiplication,
measurement
division, or
partitive division).
Additional
questions will focus
on whether the
problem is an
example or non-
example of a
particular type of
problem.
Teachers will receive
feedback from
Kahoot and the
instructor.
Teachers reflect on
how these terms
might inform their
practice.
174
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Table groups will
have an opportunity
to share something
that they learned or
questions they
might have from
this activity.
Learning
guidance:
Learning
context
• Lecture
• Demo
70 Observe students
using various
strategies to solve
multiplication,
measurement
division, and
partitive division
problems.
Write a
multiplication,
measurement
division, and a
partitive division
problem.
Complete the five-
step planning
template for
multiplication,
measurement
division, and
partitive division
problems.
Instructor highlights
the different levels
of strategies
students use when
solving
multiplication,
measurement
division, and
partitive division
problems.
Instructor
demonstrates and
points out the
process for
unpacking the
problem for
multiplication and
division problems.
Instructor
demonstrates and
points out how to
elicit student
thinking through
the video clips
showing
multiplication,
measurement
division, and
partitive division
problems.
Teachers watch video
clips of students
solving various
multiplication and
division problems.
Teachers take notes
on the levels of
strategies students
use while solving
multiplication,
measurement
division, and
partitive division
problems.
Teachers discuss
instructional moves
that the
teacher/interviewer
made to unpack the
problem and elicit
student thinking.
Teachers
collaboratively
write each of the
three types of
problems
(multiplication,
measurement
division, and
partitive division)
175
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Instructor
demonstrates the
process for writing
multiplication,
measurement
division, and
partitive division
problems.
Instructor
demonstrates the
process for
completing the
planning template
for a multiplication,
measurement
division, and
partitive division
problems.
Instructor facilitates a
discussion about
the benefits and
importance of
distinguishing the
three types of
problems.
Instructor prompts
teachers to discuss
what they learned
with their table
group.
with the support of
the facilitator.
Teachers will take
notes of the
planning template
to have a worked
example for a
multiplication,
measurement
division, and
partitive division
problem.
Teachers reflect on
what they learned
about teaching
multiplication,
measurement
division, and
partitive division
problems.
Teachers have an
opportunity to ask
clarifying questions
as instructor
monitors and
checks for
understanding.
Practice and
feedback:
The
learning
context
35 Write additional
problems of each of
the types:
multiplication,
measurement
division, and
partitive division
problems.
Instructor asks
teachers to write
additional
multiplication,
measurement
division, and
partitive division
problems.
Teachers work with
their table group to
write additional
multiplication,
measurement
division, and
partitive division
problems.
176
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Select a problem and
complete the
planning template
for a multiplication,
measurement
division, or
partitive division
problem.
Instructor prompts
teachers to select
one of the problems
they wrote to use to
complete the
planning template.
Instructor asks
teachers to put their
plan on a chart.
Facilitate a gallery
walk.
Instructor leads a
discussion about
the similarities and
differences
between the
posters.
Instructor leads a
discussion about
the implications
(benefits) and
connections of
notetaking to
classroom practice.
Instructor provides
feedback and gives
teachers
opportunities to ask
clarifying
questions.
Teachers select a
problem to develop
a lesson by
completing the
planning template.
Teachers make a
poster with their
planning template
to share with the
whole group.
Teachers walk
around, examine
the other posters,
discuss the
different posters,
and reflect on their
learning.
Teachers engage in a
discussion about
what they saw and
noticed.
Teachers have the
opportunity to ask
clarifying
questions.
Authentic
assessment
15
Plan a lesson using
the planning
template for a
multiplication,
measurement
Instructor asks
teachers to
complete a
planning template
for a multiplication,
Teachers will
demonstrate their
knowledge of the
different strategies
for solving
177
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
division, or
partitive division
problem.
measurement
division, or
partitive division
problem.
multiplication,
measurement
division, or
partitive division
problems by
completing a plan
for one type of
problem.
Teachers may ask
questions or seek
assistance as
necessary.
Retention
and
transfer
5 Learners reflect on
the objectives for
the lesson and
identify and report
on what they have
learned.
Instructor asks
teachers to consider
the implications of
this knowledge on
math instruction in
their classrooms.
Instructor prompts
teachers to reflect
on their practice
and the
implications of this
learning on their
practice.
Teachers reflect on
the objectives for
this module and the
implications for
their classroom.
Big ideas
5 The structure of
multiplication,
measurement
division, or
partitive division
problems, and the
strategies for
solving each type
of problem are
presented, and the
five-step planning
process is
practiced.
Instructor asks
teachers to state
what they have
learned about the
strategies students
use to solve
multiplication,
measurement
division, or
partitive division
problems, and the
five-step planning
process.
Teachers state what
they have learned
about the structure
of, and strategies
students use when
solving
multiplication,
measurement
division, and
partitive division
problems, and the
five steps for
planning lessons.
178
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Instructor prompts
teachers to share
impressions and
wonders and ask
any questions they
may have so the
instructor may
address them.
Teachers finish the
sentence prompts:
“Something that left
an impression on
me during this
module was…” and
“One thing I am
wondering about
is…”
Teachers (volunteers)
share their
impressions and
wonders with the
whole group.
Advance
organizer
for the
next
module
2 Refer to visual
representation of
the modules to
show the module
that was completed
and the topic of the
module that will
follow.
Instructor asks
teachers to voice
any questions that
are lingering.
Given the topic of the
next module,
“Multi-Digit
Addition Structure
of Multiplication
and Division
Problems,” ask
teachers what they
are wondering and
what questions
might have been
triggered by the
topic.
Instructor will record
questions on a
chart.
Teachers consider
questions they
might have
concerning the
strategies that
children use when
solving problems
with larger
numbers.
Total time 180
Module 6, Structure of Multi-Digit Addition Problems
Figure A7
Module 6, Structure of Multi-Digit Addition Problems
Module Duration: approximately 180 minutes synchronous engagement
Introduction:
This is the sixth module of an 8-module course to teach teachers how to implement cognitively guided instruction (CGI) in their
classrooms. Throughout the course, teachers will learn about the different problem types and strategies that students use. They will
also learn about a process for strategically planning lessons and robust mathematical discussions. The purpose of the sixth module is
to teach teachers the strategies children use when solving multi-digit addition problems, and to plan a lesson.
