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Moving from great to greater: Math growth in high achieving elementary schools - A gap analysis

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Moving from great to greater: Math growth in high achieving elementary schools - A gap analysis

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Content Running Header: Math Growth in High Achieving Elementary Schools MOVING FROM GREAT TO GREATER: MATH GROWTH IN HIGH ACHIEVING ELEMENTARY SCHOOLS – A GAP ANALYSIS by Susan Joy Shaw ____________________________________________________________________ A Dissertation Presented to the FACULTY OF THE USC ROSSIER SCHOOL OF EDUCATION UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF EDUCATION August 2018 Copyright 2018. Susan Joy Shaw Math Growth in High Achieving Elementary Schools ii Acknowledgements I am grateful for the assistance and support from the following people who enabled me to cross the finish line: • Program of Study Chair and Committee members: Larry Picus, Ruth Chung, and Kenneth Yates. • Colleagues who signed up and were willing to help with my research. • Cohort colleagues whose support certainly helped me, and we all made it through! • Friends who supported me throughout a long three years – you know who you are… • My three children who received a lesson in what life-long learning really means. • Most importantly, Tiri, my rock – thank you for picking me up when I was exhausted, for holding my hand when I was discouraged, and especially for keeping me sane all the way through. In Tiri’s words, “You have to earn it – they’re not going to give it to you!” Math Growth in High Achieving Elementary Schools iii TABLE OF CONTENTS Acknowledgements ............................................................................................................. ii List of Tables ...................................................................................................................... vi List of Figures ..................................................................................................................... vii Abstract ............................................................................................................................... viii CHAPTER ONE: INTRODUCTION ................................................................................. 1 Problem of Practice ....................................................................................................... 1 Related Literature.......................................................................................................... 3 Importance of Addressing the Problem ........................................................................ 6 Organizational Context and Mission ............................................................................ 8 Organizational Performance Status............................................................................... 10 Organizational Performance Goal................................................................................. 12 Stakeholders and Stakeholders’ Performance Goals .................................................... 12 Purpose and Research Questions .................................................................................. 12 Conceptual and Methodological Framework ................................................................ 14 Definitions..................................................................................................................... 15 Organization of the Dissertation ................................................................................... 16 CHAPTER TWO: LITERATURE REVIEW ..................................................................... 17 Assumed Knowledge Influences on Successful Teachers ............................................ 18 Assumed Declarative Knowledge Causes .............................................................. 19 Assumed Procedural Knowledge Causes................................................................ 22 Assumed Metacognitive Knowledge Causes .......................................................... 25 Assumed Motivational Influences on Successful Teachers .......................................... 29 Necessity for Effective Math Instruction ................................................................ 30 Assumed Organizational Influences on Successful Teachers ....................................... 34 Professional Learning Communities ....................................................................... 35 Summary ....................................................................................................................... 36 CHAPTER THREE: METHODOLOGY ........................................................................... 38 Conceptual and Methodological Framework ................................................................ 38 Assessment of Performance Influences ........................................................................ 40 Knowledge Assessment .......................................................................................... 40 Motivation Assessment ........................................................................................... 41 Organization Assessment ........................................................................................ 42 Participating Stakeholders and Sample Selection ................................................... 43 Sampling ........................................................................................................... 43 Recruitment ....................................................................................................... 45 Math Growth in High Achieving Elementary Schools iv Data Collection ............................................................................................................. 45 Interviews ................................................................................................................ 45 Observations ........................................................................................................... 46 Data Analysis ................................................................................................................ 47 Trustworthiness of Data .......................................................................................... 47 Role of Investigator................................................................................................. 47 Limitations and Delimitations....................................................................................... 48 Limitations .............................................................................................................. 48 Delimitations ........................................................................................................... 49 CHAPTER FOUR: FINDINGS .......................................................................................... 50 Research Question 1 – Influences ................................................................................. 50 Stakeholders’ Critical Behaviors ............................................................................ 53 Summary ........................................................................................................... 55 Assumed and Validated Knowledge and Skills Causes .......................................... 57 Assumed and Validated Motivational Causes ........................................................ 65 Assumed and Validated Organizational Causes ........................................................ 73 Summary of Gaps Found ................................................................................................. 77 Knowledge and Skills ................................................................................................ 77 Motivation .................................................................................................................. 78 Organization ............................................................................................................... 78 Summary .......................................................................................................................... 78 CHAPTER FIVE: SOLUTIONS, IMPLEMENTATION, AND EVALUATION ................ 80 Research Question 2 – Solutions ..................................................................................... 80 Knowledge and Skills ................................................................................................ 80 Motivation .................................................................................................................. 82 Cultural/Organizational Barriers ................................................................................ 84 Critical Behaviors ...................................................................................................... 86 Implementation ................................................................................................................ 89 Feedback .................................................................................................................... 89 Learning Community Time........................................................................................ 91 MAP Data .................................................................................................................. 91 Professional Development ......................................................................................... 91 Evaluation ........................................................................................................................ 92 Evaluating Level 4: Results ....................................................................................... 93 Evaluating Level 3: Behavior .................................................................................... 94 Evaluating Level 2: Learning..................................................................................... 94 Evaluating Level 1: Reaction ..................................................................................... 96 Summary .................................................................................................................... 96 Future Research ............................................................................................................... 98 Conclusion ....................................................................................................................... 99 Math Growth in High Achieving Elementary Schools v References ..............................................................................................................................101 Appendices A: Email Requesting Permission to Conduct Research ..................................................112 B: Interview Protocol ......................................................................................................113 C: Interview Questions Listed by Knowledge, Motivation, and Organizational Influences ..........................................................................................115 D: Observation Protocol .................................................................................................117 Math Growth in High Achieving Elementary Schools vi List of Tables Table 1. Grade 4-5 Math MAP Scores: Fall 2015-Spring 2016 .................................... 11 Table 2. Organizational Mission, Global Goal, and Stakeholder Performance Goals .. 13 Table 3. Summary of Assumed Needs of Teachers....................................................... 37 Table 4. Summary of Strategies for Validating and Assessing the Assumed Influences 44 Table 5. Stakeholders’ Critical Behaviors ..................................................................... 56 Table. 6. Assessment, Measurement, and Validation of Knowledge and Skills Influences ........................................................................................................ 66 Table 7. Assessment, Measurement, and Validation of Motivational Influences ......... 74 Table 8. Assessment, Measurement and Validation of Organizational Influences ....... 77 Table 9. Summary of Solutions Needed for Knowledge, Motivational, and Organizational Influences ................................................................................ 86 Table 10. Stakeholder Critical Behaviors: Current and Future Performance Indicators . 90 Table 11. Solution Themes and Implementation Plan ..................................................... 93 Table 12. Goals, Outcomes, and Methods According to Levels of Evaluation .............. 97 Math Growth in High Achieving Elementary Schools vii List of Figures Figure 1. Effective Mathematics Teaching Practices ....................................................... 32 Figure 2. Gap Analysis Overview .................................................................................... 40 Math Growth in High Achieving Elementary Schools viii Abstract This study examined the performance gap experienced by high achieving grade four and five students at the Kallang International Day School (KIDS) who were not demonstrating growth in math results from fall to spring as measured by the Measures of Academic Progress (MAP) test (NWEA, 2017). The purpose of this study was to apply the gap analysis problem- solving framework (Clark & Estes, 2008) to identify root causes of the knowledge, motivation and organizational factors that prevented teachers from effectively teaching 100% of the math content required for greater numbers of high performing students to demonstrate growth in the MAP test. The subjects of this study were teachers of grade four and five at the Kallang International Day School. Interview and observational data were collected and analyzed and used to validate and inform possible solutions for the knowledge, motivation and organization influences. Research-based solutions were recommended to close the knowledge, motivation and organization gaps, and included providing teachers with frequent, accurate, specific and timely feedback regarding their use of differentiated mathematics instruction for student groupings and using learning community time to collaboratively plan specific conceptual and skills based math lessons differentiated for high performing students. Critical behaviors of the stakeholders were also examined and recommendations included generating student math goals based on fall MAP results, teaching with the Effective Mathematical Teaching Practices, (NCTM, 2014) and using them to create plans for modifying classroom instruction to enhance high performing students’ progress. The outcomes of this study may be used by this school and others to improve the growth in math achievement for high performing students. Math Growth in High Achieving Elementary Schools 1 CHAPTER ONE: INTRODUCTION Problem of Practice Research on mathematical achievement of K-12 students in the United States has indicated there is a problem with elementary school students’ achievement in mathematics. The National Assessment of Educational Progress (NAEP) – the largest nationally administered assessment that tests what U.S. students know and can do – has communicated the results of student assessments and educational progress in grades 4, 8 and 12 in reading, math, science, writing and other subjects since 1973 (NAEP, 2012). Measuring the performance of over 50,000 students nationwide, the 2012 NAEP report revealed that students aged 9 and 13 made gains in math knowledge and skills compared with results since 1973; however, there were no significant changes in the performance of 17 year-olds. When compared with results from the previous Report Card in 2008, 9 year-olds and 17 year-olds did not make gains in math. Thirteen year-olds were the only group to make gains in math (NAEP, 2012, p. 29). Reports for 17 year-olds revealed that this age group has not made any significant gains in math since 1973. This represents nearly 40 years of testing with no progress. Results also indicated there is a common pattern of improvement for the lower performing students, although no pattern or trend for higher performing students was reported. It is clear from these statistics that there is a need to raise the levels of student achievement in the US across different age groups in mathematics. Further evidence for the need to raise the level of student achievement in mathematics in the U.S. can be found in the Trends in International Math and Science Study (TIMSS). TIMSS compares international results for 4 th and 8 th grade students’ achievement in math and science, and cognitive skills such as knowing, applying, and reasoning are measured Math Growth in High Achieving Elementary Schools 2 (Provasnik et al., 2016). The description of TIMSS International mathematics benchmarks are separated into four different categories: Advanced (625), High (550), Intermediate (475), and Low (400), with top scores indicated in parentheses. The U.S. TIMSS scores for 4 th graders, while improving from 518 to 539 between 1995 and 2015, fell in the TIMSS “intermediate” range of the international benchmarks (Provasnik et al., 2016, p. 9). The “high” benchmark was rated at 550. The difference between both benchmarks lies in students being able to apply basic math knowledge at the “intermediate” level, whereas in the “high” level students can apply their knowledge and understanding to solve problems. This U.S. score of 539 for 4 th graders was higher than 34 other global education systems and lower than the average scores of students in 10 other global education systems. If U.S. 4 th graders are not performing in the high benchmark at 550, then this indicates that the ability to apply mathematical knowledge and understanding is still beyond the grasp of this population of students. These data suggest that the U.S. is currently unsuccessful both at a national and international level in significantly raising the level of achievement in math in elementary students. The high performing 4 th grade students who demonstrated no growth in math in the past ten years, as indicated by the NAEP Report Card (2012) are representative of the wealth of mathematical talent in the US that is currently not being realized (Provasnik et al., 2016). A focus on the lack of achievement in math is most often directed towards learners who have learning deficits, come from low socioeconomic backgrounds, and/or are minorities (Becker & Luthar, 2002; Hoff, 2013; Jacob & Ludwig, 2008). However, there is a population of students who are high achievers from relatively high socio-economic backgrounds who also Math Growth in High Achieving Elementary Schools 3 do not achieve their potential in math; the achievement of these students was the problem of practice in this research study. Related Literature The impact of addressing this problem of practice is significant in terms of the societal influence mathematics has and the associated sense of urgency that comes with it. The need for more mathematicians in society has been well documented (National Research Council, 2013; Wagner, 2012). The National Academy of Sciences has been charged with providing advice on scientific and technical matters to the federal government, and one of their findings follows: Mathematical sciences work is becoming an increasingly integral and essential component of a growing array of areas of investigation in biology, medicine, social sciences, business, advanced design, climate, finance, advanced materials, and many more. This work involves the integration of mathematics, statistics, and computation in the broadest sense and the interplay of these areas with areas of potential application. All of these activities are crucial to economic growth, national competitiveness, and national security, and this fact should inform both the nature and scale of funding for the mathematical sciences as a whole. (National Research Council, [NRC] 2013, p. 2) Three more compelling reasons to ensure all students in schools are achieving in math are that economic growth, national competitiveness, and national security are dependent on mathematics as stated above. One of the recommendations from the NRC is: Recommendation 5-2: In order to motivate students and show the full value of the material, it is essential that educators explain to their K-12 and undergraduate Math Growth in High Achieving Elementary Schools 4 students how the mathematical science topics they are teaching are used and the careers that make use of them. (NRC, 2013, p. 127) Over the period of a normal 24-hour day, math is used in ways the regular human being does not understand or even consider. For example, predicting the weather, online shopping, and special effects in movies all involve the use of math (NRC, 2013). Students are curious and presenting math in ways which connect to their world helps connect them to mathematics. The limitations placed upon students is due to the lack of quality in K-12 mathematics education. If the preparation in mathematical sciences in K-12 is weak then the flow of talented students into the science, technology, engineering, and mathematical fields (STEM) is also limited. These limitations flow into the industry when numbers of experts in STEM are not being prepared which thereby restricts much needed innovations in these professions (NRC, 2013). The NRC recognizes that innovations in teaching practices are taking place in K-12 education and recommends that wider mathematical communities support these efforts in schools. Wagner (2012) supports these efforts in schools espousing the need for greater work in developing STEM education. Holding teachers accountable and having teacher evaluation based on evidence of improvement in students work over a school year is essential to improving access to STEM education for students (Wagner, 2012). Indeed, several countries including Chile, England, Canada, Portugal and Singapore have systems in place for teacher evaluation based on student achievement throughout the year (OECD, 2009). The Networking and Information Technology Research and Development (NITRD) program prepares periodic reviews involving government agencies and experts in the field to report to the President and Congress regarding research and development in Information Math Growth in High Achieving Elementary Schools 5 Technology (IT) as well as preparation of the future workforce. The last report (Executive Office of the President, 2015) stated that more than half of all new STEM jobs will be associated with information technology according to the Bureau of Labor Statistics (Executive Office of the President, 2015, p. 1). In the next seven years, more than 65% of the new jobs in STEM will be computer related creating a huge need for IT workers and by association a huge need to educate more children in STEM and IT through programs in schools (Executive Office of the President, 2015, p. 43). The report recommended leveraging investments in IT by ensuring educators are made aware of the research on the effectiveness of STEM and IT education. As part of the ninth recommendation in the report, the need for mathematicians and especially for high performing learners is highlighted. It is important to develop and implement model programs that will “attract the most talented young people to study IT. These will be the future innovators and leaders” (Executive Office of the President, 2015, p. 44). Gifted education is an area of research where some guidance for nurturing high performing learners in math to become future innovators and leaders may be found. Gifted education has focused on the need to place greater attention on high ability students and the impact these students may have on the future of our planet. The specific problem in this study focused on high performing students. These are the students who are highly capable with great potential and, yet, are not usually considered within the top 5-10% of the population as gifted (Gagne, 1993). There is an absence of research for this population of high performing students; therefore, reference to research with gifted students may be appropriate. Nevertheless, scholars in the field of gifted education have long espoused the importance of ensuring the brightest students have access to education that is challenging and Math Growth in High Achieving Elementary Schools 6 stimulating (Colangelo, Assouline, & Gross, 2004; Davis, & Rimm, 2004; VanTassel-Baska, 2003), although accompanying actions have not been forthcoming. A Nation at Risk, published in 1983, highlighted the failure of American students to match the achievement levels of similar international students. Ten years later, a new report: National Excellence: The Case for Developing America’s Talent (1993), outlined how America’s most talented youth continued to be neglected, and another ten years after that a national research-based report for advanced learners, A Nation Deceived: How Schools Hold Back America’s Brightest Students (Colangelo et al., 2004), was released. There is a wealth of information regarding best practice for teaching and instructing math with gifted students and even whole curriculums designed especially for gifted learners in math (Gavin et al., 2007; VanTassel-Baska, 2003). Within the group of learners who are high performers with high potential but who are not generally considered to be gifted, there is a distinct lack of research. Importance of Addressing the Problem Francis Su, former president of the Mathematical Association of America, believes everyone should experience the joy of mathematics. He also posits math is essential for human flourishing (Su, 2017). The business of schools is about human flourishing and building a love of mathematics in students should be an essential standard in every school. At the Kampong International Day School (KIDS) where this study took place, every student is expected to achieve their potential capability (KIDS 1 System Focus, 2015). The impact of an effective mathematical program is felt throughout the school as students’ progress, and the 1 Kampong International Day School (KIDS) is a pseudonym. An actual URL is not provided in order to protect the organization’s identity. Math Growth in High Achieving Elementary Schools 7 skills and concepts mastered in elementary school impact student success in mathematics throughout middle and high school. Recent researchers have argued that mathematics has become a domain of the elite: those who teach mathematics deliberately make it seem hard and high school math teachers want students to fail so students understand they might not have what it takes to be like their teachers – successful in math (Boaler, 2016; Su, 2017). The message is that only those who are exceptionally bright are capable of studying mathematics; therefore, Boaler (2016) and Su (2017) have challenged those who teach math in high schools and universities to make mathematics more accessible to everyone who is interested. They argue that mathematics is not hard; anyone can study math and should be able to do it. All students, young and old, are capable of experiencing the joy and flow that comes with finding success in a subject. Not only should anyone be able to study math and be successful but it is also imperative that they do (Boaler, 2016; Su, 2017). As long as mathematics remains a domain of the elite and inaccessible to the majority, future problem solvers are not being cultivated. If students believe math is too hard for them it closes the door on many avenues of study, including the science, technology, engineering, and medical professions. Successful schools are what we want for every child (Wagner, 2012). The McKinsey Report studied high performing school systems in an effort to identify what made the system successful (Barber & Mourshed, 2007). The findings revealed that three things matter most: 1. Getting the right people to become teachers, 2. Developing them into effective instructors, and 3. Ensuring that the system is able to deliver the best possible instruction for every child. (Barber & Mourshed, 2007, p. 2) Math Growth in High Achieving Elementary Schools 8 When followed, these best practices could be applied regardless of culture and location, may take place over a relatively short period time, and will potentially have a great impact when applied in failing school system settings. The implications of addressing this problem of practice are clear for educators at the Kampong International Day School (KIDS) and the wider educational profession – there is a significant need to increase the effectiveness and quality of instruction in math so that all students, including the high performing students, can make associated gains in achievement. Greater numbers of future mathematicians, scientists, engineers, IT experts, etc., can be created, especially if the focus changes to a universal message that mathematics can be learned, experienced, and achieved by anyone regardless of ability. Mathematics is not simply a pursuit for the elite. Organizational Context and Mission The vision of the Kampong International Day School (KIDS) is to be “…a model of global excellence where learners excel and are challenged to become innovative thinkers and productive, compassionate citizens ready to lead tomorrow’s world (KIDS 2 System Focus, 2015). Situated in South East Asia, KIDS is attended by approximately 3,800 students who are 3-18 years of age and hail from more than 50 different countries. KIDS is a coeducational, non-profit day school offering an American style of curriculum that includes an international perspective. The clientele consists of expatriates; 60% of whom hold U.S. passports. Many others are from Asia and Europe, and some from South America and 2 The actual URL is not provided in order to protect the identity of the organization. Math Growth in High Achieving Elementary Schools 9 Oceania. The students’ families are predominantly wealthy, in top-level jobs, and can afford more than $30,000 in school fees per child per year. The U.S. statistics quoted previously from the NAEP and TIMSS are appropriate for consideration in the Kampong International Day School (KIDS) context because KIDS students are high performing in many subject areas and are measured against U.S. benchmarks. The benchmarks at the grade four and five level occur through the Measures of Academic Progress (MAP) tests which are administered twice yearly across grades three to nine. Since one third of students attending KIDS transfer to other schools every three years and most of those move to another American school or return to the U.S. (KIDS Admissions, 2015), it is important for parents to know how competent their children are when compared to students in the U.S. It is also important for the school to know that KIDS students are high performing across all subject areas when compared with students in the U.S. as this is a major marketing tool for the school. The Kampong International Day School takes its position in South East Asia very seriously as it competes to provide an education that includes three school pillars to outline the school’s guaranteed outcomes for student learning: Pillar 1: Learners excel Pillar 2: Learners have talents Pillar 3: Learners are nurtured. (KIDS System Focus, 2015) In theory, these school pillars apply directly to high performing learners. However, in practice those learners may be missing out on the challenging learning experiences due to them. Much has been written about the lack of challenge in schools for high performing children and how they are perceived as a lost generation because educators are not tapping Math Growth in High Achieving Elementary Schools 10 into their capacity for high levels of learning (Kell, Lubinski, & Benbow, 2013). Within this relatively privileged group of students enrolled at KIDS, there is also room for doubt about how effectively the needs of its high performing students are being met. Organizational Performance Status The Measures of Academic Progress (NWEA, 2017) test is a norm-referenced computer adaptive test used in many schools to measure student growth and achievement from fall to spring. Being a computer adaptive test means that as each student answers a question correctly, the computer adapts the test to ask the student a more difficult question. Conversely, if the student answers the test incorrectly the computer adapts the test to ask the student a less difficult question. The Measures of Academic Progress (MAP) test is used at KIDS to measure not only students’ individual achievement but also compare KIDS students’ achievement with other students in the U.S. or in similar school systems. This comparison is essential to demonstrate that students at KIDS can excel. The organization’s problem of practice is how to improve growth in math performance for high performing students in grades 4-5 where there is currently a decline in math achievement levels as measured by the Measures of Academic Progress (MAP) test after 32 weeks of instruction. This decline in math achievement is outlined in Table 1. Although 60% of the grade 5 students scored in the 80 th percentile or above during fall and then again in spring, this issue is still related to the problem of practice. No growth is indicated if the number of students scoring in the 80 th percentile and above remains the same after 32 weeks of instruction. Math Growth in High Achieving Elementary Schools 11 Table 1. Grade 4-5 Math MAP Scores: Fall 2015-Spring 2016. Math MAP Scores Percentile Ranking Fall 2015 (Percentage of Students) Spring 2016 (Percentage of Students) Difference Between Fall and Spring 4 th grade >80 65 49 -16 5 th grade >80 61 62 +1 The status of performance related to the problem of practice is that 100% of the students who score in the 80 th percentile and above in fall should still be scoring in the 80 th percentile and above in the spring after 32 weeks of instruction. Thus, the numbers of students scoring in the upper percentile bands should increase after 32 weeks of instruction. As part of the organizational performance status it is important to ensure teachers are teaching the mathematics that students need to demonstrate growth on the MAP test. The impact of the performance on achieving the organization’s mission cannot be underestimated. In order to provide for all learners achieving excellence, KIDS needs to ensure high performing students are being challenged in math and are able to demonstrate growth between fall and spring. The school’s vision—ensuring all students excel, are challenged to become innovative thinkers and productive citizens—is currently not being realized (KIDS System Focus, 2015). Elementary-aged students need to achieve at high levels as evidence of the school’s vision to create innovative thinkers who are ready with the tools necessary for their world. One could argue that the current performance at the upper tiers of the MAP test does indicate high levels of achievement; however, if students start the year within those high bands then they should finish the year demonstrating stronger growth. Failure to achieve its vision could result in KIDS losing clientele, and without the clientele there is no school. There are many schools in South East Asia that provide an international education. If KIDS wants to retain its status as one of the top performers, then Math Growth in High Achieving Elementary Schools 12 demonstrating growth in math for high performing students between fall and spring is not a luxury; rather, it is a necessity. Organizational Performance Goal The KIDS’s organizational performance goal for May 2018 was to ensure, that all grade 4–5 students will score at higher levels in the MAP test after 32 weeks of math instruction. The elementary school deputy principal (i.e., the current researcher) for grades four and five established this goal based on the MAP results and discussions with teachers and administrators regarding these results. The need for this work arose from the KIDS schoolwide goal to ensure all students attain excellence. The achievement of this goal will be measured by the results in MAP testing from fall 2018 to spring in May 2019. Stakeholders and Stakeholders’ Performance Goals The stakeholders at KIDS include teachers, students, parents, and administrators. The stakeholders of focus for this study were the teachers at KIDS. Teachers contribute to the achievement of KIDS’s goal by teaching the students. They are responsible for instructing the students using effective and appropriate methods to improve the growth of all learners in math. The organizational goal is that by May 2019, all students will demonstrate growth at higher levels in math after 32 weeks of instruction as measured by the fall and spring MAP test results. Table 2 illustrates the alignment of organizational vision, performance goal, stakeholder goal, and stakeholders’ critical behaviors at KIDS. Purpose and Research Questions The purpose of this research was to conduct a gap analysis to identify root causes of the knowledge, motivation, and organizational factors that prevent teachers from achieving Math Growth in High Achieving Elementary Schools 13 Table 2. Organizational Vision, Performance Goal, Stakeholder Goal, and Stakeholders’ Critical Behaviors Organizational Vision To be a model of global excellence where learners excel and are challenged to become innovative thinkers and productive, compassionate citizens ready to lead tomorrow’s world. Organizational Performance Goal By May 2019, all grade 4-5 students will achieve at higher levels in math after 32 weeks of instruction as measured by the biannual Measures of Academic Progress (MAP) test results. Stakeholder Goal: Teachers By May 2019, all grade 4-5 mathematics teachers will effectively teach 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) test. Stakeholders Critical Behaviors For teachers to achieve the stakeholder goal above, they need to demonstrate performance of the following critical behaviors: Critical Behavior 1. Grade 4-5 classroom teachers will use student fall MAP results to generate math goals for each student to improve their growth in math over the year. Critical Behavior 2. Grade 4-5 classroom teachers will effectively teach the Common Core State Standards in mathematics. Critical Behavior 3. Grade 4-5 classroom teachers will meet in PLCs to discuss student results and create plans for modifying classroom instruction to enhance student progress. the stakeholder performance goal of effectively teaching 100% of the math content required for greater numbers of students to demonstrate growth in the Measures of Academic Progress (MAP) test. This gap analysis was conducted to ascertain the knowledge, motivational, and organizational influences that interfere with achieving the organizational performance goal at KIDS: By May 2019, all students will achieve at higher levels in math after 32 weeks of instruction as measured in the biannual Measures of Academic Progress (MAP) tests. This analysis focuses on causes for this problem due to gaps in teachers’ knowledge and skill, motivation, and organizational influences related to students demonstrating higher levels of growth in mathematics. First, a list is generated of possible or assumed influences that will be examined systematically. Second, focus is placed on actual or validated influences that are Math Growth in High Achieving Elementary Schools 14 interfering with achieving the stakeholder goal. While a complete gap analysis would generally focus on all KIDS stakeholders, for practical purposes, the stakeholders of focus in this analysis were the grade 4–5 teachers at KIDS because teachers have the greatest potential for influencing student achievement. The following research questions informed this gap analysis to address knowledge and skills, motivation, and organizational causes and solutions for teachers: 1. What are the knowledge, motivation, and organization influences for teachers not meeting their goal of effectively teaching 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) test? 2. What are the recommended knowledge, motivation and organizational solutions teachers might implement to effectively teach 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) test? Conceptual and Methodological Framework Developed by Clark and Estes (2008), gap analysis is a systematic analytical method that helps the user to clarify organizational goals and identify holes (gaps) of missing entities. The gaps between the actual performance level and the preferred level of performance within an organization formed the conceptual framework used in this study. The methodological framework used was a qualitative case study that utilized descriptive statistics. Personal knowledge and current literature informed the assumed influences in knowledge and skills, motivation, and organization that prevent the achievement of the organizational goal. The methods used to assess these influences were: student test scores, teacher interviews, and Math Growth in High Achieving Elementary Schools 15 observations as well as a comprehensive literature review. The data collected included: assessment results from MAP testing, answers given by teachers in their interviews, and observation notes. The focus for final recommendations and evaluation was on research- based solutions that were selected in a comprehensive manner. Definitions The following terms were defined for use in the study: Assessment: Methods used to evaluate student achievement, readiness, and educational needs of students. Adaptive. Adaptive assessments are administered by computer. The computer adapts the level of difficulty of the questions offered to the student by giving them a more difficult question if they answer correctly and a less difficult question if they answer incorrectly. Benchmark tests. Assessments given regularly to check that students are making growth towards academic standards. Formative. Assessments given frequently throughout a unit of study so instruction can be adjusted to meet student needs. Formative assessments are “for” learning. Summative. Assessments given at the end of a unit of study or period such as the end of a semester or year to measure student achievement. Summative assessments are “of” learning. Gap analysis: Comparing actual performance with desired performance. Growth: The amount to which students met or exceeded the projected growth target between two testing periods Math Growth in High Achieving Elementary Schools 16 High performing students: High ability students that perform at high levels (e.g. Students that achieve at the 80 th percentile and above on the Measures of Academic Progress (MAP) test) MAP: Measures of Academic Progress (MAP) is a computer adaptive assessment for measuring academic growth of students in different content areas. Professional Learning Communities (PLCs): Working groups designed to foster collaboration among teachers. Organization of the Dissertation This dissertation is organized into five chapters. Chapter One focused on introducing the problem of practice. A review of current literature is presented in Chapter Two. Within this chapter key topics are addressed, including the need to raise levels of student achievement for high performing students, the role of standardized testing, and the importance of math curriculum, math instruction, and grouping modalities and their impact on student achievement. The Methodology is presented in Chapter Three, which includes a discussion of the participants, data collection, and analysis of the data. The data and results are assessed and analyzed in Chapter Four. Finally, Chapter Five provides research-based solutions for closing the perceived gap as well as recommendations for an implementation and evaluation plan for use in the future. Math Growth in High Achieving Elementary Schools 17 CHAPTER TWO: LITERATURE REVIEW Much of the current literature on student achievement in mathematics in elementary education focuses on low ability, low achieving students in mathematics, and how to improve their levels of achievement. The literature on mathematics achievement does not appear to answer questions about high achieving math learners in elementary school. How does one identify a higher achievement bar for high performing students in math, and ensure these students receive effective instruction to enable them to reach that bar? In particular, what steps are necessary to take a great school that already has high achievement results for elementary school students in mathematics and make it even greater? The focus of this problem of practice as outlined in Chapter One is on increasing the growth of high achieving elementary school students in mathematics by identifying performance gaps that prevent teachers from teaching the material in an effective way. The three most important causes of performance gaps involve: (1) the knowledge and skills of the people concerned (in this case, the teachers) (2) their motivation to achieve the goal; and (3) organizational obstacles that may prevent the goal of higher performance from being achieved (Clark & Estes, 2008). The key areas that are explored and analyzed in this chapter include the teachers’ knowledge, motivational, and organizational influences that impact their ability to achieve higher levels of mathematical performance in their students throughout the school year. A review of the research includes: teachers’ knowledge of the CCSSM and how to use test data to inform instruction, the importance of rigorous math curriculum and its relationship to student math achievement, and different grouping modalities and the research regarding their effectiveness and impact on student achievement. An assumed metacognitive influence is the Math Growth in High Achieving Elementary Schools 18 effect of standardized testing on teaching and learning, which is also reviewed in this chapter. Teachers’ motivational influences are explored and analyzed regarding the necessity for effective math instruction, and include the level of confidence teachers have in their ability to teach CCSSM, the provision of expert and knowledgeable practitioners motivated to deliver math instruction to elementary students, as well as the correlation between understanding the mathematical content they teach and their students’ achievement in math. Finally, organizational influences on teachers include the feedback process teachers experience related to their goal and the atmosphere of competition that may exist in a high performing institution. These areas are also examined regarding the role of professional learning communities and their impact on student achievement. Assumed Knowledge Influences on Successful Teachers The intent of the knowledge analysis was to explore the assumed causes that may be the source of the knowledge gap related to this problem. As part of a thorough knowledge analysis, it is important to validate specific causes of the performance gap by considering personal knowledge, interviews, learning, motivation, and organization theory, through a review of the literature. Rueda (2011) cautioned researchers that: “Acting on assumed causes when they are in fact incorrect only compounds the problem, (p. 78). Therefore, principle- based solutions are presented only once as the causes of the performance gap are validated. Four major types of knowledge were identified by Krathwohl (2002): declarative, conceptual, procedural, and metacognitive. Declarative knowledge includes factual and conceptual knowledge. Factual knowledge encompasses the ability to recognize and recall basic facts and details pertaining to a given topic whereas conceptual knowledge pertains to knowledge of categories or principles. Procedural knowledge assesses how to do subject Math Growth in High Achieving Elementary Schools 19 specific skills and techniques, and metacognition involves awareness of one’s own cognition coupled with awareness of cognition in general (Krathwohl, 2002). Assumed Declarative Knowledge Causes The assumed factual knowledge cause in this problem of practice is that successful teachers need to know the math they are expected to teach and also know the Common Core State Standards for Mathematics (CCSSM). The assumed conceptual knowledge cause is that successful teachers know and understand the importance of a rigorous math curriculum and its relationship to student math achievement for high performing students. Math knowledge of teachers. There is a direct correlation in the research literature between teachers understanding of the mathematical content they teach and their students mathematical achievement (Ball, Hill, & Bass, 2005; Ball, Thames & Phelps, 2008; Hill, Rowan, & Ball, 2005; Le Sage, 2012). According to the National Research Council (2010): “…successful mathematics teachers need preparation that covers knowledge of mathematics, of how students learn mathematics, and of mathematical pedagogy that is aligned with the recommendations of professional societies” (as cited in LeSage, 2012, p. 16). Ball et al. (2005) conducted a qualitative study analyzing teacher’s work which included writing and grading assignments, communicating with parents, and creating homework to explore what teachers need to know about mathematics so they could be successful in the classroom. They found that the more specific mathematical knowledge teachers had significantly predicted the gains their students could make. When teachers in the top quartile were compared with those with average scores, two to three weeks of instruction were gained by the students of teachers in the top quartile (Ball et al., p. 44). The opportunity to gain two to three weeks of instruction during the year could have a significant impact on Math Growth in High Achieving Elementary Schools 20 student learning. Further research was recommended by the researchers to dispel the possibility that aptitude for teaching or teachers’ general knowledge leads to the gains in student achievement rather than the specific mathematical content knowledge of teachers. The importance of a rigorous math curriculum. The National Council of Teachers of Mathematics (NCTM, 2014) described a rigorous