Module 1a:
Overview
Module 1b:
Designing the
Classroom
Space
Module 2:
Structure of
the Problems
Module 3:
Noticing
Strategies
(Join)
Module 4:
Noticing
Strategies
(Separate)
Module 5:
Structure of
Multiplication
& Division
Problems
Module 6:
Multi-Digit
Addition
Problems
Module 7:
Multi-Digit
Subtraction
Problems
Module 8:
Base-Ten
Number
Concepts
179
180
Learning Objective(s)
Terminal Objective:
● Given the background information of CGI and the Common Core State Standards,
teachers will identify strategies that children use when solving multi-digit addition
problems, write problems and create culturally responsive lesson plans to implement in
their classroom per the rubric in the lesson template.
Enabling Objective(s):
● For a given grade level, teachers will:
○ Know the meaning of direct modeling strategies for multi-digit addition problems
○ Know the meaning of counting strategies for multi-digit addition problems
○ Know the meaning of invented algorithms for multi-digit addition problems
○ Identify and analyze strategies that children use to solve multi-digit addition
problems
○ Synthesize strategies that students are using to create (write) problems and lessons
that respond to students’ lived experiences
○ Design lessons using the 5-step planning template for a multi-digit addition
problem
Learning Activities
● After introductions and attention activities and learning objectives, assess prior
knowledge of the structure of and strategies that children use to solve multi-digit addition
problems.
● Teach any necessary prerequisite knowledge by providing definitions and examples and
nonexamples.
● Provide opportunities for learners to generate their own examples and nonexamples.
● Model the procedure for identifying the structure of and strategies that children use to
solve multi-digit addition problems.
● Provide practice and feedback on identifying the structure of and strategies that children
use to solve multi-digit addition problems.
● Model the procedure for planning a lesson for multi-digit addition problems.
● Provide practice and feedback for planning a lesson for multi-digit addition problems.
● Provide opportunities to transfer knowledge to the teachers’ own classroom practice.
Summative Assessment
● Given the various examples of CGI classrooms, teachers will identify the strategies that
children use to solve multi-digit addition problems and will write problems and create
lesson plans to implement in their classroom per the rubric in the lesson template.
Lesson Materials
● Written resource: Math Common Core State Standards, grades K-5 (from Module 1a)
● Written resource: Appendix for Math Common Core State Standards (from Module 1a)
● Video clips from Children’s Mathematics: Cognitively Guided Instruction
● Presentation slides
● Handout with key slides
● Access to Kahoot site on internet (via phone or a computer)
181
● Handout with “Lesson Planning Template”
● Handout with graphic organizer: “Multi-Digit Problems”
● Blank chart paper
● Charts/Posters
○ Agenda: Day 3
○ Learning Goals for Module 6
○ Burning Questions
Learner Characteristic Accommodations
Learners will complete this module in person. Teachers will sit in table groups to support small
group conversations and collaboration. Slides will be projected so that all may access the
materials. Teachers will also receive handouts with key slides on them to facilitate listening and
note taking. If needed, teachers may request further accommodations.
Facilitator’s Notes
This opening session of the course will present the background information about CGI. It is often
reassuring to know that this way of teaching math is not a fad. In fact, it is grounded in research
over the last 40 years and the research is continuing today. Teachers will also examine the Math
Common Core State Standards and find that problem types are weaved into the standards. Thus,
a change is warranted. Module 6 examines the strategies that children use to solve multi-digit
addition problems. Throughout the course, teachers will have opportunities to observe children,
classrooms, and discuss ideas with colleagues. All sessions will be synchronous and face-to-face.
Teachers will be asked to present problems to their class and return to subsequent sessions with
student work.
Table A7 provides the instructional activities and details the instructional sequence, duration,
descriptions of the learning activities, instructor actions/decisions (supplantive events of
instruction) and learner actions/decisions (generative events of instruction).
182
Table A7
Instructional Activities, Module 6
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Gain
attention
10
Discuss student work
samples to
determine the
benefits and
potential
conceptual
challenges of the
strategies that
students use.
Instructor presents
three samples of
student work: one
using the traditional
algorithm, one
using a buggy
algorithm, one
using a place value-
based algorithm.
Instructor facilitates a
discussion about
what the student
work shows that
students know, as
well as the benefits
and potential
conceptual
challenges of the
strategies.
Teachers examine the
three student
strategies.
Teachers discuss
what the students
know based on the
student work.
Teachers discuss the
benefits and
potential
conceptual
challenges of the
strategies.
Learning
objectives
4 The learning
objectives for the
module are posted
and reviewed.
In order to establish
relevance and to
facilitate learning,
ask learners to read
the terminal and
enabling learning
objectives.
Ask teachers to share
their personal
learning objectives
with a partner.
Teachers read the
terminal and
enabling learning
objectives for
themselves.
Teachers will share
their personal
learning objectives
with a partner.
Purpose for
learning
• Benefits
• Risks
3 Benefits:
• It is important to
know the various
strategies children
Instructor asks
teachers to discuss
the areas they
would like to learn
Teachers participate
in conversations at
their table and
reflect on their
183
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
use to solve multi-
digit addition
problems.
• This will support
teachers to support
students as they
develop strategies
to solve multi-digit
addition problems
in a way that makes
sense to them.
• Teachers will be
able to support
students to make
progress towards
more sophisticated
strategies.
• Pre-planning
lessons (using the
five practices
template) supports
teachers to feel
more confident and
prepared to teach
problem solving
lessons.
Risks:
A teacher who does
not understand how
children solve
multi-digit addition
problems may
inadvertently
confuse students
and be confused by
students when
teaching word
problems.
more regarding
teaching with
multi-digit addition
problems.
Instructor invites
teachers to record
any burning
questions on a post-
it and add it to the
“Burning
Questions” chart.
The instructor will
peruse the
questions and make
notes to address
these throughout
the sessions.
experiences,
practice and areas
for
growth/learning.
Teachers may record
burning questions
on a post-it and add
it to the chart.
Overview 12 Activate prior
knowledge about
Instructor reviews the
different types of
Teachers make note
of strategies that
184
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
• Prior
knowledge
• New
knowledge
• Learning
strategies
(What you
already
know,
what you
are going
to learn,
and how
you are
going to
learn it.)
teaching join
problem types by
reviewing the
strategies listed
under the
“Operations and
Algebraic
Thinking”
standards as well as
the “Number and
Operations in Base
Ten.” Make special
note that the
standard algorithm
is expected at
fourth grade and
not in any earlier
grade.