math curriculum as: …a coherent sequencing of core mathematical ideas that are well articulated across the grades. Such an effective curriculum incorporates problems in contexts from everyday life and other subjects whenever possible. These tasks engage students and generate interest and curiosity in the topics under investigation. (p. 4) The CCSSM fit the aforementioned description and have been adopted by many states in the U.S. with the belief that, if implemented with fidelity, these standards have the potential to greatly increase student achievement in math (Johnsen & Sheffield, 2013). In addition to the CCSSM, the National Council for Mathematics (NCTM) has also developed eight Standards for Mathematical Practice which are designed to be used alongside the content standards. However, while the CCSSM sets high standards in terms of the expectations for students, this curriculum is not challenging enough for mathematically talented students. According to the Center for Gifted Education (2004), these students “…need curriculum that moves at a faster pace, is less repetitive, goes deeper into important thought processes, and covers a broader set of ideas than that encountered in a commonly used textbook series,” (as cited in Assouline & Lupkowski-Shoplik, 2011, p. 223). Tomlinson (2005) posited there is a need for curriculum for gifted students that is “…meaning-making, rich and high level” (p. 160). Tieso (2005) noted there is an urgent need Math Growth in High Achieving Elementary Schools 21 for teachers to inspect common curricular applications to ensure learning tasks are “…authentic, original, and challenging,” (p. 82). Curriculum compacting is a method commonly used with high performing math students to better meet their needs in the regular classroom (Reis et al., 1992). Students are often pretested before a unit so they can demonstrate prior mastery of concepts. The remaining unknown curriculum is then compacted so students learn new concepts and skills as opposed to practicing concepts and skills they have already mastered. Curriculum compacting is a highly effective practice that ensures students have the opportunity to engage with new material and keep their interest level high (Reis et al.). Time is often saved since these learners do not need the same amounts of time on basic curriculum that their classmates do. This “saved” time can be used on individual math projects of interest, which are another effective way of supplementing the curriculum for high ability students (Reis et al.). Hook, Bishop, and Hook (2006) introduced a high-quality curriculum into low performing schools (and one high) with the expectation that student achievement as measured by National Performance Rankings (NPR) in each school would improve. Findings of their study revealed that all participating schools improved on average from 22.9 to 26.9 NPR points over the course of the study through the use of the high-quality curriculum. The high performing school improved from 74 to 92 NPR, which suggests a quality curriculum can also benefit high performing districts. Some limitations of the study included the focus on a curriculum from 1998 which was before the Common Core; nevertheless, the finding that a high-quality curriculum improved performance while a poorly designed, inadequate curriculum could not be compensated for by simply providing professional development for Math Growth in High Achieving Elementary Schools 22 teachers (Hook et al)., adds to the body of literature emphasizing the importance of a high- quality curriculum for all students. Slavin and Lake (2008) conducted a best evidence synthesis examining research on different math programs so educators could make decisions on programs likely to impact students learning. Findings of their study revealed a compelling lack of evidence that the type of textbook used makes a difference on student achievement, and they recommended more research is needed in this area. The type of instruction, not the curriculum or program, was found to have the highest impact on student learning (Slavin & Lake). This emphasis on the type of instruction indicates an awareness of procedural knowledge necessary for teachers to deliver high-quality instruction. Assumed Procedural Knowledge Causes The assumed procedural knowledge issue is that successful teachers know how to use math test data to inform instruction for students, are also knowledgeable about the effectiveness and impact of grouping on student achievement, and know how to group all students accordingly. Ability grouping practices in mathematics instruction. For the purpose of this literature review, ability grouping is defined as an instructional practice whereby students are grouped by ability based on some form of pretest or similar with the purpose of learning in a group more suited to their ability level and that the placement in the group is not permanent (Steenbergen-Hu, Makel, & Olszewski-Kubilius, 2016). Research on ability grouping is substantial, with most studies indicating gains in achievement through its use (Kulik & Kulik, 1992; Slavin, 1988; Steenbergen-Hu et al., 2016; Tieso, 2005). Ability grouping practices used for mathematical instruction have the capacity to enhance student achievement Math Growth in High Achieving Elementary Schools 23 for all learners according to several researchers (Kulik & Kulik, 1992; Slavin, 1988; Steenbergen-Hu et al., 2016; Tieso, 2005). Research on different types of grouping used with high ability students indicate that, by adapting pedagogy from the gifted education field, substantial gains in achievement can be the result for high performing learners with increases in achievement for average and low ability students as well (Kulik & Kulik, 1992; Lou et al., 1996; Slavin, 1988; Tieso, 2005). Grouping by ability means that all students can learn at their own pace; high achieving students move more quickly through the curriculum and low achievers are able to receive extra support (Slavin, 1988). Research findings suggest the need for ability grouping to be flexible so that students are able to move in and out of groups based on their prior knowledge. Adaptation of the curriculum, instructional methods and materials to meet the needs of the group members are also necessary to ensure student performance is enhanced (Kulik & Kulik, 1992; Lou et al., 1996; Slavin, 1987; Steenbergen-Hu et al., 2016; Tieso, 2005). Specific ability grouping practices reviewed in the current research included: between-class, within class, cross grade subject grouping, and clustering. Between-class grouping is defined as placing students in the same grade into high, medium and low groups based on their pretest achievement. Effects of this type of grouping were conflicting. Kulik and Kulik (1992) found achievement gains for high ability learners. Steenbergen-Hu et al. (2016) found minimal effects on achievement with between-class grouping but suggested that previous research may have underestimated the effects of between-class grouping on achievement. Steenbergen-Hu et al. (2016) recommended further research on this grouping type and its effects on mathematics achievement. Math Growth in High Achieving Elementary Schools 24 Within-class grouping is similar to between-class grouping, except that the ability groups are contained within the regular classroom. This type of grouping was found to have positive effects on student learning (Kulik & Kulik, 1992) particularly in content areas such as science and math (Lou et al., 1996). Small groups of 3-4 members were found to be more effective than larger groups (Lou et al., 1996). Slavin’s (1987) synthesis of grouping research continues to inform more recent research, having been cited nearly 1,500 times in a 30-year period. Slavin found that within-class grouping had significant impact on all students’ achievement in mathematics and interestingly, the effect was greatest for low achievers. More recently, Steenbergen-Hu et al’s (2016) meta analyses revealed a small but significant impact on student achievement for all levels of ability, reinforcing the findings of Slavin (1987). Cross grade subject grouping was also found to have significant effects on student achievement (Kulik & Kulik, 1992; Steenbergen-Hu et al., 2016). This grouping practice involves pre-testing students then grouping them in high, medium and low ability groups across an entire grade level for one subject at a time. Researchers recommended more study in this area is needed to further generalize the results. Steenbergen-Hu et al. (2016) listed cluster grouping as a form of within class grouping. Cluster grouping is defined as a cluster of several high performing students placed into one classroom in order to benefit from a teacher with experience teaching high performing students (Pierce et al., 2011). Rogers (1991) analyzed syntheses of research and found that “…the effect size for cluster grouping was .62, which translated to an estimated gain of approximately 6 academic months for gifted students in cluster groups” (as cited in Pierce et al., 2011, p. 573). Math Growth in High Achieving Elementary Schools 25 There is a preponderance of research that supports that ability grouping has positive effects on student achievement not only for high ability students but also for students of all abilities (Kulik & Kulik, 1992; Slavin, 1988; Steenbergen-Hu et al., 2016; Tieso, 2005). Nevertheless, opponents have argued that ability grouping ensures low achievers have poor peer models, low teacher expectations and slow instructional pace (Slavin, 1988) and some studies indicated harmful effects on achievement (Braddock & Slavin, 1992). Feldhusen and Moon (1992) posited: Appropriate grouping, acceleration of instruction to the students’ level of readiness, teachers who can create truly challenging classroom instructional activities and help students rise to the challenge, and association with peers of equal ability in a warmly supportive educational climate free of negative peer pressures – these are the ingredients of excellent instruction for our most able students. (p. 66) Pierce et al. (2011) also highlighted the dearth of research into the effects of ability grouping on mathematical achievement and recommended further research in this area. Assumed Metacognitive Knowledge Causes The assumed metacognitive knowledge issue is that successful teachers understand and can reflect on their impact on student learning through the role of standardized testing in education and its relationship to student achievement in math. The effects of standardized testing on teaching and learning with high performing students. In terms of real world relevance, basic proficiency in mathematics as demonstrated by standardized test results in the elementary school is essential so the skills and concepts can be built upon when moving through middle and then high school. Later in college, mathematics proficiency skills are required and are the basis for many tasks as an Math Growth in High Achieving Elementary Schools 26 adult. “Citizens who cannot reason mathematically are cut off from whole realms of human endeavor. Innumeracy deprives them not only of opportunity but also of competence in everyday tasks” (National Research Council, 2001). Assessment has the power to change teaching and learning (Earl, 2013); therefore, successful teachers ensure students develop a solid mathematics skills and concept foundation by utilizing standardized test results and classroom assessments to demonstrate growth in student achievement. The effect of the U.S. state standardized testing programs on instructional practices of elementary teachers with gifted students revealed that the instructional practices used does not assist gifted students to achieve their potential (Moon, Brighton, & Callahan, 2003). Although this problem of practice is not specifically about gifted students, the lack of research regarding high performing students means some answers and direction may be found and considered by examining research for gifted learners. Findings of research conducted by Moon et al. (2003) revealed that, as the pressure to obtain better test scores on yearly standardized tests in the U.S. increased, the level of instructional practice and by association, effective teaching, decreased. “Teachers seem to believe that the best method for preparing students for the state test is to simulate the testing experience in classroom instruction. Consequently, classroom lessons focus on isolated skills and tend to emphasize facts and rules” (p. 52). Teachers reported teaching to the tests by adjusting the curriculum sequence to better suit the timing of the tests, increasing the use of worksheets, teaching test-taking strategies, and requiring students to review and practice similar test items. This skills development focus is to the detriment of greater depth and integration of concepts within the curriculum and the development of higher order thinking skills necessary for all students but particularly Math Growth in High Achieving Elementary Schools 27 gifted students, (Krathwohl, 2002; Moon et al., 2003). Often high ability students sabotage the tests, deliberately underperforming due to frustration with the limited instruction from teachers (Moon et al., 2003). As a result of standardized testing, similar instructional practices were reported by teachers indicating movement towards a one-size-fits-all curriculum (Moon et al.). Therefore, standardized tests results should be treated with caution for high ability learners since they may not adequately reflect the ability level of students. Moon et al. (2003) cautioned, “Teachers’ lack of freedom to experiment with curricula may also affect their willingness or ability to explore innovative instructional strategies for use in developing this nation’s talent pool” (p. 50). If students are not learning at high levels then they will not be graduating with the expertise to produce, create and innovate for the benefit of their fellow American and global citizens as was mentioned in Chapter One. A plethora of research is available regarding the negative effects of standardized testing on student achievement where students have learning disabilities, are minorities and/or from socioeconomically disadvantaged backgrounds (Horn, 2003; Steele & Aronson, 1995). In contrast, there is little research available regarding the effects of standardized testing on student achievement regarding students who are relatively wealthy, privileged, and achieve at high levels. A study of the effects of standardized testing on this specific group of learners is recommended by this researcher as an area for further exploration. MAP testing. The computer adaptive MAP test is used to compare the achievement levels of KIDS students with their U.S. counterparts. Described on the KIDS website, “MAP is a formative assessment, meaning that the score does not count towards your child’s grade Math Growth in High Achieving Elementary Schools 28 but rather informs our teachers about the instructional levels of the students in their classes” (KIDS, MAP FAQ, 2017). The MAP test addresses the 2010 CCSSM and aligns well with the KIDS math program for the upper elementary school. Teachers at KIDS are expected to teach to the CCSSM and therefore it would seem that students should be making more growth than outlined in the definition of this problem of practice. KIDS teachers are also expected to differentiate their instruction so each student, no matter at what level of performance, demonstrates growth over the course of the school year. This growth is observable on grades’ 4 – 5 common assessments based on the CCSSM designed by classroom teachers however that same growth is limited on the MAP assessments results. Bjorklund-Young and Borkoski (2016) questioned the research base of the MAP tests and noted there are no empirically reviewed articles available regarding MAP data. Additionally, Bjorklund-Young and Borkoski pointed out that the paucity of research that is available regarding the validity of the assessments is sponsored by the Northwest Evaluation Association (NWEA) which designs the tests, indicating a conflict of interest with the results. In communication with NWEA, it was difficult to gain specific information regarding content validity. Nevertheless, a 2016 study of the MAP alignment with CCSSM by WestEd revealed that “…overall, across all grades, 99 percent of the items in the NWEA CC item pool sampled for Mathematics were aligned (Strong + Partial) to the CCSS in mathematics at or immediately adjacent to the target grade” (personal communication, December 2017). A review of the MAP informational materials available online revealed that the MAP tests have been utilized in schools for more than 40 years, which amounts to a substantial quantity of data related to students tested over the 40-year period. While aligned to the Math Growth in High Achieving Elementary Schools 29 CCSSM, MAP tests are available that are also aligned to individual U.S. state standards. Testing support includes documentation informing the user of ways in which to utilize the test data to measure student growth. NWEA cites seven points used in the MAP test to ensure student growth is measured accurately and fairly: 1. Aligns test questions to content standards 2. Uses a vertical scale of measurement 3. Matches question difficulty level to student ability 4. Uses a deep pool of questions to increase precision 5. Ensures fairness through empirical bias and sensitivity reviews 6. Defines the purpose of the assessment 7. Provides context for growth. (NWEA, 2017, p. 8) Significant support is also offered through the NWEA website, including instructions for teachers on how to read student profiles and use them to set goals for individual students, information for parents, and access information for administrators reviewing school-wide data (NWEA, 2017). Assumed Motivational Influences on Successful Teachers Utility value described by Eccles and Wigfield (1995) is one of three values assumed to have an influence on motivation (as cited in Clark & Estes, 2008, p. 95). Connecting values to work goals enhances teacher’s commitment and the utility value in this case is that successful teachers are able to recognize the value in providing effective math instruction to students. High ability students are already performing at high levels in math, therefore, successful teachers see the value of and are confident in their ability to teach CCSSM to those high ability learners. Math Growth in High Achieving Elementary Schools 30 In addition, successful teachers have high levels of self-efficacy which is another assumed influence on their motivation (Bandura, 1994). Teachers need to be confident in their ability to be expert and knowledgeable practitioners. It is important that teachers believe that effective instruction can make a difference to their students’ growth and achievement in mathematics. Necessity for Effective Math Instruction Changes needed in instructional practice for 21 st century learners. With the rapid growth in technology as well as access to the internet and social media, students have greater access to information today than in the past. It is hard to credit that twenty years ago less than one percent of the global population were internet users. Fast forward twenty years and more than half of the global population now use the internet (Internet World Statistics, 2017). “Perhaps the most challenging dilemma for teachers today is that routine cognitive skills, the skills that are the easiest to teach and easier to test, are also the skills that are easiest to digitize, automate and outsource” (Schleicher, 2012, p. 11). The line between procedural knowledge and motivation can become blurred when the teaching skills or procedures affect motivational issues of classroom teachers. Today’s teachers need to be able to use procedural knowledge skills to personalize learning and deliver content to a diverse range of students. Most importantly, teachers must have the motivation to recognize the “…need to place much greater emphasis on enabling individuals to become lifelong learners, to manage complex ways of thinking and complex ways of working that computers cannot take over easily” (p. 11). These are massive changes for teachers to take on when half of the teaching profession began their career during the time when internet use was unknown (National Centre for Education Statistics, 2014). Math Growth in High Achieving Elementary Schools 31 Changing instructional practices is not as easy as it sounds. Brighton et al. (2005) found that for practices to change “…individual and peer reflection, an informed, supportive educational community, and the self-evaluation of prior assumptions about teaching and learning” were necessary (p. xiv). Teachers who are already in the process of changing their instructional practices benefit from “consistent, differentiated coaching, substantial time, and honest, informed feedback about their efforts” (p. xiv). Thus, teachers who collaborate regarding instruction in teams demonstrate improved student achievement (Quintero, 2017; Ronfeldt, Farmer, McQueen, & Grissom, 2015). What has not changed, however, is the need for teachers passionate about education, who are experts and knowledgeable in their field. The most important qualities of a good mathematics teacher are: someone who is: “(a) passionate about teaching mathematics; (b) responds to students’ individual needs; (c) gives clear explanations; (d) uses scaffolding rather than providing answers; (e) encourages positive attitudes towards mathematics; (f) shows an awareness of each student’s prior knowledge” (Attard, 2011, p. 375). According to Anthony and Walshaw (2009), and Hattie (2003), the most powerful influence on student’s engagement in mathematics is the effectiveness of the teacher (as cited in Attard, 2011, p. 366). Effective teaching. The National Council of Teachers of Mathematics (NCTM) defines effective teaching as: “…teaching that engages students in meaningful learning through individual and collaborative experiences that promote their ability to make sense of mathematical ideas and reason mathematically” (NCTM, 2014, p. 7). To further assist teachers, the NCTM also provides a list of Effective Mathematics Teaching Practices (see Figure 1). Although procedural in nature, these are the skills teachers need to demonstrate Math Growth in High Achieving Elementary Schools 32 Source: National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. Figure 1. Effective Mathematics Teaching Practices Math Growth in High Achieving Elementary Schools 33 that lead to greater self-efficacy in teachers when teachers know they can teach high performing students well and are motivated to teach this section of the school population. It is important that teachers teach through the lens of these mathematics practices so they can: (1) create the conditions necessary to support the effective teaching practices; (2) implement strategies designed to attain higher levels of mathematics achievement for all students; (3) be educated on the difference between unproductive and productive beliefs in teaching mathematics; and (4) utilize strategies that engage students in mathematical thinking, reasoning, and sense making to significantly strengthen teaching and learning (NCTM, 2014). A greater emphasis on shifting from teaching only content knowledge to the blended and explicit teaching of content knowledge and math practices has taken place in recent years. Findings of research by Hattie et al. (2017) on visible learning in mathematics adds credibility to the need for effective teachers, recommending that teachers at the very minimum do the following: (1) indicate what the goals for the lesson are and the measures for achieving them, and (2) focus on the learning and change instruction if student learning does not occur. Effective practitioners make mathematics learning visible for students by continuing to refer to the goals of the lesson throughout the lesson to underscore to students the what, why and how of the lesson. This, in turn, allows students to take ownership of their learning (Hattie et al., 2017). Defining the different phases of learning and understanding how to scaffold these for different learners are also tasks expert and knowledgeable mathematical practitioners do. Instructional goals for every lesson should include a combination of surface learning, deeper learning and transfer learning. Beginning with surface learning ensures Math Growth in High Achieving Elementary Schools 34 students have a foundation on which to build as they move toward deeper learning of concepts. This deeper learning is then consolidated and applied in new and different ways as transfer learning (Hattie et al.). Assumed Organizational Influences on Successful Teachers It is important that organizational processes are defined along with material resources as the absence of these can prevent performance goals from being achieved (Clark & Estes, 2008). In addition, Gallimore and Goldenberg (2001) suggested that cultural models and cultural settings influence achievement of performance goals: “Cultural models are so familiar they are often invisible and unnoticed by those who hold them” (p. 47). The organization needs to ensure teachers know what the goal is and that teachers have the resources needed to achieve the goal. The assumed cause related to the cultural model in this problem of practice is that KIDS has an exceptionally high achieving faculty who may contribute to an atmosphere of competition and a fear of failure related to teaching high performing students. This barrier may prevent teachers from reaching out to their colleagues for help when needed or sharing their expertise amongst the faculty. Cultural settings are also important and are defined as occurring “whenever two or more people come together, over time, to accomplish something” (Sarason, 1972, as cited in Gallimore & Goldenberg, 2001, p. 47). The assumed cause related to the cultural setting is that the organization has an effective communication process to share necessary information with teachers. Processes also need to be in place to ensure teachers get timely, concrete feedback about their performance with respect to the goal. Math Growth in High Achieving Elementary Schools 35 Professional Learning Communities An organizational construct helpful in many schools for breaking down barriers and creating systems for collaborative practice is that of professional learning communities (DuFour, Eaker & DuFour, 2005) where teachers work in collaborative teams. Professional learning communities (PLCs) when implemented well and supported through resources have the potential to help students make significant improvements in achievement (DuFour et al., 2005; Jacobson, 2010; Schmoker, 2004, 2009; Wiliam, 2007). PLCs are most effective when teachers “build effective curriculum-based lessons and units together which they routinely refine together based on common assessment data,” (Hiebert & Stigler, 2017; Schmoker, 2009). PLCs, hereafter referred to as Learning Communities (LCs), have been in place in the elementary school at KIDS for the past several years. They have contributed to a greater alignment of curriculum and teaching practices across the school. LCs in the elementary school also create an annual goal based on a core content area such as reading, writing, or math to achieve as a team and this process is fully established in the elementary school. One of the problems with LCs is that the perceived benefits are not realized if they are not implemented properly and given adequate support in terms of funding, time and administrative support. Recent research has focused on the difference between LCs that are high performing and LCs that are not performing (Wiliam, 2007). One may form LCs of teachers and have them meet; however, it cannot be assumed that this meeting will result in data scrutiny, collaboratively produced lessons, and alteration in teaching practices for the result of higher levels of student achievement. In the elementary school, particularly in grades four and five which are the focus for this study, it is essential that the effectiveness of LCs is explored as an organizational influence on successful teachers. LCs can be used as an Math Growth in High Achieving Elementary Schools 36 existing organizational construct to ensure teachers receive the timely, concrete feedback about their performance teaching CCSS in mathematics and have opportunities to reflect, share and improve their practice. Summary This chapter explored and analyzed the key areas that impact effective math instruction and the ability of students to achieve higher levels of mathematical performance throughout the school year. A summary of sources found in the literature regarding the assumed needs of teachers, is provided in Table 3. Chapter Three will discuss the Methodology applied to carry out the gap analysis based on personal knowledge of the organization (KIDS) and the related literature. Math Growth in High Achieving Elementary Schools 37 Table 3. Summary of Assumed Needs of Teachers Assumed Needs Research Literature Knowledge (Declarative) • Teachers know the Common Core State Standards (CCSS) in mathematics. (Factual knowledge) • Teachers know how to use math test data to inform instruction for students. (Factual knowledge) • Teachers know the importance of rigorous math curriculum and its relationship to student math achievement. (Conceptual knowledge) Assouline & Lupkowski-Shoplik, 2011; Hook, Bishop & Hook, 2006; Johnsen & Sheffield, 2013; NCTM, 2014; Reis et al., 1992; Slavin & Lake, 2008; Tieso, 2005; Knowledge (Procedural) • Teachers are knowledgeable about the effectiveness and impact of grouping on student achievement and can group all students accordingly. Braddock & Slavin, 1992; Feldhusen & Moon, 1992; Kulik & Kulik, 1992; Lou et al., 1996; Pierce et al., 2011; Rogers, 1991; Slavin, 1988; Steenbergen-Hu et al., 2016; Tieso, 2005 Knowledge (Metacognitive) • Teachers understand how their behavior impacts student learning. Horn, 2003; Krathwohl, 2002; Moon, Brighton & Callahan, 2003; Steele & Aronson, 1995 Motivation (Self-efficacy) • Successful teachers are confident in their ability to teach CCSS in mathematics to students. Eccles, 2009; Pintrich 2003; Schleicher, 2012; Motivation (Expectancy value) • Successful teachers believe they are capable of effective instruction that can make a difference to their students’ growth and achievement in mathematics. • Successful teachers believe students should demonstrate growth in math achievement. • Successful teachers understand and value the correlation between understanding the mathematical content they teach and their students’ mathematical achievement. Anthony & Walshaw, 2009; Attard, 2011; Ball & Forzani, 2010; Ball, Hill & Bass, 2005; Ball, Thames & Phelps, 2008; Brighton et al. 2005; Hattie, 2003; Hattie et al., 2017; Hill, Rowan & Ball, 2005; Le Sage, 2012; National Research Council, 2010; NCTM, 2014; Ronfeldt, McQueen & Grissom, 2015; Organizational Barriers (Cultural models) • Working in a high achieving school may create atmosphere of competition and fear of failure amongst teachers. Gallimore & Goldenberg, 2001; Kirschner et al., 2009 Organizational Barriers (Cultural settings) • There is a process that ensures that teachers get timely, concrete feedback about their performance with respect to the goal. Anderman and Anderman, 2009; Clark & Estes, 2008; DuFour, Eaker & DuFour, 2005; Gallimore & Goldenberg, 2001; Hiebert & Stigler, 2017; Jacobson, 2010; Reeves 2016; Schmoker, 2004, 2009; Wiliam, 2007. Math Growth in High Achieving Elementary Schools 38 CHAPTER THREE: METHODOLOGY Purpose and Research Questions The purpose of this study was to conduct a gap analysis to ascertain the knowledge and skills, motivation, and organizational needs necessary to ensure that high performing upper elementary school students at the Kampong International Day School are making progress in mathematics growth as measured by the results of the MAP test which is administered twice each year. A list of assumed causes of the problem was created through an review of related literature, followed by a systematic examination of actual or validated causes. While a complete gap analysis would focus on all stakeholders, for practical purposes the stakeholders in this analysis were the teachers of students in grade four and five. The research questions that informed the gap analysis focused on knowledge and skills, motivation, and organizational causes and solutions for teachers: 1. What are the knowledge, motivation, and organization influences for teachers not meeting their goal of effectively teaching 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) test? 2. What are the recommended knowledge, motivation and organizational solutions teachers might implement to effectively teach 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) test? Conceptual and Methodological Framework Gap analysis is an eight-step, systematic, analytical framework developed by Clark and Estes (2008). Gap analysis enables an organization to clarify goals and identify the gap Math Growth in High Achieving Elementary Schools 39 between the actual performance of an organization and its ideal performance. Application of the gap analysis in this study utilized assumed causes of the performance gap based on personal knowledge of the organization (KIDS) and the related literature. Validation of the assumed causes was conducted by utilizing surveys, personal interviews, a review of related literature, and content analysis. Then, solutions based on research were recommended and evaluated comprehensively. The first step in the gap analysis method is to define the broad organizational goals that drive the organization (Clark & Estes 2008). While the goals must be stated in broad terms, there also needs to be adequate specificity to guide the organization’s daily functioning. In the second step, stakeholder goals are developed which relate directly to the organizational goal. Step three explores the stakeholder goals by analyzing the gap in performance while considering the change necessary to achieve the organization and stakeholder goals. Three factors affect organizational performance: knowledge, motivation, and organizational factors. Categorizing performance problems into one of these three groups enables organizations to identify the specific problems affecting stakeholders while remaining within the context of the overarching performance goal of the organization (Clark & Estes 2008). Step four involves identifying assumed causes of the gaps from step three, which were framed as knowledge, motivation or organizational factors. Step five involves the validation of the root causes of the identified gaps. Step six occurs when the causes are validated so that solutions can be identified. Once possible solutions are identified, they are implemented as step seven. Step eight enables one to evaluate the solutions. A flow chart of gap analysis is provided in Figure 2. Math Growth in High Achieving Elementary Schools 40 Source: Clark, R. E., & Estes, F. (2008). Turning research into results: A guide to selecting the right performance solutions. Atlanta, GA: CEP Press. Figure 2. Gap Analysis Overview In the current study, the final step in the gap analysis process—evaluation—was enhanced by applying Kirkpatrick and Kirkpatrick’s (2008) four levels of training evaluation: reaction, learning, behavior, and results. The four levels were helpful for identifying the goals, indicators, and assessment methods necessary to determine whether the high performing students in grades four and five were demonstrating growth in their learning of mathematics as measured by the MAP test. Assessment of Performance Influences Knowledge Assessment A review of the literature revealed five possible knowledge influences, as displayed previously in Table 2. The first knowledge influence—Teachers know the Common Core Math Growth in High Achieving Elementary Schools 41 State Standards (CCSS) in mathematics—was assessed through interview questions asking participants about the CCSSM. The CCSSM were also assessed as part of the classroom lesson observations. The second knowledge influence—Teachers know how to use math test data to inform instruction for students—was addressed through interview questions focusing on how to use test data to inform instruction for students. PLC observations were used to assess this influence. Both assumed influences represented factual knowledge. Conceptual knowledge was another possible knowledge influence—Successful teachers know the importance of rigorous math curriculum and its relationship to student math achievement. This was assessed through interview questions related to math curriculum and its influence on student achievement. One of the possible knowledge influences was procedural knowledge—Successful teachers are knowledgeable about the effectiveness and impact of grouping on student achievement and can group all students accordingly. To validate this influence, interview questions focused on the effectiveness and impact of grouping on student achievement and questions related to teachers’ grouping methods for students in math. Classroom lesson observations were also used to assess this influence. The final possible knowledge influence was metacognitive knowledge—Successful teachers understand how their behavior impacts student learning. This was assessed through interview questions and classroom lesson observations. Motivation Assessment Four possible motivational influences were revealed in the literature; one regarding self-efficacy and three regarding expectancy value. The assumed influence in self-efficacy is teachers are confident in their ability to teach CCSSM to students. This was assessed through Math Growth in High Achieving Elementary Schools 42 interview questions about the teachers’ level of confidence in this area as well as through classroom lesson observations. In terms of expectancy value, the first influence—Successful teachers believe they are capable of effective instruction that can make a difference to their students’ growth and achievement in mathematics—was measured through interview items, such as: “In what ways do you believe your instruction makes a difference to your students’ achievement in mathematics?” The second motivational influence related to expectancy value—Successful teachers believe students should demonstrate growth in math achievement—was validated by interview questions asking teachers about the importance of students’ demonstrating growth in math achievement. The third motivational influence related to expectancy value was that teachers understand and value the correlation between understanding the mathematical content they teach and their students’ mathematical achievement. This was assessed through interview questions. Organization Assessment The literature review conducted in chapter two revealed two possible organizational influences; one organizational barrier for cultural models and one for cultural settings. The assumed influence for the cultural model is that working in a high achieving institution may create an atmosphere of competition and a fear of failure and therefore teachers might not ask for help when needed. Assessment of this influence took place through interview questions. The assumed cultural setting barrier was that there is a process that ensures that teachers get timely, concrete feedback about their performance with respect to the goal. Assessment of this influence was through interview questions about whether teachers believe there is a Math Growth in High Achieving Elementary Schools 43 process for receiving feedback about their CCSSM teaching performance. A summary of the assumed influences and assessment strategies is in Table 4. Participating Stakeholders and Sample Selection Grade four and five teachers of mathematics were selected as the stakeholders of focus for this research study because they were responsible for the mathematics instruction for students in grade four and five. Purposeful sampling was the method used because these teachers were the ones who could give the information necessary for this research study. At present, there are 13 grade four mathematics teachers and 14 grade five mathematics teachers, for a total of 27 teachers. Of these 27 teachers only those who have taught mathematics to grade four or five students at KIDS for two or more years were invited to participate. This researcher deemed it important that teachers have sufficient experience teaching mathematics in the grade level at KIDS to participate in the study. In this way, the researcher could ensure all participants were familiar with the KIDS curriculum for their grade level and familiar with the administration of the MAP test. Sampling To effectively determine the knowledge, motivation and organizational culture issues that lead to high performing students in grades four and five not making expected levels of growth in mathematics, the sample for this study was grade four and five classroom teachers at KIDS. The data collection strategies proposed included interviews and observations and the criteria for sampling was the same for each data collection strategy listed as follows: Math Growth in High Achieving Elementary Schools 44 Table 4. Summary of Strategies for Validating and Assessing the Assumed Influences Assumed Needs of Successful Teachers Validation Strategies Knowledge (Declarative) • Teachers know the Common Core State Standards (CCSS) in mathematics. (Factual knowledge) • Teachers know how to use math test data to inform instruction for students. (Factual knowledge) • Teachers know the importance of rigorous math curriculum and its relationship to student math achievement. (Conceptual knowledge) Observation: Evidence of knowing CCSSM in classroom lesson. Interview: Teachers answered questions about the CCSS in mathematics. Observation: Evidence of using math data to inform instruction during classroom lesson. Interview: Teachers answered questions about how to use MAP results to inform instruction for students. Interview: Teachers answered questions related to the importance of rigorous math curriculum and its relationship to student achievement. Knowledge (Procedural) • Teachers are knowledgeable about the effectiveness and impact of grouping on student achievement and can group all students accordingly. Observation: Evidence that teachers are knowledgeable about grouping during classroom lesson. Interview: Teachers answered questions related to the effectiveness and impact of grouping on student achievement. Teachers answered questions related to their grouping methods for students in math. Knowledge (Metacognitive) • Teachers understand how their behavior impacts student learning. Observation: Evidence that teachers understand how their behavior impacts student learning. Interview: Teachers were asked to self-evaluate how their behavior impacts student learning in math. Motivation (Self-efficacy) • Teachers are confident in their ability to teach CCSS in mathematics to students. Observation: Behavior indicates teachers’ confidence in their ability to teach mathematics during classroom lessons. Interview: Teachers answered questions regarding their confidence in their ability to teach CCSSM to students. Motivation (Expectancy value) • Teachers believe they are capable of effective instruction that can make a difference to their students’ growth and achievement in mathematics. • Successful teachers believe students should demonstrate growth in math achievement. • Successful teachers understand and value the correlation between understanding the mathematical content they teach and their students’ mathematical achievement. Interview: Teachers answered questions regarding their capability of delivering effective instruction that makes a difference to students’ growth and achievement in mathematics. Interview: Teachers answered questions about the importance of their students demonstrating growth in math achievement. Interview: Teachers answered questions about correlations between their understanding of the mathematical content they teach and their students’ mathematical achievement. Organizational Barriers (Cultural models) • Working in a high achieving school may create atmosphere of competition and fear of failure amongst teachers. Interview: Teachers answered questions regarding their perceptions of an atmosphere of competition and fear of failure within grades 4-5. Organizational Barriers (Cultural settings) • There is a process that ensures that teachers get timely, concrete feedback about their performance in teaching CCSSM. Interview: Teachers answered questions about whether they believe there is a process that ensures that they get timely, concrete feedback about their performance in teaching CCSSM. Math Growth in High Achieving Elementary Schools 45 Criterion 1. Teachers who teach mathematics to grade four or five students (n=27) Criterion 2. Teachers who have taught mathematics to grade four or five students at KIDS for two or more years (n=21/27). Recruitment Participants were recruited by invitation and sample emails provided in Appendix A. All mathematics teachers of grade four and five students who met the criteria outlined above were contacted for possible participation in the interviews and observations. All participants were expected to participate in both data collection methods: interviews and observations. At the time of recruitment, teachers were informed that this researcher, as their supervisor, would not be conducting the interviews and observations. Data Collection Once the sample population was selected the two data collection instruments used were interviews and observations. Use of two data collection instruments ensured triangulation of data for purposes of trustworthiness. Permission was sought from the University of Southern California’s Institutional Review Board (IRB) before data collection began. Interviews Interview Protocol. The interview protocol followed a combination of a semi structured and a highly-structured format as defined by Merriam and Tisdell (2016). The interview was semi-structured in that specific data were required from the respondents and most of the interview was guided by a list of questions (see Appendix B). A copy of the interview questions listed by critical areas, knowledge, motivation and organizational influences is provided in Appendix C. The interview was highly structured in that the Math Growth in High Achieving Elementary Schools 46 wording and order of the questions was predetermined. The rationale for the interview protocol was to begin with an introduction to the study that gave the respondents enough information to understand the importance of their responses and to ensure they felt comfortable before they were asked to answer questions. Respondents were given an opportunity to ask clarifying questions before the interview began. Data collection: Interviews were conducted on an individual basis at a time of the participant’s choosing. Each interview was approximately 50 minutes in duration and conducted by a nominee who was not a direct supervisor of the participant. Permission was sought to audio record and to take notes during the interview. Following each interview, the interviewer made notes analyzing the interview process. These notes were later used to inform the data analysis. All interviews were recorded on a smart phone using the Voice Recorder application, uploaded into Speechmatics, and then transcribed for data analysis. Observations Observation protocol. The observation protocol was created with the purpose of observing differences in teaching in grade four and five mathematics classes (see Appendix D). A simple structure whereby observations were recorded on one side of the page and comments on the other allowed the researcher to write exactly what was observed and make comments at the same time. The comments helped inform later analysis of the data. Each observation was marked in time by recording which part of the lesson – beginning, middle or end of the observation time was observed. The researcher’s assistant completed the observations. Data collection. Each observation took place during fourth and fifth grade classrooms mathematical instruction time. The duration of the observations was Math Growth in High Achieving Elementary Schools 47 approximately 25 minutes with the researcher’s assistant taking notes during that time. All possible identification marks were removed before sharing the observation data with the researcher. Data Analysis Different strategies were used to analyze interview and observation data. Interviews were transcribed and coded according to the knowledge, motivation and organizational categories. The observation analysis was useful for triangulating the interview data and to check if the theory as outlined through the interview transferred to practical use of mathematical instructional methods for students in the classroom. The interview and observation analysis helped validate and inform possible solutions for the knowledge, motivation and organization influences. Trustworthiness of Data It was of paramount importance to ensure that the data collected was trustworthy. Three ways in which this researcher ensured the data was trustworthy were: (1) triangulation of data between interviews and observations; (2) assurance of anonymity and confidentiality; and (3) member checks. Role of Investigator As a deputy principal in the elementary school it was imperative that teachers did not feel compromised by their supervisor conducting this type of research study. The researcher’s role in this gap analysis was to conduct a problem-solving investigation to improve the organization’s performance. Potential interests this researcher had in the results of this study were to find out what practices were leading to high performance students reaching their potential in mathematics. Any potential confusion by members of KIDS of the Math Growth in High Achieving Elementary Schools 48 dual roles of this researcher in this