Provide brief
overview of lesson:
• Examine strategies
that students use by
watching videos of
students who use a
variety of strategies
to solve multi-digit
addition problems.
• Demonstrate how
to write multi-digit
addition problems.
• Demonstrate how
to plan a lesson for
multi-digit addition
problems that starts
with strategies
students currently
use and helps them
progress to more
sophisticated
strategies.
join problems
presented in
Module 3.
Instructor reminds
teachers that the
multi-digit addition
problems are join
problems, with
larger numbers.
Instructor provides an
overview of the
lesson explaining
how the different
activities support
the module goals.
Instructor prompts
teachers to
articulate what they
will focus on
during this module.
are called out in the
standards.
Teachers review their
knowledge of join
problems.
Teachers discuss their
learning focus for
the module at their
table.
Teachers have an
opportunity to ask
for clarification or
elaboration on any
items.
185
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Prerequisite
knowledge
18 Present the definition
of the different
levels of student
strategies for multi-
digit addition
problems.
Check knowledge of
the different types
of and levels of
strategies and
distinguish
examples versus
non-examples
through an activity
on Kahoot.
Instructor explains
different levels of
student strategies
for multi-digit
addition problems.
Instructor identifies
gaps in knowledge
through the Kahoot
activity which will
check the level of
student strategies
for multi-digit
addition problems
and ask teachers to
distinguish between
an example and a
non-example.
Instructor gives
teachers an
opportunity to ask
clarifying questions
and state something
they have learned.
Instructor addresses
questions.
Teachers record notes
on the different
types of and levels
of strategies for
multi-digit addition
problems.
Teachers engage in
the Kahoot activity.
Teachers answer
questions that ask
them to identify the
level of student
strategies for multi-
digit addition
problems.
Teachers will also
answer questions
that distinguish
examples versus
non-examples.
Teachers will receive
feedback from
Kahoot and the
instructor.
Teachers reflect on
how these terms
might inform their
practice.
Table groups will
have an opportunity
to share something
that they learned or
ask questions they
might have.
186
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Learning
guidance:
learning
context
• Lecture
• Demo
70 Observe students
solving various
multi-digit addition
problems.
Revisit and refine
notetaking
strategies while
watching video
clips that depict
distinguishing
characteristics of
different strategies.
Write three multi-
digit addition
problems.
Complete the five-
step planning
template for a
multi-digit addition
problem using the
problems that the
group
collaboratively
wrote.
Instructor
demonstrates the
steps for
identifying the
level and types of
strategies students
use while watching
video clips.
Instructor names and
distinguishes the
different levels of
and types of
strategies students
use to solve multi-
digit addition
problems.
Instructor reviews
techniques for
taking notes of
strategies students
use while solving
multi-digit addition
problems.
Instructor
demonstrates the
steps for writing
multi-digit addition
problems by
focusing on the
structure and key
characteristics of
these problems.
Instructor
demonstrates the
process for
completing the
planning template
for multi-digit
Teachers take notes
as they watch video
clips of students
solving various
multi-digit addition
problems.
Teachers discuss the
levels of and types
of strategies that
students used to
solve multi-digit
addition problems.
Teachers will revisit
the purpose of and
techniques for
taking notes while
observing students.
Teachers
collaboratively
write multi-digit
addition problems
with the support of
the facilitator.
Teachers will take
notes of the
planning template
to have a worked
example for multi-
digit addition
problem.
Teachers discuss
possible closing
discussion themes
for the problem.
Teachers reflect on
what they learned
187
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
addition problems
with an emphasis
on designing the
closing discussion.
As the demonstration
lesson proceeds,
the instructor will
refer back to the
notes on student
strategies to
support the
planning process
for the current
lesson as well as
future lessons.
Instructor prompts
teachers to discuss
what they learned
with their table
group.
about teaching
multi-digit addition
problems.
Teachers have an
opportunity to ask
clarifying questions
as instructor
monitors and
checks for
understanding.
Practice and
feedback:
The
learning
context
35 Write additional
multi-digit addition
problems.
Select a problem,
complete the
planning template
for that problem,
and display it for a
gallery walk.
Instructor prompts
teachers to write
multi-digit addition
problems with their
table group.
Instructor asks table
groups to select a
problem to use to
complete the
planning template.
Instructor asks
teachers to display
it to the group.
Facilitate a gallery
walk.
Teachers
collaboratively
write multi-digit
addition problems
with their table
group.
Teachers complete
the planning
template to plan
one of the problems
they wrote.
Teachers make notes
of the levels of and
types of strategies
on the planning
template and
articulate possible
188
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Instructor facilitates a
discussion about
the different
problems, focusing
on the themes for
the closing
discussion.
Provide feedback and
give teachers
opportunities to ask
clarifying
questions.
themes for the
closing discussion.
Teachers make a
poster with their
plan to share with
the whole group.
Teachers walk
around, examine
the other posters,
discuss the
different posters,
and reflect on their
learning.
Teachers engage in a
discussion about
what they saw and
noticed.
Teachers have the
opportunity to ask
clarifying
questions.
Authentic
assessment
16
Plan a lesson using
the planning
template for a
multi-digit addition
problem including
the potential
storylines for the
closing discussion.
Instructor asks
teachers to
complete a
planning template
for a multi-digit
addition problem.
Teachers will
demonstrate their
knowledge of a
multi-digit addition
problem and the
process for
planning those
problems by
completing a plan
for a problem.
Teachers may ask
questions or seek
assistance as
necessary.
189
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Retention
and
transfer
5 Learners reflect on
the objectives for
the lesson and
identify and report
on what they have
learned.
Instructor asks
teachers to consider
the implications of
this knowledge on
math instruction in
their classrooms.
Instructor prompts
teachers to reflect
on their practice
and the
implications of this
learning on their
practice.
Teachers reflect on
the objectives for
this module and the
implications for
their classroom.
Big ideas
5 The levels of and
types of strategies
used for solving
multi-digit addition
problems are
presented, and the
five-step planning
process is
practiced.
Instructor asks
teachers to state
what they have
learned about the
levels and types of
strategies students
use to solve multi-
digit addition
problems, and the
five-step planning
process.
Instructor prompts
teachers to share
impressions and
wonders and ask
any questions they
may have so the
instructor may
address them.
Teachers state what
they have learned
about the levels and
types of strategies
students use to
solve multi-digit
addition problems,
and the five steps
for planning
lessons.
Teachers finish the
sentence prompts:
“Something that left
an impression on
me during this
module was…” and
“One thing I am
wondering about
is…”
Teachers (volunteers)
share their
impressions and
wonders with the
whole group.