study were addressed through LC meetings, email, phone calls, face-to-face meetings and other methods as appropriate. It was important there was minimal potential for the participants in this investigation who were in a subordinate role to feel coerced or pressured to participate. For example, teachers might feel their decision might affect performance evaluations or job advancement. To ensure that teachers understood this researcher’s role as an investigator rather than as an employer the following steps were taken. This researcher: • Ensured that steps were in place to preserve the anonymity of all participants. • Arranged to have another person obtain informed consent from subjects who were subordinates. • Avoided situations where there was access to teachers’ identifying information. • Ensured that the voluntary nature of participation and the right to not participate in the project was clearly understood. • Ensured complete confidentiality of information, identity and data. • Obtained permission to use documentation or data that was produced for other institutional purposes. Limitations and Delimitations Limitations One limitation of this study was its small sample size of teachers. Due to the nature of the criteria for sampling: teachers who teach mathematics to grade four or five students, and teachers who have taught mathematics to grade four or five students at KIDS for two or more years, only six teachers of grades four and five mathematics greed to participate. However, since all the teacher respondents met the aforementioned requirements, this researcher Math Growth in High Achieving Elementary Schools 49 perceived the six teachers provided a reasonable sampling ratio and a valid cross section of teacher responses. Another limitation was that teachers volunteered; therefore, they may have already been willing to share their experiences and had positive perceptions towards high performing students and their achievement in mathematics. They might also have been teachers who felt confident in their teaching ability and content knowledge of mathematics. A third limitation of this study was that it was conducted over a two-month period. A longer time frame in which to study the impact of instructional methods, content knowledge and ability of teachers to utilize effective mathematical teaching practices towards improving their students’ end of year MAP scores might have allowed for a more in-depth study. Delimitations A delimitation of this study was the location of the school and the specific demographics. Participants came from the same international school and their students represented a generally high income, high achieving demographic. Caution is recommended when the results of this study are generalized to other settings. However, this researcher believes that there are international schools and private schools with a similar demographic of relatively wealthy, high achieving students who may benefit from this research in ensuring that their high performing students are indeed demonstrating growth in mathematics throughout the year. Math Growth in High Achieving Elementary Schools 50 CHAPTER FOUR: FINDINGS The purpose of this chapter is to report the results and findings from analysis of the data collected. The Clark and Estes Gap Analysis Framework (2008) was used to help identify whether knowledge, motivation or organizational factors were barriers to the lack of growth in math for high ability students. Quantitative and qualitative data were collected through classroom observations and interviews. Then the data were triangulated and analyzed in order to better understand both the knowledge, motivation, and organizational factors as well as the stakeholder critical behaviors that influence the math achievement of high performing students. Research Question 1 – Influences What are the knowledge, motivation, and organization influences for teachers not meeting their goal of effectively teaching 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) test? In order to investigate reasons for teachers not meeting their goal of effectively teaching 100% of the math content required for greater numbers of high performing students to demonstrate growth on the MAP test, observations and interviews were conducted. Six teachers voluntarily participated in the observations and interviews to establish if the root cause for the teachers not meeting their goal was due to knowledge, motivation, and/or organizational barriers. For the observations, four 4 th grade and two 5 th grade teachers were observed teaching a math lesson for a period of approximately 25 minutes each. The purpose of the observation was to provide evidence that teachers: Math Growth in High Achieving Elementary Schools 51 • Know the Common Core State Standards (CCSS) in mathematics (Knowledge #1) • Know how to use math test data to inform instruction for students (Knowledge #2) • Know the effectiveness and impact of grouping on student achievement and can group all students accordingly. (Knowledge #4) • Understand how their behavior impacts student learning. (Knowledge #5) • Are confident in their ability to teach CCSSM to students (Motivation #1) Each observation took place during a regularly scheduled math lesson and the observer was asked to be as unobtrusive as possible throughout the lesson. The observer was asked to take note of the following items during the observation (adapted from Merriam & Tisdell, 2016, p. 141): 1. The physical setting: What is the physical setting like for this lesson? How is space allocated? What objects, resources, technologies are in the setting? 2. The participants: Describe who is in the scene, how many people and their roles. What are the ways in which the participants organize themselves? 3. Activities and interactions: What is going on? Is there a definable sequence of activities? How do the students interact with the activity and one another? What norms or rules structure the activities and interactions? When did the activity begin? How long does it last? Is it a typical activity or unusual? 4. Conversation: what is the content of the conversations in this setting? Who speaks to whom? Who listens? Quote directly, paraphrase and summarize conversations. 5. Your own behavior: How is your role affecting the scene you are observing? What thoughts are you having about what is going on? Record these in the observer comments section. Math Growth in High Achieving Elementary Schools 52 6. After the observation take time to record as many field notes as possible about the observation. The observer attended each observation lesson during the same period to ensure consistency of the observation of the math content delivered within the observed 25 minutes of instruction. The duration of each math lesson is 70 minutes, however, the observer was not available for this length of time. Twenty-five minutes were deemed a reasonable length of time to be able to observe evidence of the indicators above. The observer removed any indicators that might enable identification of any participants and then emailed the observations to this researcher for coding. After coding, general themes were established and are discussed below through the KMO framework (Clark & Estes, 2008). For the interviews a protocol was created utilizing suggestions from Patton (as cited in Merriam & Tisdell, 2016, p. 118). These included experience and behavior questions, opinion and values questions, feeling questions and knowledge questions. The six respondents from the observations were asked two warm up questions. These were “feeling questions” designed to help respondents relax in preparation for the rest of the interview. Fourteen more questions were asked with seven being knowledge questions, three were opinion and values questions and experience and behavior questions, with the remaining interview questions focused on the stakeholders critical behaviors. Interviews took place at a time and date of the respondents choosing at the school site. Teacher anonymity was protected by emailing the recorded interviews to an independent source for transcription and identifying markers were removed before returning the data to the researcher. Analysis of the interview data revealed that one question had been omitted. The research assistant contacted the interviewees and asked each one the remaining question Math Growth in High Achieving Elementary Schools 53 which was then added to the interview data. It is important to note that the use of this data outside of the original interview process means the results should be interpreted with caution. Stakeholders’ Critical Behaviors In addition to exploring the knowledge, motivational and organizational influences, which are discussed later in this chapter, it is important to also remember the stakeholder goal mentioned in Chapter One—All grade 4-5 mathematics teachers will effectively teach 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) test. For teachers to achieve the stakeholder goal they need to demonstrate performance in three critical areas as outlined below. Stakeholder Behavior 1: Grade 4-5 classroom teachers will use student fall MAP results to generate math goals for each student to improve their growth in math over the year. Interview data revealed that all teachers use goal setting with their students however these goals may not necessarily include a math goal and not all of the goals are based on the MAP data. Some teachers develop math growth goals with students, some set their own math goals for students and some teachers base the students’ goals on learning progressions instead of the CCSSM standards. Teacher D shared that high performing students will often have a goal to challenge themselves and when they do the teacher will work with the students and the Math Enrichment teacher to access resources that will help those students reach their goal (personal communication, November 14, 2017). This is an area for further emphasis with grades 4-5 at KIDS discussed in chapter five to ensure teachers know how to use student Math Growth in High Achieving Elementary Schools 54 MAP data to set math goals for students to demonstrate their growth in math throughout the year. Stakeholder Critical Behavior 2: Grade 4-5 classroom teachers will effectively teach the Common Core State Standards in Mathematics. Interview questions focused on the use of students’ prior knowledge to teach the CCSSM. Analysis suggested that teachers are able to use pre-assessment data to know which areas of math instruction students need. Teacher B commented that pre-assessment data helped to identify lessons that are essential and may require an extension of time to teach, some lessons that may not require a lot of time, and some lessons that can be omitted altogether when the pre-assessment data indicates students have already mastered that content (personal communication, November 5, 2017). Combined with responses to knowledge and motivational influences mentioned earlier in this chapter, teachers are confident in their knowledge of the CCSSM however whether this makes them effective at teaching the CCSSM was not evidenced by the data collected by this researcher and is an area for further discussion in chapter five. Stakeholder Critical Behavior 3. Grade 4-5 classroom teachers will meet in LCs to discuss student results and create plans for modifying classroom instruction to enhance student progress. Analysis of interview data indicated that teachers meet regularly in their learning communities and often discuss student results. Teachers use their learning communities to identify effective strategies however the data did not reveal whether teachers then created plans to modify their instruction to enhance student progress or shared their own effective Math Growth in High Achieving Elementary Schools 55 strategies as part of this process. Teacher E mentioned that collective responsibility for all students was essential when discussing student data and that the learning community discussions provide opportunities to learn from other faculty ways in which they meet students’ needs in mathematics (personal communication, November 5, 2017). Interview analysis made clear that teachers valued their meeting time in learning communities and take responsibility for individuals needing to be willing to share and learn. Five of the six teachers shared that the learning community was a place to share successes, failures, give feedback to others, share ideas and be receptive to ideas from others. Teacher F commented that their responsibility is “…to bring student data, to talk about my students, to listen to my peers, and hear what they’re doing in the classroom and how I can learn to do it better” (personal communication, November 5, 2017). Summary The interview data revealed that teachers do not use student fall MAP results to generate math goals for each student to improve on their growth in math over the year. The data also revealed that teachers use pre-assessment data to guide their math instruction however this does not necessarily support teachers’ ability to effectively teach the CCSSM. When combined with the findings regarding knowledge and motivational influences this is an area for further investigation as mentioned previously in this section. Finally, analysis of the interview data suggests that teachers meet regularly in their learning communities to discuss student results. What is unclear is whether that discussion has the resulting effect of teachers creating plans for modifying classroom instruction to enhance student progress. A summary of the stakeholders’ critical behaviors and their current performance indicators based on interview data analysis is shared in Table 5. Math Growth in High Achieving Elementary Schools 56 Table 5. Stakeholders Critical Behaviors Organizational Vision To be a model of global excellence where learners excel and are challenged to become innovative thinkers and productive, compassionate citizens ready to lead tomorrow’s world. Organizational Performance Goal By May 2019, all grade 4-5 students will achieve at higher levels in math after 32 weeks of instruction as measured by the biannual Measures of Academic Progress (MAP) test results. Stakeholder Goal: Teachers By May 2019, all grade 4-5 mathematics teachers will effectively teach 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) standardized test. Stakeholders Critical Behaviors For teachers to achieve the stakeholder goal above, they need to demonstrate performance of the following critical behaviors. Critical Behavior 1: Grade 4-5 classroom teachers will use student fall MAP results to generate math goals for each student to improve their growth in math over the year. Current Performance Indicators (based on interview data): • Teachers do not use student MAP results to generate math growth goals. Critical Behavior 2: Grade 4-5 classroom teachers will effectively teach the Common Core State Standards in Mathematics. Current Performance Indicators (based on interview data): • Teachers use pre-assessment data based on CCSSM standards to know which areas to focus math instruction on. Critical Behavior 3: Grade 4-5 classroom teachers will meet in PLCs to discuss student results and create plans for modifying classroom instruction to enhance student progress. Current Performance Indicators (based on interview data): • Teachers discuss student data in teams and use the data to identify strengths and weaknesses. • Teachers use their learning communities to identify effective teaching strategies. Math Growth in High Achieving Elementary Schools 57 Assumed and Validated Knowledge and Skills Causes Krathwohl’s taxonomy for learning (2002) was the basis for assessing the knowledge domain through questions focusing on factual, conceptual, procedural and metacognitive types of knowledge. These were linked to the organizational vision performance goal as well as the stakeholder goal. Assumed Knowledge Need of Successful Teachers #1 Factual: Teachers know the Common Core State Standards in Mathematics (CCSSM). Observation results. Evidence of teachers having the factual knowledge of the Common Core State Standards in Mathematics (CCSSM) was present in the observational data. One of the main themes from the observations was that teachers’ instruction was related directly to the CCSSM. For example, the use and reference to mathematical terms and concepts such as composition, arrays, distributive property, inverse operations and patterns in numbers are all math topics taught through the CCSSM and these were present throughout the observations. Further evidence of teachers having the factual knowledge of the CCSSM necessary to demonstrate knowledge of the CCSSM was the consistent use of relevant mathematical vocabulary related to CCSSM throughout the lesson. For example, benchmark numbers, decompose, array, square units, multiplication, distributive property, equation, model, factors, division, strategies, addition, subtraction, inverse operations, opposites, patterns, comparison, bar models, box models, and exponents were used interchangeably and appropriately throughout the grade four and five math lessons. Interview results. Evidence of teachers having the factual knowledge of the Common Core State Standards in Mathematics (CCSSM) was present in the interview Math Growth in High Achieving Elementary Schools 58 responses. One of the main themes from the interview responses was that teachers’ felt comfortable with their knowledge of the CCSSM. For example, some teachers had been on previous math teams which were responsible for developing curriculum units based on the CCSSM and felt this had given them a good level of knowledge regarding the CCSSM. Some believed that the work their teams had completed the previous year on their team math goal had also helped them to know the CCSSM. Teacher A responded, “We worked on our power standards and you know broke down each standard. So that helped a lot to really get to know the standards” (personal communication, November 10, 2017). Teacher F’s response was, “We have to know them in order to be able to know what to teach the students” (personal communication, November 5, 2017). One of the themes from the CCSSM probing question, “What changes have you made to your math instruction as a result of teaching the CCSSM?” was teachers now teach with a deeper level of exposure to the mathematical concepts, students do more problem solving and students are expected to demonstrate their thinking. Teachers also believed their instruction has become more targeted through the use of power standards from the CCSSM. Teacher B mentioned that the power standards contributed to a greater math knowledge of students (personal communication, November 5, 2017). Teacher C shared how important it is to be more aware of the math practices rather than simply focusing on mathematical skills (personal communication, November 10, 2017). Synthesis of observation and interview results. The observation and interview results validated the assumed knowledge cause of successful teachers knowing the CCSSM. Observational information was mainly factual relating to the use of and reference to mathematical terms and concepts and the consistent use of mathematical vocabulary Math Growth in High Achieving Elementary Schools 59 throughout the lesson. In the interview setting, teachers said they felt comfortable with their knowledge of the CCSSM and their previous experience on math teams helped expand this knowledge. Teachers also felt that they teach with a deeper level of exposure to mathematical concepts, and their instruction is more targeted due to the focus of CCSSM. Assumed Knowledge Need of Successful Teachers #2 Factual: Teachers know how to use math test data to inform instruction for students. Observation results. Observational data related to the knowledge influence of successful teachers knowing how to use math test data to inform instruction for students revealed that unless a math test was taking place it was difficult to identify ‘test’ data as described. The decision was made by the researcher to eliminate the word test from the description revising it to: Teachers know how to use math data to inform instruction for students. In this way, themes such as teachers asking students to explain their thinking, teachers refocusing or redirecting incorrect thinking by repeating math instruction could be considered evidence of teachers using math data to inform instruction. Interview results. Teachers use data to inform math instruction for high performance students in different ways as evidenced by the interview responses. The use of pre- assessment data was mentioned by all respondents as a way to help form groups. One teacher mentioned the use of formative assessments and three teachers said they use MAP test data particularly at the beginning of the year to differentiate classroom instruction. The use of MAP test data to inform classroom instruction was mentioned by five teachers as one way to identify general trends across the class including potential areas of strength and challenge. The MAP test data was referred to by Teacher C as “a ball park overview for student needs” (personal communication, November 10, 2017). Math Growth in High Achieving Elementary Schools 60 Synthesis of observation and interview results. The observation and interview results validated the assumed cause of teachers knowing how to use math data to inform instruction for students. Evidence of this was the use of formative assessments and pre- assessment data by all respondents to inform student grouping in mathematics. MAP data was also referenced in the interviews as a way to inform classroom instruction by identifying general areas of strength and challenge. Assumed Knowledge Need of Successful Teachers #3 Conceptual Knowledge: Teachers know the importance of rigorous math curriculum and its relationship to student math achievement. Interview results. The final assumed need of successful teachers in terms of declarative knowledge was whether teachers know the importance of a rigorous math curriculum and its relationship to student math achievement. Responses from teachers demonstrated the themes of ensuring students feel appropriately challenged and that a rigorous math curriculum keeps students interested and engaged in their learning. Teacher C mentioned a direct correlation between the importance of a rigorous math curriculum and its relationship to student math achievement by saying that to increase student achievement the math curriculum needs to push students to the next level (personal communication, November 10, 2017). Teacher D believed that a rigorous curriculum is really important for students to achieve what they are capable of and also for teachers to see what students are capable of (personal communication, November 14, 2017). The interview results validated the assumed knowledge need that teachers know the importance of rigorous math curriculum and its relationship to student achievement. It is Math Growth in High Achieving Elementary Schools 61 important to note that this question was omitted from the original interview. The interviewer went back to each respondent to ask this additional question after the researcher realized it had been omitted from the interview protocol. Assumed Knowledge Need of Successful Teachers #4 Procedural Knowledge: Teachers are knowledgeable about the effectiveness and impact of grouping on student achievement and can group all students accordingly. Observation results. Themes from the observational data that were evidence of teachers having procedural knowledge related to the effectiveness and impact of grouping on student achievement, and whether teachers could group all students accordingly, included the use of groupings across classes and the range of activities used for each group. Groupings varied in name from math stations, rotations and centers. Some student groups worked with the Instructional Assistant, some worked with the teacher and some groups worked independently. Activities conducted in independent math groups included math games, independent worksheets, iPad websites and apps and collaborative work with partners. What was unclear through the observations was whether the groupings of students received differentiated instruction. It was also unclear whether the math apps and websites used were differentiated according to student need. Interview results. In terms of procedural knowledge, teachers were asked about their knowledge of the effectiveness and impact of grouping on students’ achievement and if they were able to group all students accordingly. The first question asked teachers if they grouped students for math instruction. Three teachers indicated that they group students based on MAP and preassessment unit data; two teachers indicated that they divide the students Math Growth in High Achieving Elementary Schools 62 according to the preassessment data and group students across multiple classes in differentiated groups. When asked how teachers decide which groups each student should be in, five responses indicated that teachers use their preassessment data; five also responded that they use exit slips, whereby students demonstrate their understanding of concepts as they ‘exit’ the lesson, as formative assessments for grouping their students. Teacher C responded that sometimes students are grouped according to how well they have achieved a particular standard and “the groups can be kind of flexibly grouped based on how well students are showing their current understanding of the math content” (personal communication, November 10, 2017). Teachers acknowledged that the effect of grouping students had been largely positive. Four teachers mentioned the growth over time particularly for struggling students. Teacher C said that grouping “helps some students feel successful - students who might not normally feel like so successful” (personal communication, November 10, 2017). Teacher A mentioned the high performing students “and then my higher kids like today that are working on some problems and sharing their ideas: well, let me listen to your theory. And I don't believe that your theory is going to work - like carrying these conversations - like this is how adults talk you know” (personal communication, November 10, 2017). Synthesis of observation and interview results. Observational data indicated that teachers are knowledgeable about the effectiveness and impact of grouping on student achievement and can group all students accordingly. Math groupings were used in and across classrooms but it was unclear whether students in these groups received differentiated Math Growth in High Achieving Elementary Schools 63 instruction. Due to insufficient data gathered in the observations this cause could not be validated. Interview data revealed that preassessment data were used for deciding which groups students would be in and the effects of grouping on student achievement were perceived as positive by their teachers. Interview data validated the assumed need that teachers are knowledgeable about the effectiveness and impact of grouping on student achievement and can group all students accordingly. However, it is one skill to be able to group learners and another to be able to differentiate for those learners once they are grouped. It was unclear from the interview results if teachers were differentiating for their learners even though they were grouped and more specific questions relating to the differentiated instruction for each group would need to be asked. This would be a possible area for further study. Assumed Knowledge Need of Successful Teachers #5 Metacognitive Knowledge: Teachers understand how their behavior impacts student learning. Observation results. The final assumed knowledge need for successful teachers is metacognitive and teachers understanding how their behavior impacts student learning was used as evidence of metacognitive knowledge in the observations. One of the themes from the data was that teachers gave reminders and prompts by referring to prior learning from previous lessons. This was evidence that teachers were supportive of their students and did not expect them to have memorized everything from a previous lesson. More evidence of teachers understanding how their behavior impacts student learning was through teachers regularly inviting students to share their thinking, their work and/or their explanations as the lesson progressed. Math Growth in High Achieving Elementary Schools 64 Further evidence of this metacognitive knowledge was the regularity with which teachers referenced Common Core Math Practice 5: Construct viable arguments and critique the reasoning of others. This happened when teachers asked students to explain the different strategies they had used in their work demonstrating that there is more than one way to solve a math problem. This was also evident when teachers called on different students for responses thereby expecting that all students would participate by constructing their own arguments or critiquing others. Additional evidence of teachers understanding how their behavior impacts learning was the observational theme of positive and supportive language used by teachers within the math lessons. This theme included teachers’ positive body language and gestures indicating an expectation of success by teachers of all students. Finally, teacher behavior impacting learning was also evident in that teachers moved around the classroom during instruction and independent work time giving further opportunities to reinforce expectations as well as check on the progress of all students and intervene where necessary. Interview results. In the interview teachers were asked to self-evaluate how their behavior impacts student learning in math. Themes uncovered from answering this question were that four teachers out of six believed that if they were positive about math then their students would also be positive about math. Teachers referred to the need to make math fun, believe in students and that will lead to confidence; building relationships with students also helps their attitudes to math. Teacher E said, “I think just having a positive mindset and being encouraging and kind of building like a positive math environment I think is important” (personal communication, November 5, 2017). Math Growth in High Achieving Elementary Schools 65 Teacher F enthusiastically stated “Oh my gosh, if I’m positive they're positive. If I’m not so positive or telling them it’s going to be really difficult and hard, then they're going to think it’s really difficult and hard. If I expect them to not model and to not estimate, then they're not going to model and estimate it. So, I'm training mathematicians. And so, I’m just as concerned about the behaviors and the learning process as I am about the content and their mastery” (personal communication, November 5, 2017). Synthesis of observation and interview results. The observation and interview results validated the assumed cause of teachers understanding how their behavior impacts student learning. Observational data indicated that teachers gave reminders and prompts, referred to prior learning and used positive and supportive language within math lessons. This was supported by the evidence in the interview data whereby teachers shared that if they were positive about math then the students would be too. Further, teachers believed that making math fun and believing in students’ abilities would boost students’ confidence levels in math. Table 6 provides a summary of each assumed knowledge need, how it was assessed, and the evidence for validating each need. Assumed and Validated Motivational Causes One of the assumed motivational causes is self efficacy (Eccles, 2009). Self-efficacy in this context means that teachers believe they are confident in their ability to provide instruction for high performing learners in mathematics. These students are already performing at high levels which may make it difficult for teachers to improve on what these students are able to achieve. This could lead to low levels of teacher self efficacy as it may be difficult for teachers to believe they can provide appropriate instruction to high performing students to enable them to achieve at higher levels. Math Growth in High Achieving Elementary Schools 66 Table 6. Assessment, Measurement and Validation of Knowledge and Skills Influences Organizational Vision To be a world leader in education cultivating exceptional thinkers prepared for the future. Organizational Performance Goal By May 2018, all grade 4-5 students will achieve at higher levels in math after 32 weeks of instruction as measured by the biannual Measures of Academic Progress (MAP) test results. Stakeholder Goal: Teachers By May 2018, all grade 4-5 mathematics teachers will effectively teach 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) test. Assumed Knowledge Needs of Successful Teachers Declarative Knowledge Knowledge Need #1 Factual: Teachers know the Common Core State Standards (CCSS) in mathematics How Was It Assessed? Was it Validated? Observation: • Evidence of knowing CCSSM in classroom lesson. Validated - Yes Interview: • Teachers answered questions about the CCSS in mathematics. Validated - Yes Evidence Themes Observation: • Teacher’s instruction is related to CCSSM. • Teacher uses vocabulary common to CCSSM. Interview: • Teachers are comfortable with the CCSSM. • Teachers have past experience with CCSSM. • Teachers know there is a deeper level of exposure in concepts with CCSSM. Knowledge Need #2 Factual: Teachers know how to use math test data to inform instruction for students. How Was It Assessed? Was it Validated? Observation: • Evidence of using math data to inform instruction during classroom lesson. Validated - Yes Interview: • Teachers answered questions about how to use MAP results to inform instruction for students. Validated - Yes Evidence Themes Observation: • Teacher asks students to explain their thinking. • Teacher refocuses/redirects incorrect thinking by instructing again. • Formative assessment used throughout lesson. • Teacher explains to students as inaccuracies are noted. Interview: • Teachers use data to identify general trends in student strengths and weaknesses. Conceptual Knowledge: Knowledge Need #3 Teachers know the importance of rigorous math curriculum and its relationship to student math achievement How Was It Assessed? Was it Validated? Interview: • Teachers answered questions related to the importance of rigorous math curriculum and its relationship to student achievement. Validated – Yes Evidence Themes Interview: • Ensuring students feel appropriately challenged. • A rigorous math curriculum keeps students interested and engaged in their learning. Math Growth in High Achieving Elementary Schools 67 Table 6. (Continued) Procedural Knowledge: Knowledge Need #4 Teachers are knowledgeable about the effectiveness and impact of grouping on student achievement and can group all students accordingly. How Was It Assessed? Was it Validated? Observation: • Evidence that teachers are knowledgeable about grouping during classroom lesson. Validated - No Interview: • Teachers answered questions related to the effectiveness and impact of grouping on student achievement. • Teachers answered questions related to their grouping methods for students in math. Validated - Yes Evidence Themes Observation: • Teachers using groupings. • Teachers have grouped students in different ways. • Unclear if groups of students received differentiated instruction. Interview: • Teachers use data from multiple measures to group students. Metacognitive Knowledge: Knowledge Need #5 Teachers understand how their behavior impacts student learning. How Was It Assessed? Was it Validated? Observation: • Evidence that teachers understand how their behavior impacts student learning. Validated – Yes Interview: • Teachers were asked to self-evaluate how their behavior impacts student learning in math. Validated – Yes Evidence Themes Observation: • Teachers give reminders and prompts Modeling Math Practices: • Teacher references different strategies used by students in their work. • Teacher invites all students to share their thinking. • Teacher uses positive and supportive language. • Teacher moves around the room during instruction and independent work time. Interview: • Teachers are positive about math. • Teachers make math fun for students. • Teachers belief in students contributes to confidence. Another assumed motivational cause is that of utility value or “Do I want to do the task?” as expressed within Eccles’s (2009) Expectancy Value Motivational Theory. Utility value reflects how well a task meets psychological needs or fits within a person’s long term goals. In this sense, the motivational needs identified in this setting assume that successful teachers place a value on their work and believe they want to do this work. Math Growth in High Achieving Elementary Schools 68 Assumed Motivational Need of Successful Teachers #1 Self Efficacy: Teachers are confident in their ability to teach CCSSM to students. Observation results. Evidence of teachers demonstrating motivational influences in self efficacy through having confidence in their ability to teach CCSSM to students was not explicitly noted in the observational data. Confidence is difficult to observe however, persistence and mental effort are motivational constructs that may be evidence of confidence in work places (Clark & Estes, 2008). Successful teachers demonstrate persistence by focusing on their most important goals and not becoming distracted. Mental effort is displayed when successful teachers decide how much mental effort they will put into achieving their goals. Teacher A persisted in ensuring students understood decomposition in math by making connections for students between the math content and the math video explaining “the narrator kept the three rows the same but changed the units” (personal communication, October 3, 2017). Teacher B focused on instructing students in solving bar models with comparisons. They instructed first, referred to the video for more information on creating bar models and then students were asked to show their work on whiteboards and explain their work to a partner. Persistence was demonstrated through students needing to include drawings and equations in their work. This additional mental effort on the part of the teacher ensured a focus on really understanding the concept before moving to the next one. (personal communication, October 5, 2017). Reviewing the observations for specific signs of persistence and mental effort as evidence of teachers’ confidence in their ability to teach CCSSM to students revealed some signs of confidence. However, this researcher believes there was not enough evidence for strong conclusions to validate this motivational cause. It Math Growth in High Achieving Elementary Schools 69 could be assumed teachers have confidence in their ability by reflecting on the teacher behaviors and knowledge indicated in the knowledge areas above however it is difficult to make assertions because of a lack of compelling evidence in this area. Interview results. Teachers were very confident in their ability to teach CCSSM as evidenced and validated by the interview results. Several teachers believed that their confidence was a result of prior study of the math standards in recent years writing curriculum units based on the CCSSM. Teachers felt that an area of challenge was knowing what math fluency looked like in other grade levels and believed that they needed help to better challenge the higher performing students in math. Teacher A commented that they had previously experienced many workshops focusing on math fluency and proficiency in earlier grades but very few that focused on higher grade levels. Teacher A went on to mention that this focus on the higher grade levels would be good to have in order to improve their instruction for higher performing students (personal communication, November 10, 2017). Further evidence of the need for professional development in challenging high performing math students was given by Teacher F “I’m confident teaching the basic common core but I would love to figure out how to enhance our program, particularly because we have such a strong math population of students here” (personal communication, November 5, 2017). Working with groups and integrating math practices were also identified as areas needing strengthening. Teacher B commented “Even this year I'm thinking about trying something new just with groupings and rotations of stations and things like that” (personal communication, November 5, 2017). Math Growth in High Achieving Elementary Schools 70 Teacher C was concerned with math practices saying “I think the part of the standards that’s always more difficult in understanding is to how to see the math practices that we teach in addition to just the standards” (personal communication, November 10, 2017). Summary of Observation and Interview Results. Observational data revealed that teachers’ confidence in teaching the CCSSM was inconclusive however the interview data indicated teachers felt very confident in their ability to teach CCSSM. This discrepancy in validating the two data sets suggests that although teachers say they are confident teaching the CCSSM their behavior as observed in the observations indicates they are not. Specific exploration of persistence and mental effort as indicators of confidence in the future may provide additional evidence of teachers’ confidence when teaching the CCSSM. If teachers lack confidence in teaching the CCSSM (as evidenced through the observations) then this could be one of the reasons high performing students are not demonstrating higher levels of growth in the math achievement data. In addition, the interview data indicates that although teachers feel confident teaching the CCSSM their comments also reveal the need for further professional development in the area of teaching high performing students. If teachers are asking for this professional development then it is difficult to assume they feel confident in this area even though they say they are. Assumed Motivational Need of Successful Teachers #2 Expectancy Value: Teachers believe they are capable of effective instruction that can make a difference to their students’ growth and achievement in mathematics. Interview results. All interview respondents believed they are able to provide effective instruction that makes a difference to their students’ growth and achievement in Math Growth in High Achieving Elementary Schools 71 math. Teacher D shared “I feel like we are using the expertise of each other so much more and the focus on data informs my teaching” (personal communication, November 14, 2017). Teacher A mentioned always wanting to do better particularly for the students needing extension: “But I always want to do better. You know so like for the enrichment kids like I’d always like a better project for them and I'd always like more time with just them” (personal communication, November 10, 2017). A theme emerged through the interview data of teachers desire to provide better enrichment opportunities for high performing students. This was deemed an area teachers felt they could benefit from further professional development in order to find alternative ways to meet the needs of those students. Assumed Motivational Need #3 Expectancy Value: Successful teachers believe students should demonstrate growth in math achievement from the beginning to the end of the school year. Interview results. The interview data validated the need for successful teachers believing students should demonstrate growth in math achievement. This was achieved by teachers using data to inform their students of their growth in mathematics. Teachers also provide feedback to students about their growth in math achievement. Teacher E shared “…like assessing how they’re doing and being able to provide feedback to them” (personal communication, November 5, 2017). Although the need for professional development in meeting the needs of higher performing students was mentioned earlier in this chapter, teachers also talked about finding ways to extend the thinking of their high performing students in math. Teacher E explained Math Growth in High Achieving Elementary Schools 72 “If they’re getting a concept really quickly then I find ways to help them extend their thinking” (personal communication, November 5, 2017). Assumed Motivational Need of Successful Teachers #4 Expectancy Value: Successful teachers value the correlation between understanding the mathematical content they teach and their students’ mathematical achievement. Interview results. Five teachers responded with a ‘yes’ in terms of motivation through expectancy value indicating there was a correlation between their understanding of the CCSSM and their students’ achievement in math. However, Teacher D was the only teacher who expanded on their answer, saying: “I do think there's a correlation because just to focus on the power standards like I keep saying and to make sure they know those really well and then they're successful say on the formative assessment and also the end of the unit test” (personal communication, November 14, 2017). Teacher C responded that it is helpful to have a solid understanding of the standards as it helps teachers to know exactly what they need to teach so students can demonstrate their understanding of that standard (personal communication, November 10, 2017). Teacher F believed it was “…helpful if teachers know the next grade level of CCSSM too so they can meet the needs of their current students knowing what they will be taught next” (personal communication, November 5, 2017). Summary. Successful teachers do understand and value the correlation between understanding the mathematical content they teach and their students’ mathematical achievement as evidenced by the interview data therefore this need was validated. In addition, a need for professional development for teachers to feel more successful with teaching their high performance learners was voiced. One could argue that if successful Math Growth in High Achieving Elementary Schools 73 teachers do understand and value the correlation between understanding the mathematical content they teach and their students’ mathematical achievement and if the teachers at KIDS feel there is a knowledge gap when teaching their high performance students then perhaps the assumed need is not validated. Table 7 provides a brief summary of each assumed motivational need, how it was assessed, and includes the evidence for validating each need. Assumed and Validated Organizational Causes Gallimore and Goldenberg (2001) suggest that cultural models and cultural settings influence achievement of performance goals. “Cultural models are so familiar they are often invisible and unnoticed by those who hold them,” (Gallimore and Goldenberg, 2001, p. 47). Cultural settings are also important and are defined as occurring “whenever two or more people come together, over time, to accomplish something,” (Sarason, 1972, as cited in Gallimore and Goldenberg, 2001, p. 47). Assumed Organizational Need of Successful Teachers #1 Cultural Models: Working in a high achieving school may create an atmosphere of competition and fear of failure amongst teachers. In response to the interview question regarding working in a high achieving school such as KIDS an atmosphere of competition and fear of failure amongst teachers may be created, five of the respondents said this was false for their experience. Teacher D was the only one who responded to the part of the questions about the atmosphere of competition among teachers replying “I don't think so at all. No” (personal communication, November 14, 2017). Math Growth in High Achieving Elementary Schools 74 Table 7. Assessment, Measurement and Validation of Motivational Influences Organizational Vision To be a world leader in education cultivating exceptional thinkers prepared for the future. Organizational Performance Goal By May 2018, all grade 4-5 students will achieve at higher levels in math after 32 weeks of instruction as measured by the biannual Measures of Academic Progress (MAP) test results. Stakeholder Goal: Teachers By May 2018, all grade 4-5 mathematics teachers will effectively teach 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) test. Assumed Motivational Needs of Successful Teachers Self Efficacy Motivational Need #1: Teachers are confident in their ability to teach CCSS in mathematics to students. How Was It Assessed? Was it Validated? Observation: • Persistence and mental effort used as indicators of teachers’ confidence in their ability to teach mathematics during classroom lessons. Validated – No Interview: • Teachers answered questions regarding their confidence in their ability to teach CCSSM to students. Validated – Yes Evidence Themes Observation: • Persistence: Teachers focused on their most important goals and did not become distracted. • Mental effort: Teachers decide how much mental effort they will put into achieving their goals. • Teacher’s instruction is related to CCSSM topics. • Teacher uses vocabulary common to CCSSM. Interview • Teachers state they are confident in their ability to teach the CCSSM. • Teachers seek out new knowledge on how to further improve their teaching for additional student achievement. Expectancy Value Motivational Need #2: Teachers believe they are capable of effective instruction that can make a difference to their students’ growth and achievement in mathematics. How Was It Assessed? Was it Validated? Interview: • Teachers answered questions regarding their capability of delivering effective instruction that makes a difference to students’ growth and achievement in mathematics. Validated – Yes Evidence Themes: Interview: • Teachers believe they are capable of delivering effective instruction that makes a difference to students’ growth and achievement in mathematics. • Teachers want to provide better enrichment opportunities for high performing students. Motivational Need #3: Successful teachers believe students should