190
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Advance
organizer
for the
next
module
2 Refer to visual
representation of
the modules to
show the module
that was completed
and the topic of the
module that will
follow.
Instructor asks
teachers to voice
any questions that
are lingering.
Given the topic of the
next module,
“Multi-Digit
Subtraction
Problems,” ask
teachers what they
are wondering and
what questions
might have been
triggered by the
topic.
Instructor will record
questions on a
chart.
Teachers consider
questions they
might have
concerning the
strategies that
children use with
multi-digit
subtraction
problems.
Total time 180
Module 7, Multi-Digit Subtraction Problems
Figure A8
Module 7, Multi-Digit Subtraction Problems
Module Duration: approximately 180 minutes synchronous engagement
Introduction:
This is the seventh module of an 8-module course to teach teachers how to implement cognitively guided instruction (CGI) in their
classrooms. Throughout the course, teachers will learn about the different problem types and strategies that students use. They will
also learn about a process for strategically planning lessons and robust mathematical discussions. The purpose of the seventh module
is to teach teachers the strategies children use when solving multi-digit subtraction problems, and to plan a lesson.
Module 1a:
Overview
Module 1b:
Designing the
Classroom
Space
Module 2:
Structure of
the Problems
Module 3:
Noticing
Strategies
(Join)
Module 4:
Noticing
Strategies
(Separate)
Module 5:
Structure of
Multiplication
& Division
Problems
Module 6:
Multi-Digit
Addition
Problems
Module 7:
Multi-Digit
Subtraction
Problems
Module 8:
Base-Ten
Number
Concepts
191
192
Learning Objective(s)
Terminal Objective:
• Given the background information of CGI and the Common Core State Standards,
teachers will identify features of and strategies that children use when solving multi-digit
subtraction problems, write problems and create culturally responsive lesson plans to
implement in their classroom per the rubric in the lesson template.
Enabling Objective(s):
• For a given grade level, teachers will:
o Know the meaning of direct modeling strategies for multi-digit subtraction
problems
○ Know the meaning of counting strategies for multi-digit subtraction problems
○ Know the meaning of invented algorithms for multi-digit subtraction problems
○ Identify and analyze strategies that children use to solve multi-digit subtraction
problems
○ Synthesize strategies that students are using to create (write) problems and lessons
that respond to students’ lived experiences
○ Design lessons using the 5-step planning template for a multi-digit subtraction
problem
Learning Activities
● After introductions, attention activities and learning objectives, assess prior knowledge of
the structure of and strategies that children use to solve multi-digit subtraction problems.
● Teach any necessary prerequisite knowledge by providing definitions and examples and
nonexamples.
● Provide opportunities for learners to generate their own examples and nonexamples.
● Model the procedure for identifying the strategies that children use to solve multi-digit
subtraction problems.
● Provide practice and feedback on identifying strategies that children use to solve multi-
digit subtraction problems.
● Model the procedure for planning a lesson for multi-digit subtraction problems.
● Provide practice and feedback for planning a lesson for multi-digit subtraction problems.
● Provide opportunities to transfer knowledge to the teachers’ own classroom practice.
Summative Assessment
● Given the various examples of CGI classrooms, teachers will identify the strategies that
children use to solve multi-digit subtraction problems and will write problems and create
lesson plans to implement in their classroom per the rubric in the lesson template.
Lesson Materials
● Written resource: Math Common Core State Standards, grades K-5 (from Module 1a)
● Written resource: Appendix for Math Common Core State Standards (from Module 1a)
● Video clips from Children’s Mathematics: Cognitively Guided Instruction
● Presentation slides
● Handout with key slides
● Access to Kahoot site on internet (via phone or a computer)
193
● Handout with “Lesson Planning Template”
● Handout with graphic organizer: “Multi-Digit Problems”
● Blank chart paper
● Charts/Posters
○ Agenda: Day 4
○ Learning Goals for Module 7
○ Burning Questions
Learner Characteristic Accommodations
Learners will complete this module in person. Teachers will sit in table groups to support small
group conversations and collaboration. Slides will be projected so that all may access the
materials. Teachers will also receive handouts with key slides on them to facilitate listening and
note taking. If needed, teachers may request further accommodations.
Facilitator’s Notes
This opening session of the course will present the background information about CGI. It is often
reassuring to know that this way of teaching math is not a fad. In fact, it is grounded in research
over the last 40 years and the research is continuing today. Teachers will also examine the Math
Common Core State Standards and find that problem types are weaved into the standards. Thus,
a change is warranted. Module 7 examines the strategies that children use to solve multi-digit
subtraction problems. Throughout the course, teachers will have opportunities to observe
children, classrooms, and discuss ideas with colleagues. All sessions will be synchronous and
face-to-face. Teachers will be asked to present problems to their class and return to subsequent
sessions with student work.
Table A8 provides the instructional activities and details the instructional sequence, duration,
descriptions of the learning activities, instructor actions/decisions (supplantive events of
instruction) and learner actions/decisions (generative events of instruction).
194
Table A8
Instructional Activities, Module 7
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Gain
attention
10
Discuss student work
samples to
determine the
benefits and
potential
conceptual
challenges of the
strategies that
students use.
Instructor presents
three samples of
student work: one
using the traditional
algorithm, one
using a buggy
algorithm, one
using an invented
algorithm.
Instructor facilitates a
discussion about
what the student
work shows that
students know, as
well as the benefits
and potential
conceptual
challenges of the
strategies.
Teachers examine the
three student
strategies.
Teachers discuss what
the students know
based on the student
work.
Teachers discuss the
benefits and
potential conceptual
challenges of the
strategies.
Learning
objectives
4 The learning
objectives for the
module are posted
and reviewed.
In order to establish
relevance and to
facilitate learning,
ask learners to read
the terminal and
enabling learning
objectives.
Ask teachers to share
their personal
learning objectives
with a partner.
Teachers read the
terminal and
enabling learning
objectives for
themselves.
Teachers will share
their personal
learning objectives
with a partner.
Purpose for
learning
• Benefits
• Risks
3 Benefits:
• It is important to
know the various
strategies that
Instructor asks
teachers to discuss
areas they would
like to learn more
Teachers participate
in conversations at
their table and
reflect on their
195
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
children use to
solve multi-digit
subtraction
problems.
• This will support
teachers to
understand the
progression of
strategies that
children use as
they solve multi-
digit subtraction
problems in a way
that makes sense
to them.
• Teachers will be
able to support
students to make
progress towards
more sophisticated
strategies.
• Pre-planning
lessons supports
teachers to feel
more confident
and prepared to
teach problem
solving lessons.
Risks:
A teacher who does
not understand
how children solve
multi-digit
subtraction
problems may
inadvertently
confuse students
and be confused
by students when
teaching word
problems.
about regarding
multi-digit
subtraction
problems.
Instructor invites
teachers to record
any burning
questions on a post-
it and add it to the
“Burning
Questions” chart.
The instructor will
peruse the questions
and make notes to
address these
throughout the
sessions.
experiences,
practice and areas
for growth/learning.
Teachers may record
burning questions
on a post-it and add
it to the chart.
196
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Overview
• Prior
knowledge
• New
knowledge
• Learning
strategies
(What you
already
know,
what you
are going
to learn,
and how
you are
going to
learn it.)
12 Activate prior
knowledge about
teaching join
problem types by
reviewing the
strategies listed
under the
“Operations and
Algebraic
Thinking”
standards as well
as the “Number
and Operations in
Base Ten.” Make
special note that
the standard
algorithm is
expected at fourth
grade and not in
any earlier grade.
Provide brief
overview of
lesson:
• Examine strategies
that students use
by watching
videos of students
who use a variety
of strategies to
solve multi-digit
subtraction
problems.
• Demonstrate how
to write multi-digit
subtraction
problems.
• Demonstrate how
to plan a lesson for
multi-digit
subtraction
problems that
Instructor reviews the
different types of
separate problems
presented in
Module 4.
Instructor reminds
teachers that multi-
digit subtraction
problems are
separate problems,
with larger
numbers.
Instructor provides an
overview of the
module and
reminds teachers
that the key slides
are available in the
handout.
Instructor prompts
teachers to
articulate what they
will focus on during
this module.
Teachers review the
strategies that are
called out in the
standards.
Teachers review their
knowledge of
separate problems.
Teachers discuss their
learning focus for
the module at their
table.
Teachers have an
opportunity to ask
for clarification or
elaboration on any
items.
197
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
starts with
strategies students
currently use and
helps them
progress to more
sophisticated
strategies.
Prerequisite
knowledge
18 Present the definition
of the different
types of and levels
of student
strategies for
multi-digit
subtraction
problems.
Check knowledge of
the different types
of and levels of
strategies and
distinguish
examples versus
non-examples
through an activity
on Kahoot.
Instructor explains
different levels of
student strategies
for multi-digit
subtraction
problems.
Instructor identifies
gaps in knowledge
through the Kahoot
activity which will
check the level of
student strategies
for multi-digit
subtraction
problems and ask
teachers to
distinguish between
an example and a
non-example.
Instructor gives
teachers an
opportunity to ask
clarifying questions
and state something
they have learned.
Instructor addresses
questions.
Teachers record notes
on the different
types of and levels
of strategies for
multi-digit
subtraction
problems.
Teachers engage in
the Kahoot activity.
Teachers answer
questions that ask
them to identify the
levels of strategies
for multi-digit
subtraction
problems.
Teachers will also
answer questions
that distinguish
examples versus
non-examples.
Teachers will receive
feedback from
Kahoot and the
instructor.
Teachers reflect on
how these terms
might inform their
practice.
198
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Table groups will
have an opportunity
to share something
that they learned or
questions they
might have.
Learning
guidance:
Learning
context
• Lecture
• Demo
70 Observe students
solving various
multi-digit
subtraction
problems.
Revisit and fine tune
notetaking
strategies while
watching video
clips that depict
distinguishing
characteristics of
different
strategies.
Write three multi-
digit subtraction
problems.
Complete the five-
step planning
template for a
multi-digit
subtraction
problem using one
of the problems
that the group
collaboratively
wrote.
Instructor
demonstrates the
steps for identifying
the level and types
of strategies
students use while
watching video
clips.
Instructor names and
distinguishes the
different levels of,
and types of
strategies students
use to solve multi-
digit subtraction
problems.
Instructor reviews
techniques for
taking notes of
strategies students
use while solving
multi-digit
subtraction
problems.
Instructor
demonstrates the
steps for writing
multi-digit
subtraction
problems by
focusing on the
Teachers take notes as
they watch video
clips of students
solving various
multi-digit
subtraction
problems.
Teachers discuss the
levels of and types
of strategies that
students use to
solve multi-digit
subtraction
problems.
Teachers review the
purpose of and
techniques for
taking notes while
observing students.
Teachers
collaboratively
write each of the
three multi-digit
subtraction
problems with the
support of the
facilitator.
Teachers will take
notes of the
planning template
199
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
structure and key
characteristics of
these problems.
Instructor
demonstrates the
process for
completing the
planning template
for multi-digit
subtraction
problems with an
emphasis on
designing the
closing discussion.
As the demonstration
lesson proceeds, the
instructor will refer
back to the notes on
student strategies to
support the
planning process
for the current
lesson as well as
future lessons.
Instructor prompts
teachers to discuss
what they learned
with their table
group.
to have a worked
example for a multi-
digit subtraction
problem.
Teachers discuss
possible closing
discussion themes
for the problem.
Teachers reflect on
what they learned
about multi-digit
subtraction
problems.
Teachers have an
opportunity to ask
clarifying questions
as instructor
monitors and
checks for
understanding.
Practice and
feedback:
The
learning
context
35 Write additional
multi-digit
subtraction
problems.
Select a problem,
complete the
planning template
for that problem,
Instructor asks
teachers to write
additional multi-
digit subtraction
problems.
Instructor prompts
table groups to
select one of the
Teachers
collaboratively
write multi-digit
subtraction
problems with their
table group.
Teachers complete the
planning template
200
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
and display it for a
gallery walk.
problems they
wrote to use to
complete the
planning template.
Instructor asks
teachers to put their
plan on a chart.
Facilitate a gallery
walk
Instructor facilitates a
discussion about the
different problems,
focusing on the
themes for the
closing discussion.
Instructor provides
feedback and gives
teachers
opportunities to ask
clarifying
questions.
to plan one of the
problems they
wrote.
Teachers make notes
of the levels of and
types of strategies
on the planning
template and
articulate possible
themes for the
closing discussion.
Teachers make a
poster with their
plan to share with
the whole group.
Teachers walk around
the room with their
table group and
discuss what they
see on the different
posters.