demonstrate growth in math achievement. How Was It Assessed? Was it Validated? Interview: • Teachers answered questions about the importance of their students demonstrating growth in math achievement. Validated – Yes Evidence Themes Interview: • Teachers use data to inform their students growth in mathematics. • Teachers provide feedback to students about their growth in math achievement. • Teachers find ways to extend the thinking of high performing students in math. Motivational Need #4: Successful teachers understand and value the correlation between understanding the mathematical content they teach and their students’ mathematical achievement. How Was It Assessed? Was it Validated? Interview: • Teachers answered questions about correlations between their understanding of the mathematical content they teach and their students’ mathematical achievement. Validated – Yes Evidence Themes Interview: • Teachers understand the correlation between the mathematical content they teach and their students’ achievement. Math Growth in High Achieving Elementary Schools 75 Teacher E spent some time considering the question, replying: “I think at times there's definitely that idea like fear of failure and so sometimes I feel maybe people are scared to try things or to admit if things don’t go well. Like new initiatives or new content, new curriculum and things like that that are happening. And so maybe people might be less willing to try some of those things or be open about those things. But I think it’s really important that kind of comes into play of like building that LC community. And like if there is that trust and stuff then I think people are willing to be like – hey I tried this lesson or I tried this and it just didn’t go well at all like I feel like kids bombed it. I didn’t do a good job teaching this – like what can I do to make it better which I think would be the ideal thing.” (personal communication, November 5, 2017) Teacher F responded with “…my fear is not from admin or from KIDS or from not meeting the expectation of my peers. My fear is from not doing everything I possibly can to help my students” (personal communication, November 5, 2017). The organizational cause related to the cultural model in this gap analysis was that working in a high achieving school may create atmosphere of competition and fear of failure among teachers. From the data analysis, this cause was not validated as teachers do not believe there is an atmosphere of competition or the fear of failure. Rather, the fear of failure that was reported by three teachers was that of failing their students if their instruction meant students did not make adequate progress. This also supports a motivational finding that teachers take responsibility for their students achievement. Math Growth in High Achieving Elementary Schools 76 Assumed Organizational Need of Successful Teachers #2 Cultural Settings: There is a process that ensures that teachers get timely, concrete feedback about their performance in teaching CCSSM. Interview results. Three teachers stated that feedback about their performance teaching CCSSM was based on how their students perform. Teacher A shared, “I feel like most of the feedback I get is just based off what the students are doing. It’s like the assessments – like do they get it?” (personal communication, November 10, 2017). Teachers observed that feedback from administration or directly from peers was rare regarding the CCSSM. Teacher A pointed out: “I think that I do. It’s, I feel like the feedback is a bit limited though – it’s kind of hit and miss” (personal communication, November 10, 2017). Teacher D shared, “On occasion. Probably not very often, but if you know there's an observation, then I would. But that just depends on if there is an observation. And so not yet. I think we’re headed in that direction. Like for example, our small LC is working to schedule observations of each other so we can just help each other improve by giving feedback about how the lesson went. So, I think that’s something that will help in that area” (personal communication, November 14, 2017). Teacher E’s response was: “…not in particular about the Common Core. I mean I think I would get feedback in general about math maybe when people come to observe and things like that but not necessarily targeted with that lens – I don’t think I do” (personal communication, November 5, 2017). The interview results did not validate the assumed organizational cause of a process existing that ensures teachers get timely, concrete feedback about their performance in Math Growth in High Achieving Elementary Schools 77 teaching CCSSM. Table 8 provides a summary of each assumed organizational need, how it was assessed, and includes the evidence for validating each need. Summary of Gaps Found Knowledge and Skills The observation and interview results validated that there were few gaps in teachers’ knowledge and skills. Teachers are knowledgeable of the CCSSM and know how to use data to inform instruction. Teachers are knowledgeable about grouping and can group accordingly –whether these groupings are then taught differentiated lessons was not made clear in this study Table 8. Assessment, Measurement and Validation of Organizational Influences Organizational Vision To be a world leader in education cultivating exceptional thinkers prepared for the future. Organizational Performance Goal By May 2018, all grade 4-5 students will achieve at higher levels in math after 32 weeks of instruction as measured by the biannual Measures of Academic Progress (MAP) test results. Stakeholder Goal: Teachers By May 2018, all grade 4-5 mathematics teachers will effectively teach 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) test. Assumed Organizational Needs of Successful Teachers Cultural Models Organizational Need #1: Working in a high achieving school may create an atmosphere of competition and fear of failure among teachers. How Was It Assessed? Was it Validated? Interview: • Teachers answered questions regarding their perceptions of an atmosphere of competition and fear of failure within grades 4-5. Validated - No Evidence Themes Interview: • Teachers do not have a fear of failure. • There is no atmosphere of competition among teachers. Cultural Settings Organizational Need #2: There is a process that ensures that teachers get timely, concrete feedback about their performance in teaching CCSSM. How Was It Assessed? Was it Validated? Interview: • Teachers answered questions about whether they believe there is a process that ensures that they get timely, concrete feedback about their performance in teaching CCSSM. Validated - No Evidence Themes Interview: • Teachers reported that they do not receive much feedback directly related to the CCSSM. • Some feedback received if a math lesson is observed by an administrator – not very often. Math Growth in High Achieving Elementary Schools 78 study. Teachers also know the importance of math curriculum and its relationships to student math achievement. Finally, teachers understand how their behavior impacts student learning. Motivation The observation data did not support the findings in the interviews that teachers were confident in their ability to teach CCSSM. Interview data validated the motivational needs of successful teachers who believe they are capable of effective instruction that can make a difference to their students’ growth and achievement in mathematics. Interview data also supported that teachers believe students should demonstrate growth in math achievement. Teachers also understand and value the correlation between understanding the mathematical content they teach and their students mathematical achievement. Organization The interview results highlighted that there is no organizational process to ensure teachers get timely, concrete feedback about their performance teaching CCSSM. The organizational cause of working in a high achieving school may create an atmosphere of competition and a fear of failure among teachers was also not validated. Summary When integrating the data across the knowledge, motivation and organizational gaps it appears that teachers believe they know more than they actually do. It also becomes clear that teachers do not receive the support they need in order to identify and rectify the gaps in their knowledge. The stakeholder critical behavior data analysis suggests that more specific expectations for faculty are necessary including the need to use students’ fall MAP data to establish growth goals in math and to modify instruction based on learning community discussions. Math Growth in High Achieving Elementary Schools 79 This chapter included analysis of the stakeholders’ critical behaviors data in order to ensure the stakeholder goal from chapter one can be met. Also included was a comprehensive analysis of the knowledge, motivational and organizational assumed causes data for Research Question One: Teachers not meeting their goal of effectively teaching 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) test. Chapter five will focus on Research Question Two by identifying the recommended knowledge, motivation and organizational solutions teachers might implement to effectively teach 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) test. Solutions to the problems identified through analysis of the stakeholders’ critical behaviors will also be presented. Math Growth in High Achieving Elementary Schools 80 CHAPTER FIVE: SOLUTIONS, IMPLEMENTATION, AND EVALUATION The purpose of this chapter is to outline research-based solutions that address the validated knowledge, motivation and organization causes as well as the critical behaviors of stakeholders as identified in the data analysis in chapter four. Research based solutions will be presented first followed by a comprehensive plan for implementing and evaluating the solutions within the institution. In conclusion, a brief summary including recommendations for future practice and implications for other elementary schools will be presented. Research Question Two – Solutions What are the recommended knowledge, motivation and organizational solutions teachers might implement to effectively teach 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) test? Knowledge and Skills Five knowledge and skill influences that teachers need to exhibit in their practice were investigated through this study; two focused on declarative knowledge, with the remaining focus on conceptual, procedural, and metacognitive knowledge and skills influences. Four of the knowledge and skills influences were validated and did not require a solution. The fourth procedural knowledge, influence: Teachers are knowledgeable about the effectiveness and impact of grouping on student achievement and can group all students accordingly, was not validated in the observational data but was validated in the interview data. This discrepancy is important and solutions are explored next. As outlined in Chapter Four, observational data indicated that teachers are knowledgeable about the effectiveness and impact of grouping on student achievement and Math Growth in High Achieving Elementary Schools 81 can group all students accordingly; however, it was unclear if these groupings were by ability with differentiated instruction received by the students. In contrast, the interview data revealed that teachers were using student preassessment data to form instructional groups and teachers believed there was a positive effect on student achievement as a result. Nevertheless, the emphasis on differentiating instruction for each grouping was not clearly defined. One solution for this discrepancy would be to provide professional development regarding the use of curriculum compacting mentioned in the research literature. A focus on pretesting is already in place at KIDS, implementing curriculum compacting as a strategy for high performing students whereby the focus is on teaching the curriculum that is unknown at the time of pretesting is supported by research (Reis et al., 1992). Another suggested solution for this discrepancy within the data would be observations by peers or supervisors focusing on the difference between lessons for each instructional grouping and provide feedback to teachers (Brighton et al., 2005). Using the current in-school learning community structure for teachers to share their groupings and to collaborate on specific lessons for each group is also suggested as a solution (Hiebert & Stigler, 2017; Schmoker, 2009). “Teaching up” (Tomlinson & Javius, 2012) supports this collaboration as a method where teachers first design the tasks for the high performing students then scaffold the learning experience to support all learners working with a higher level of understanding. As teachers continue working in their learning communities instructional groupings of students could be formed across several classes. Research findings have indicated that these groupings need to be flexible in order for students to move between groupings, and that appropriate adaptation of curriculum materials and instructional methods to meet the needs of Math Growth in High Achieving Elementary Schools 82 each instructional group is also important (Kulik & Kulik, 1992; Lou et al., 1996; Slavin, 1987; Steenbergen-Hu et al., 2016; Tieso, 2005). This could happen unit by unit or quarter by quarter, depending on the grade level structure. It is also important to note another limitation to this study—the question of whether teachers know how to teach the content standards appropriately or not in order to increase student achievement was not explored. Other procedural skills that were not explored by this researcher and would be important for any future study include how to differentiate instruction for each student and how to use culturally relevant padegogy in the classroom to maintain student interest. Motivation One self-efficacy and three expectancy value motivational influences were researched in this study. All were validated except for the self-efficacy motivational influence – Teachers are confident in their ability to teach CCSS in mathematics to students. This influence was not validated in the observation data but was validated in the interview data. This difference was highlighted in Chapter Four, with teachers’ belief they were confident in their ability to teach CCSSM to students; nevertheless, this was not in evidence through the observation data. If teachers believe they are confident teachers of CCSSM but there is no evidence of their confidence in the observational data, this might be the root cause of the focus of this study—why high performing students do not make the expected growth in mathematics across the course of the year. Solutions to this problem are complex, and this researcher investigated the data matching individual observation data with individual’s interview responses in an effort to reconcile why the observational data were not validated whereas the interview data were Math Growth in High Achieving Elementary Schools 83 validated. The interview data suggested that teachers are more confident than they should be; teachers think they are effective at teaching the CCSSM; however, this was not indicated in their performance as measured by the observational data. Matching individual’s observational data with their corresponding interview responses did not reveal any further insight into this problem but it did highlight limitations of this study. In general, this researcher perceived the observational data were quite weak. It is difficult to define specifically how to observe confidence in teaching math—what specifically does confidence in teaching math look like? This researcher would recommend better definitions in this area for future observations. Another limitation was that the observational data were collected by a researcher who was unfamiliar with the CCSSM. This limited the type of observations made and the ability of the researcher to observe the finer details of a math lesson such as might be possible with a math practitioner. In addition, the observations were only 25 minutes in length instead of a full 70-minute math lesson which may have provided a richer set of data. Typically, math lessons are formally observed by the teachers’ supervisor over a 15-20 minute period. Therefore, a recommendation would be for the supervisor to extend the observations to the entire math lesson several times a year. For many administrators, this would be unrealistic due to the time involved; however, if a system were developed for gathering specific data through peer and supervisor observations, this practice could inform changes in math instruction for individual teachers and directly impact student achievement in math (Brighton et al., 2005). If, as the data suggest, teachers are overconfident in their ability to effectively teach math then different solutions are necessary. NCTM’s Effective Mathematics Teaching Math Growth in High Achieving Elementary Schools 84 Practices (NCTM, 2014) is one resource that could be used to assist in building capacity within faculty for effective mathematics instruction. “Teaching mathematics requires specialized expertise and professional knowledge that includes not only knowing mathematics but knowing it in ways that make it useful for the work of teaching” (Ball & Forzani, 2011; Ball, Thames & Phelps, 2008; as cited in NCTM, 2014, p. 11). Different ways to implement the NCTM’s Effective Mathematics Teaching Practices (2014) as a possible solution to this problem are further discussed in this chapter as part of the implementation and evaluation plan. Cultural/Organizational Barriers The two organizational needs explored through this study were related to cultural models and cultural settings. However, both were not validated; therefore, solutions for these problems were sought. The first organizational need related to cultural models – working in a high achieving school may create an atmosphere of competition and fear of failure among teachers – was not validated. The interview data shared in Chapter Four revealed that teachers were concerned with a fear of failing their students but were not concerned with an organizational fear of failing or working in a competitive atmosphere. Interview data indicated that the impact of working in learning communities over several years has led to a working environment devoid of competition and a fear of failure through a focus on collective responsibility for students (DuFour et al., 2005). Interview data also revealed teachers’ belief that they are not on their own when working through academic and social emotional issues with students; instead, by collaborating with their colleagues the students belong to all faculty as well as the responsibility for their achievement (DuFour, et al., 2005; Jacobson, 2010; Schmoker, 2004, Math Growth in High Achieving Elementary Schools 85 2009; Wiliam, 2007). Interestingly, this collaborative focus has not yet resulted in expected growth for high performing students which is where the need for solutions to this problem is necessary. Potential solutions include continuing the focus on collaborative practices by using the existing learning community structure to look at student data specifically related to high performing students. By involving the instructional coach so teachers receive differentiated coaching as well as specific, relevant, and timely feedback on their instructional methods directly related to their high performing students, the result will be higher levels of growth in mathematics for those students (Brighton et al., 2005; Quintero, 2017; Ronfeldt et al., 2015). The second organizational influence, which is related to cultural settings was – there is a process that ensures that teachers get timely, concrete feedback about their performance in teaching CCSSM. As discussed in Chapter Four, teachers do not currently receive specific, relevant and timely feedback necessary to revise their instruction to have a greater impact on student achievement in mathematics. Teachers receive feedback but it is sporadic and not focused on improving their instructional practices. The importance of feedback as a method of improving practice has been well researched in recent years and is included as part of the learning community structure (DuFour, DuFour & Eaker, 2008). Implementing a school-wide system of providing regular feedback related to mathematics instruction is one suggested solution. The current evaluation system in place at KIDS could continue to be used, but with greater focus on feedback directly related to mathematics instruction. How teachers use feedback could be highlighted through learning community discussions and become part of the yearly goal-setting practice for teachers. As indicated in the motivational influences section, ensuring teachers receive Math Growth in High Achieving Elementary Schools 86 feedback for a full math lesson rather than the current 15-20 minutes could also provide more relevant feedback for teachers. Table 9 provides a summary of the solutions needed for the knowledge, motivation and organizational influences. Critical Behaviors In order to achieve the stakeholder goal of grade 4-5 mathematics teachers effectively teaching 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) test, stakeholders must demonstrate performance of three critical behaviors: 1. Grade 4–5 classroom teachers will use student fall MAP results to generate math goals for each student to improve their growth in math over the year. Table 9. Summary of Solutions Needed for Knowledge, Motivational, and Organizational Influences Knowledge Influences Cause Solution Declarative Knowledge Need #1 Factual: Teachers know the Common Core State Standards (CCSS) in Mathematics. Validated No need Knowledge Need #2 Factual: Teachers know how to use math test data to inform instruction for students. Validated No need Conceptual Knowledge Need #3 Teachers know the importance of rigorous math curriculum and its relationship to student math achievement. Validated No need Procedural Knowledge Need #4 Teachers are knowledgeable about the effectiveness and impact of grouping on student achievement and can group all students accordingly. Not validated in observational data Validated in interview data Administration provides professional development in use of curriculum compacting. Observations by peers and/or supervisors focus on the difference between the lessons for each instructional grouping and provide feedback to teachers. Use learning community time for teachers to share their groupings and to collaborate on specific lessons for each group. Develop flexible instructional groupings of students for math across learning communities (3-4 classes). Metacognitive Knowledge Need #5 Teachers understand how their behavior impacts student learning. Validated No need Math Growth in High Achieving Elementary Schools 87 Table 9. (Continued) Motivational Influences Cause Solution Self Efficacy Motivational Need #1: Teachers are confident in their ability to teach CCSS in mathematics to students. Not validated in observational data Validated in interview data Coaches and teachers provide professional development for NCTM’s Effective Mathematics Teaching Practices (2014) to build capacity and confidence within faculty for more effective mathematics instruction. Supervisor, coach and peers to observe full math lessons several times a year to provide feedback on effectiveness of teaching CCSSM. Expectancy Value Motivational Need #2: Teachers believe they are capable of effective instruction that can make a difference to their students’ growth and achievement in mathematics. Validated No need Motivational Need #3: Successful teachers believe students should demonstrate growth in math achievement. Validated No need Motivational Need #4: Successful teachers understand and value the correlation between understanding the mathematical content they teach and their students’ mathematical achievement. Validated No need Organizational Influences Cause Solution Cultural Models Organizational Need #1: Working in a high achieving school may create an atmosphere of competition and fear of failure among teachers. Not validated Use the existing learning community structure to look at student data specifically related to high performing students. Involve the instructional coach to ensure teachers receive differentiated coaching as well as feedback on their instructional methods directly related to high performing students. Cultural Settings Organizational Need #2: There is a process that ensures that teachers get timely, concrete feedback about their performance in teaching CCSSM. Not validated Supervisory and peer observations focus on feedback directly related to mathematics instruction. Build use of feedback into learning community discussions. Build use of feedback into the yearly goal-setting practice for teachers. Ensure teachers receive feedback for a full math lesson. 2. Grade 4–5 classroom teachers will effectively teach the Common Core State Standards in mathematics as measured by MAP results and summative assessments. 