Teachers engage in a
discussion about
what they saw and
noticed.
Teachers have the
opportunity to ask
clarifying questions.
Authentic
assessment
16
Plan a lesson using
the planning
template for a
multi-digit
subtraction
problem including
the potential
Instructor asks
teachers to
complete a planning
template for a
multi-digit
subtraction
problem.
Teachers will
demonstrate their
knowledge of multi-
digit subtraction
problems and the
process for planning
those problems.
201
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
storylines for the
closing discussion.
Teachers may ask
questions or seek
assistance as
necessary.
Retention
and
transfer
5 Learners reflect on
the objectives for
the lesson and
identify and report
on what they have
learned.
Instructor asks
teachers to consider
the implications of
this knowledge on
math instruction in
their classrooms.
Instructor prompts
teachers to reflect
on their practice
and the implications
of this learning on
their practice.
Teachers reflect on
the objectives for
this module and the
implications for
their classroom.
Big ideas
5 The levels of and
types of strategies
used for solving
multi-digit
subtraction
problems are
presented, and the
five-step planning
process is
practiced.
Instructor asks
teachers to state
what they have
learned about the
levels and types of
strategies students
use to solve multi-
digit subtraction
problems, and the
five-step planning
process.
Instructor prompts
teachers to share
impressions and
wonders and ask
any questions they
may have so the
instructor may
address them.
Teachers state what
they have learned
about the levels and
types of strategies
students use when
solving multi-digit
subtraction
problems, and the
five steps for
planning lessons.
Teachers finish the
sentence prompts:
“Something that left
an impression on
me during this
module was…” and
“One thing I am
wondering about
is…”
Teachers (volunteers)
share their
impressions and
202
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
wonders with the
whole group.
Advance
organizer
for the
next
module
2 Refer to visual
representation of
the modules to
show the module
that was
completed and the
topic of the
module that will
follow.
Instructor asks
teachers to voice
any questions that
are lingering.
Given the topic of the
next module,
“Base-Ten Number
Concepts,” ask
teachers what they
are wondering and
what questions
might have been
triggered by the
topic.
Instructor will record
questions on a
chart.
Teachers consider
questions they
might have
concerning the
strategies that
children use when
solving problems
with larger
numbers.
Total time 180
Module 8, Base-Ten Number Concepts
Figure A9
Module 8, Base-Ten Concepts
Module Duration: approximately 180 minutes synchronous engagement
Introduction:
This is the eighth module of an 8-module course to teach teachers how to implement cognitively guided instruction (CGI) in their
classrooms. Throughout the course, teachers will learn about the different problem types and strategies that students use. They will
also learn about a process for strategically planning lessons and robust mathematical discussions. The purpose of the last module is
threefold. First, it is to teach teachers a way to teach students base-ten concepts through multiplication and measurement division
problems. Second, it is to familiarize teachers with the strategies children use when solving these problems, the tools that support
understanding, and for teachers to plan a lesson.
Module 1a:
Overview
Module 1b:
Designing the
Classroom
Space
Module 2:
Structure of
the Problems
Module 3:
Noticing
Strategies
(Join)
Module 4:
Noticing
Strategies
(Separate)
Module 5:
Structure of
Multiplication
& Division
Problems
Module 6:
Multi-Digit
Addition
Problems
Module 7:
Multi-Digit
Subtraction
Problems
Module 8:
Base-Ten
Number
Concepts
203
204
Learning Objective(s)
Terminal Objective:
● Given the background information of CGI and the Common Core State Standards,
teachers will identify the types of problems that support base-ten concepts and the
strategies that children use when solving multi-digit subtraction problems, write problems
and create culturally responsive lesson plans to implement in their classroom per the
rubric in the lesson template.
Enabling Objective(s):
● For a given grade level, teachers will be able to:
○ Know the meaning of direct modeling strategies for base-ten number problems
(multiplication and measurement division problems)
○ Know the meaning of counting strategies for base-ten number problems
(multiplication and measurement division problems)
○ Know the meaning of invented algorithms for base-ten number problems
(multiplication and measurement division problems)
○ Identify and distinguish the types of problems that support base-ten concepts
○ Identify and analyze strategies that children use to solve base-ten number
problems (multiplication and measurement division problems)
○ Synthesize strategies that students are using to create (write) problems and lessons
that respond to students’ lived experiences
○ Design lessons using the 5-step planning template for base-ten number problems
(multiplication and measurement division problems)
Learning Activities
● After introductions, attention activities and learning objectives, assess prior knowledge of
the types of problems that support base-ten concepts and the strategies that children use
to solve base-ten number problems.
● Teach any necessary prerequisite knowledge by providing definitions and examples and
nonexamples.
● Provide opportunities for learners to generate their own examples and nonexamples.
● Model the procedure for identifying the types of problems that support base-ten concepts
and the strategies that children use to solve base-ten number problems.
● Provide practice and feedback on identifying the types of problems that support base-ten
concepts (multiplication and measurement division problems) and the strategies that
children use to solve base-ten number problems.
● Model the procedure for planning a lesson for base-ten number problems (multiplication
and measurement division problems).
● Provide practice and feedback for planning a lesson for base-ten number problems
(multiplication and measurement division problems).
● Provide opportunities to transfer knowledge to the teachers’ own classroom practice.
Summative Assessment
● Given the various examples of CGI classrooms, teachers will identify the types of
problems that support base-ten concepts and strategies that children use to solve base-ten
number problems (multiplication and measurement division problems) and will write
205
problems and create lesson plans to implement in their classroom per the rubric in the
lesson template.
Lesson Materials
● Written resource: Math Common Core State Standards, grades K-5 (from Module 1a)
● Written resource: Appendix for Math Common Core State Standards (from Module 1a)
● Video clips from Children’s Mathematics: Cognitively Guided Instruction
● Presentation slides
● Handout with key slides
● Access to Kahoot site on internet (via phone or a computer)
● Handout with “Lesson Planning Template”
● Handout with graphic organizer: “Base-Ten Concepts”
● Charts/Posters
○ Agenda: Day 4
○ Learning Goals for Module 8
○ Burning Questions
Learner Characteristic Accommodations
Learners will complete this module in person. Teachers will sit in table groups to support small
group conversations and collaboration. Slides will be projected so that all may access the
materials. Teachers will also receive handouts with key slides on them to facilitate listening and
note taking. If needed, teachers may request further accommodations.
Facilitator’s Notes
This opening session of the course will present the background information about CGI. It is often
reassuring to know that this way of teaching math is not a fad. In fact, it is grounded in research
over the last 40 years and the research is continuing today. Teachers will also examine the Math
Common Core State Standards and find that problem types are weaved into the standards. Thus,
a change is warranted. Module 8 examines the types of problems that support base-ten concepts
and strategies that children use to solve base-ten number problems (multiplication and
measurement division problems). Throughout the course, teachers will have opportunities to
observe children, classrooms, and discuss ideas with colleagues. All sessions will be
synchronous and face-to-face. Teachers will be asked to present problems to their class and
return to subsequent sessions with student work.
Table A9 provides the instructional activities and details the instructional sequence, duration,
descriptions of the learning activities, instructor actions/decisions (supplantive events of
instruction) and learner actions/decisions (generative events of instruction).
206
Table A9
Instructional Activities, Module 8
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
Gain
attention
10 Discuss challenges in
teaching base-ten
concepts.
Discuss what base-ten
understanding
looks like in
students.
Instructor presents the
question:
“How many tens are
in the number
125?”
Instructor asks
teachers to discuss
the challenges
students face with
this question. The
point of the
conversation is the
difference between
the number in the
tens place versus
how many tens
make the number
125.
Instructor facilitates a
discussion about
challenges in
teaching base-ten
concepts.
Instructor prompts
teachers to name
what students with
base-ten
understanding
would be able to
do.
Teachers discuss the
question: “How
many tens are in
the number 125?”
Teachers participate
in a discussion
about challenges
they have
encountered in
teaching base-ten
concepts.
Teachers describe the
characteristics of a
student who has a
robust
understanding of
base-ten concepts.
Learning
objectives
4 The learning
objectives for the
module are posted
and reviewed.
In order to establish
relevance and to
facilitate learning,
ask learners to read
Teachers read the
terminal and
enabling learning
207
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
the terminal and
enabling learning
objectives.
Ask teachers to share
their personal
learning objectives
with a partner.
objectives for
themselves.
Teachers will share
their personal
learning objectives
with a partner.
Purpose for
learning
• Benefits
• Risks
3 Benefits:
• It is important to
know how to
develop base-ten
concepts through
word problems.
• This will support
teachers to develop
conceptual
knowledge by
focusing on the
number system, the
patterns of digits
that form numbers,
and the contexts
that help children
cultivate
understanding in a
way that makes
sense to them.
• Pre-planning
lessons supports
teachers to feel
more confident and
prepared to teach
problem solving
lessons.
Risks:
A teacher who does
not understand how
children develop
base-ten concepts
Instructor asks
teachers to discuss
areas they would
like to learn more
about regarding
base-ten concepts.
Instructor invites
teachers to record
any burning
questions on a post-
it and add it to the
“Burning
Questions” chart.
The instructor will
peruse the
questions and make
notes to address
these throughout
the sessions.
Teachers participate
in conversations at
their table and
reflect on their
experiences,
practice and areas
for
growth/learning.
Teachers may record
burning questions
on a post-it and add
it to the chart.
208
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
may become
frustrated by the
student’s surface
level
understanding.
Overview
• Prior
knowledge
• New
knowledge
• Learning
strategies
(What you
already
know,
what you
are going
to learn,
and how
you are
going to
learn it.)
12 Review prior
knowledge of
multiplication and
division problems.
Provide a brief
overview of the
module:
• Examine strategies
that students use by
watching videos of
students who
multiplication and
division problems
that develop base-
ten concepts.
• Demonstrate how
to write
multiplication and
division base-ten
problems.
• Demonstrate how
to plan a lesson for
base-ten problems
that utilizes
multiplication and
division problems. .
Instructor reviews the
different types of
multiplication and
division problems
that were presented
in Module 5.
Instructor explains
that multiplication
and measurement
division problems
support base-ten
concepts.
Instructor provides an
overview of the
lesson explaining
how the different
activities support
the module goals.
Instructor prompts
teachers to
articulate what they
will focus on
during this module.
Teachers review their
knowledge of
multiplication and
measurement
division problems.
Teachers discuss their
learning focus for
the module at their
table.
Teachers have an
opportunity to ask
for clarification or
elaboration on any
items.
Prerequisite
knowledge
18 Present the definition
of the different
levels of student
strategies for base-
ten problems.
Check knowledge of
the different types
of and levels of
Instructor explains
different levels of
student strategies
for base-ten
problems.
Instructor identifies
gaps in knowledge
through the Kahoot
Teachers record notes
on the different
types of and levels
of strategies for
base-ten problems.
Teachers engage in
the Kahoot activity.
209
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
strategies and
distinguish
examples versus
non-examples
through an activity
on Kahoot.
activity which will
check the level of
student strategies
for multi-digit
addition problems
and asks teachers to
distinguish between
an example and a
non-example.
Instructor encourages
teachers to ask
clarifying questions
and state something
they have learned.
Instructor addresses
questions.
Teachers answer
questions that ask
them to identify the
level of student
strategies for base-
ten problems.
Teachers will also
answer questions
that distinguish
examples versus
non-examples.
Teachers will receive
feedback from
Kahoot and the
instructor.
Teachers reflect on
how these terms
might inform their
practice.
Table groups will
have an opportunity
to share something
that they learned or
questions they
might have from
this activity.
Learning
guidance:
Learning
context
• Lecture
• Demo
70 Observe students
solving various
base-ten problems.
Revisit and fine tune
notetaking
strategies while
watching video
clips that depict
distinguishing
Instructor
demonstrates the
steps for
identifying the
level and types of
strategies students
use while watching
video clips.
Instructor names and
distinguishes the
Teachers take notes
as they watch video
clips of students
solving various
base-ten problems.
Teachers discuss the
levels of and types
of strategies that
students used to
210
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
characteristics of
different strategies.
Write three base-ten
problems.
Complete the five-
step planning
template for a base-
ten problem using
the problems that
the group
collaboratively
wrote.
different levels of,
and types of
strategies students
use to solve base-
ten problems.
Instructor
demonstrates
techniques for
taking notes of
strategies students
use while solving
base-ten problems.
Instructor
demonstrates the
steps for writing
base-ten problems
by focusing on the
structure and key
characteristics of
these problems.
Instructor
demonstrates the
process for
completing the
planning template
for base-ten
problems with an
emphasis on
designing the
closing discussion.
As the demonstration
lesson proceeds,
the instructor will
refer back to the
notes on student
strategies to
support the
planning process
solve base-ten
problems.
Teachers review the
purpose of and
techniques for
taking notes while
observing students.
Teachers
collaboratively
write base-ten
problems with the
support of the
facilitator.
Teachers will take
notes of the
planning template
to have a worked
example for a base-
ten addition
problem.
Teachers discuss
possible closing
discussion themes
for the problem.
Teachers reflect on
what they learned
about teaching
base-ten problems.
Teachers have an
opportunity to ask
clarifying questions
as instructor
monitors and
checks for
understanding.
211
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
for the current
lesson as well as
future lessons.
Instructor prompts
teachers to discuss
what they learned
with their table
group.
Practice and
feedback:
The
learning
context
35 Write additional base-
ten problems.
Select a problem,
complete the
planning template
for that problem,
and display it for a
gallery walk.
Instructor prompts
teachers to write
base-ten problems
with their table
group.
Instructor asks table
groups to select a
problem to use to
complete the
planning template.
Instructor asks
teachers to display
their plan to the
group.
Facilitate a gallery
walk
Instructor facilitates a
discussion about
the different
problems, focusing
on the themes for
the closing
discussion.
Instructor provides
feedback and gives
teachers
opportunities to ask
Teachers
collaboratively
write base-ten
problems with their
table group.
Teachers complete
the planning
template to plan
one of the problems
they wrote.
Teachers make notes
of the levels of and
types of strategies
on the planning
template and
articulate possible
themes for the
closing discussion.
Teachers make a
poster with their
plan to share with
the whole group.
Teachers walk around
the room with their
table group and
discuss what they
see on the different
posters.
212
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
clarifying
questions.
Teachers engage in a
discussion about
what they saw and
noticed.
Teachers have the
opportunity to ask
clarifying
questions.
Authentic
assessment
16
Plan a lesson using
the planning
template for a base-
ten problem
including the
potential storylines
for the closing
discussion.
Instructor asks
teachers to
complete a
planning template
for a base-ten
problem.
Teachers will
demonstrate their
knowledge of a
base-ten problem
and the process for
planning those
problems.
Teachers may ask
questions or seek
assistance as
necessary.
Retention
and
transfer
5 Learners reflect on
the objectives for
the lesson and
identify and report
on what they have
learned.
Instructor asks
teachers to consider
the implications of
this knowledge on
math instruction in
their classrooms.
Instructor prompts
teachers to reflect
on their practice
and the
implications of this
learning on their
practice.
Teachers reflect on
the objectives for
this module and the
implications for
their classroom.
Big ideas
7 The levels of and
types of strategies
used for solving
base-ten problems
Instructor asks
teachers to state
what they have
learned about the
Teachers state what
they have learned
about the levels and
types of strategies
213
Instructional
sequence
Time
(min)
Description of the
learning activity
Instructor
action/decision
(supplantive)
Learner action/
decision
(generative)
are presented, and
the five-step
planning process is
practiced.
levels and types of
strategies students
use to solve base-
ten problems, and
the five-step
planning process.
Instructor asks
teachers to reflect
on their learning
during this module
with an elbow
partner.
Instructor prompts
teachers to share
impressions and
commitments and
ask any questions
they may have so
the instructor may
address them.
students use to
solve base-ten
problems, and the
five steps for
planning lessons.
Teachers finish the
sentence prompts:
“Something that left
an impression on
me during this
module was…” and
“One thing I am
committed to
practice is…”
Teachers (volunteers)
share their
impressions and
commitments with
the whole group.
Teachers may ask any
lingering questions.
Total time 180
214
Appendix B: Materials for Module 3
Figure A10
Module 3 First Page Slide Handout
215
Figure A11
Module 3 Second Page Slide Handout
216
Figure A12
Problem Type Chart in CCSS Appendix
Figure A13
Student Strategies
217
218
Figure A14
Lesson Planning Template
Figure A15
Planning Reflection
219
220
Figure A16
Lesson Planning: Allison and Cookies
221
Appendix C: Keynote Slides for Module 3
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
Appendix D: Evaluation Administered at the End of Each Day of the Course
Open-ended questions:
1. What did you learn today? (L2)
2. What are you committed to trying? (L2)
3. What do you need to try this? (L2)
4. What lingering questions do you have? (L2)
5. What supported your learning today? (L1, L2)
6. What changes might be needed to better support your learning? (L1, L2)
Rate answers on a scale (1 = strongly disagree to 4 = strongly agree).
7. The activities supported my learning. (L1)
8. The course content is relevant to my classroom practice. (L1)
9. I would recommend this course to other teachers. (L1)
241
Appendix E: Evaluation Administered Delayed for a Period After the Program
Rate answers on a scale (1 = strongly disagree to 4 = strongly agree).
1. The course was applicable to my classroom practice. (L1)
What was (or was not relevant)
2. The time spent learning in the course was beneficial. (L1)
How might this course be improved?
3. I plan multiple problem-solving lessons each week. (L3)
What have you noticed about the planning process?
4. I (or the grade level team) feel confident writing new problems to pose. (L2)
What are some successes or challenges in writing new problems?
5. I recognize a variety of strategies that students use to solve problems in my
classroom. (L2, L3, L4)
What has impressed you about student strategies?
6. I use observations of student strategies to plan subsequent problem-solving lessons.
(L3, L4)
How have your observations of student strategies impacted your lesson planning?
7. My grade level team discusses lessons and student thinking during collaboration
meetings. (L3, L4)
What has been most beneficial during these discussions?
8. My grade level team discusses lessons and student thinking during collaboration
meetings. (L3, L4)
How has the instructional team supported you?
9. I have seen improvements in my students’ problem-solving skills. (L4)
242
In what ways have students made improvements?
Abstract (if available)
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Asset Metadata
Creator
Song, Lydia
(author)
Core Title
Bridging theory and practice: developing children’s mathematical thinking through cognitively guided instruction
School
Rossier School of Education
Degree
Doctor of Education
Degree Program
Educational Leadership
Degree Conferral Date
2022-08
Publication Date
08/03/2022
Defense Date
07/11/2022
Publisher
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(original),
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Tag
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Yates, Kenneth (
committee chair
), Hasan, Angela "Laila" (
committee member
), Hirabayashi, Kimberly (
committee member
)
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lydiason@usc.edu,songforlydia@gmail.com
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Tags
Cognitively Guided Instruction
mathematical thinking
orchestrating discussions
problem solving