3. Grade 4–5 classroom teachers will meet in LCs to discuss student results and create plans for modifying classroom instruction to enhance student progress. For the first critical behavior, the current performance by stakeholders indicated that teachers do not use MAP test data to generate math goals for each student to improve their Math Growth in High Achieving Elementary Schools 88 growth in math throughout the year. The suggested solution or future performance indicator for this problem is for teachers to begin using MAP test data to set goals for their students. Students at KIDS set goals every year so creating a math goal with students based on their MAP test data would appear to be a realistic and manageable task for teachers. As mentioned in Chapter Two, NWEA provides significant online support for teachers demonstrating the use of individual student profiles and how they can be used to set goals. In addition, the first of eight Effective Mathematics Teaching Practices identified by NCTM (2014) is goal setting. It was suggested within this practice that “…effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions and uses the goals to guide instruction” (NCTM, 2014, p. 10). The second critical behavior involves effective teaching of the CCSSM. As discussed in Chapter Four, this was difficult to verify by this researcher. Current performance indicates that teachers use pre-assessment data to plan lessons that focus on student areas of need in math. What is unclear from this practice is whether this demonstrates effective teaching of the CCSSM. Suggested future performance indicators for this critical behavior include using the Effective Mathematics Teaching Practices to ensure high performing students demonstrate growth in mathematics, (NCTM, 2014), and using student MAP and summative assessment results as evidence to demonstrate growth and thereby demonstrate evidence of effective teaching of the CCSSM. The third stakeholder critical behavior essential for achieving the stakeholder goal involves teachers meeting in learning communities to discuss student results and create plans for modifying classroom instruction to enhance student progress. Current performance indicates that teachers do meet in learning communities and discuss student data in order to Math Growth in High Achieving Elementary Schools 89 identify strengths and weaknesses, and to identify effective teaching strategies. Although it is essential that teachers collaborate by meeting in learning communities, the use of this time to more productively focus on student growth in mathematics is recommended. Suggested future performance indicators include teachers using their learning community time to focus on high performance students’ growth using MAP and summative unit data as well as utilizing the Effective Mathematical Teaching Practices to create plans for modifying classroom instruction. Table 10 provides a summary of each of the stakeholder critical behaviors by listing the current performance indicators and the future performance indicators. Implementation Emerging themes from the solutions suggested above included feedback, use of learning community time, MAP data and professional development. These themes are discussed below through the proposed implementation plan which is outlined briefly in Table 11. Feedback Feedback has been proven to be one of the most effective ways to improve the quality of instruction and achievement in schools (DuFour, Eaker & DuFour, 2008). The suggested implementation plan would begin August 2018 and involve teachers setting goals for their use of differentiated mathematical instruction for student groupings. This goal setting would focus on differentiated instruction for high performing students initially but could extend to all students. Supervisors, the instructional coach and other grade level teachers would commit to observing colleague’s lessons to provide feedback to teachers. This feedback would focus on teaching the CCSSM and strategies for teaching high performing students Math Growth in High Achieving Elementary Schools 90 Table 10. Stakeholder Critical Behaviors: Current and Future Performance Indicators Organizational Vision To be a model of global excellence where learners excel and are challenged to become innovative thinkers and productive, compassionate citizens ready to lead tomorrow’s world. Organizational Performance Goal By May 2018, all grade 4-5 students will achieve at higher levels in math after 32 weeks of instruction as measured by the biannual Measures of Academic Progress (MAP) test results. Stakeholder Goal: Teachers By May 2018, all grade 4-5 mathematics teachers will effectively teach 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) standardized test. Stakeholders Critical Behaviors For teachers to achieve the stakeholder goal above, they need to demonstrate performance of the following critical behaviors. Critical Behavior 1: Grade 4-5 classroom teachers will use student fall MAP results to generate math goals for each student to improve their growth in math over the year. Current Performance Indicators (based on interview data): • Teachers do not use student MAP results to generate math growth goals. Suggested Future Performance Indicators • Teachers will use student MAP results to generate individual student math goals to demonstrate growth. Critical Behavior 2: Grade 4-5 classroom teachers will effectively teach the Common Core State Standards in mathematics as measured by MAP results and summative assessments. Current Performance Indicators (based on interview data): • Teachers use pre-assessment data based on CCSSM standards to know which areas to focus math instruction on. Suggested Future Performance Indicators • Teachers will use the Effective Mathematical Teaching Practices to ensure high performing students demonstrate growth in math. • Teachers will use MAP data and summative assessment results as evidence to demonstrate effective teaching of the CCSSM. Critical Behavior 3: Grade 4-5 classroom teachers will meet in PLCs to discuss student results and create plans for modifying classroom instruction to enhance student progress. Current Performance Indicators (based on interview data): • Teachers discuss student data in teams and use the data to identify strengths and weaknesses. • Teachers use their learning communities to identify effective teaching strategies. Suggested Future Performance Indicators • Teachers discuss student data in teams focusing on high performance students’ growth using MAP and summative unit data. • Teachers discuss the Effective Mathematical Teaching Practices as part of their learning community work and use them to create plans for modifying classroom instruction to enhance student progress. Math Growth in High Achieving Elementary Schools 91 and would be provided once per semester from all parties leading to six observations focused on CCSSM feedback throughout the year. Learning Community Time The second theme identified in the solutions was using Learning Community time for discussions and planning. This structure is already in place at KIDS and the solutions focus on ensuring discussions involve review of the data specifically related to high performing students. Additionally, learning communities would plan their math lessons collaboratively during learning community time and revise those lessons at the beginning, middle and end of each math unit using formative and summative data. MAP Data Once the fall MAP data results are available in September 2018, students and teachers would collaboratively generate student math goals. There is a goal setting conference already established in October where students share their learning goals with parents. High performing students (and possibly all students) would be directed towards having a math goal as part of this structure. These goals are then the focus for instruction and learning with a progress review during the family conferences in March. A final review would take place after the spring MAP results in May. This focus on learning and instruction should result in growth for high performing students from the beginning to the end of the school year. Professional Development Solutions suggested within the theme of professional development included differentiated instruction, feedback and the use of the Effective Mathematics Teaching Practices (NCTM, 2014). Beginning September 2018, professional development would be offered through the Learning Community time focusing on ensuring teachers know how to Math Growth in High Achieving Elementary Schools 92 interpret MAP data. Use of the newly revised Student Profile (NWEA, 2017) gives detailed data for individual students in each math domain related specifically to the CCSSM. Using individual students’ profiles to assist with goal setting for students learning and instructional needs would be an additional way to utilize this data. Existing learning community time would be used to map out further professional development with mini-sessions from October to April focusing on teaching teachers how to differentiate instruction for high performing students in math. Another area of focus for professional development is teaching teachers how to use the Effective Mathematics Teaching Practices (NCTM, 2014). Outlined in Chapter Two, these provide sound research based practices that could have a positive impact on student achievement. Expertise for this teaching is in-house with some grade level teachers, the Math Enrichment teachers and the instructional coach being qualified thereby enabling access across the year and low additional cost to provide this professional development. The themes present in the solutions to the knowledge, motivation, and organizational influences as well as the critical behaviors necessary for success are outlined with the implementation plan in Table 11. Evaluation Evaluating and assessing the implementation plan’s effectiveness in closing the knowledge, motivation and organizational gaps is the final stage in the gap analysis framework (Clark & Estes, 2008). If the plan is successful then the desired outcome of high performing students demonstrating higher levels of growth in mathematics as measured by their MAP scores from fall to spring each year will have been achieved. Kirkpatrick and Kirkpatrick’s (2008) four levels of training evaluation was the framework used to measure the success of the implementation program, which is outlined as follows. Math Growth in High Achieving Elementary Schools 93 Table 11. Solution Themes and Implementation Plan Solution Themes Implementation Plan Feedback Provide frequent, accurate, specific and timely feedback to teachers regarding their use of differentiated mathematical instruction for student groupings through: • Supervisor, coach and peer observations • Goal setting • Learning community discussions Implement August 2018 • Teachers set instructional goals • Supervisor, coach and peer observations once per semester focused on CCSSM (Total = 6 per school year) Learning Community Structure Use Learning Community time to: • Review high performing students’ data to demonstrate growth and plan next steps at beginning, middle and end of each math unit. • Collaboratively plan specific conceptual and skills based math lessons differentiated for high performing students. Implement August 2018 Learning community discussions focused on math: • Data review focused on high performing students • Lesson planning and review at beginning, middle and end of each math unit. Map Data Use MAP data to: • Generate student math goals. • Demonstrate effective teaching of the CCSSM. Implement September 2018 • Generate student math goals September/October as part of Students Goal Setting Conferences, monitor at March during Family Conferences then review with end of year data in May. • Demonstrate growth signifying effective teaching Professional Development Provide professional development in: • Using MAP test data • Differentiated Instruction • Feedback • Effective Mathematics Teaching Practices Implement September 2018 • Focus on ensuring teachers know how to interpret MAP data Implement October - April, 2018 Mini PD sessions focused on: • Strategies for differentiating mathematical instruction • Providing frequent, accurate, specific and timely feedback • Knowledge and application of the Effective Mathematics Teaching Practices Evaluating Level 4: Results At Level 4 the emphasis is on creating desired outcomes of the implementation plan in its entirety. The goal is to ensure all grade 4-5 mathematics teachers will effectively teach 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) standardized test. The desired outcome will be that high performing students in grades 4-5 will demonstrate higher levels of growth in the MAP test after 32 weeks of math instruction. The metric used to measure whether the goal was achieved or not will be the level of growth in the MAP test. Math Growth in High Achieving Elementary Schools 94 The method will be to compare the difference in MAP results for high performing students from beginning to the end of the school year. Evaluating Level 3: Behavior Level 3 focuses on behavior and how much of the professional development teachers apply in their daily work with students. The goal in this case is for all faculty to receive professional development in analyzing, interpreting and applying student data to improve achievement in math. Within this goal is the expectation that teachers will collaboratively plan specific conceptual and skills based math lessons differentiated for high performing students. The desired outcome necessary for the goal to be achieved is that the professional development training is applied and transferred to the classroom and Learning Community discussion and practices. The metric used to measure progress towards this goal is focusing on math during Learning Community time by reviewing students’ data and planning lessons for high performing math students. Evaluating Level 2: Learning Level 2 focuses on the degree to which faculty can learn the requisite knowledge, skills, attitude, confidence and commitment based on their performance during the professional development sessions. There are three separate goals for evaluating this section of the implementation plan. The first goal is that all teachers will use targeted and timely feedback regarding their use of differentiated mathematical instruction for student groupings and their effective teaching of the CCSSM. This feedback received by teachers will result in the desired outcome of mathematical instruction for high performing students demonstrating use of differentiated instruction and effective teaching of the CCSSM. Teachers will receive this feedback through supervisor, coach and peer observations which will then be used to set Math Growth in High Achieving Elementary Schools 95 instructional goals. The teachers’ goals will provide evidence of applying the professional development and whether further professional development is needed. The second goal for evaluating Level 2: Learning is that all teachers will know how to use MAP data effectively to demonstrate growth in high performing students in math. The desired outcome for reaching this goal is that teachers will know how to interpret MAP data and then know how to use MAP’s Student Profiles to generate individual math goals. These student goals will be formulated at the beginning of the year after the fall MAP testing with progress monitoring at mid-year and final progress reviewed at the end of year. The method used to ensure this happens will be to use Learning Community time for teachers to teach others how to interpret MAP data and for teachers to share ways in which they used the Student Profiles for generating goals, modifiying goals and reviewing student progress. The final goal for evaluating level two of the implementation plan is that all teachers will know how to provide differentiated instruction for high performing students and will be able to use the Effective Mathematics Teaching Practices to enhance their practice. The desired outcome through achieving this goal is for teachers to develop confidence and competence when teaching high performing students through the use of differentiated instruction and the Effective Mathematics Teaching Practices. This professional development will be provided within the Learning Community by other faculty, the instructional coach and the administrator and will focus on the Effective Mathematics Teaching Practices as well as strategies for differentiating instruction. Discussion of the supervisor, coach and peer observations regarding differentiated instruction and use of the Effective Mathematics Teaching Practices within individual teacher meetings with their supervisor and through Learning Community time will be one way to ensure that faculty have learned the necessary Math Growth in High Achieving Elementary Schools 96 content through the professional development sessions. Data will also be collected through a Focus Group format as part of those Learning Community discussions. Evaluating Level 1: Reaction Level 1: Reaction concerns the degree to which participants enjoy the training or are satisfied that the professional development was worthwhile and relevant to their jobs. The goal for this level is that teachers will reflect on the professional development received through this training as engaging and relevant. The desired outcome necessary to achieve the goal is to ensure teachers find the professional development engaging and relevant to achieving high levels of growth for high performing students in math. Using in-house expertise for professional development – teachers, coaches, and administration shows value for existing expertise and can be specifically tailored to teachers’ levels of readiness thereby ensuring its relevance. A brief survey at end of each professional development session can be used to measure engagement and relevance. An end of the year teacher reflection regarding changes in instructional practice, student achievement and growth through MAP test results would be helpful in not only measuring levels of engagement and relevance but also in setting instructional goals for the following year. Summary A clear focus on the entire evaluation plan ensures the researcher considers the connections between easy fixes and the end goal, and helps identify potential problem spots as well as places where the training may not have been implemented successfully. Table 13 summarizes the goals, outcomes, metrics and methods related to Kirkpatrick & Kirkpatrick four levels of training and evaluation framework (2008). Math Growth in High Achieving Elementary Schools 97 Table 12. Goals, Outcomes and Methods According to Levels of Evaluation Level 4: Results – The degree to which targeted outcomes occur as a result of the training and the support and accountability package. Goal Desired Outcome Metric/Method By May 2019, all grade 4-5 mathematics teachers will effectively teach 100% of the math content required for greater numbers of high performing students to demonstrate growth in the Measures of Academic Progress (MAP) standardized test. High performing students in grades 4-5 will demonstrate higher levels of growth in the MAP test after 32 weeks of math instruction. Metric: Increase in growth performance on MAP test results. Method: Compare MAP results from fall to spring. Level 3: Behavior – The degree to which participants apply what they learned during training when they are back on the job. Goal Desired Outcome Metric/Method By May 2019, all Learning Communities will receive professional development so they know how to: a) Review high performing students’ data to demonstrate growth and plan next steps at beginning, middle and end of each math unit. b) Collaboratively plan specific conceptual and skills based math lessons differentiated for high performing students. Professional development training is applied and transferred to the classroom and Learning Community discussion and practices. Metric: Learning Community discussions will focus on math by: • Reviewing high performing students’ data. • Planning lessons for high performing math students. Method: Review high performing students’ progress and revise lesson plans at beginning, middle and end of each math unit. Level 2: Learning – The degree to which participants acquire the intended knowledge, skills, attitude, confidence, and commitment based on their participation in the training. Goal Desired Outcome Metric/Method Feedback By May 2019, all teachers will use targeted and timely feedback regarding their use of differentiated mathematical instruction for student groupings and their effective teaching of the CCSSM. Feedback received by teachers will result in mathematical instruction for high performing students demonstrating use of differentiated instruction and effective teaching of the CCSSM. Metric: • Teachers set instructional goals based on feedback • Supervisor, coach and peer observations take place once per semester (Total = 6 per school year) Method: Review application of professional development through teachers’ goals for instruction combined with observational feedback data. Map Data By May 2019 all teachers will know how to use MAP data effectively to demonstrate growth in high performing students in math. Teachers will know how to: • Interpret MAP data • Use MAP’s Student Profiles to generate individual math goals Metric: Generate student math goals at beginning of the year, monitor progress at mid-year and review progress at the end of year. Method: Use Learning Community time to • teach others how to interpret MAP data and • share ways in which Student Profiles were used in goal generating, modification and reviewing student progress. Math Growth in High Achieving Elementary Schools 98 Table 12. (Continued) Level 2: Learning – The degree to which participants acquire the intended knowledge, skills, attitude, confidence, and commitment based on their participation in the training. Goal Desired Outcome Metric/Method Professional Development By May 2019, all teachers will know how to • Provide differentiated instruction for high performing students. • Use the Effective Mathematics Teaching Practices to enhance their practice. Teachers develop confidence and competence at teaching high performing students through differentiated instruction and use of the Effective Mathematics Teaching Practices. Metric: Professional development focused on: • Effective Mathematics Teaching Practices • Strategies for differentiating mathematical instruction Method: Discussion of supervisor, coach and peer observations regarding differentiated instruction and use of the Effective Mathematics Teaching Practices. Collect data through a Focus Group format as part of Learning Community discussions. Level 1: Reaction – The degree to which participants find the training favorable, engaging and relevant to their jobs Goal Desired Outcome Metric/Method By May 2019, teachers will reflect on the professional development received through this training as engaging and relevant. To ensure teachers find the professional development engaging and relevant to achieving high levels of growth for high performing students in math. Metric: Use in-house expertise for professional development – teachers, coaches, administration Method: Quick survey at end of each professional development session. End of the year reflection regarding changes in instructional practice, student achievement and MAP test results. Set instructional goals for following year. Source: (Kirkpatrick & Kirkpatrick, 2008) Future Research Suggested areas for further research include how to better define, describe and record what confidence in teaching looks like. Observing how confident teachers are can be very subjective – how can it be made more objective? Research that describes and demonstrates how to bridge the gap between teacher’s self-confidence and their ability as an effective instructor would also be helpful. KIDS teachers had confidence in their ability to teach mathematics however the high performing mathematicians in their classes did not demonstrate requisite growth in achievement across the year indicating a discrepancy between confidence and achievement. Math Growth in High Achieving Elementary Schools 99 Another suggested area for future research (Hill et al., 2005) is that of comparing and contrasting mathematical instructional practices of teachers who are more mathematically knowledgeable with those who are less mathematically knowledgeable in order to better understand the need for specific mathematical content knowledge. Hill et al. (2005) also suggests “ongoing research on teaching, on students' learning, and on the mathematical demands of high-quality instruction can contribute to increasing precision in our understanding of the role of content knowledge in teaching” (p. 401). Research into the achievement of high performing students in mathematics and other curriculum areas would be helpful to refer to with this type of study. Much of the gifted research refers to a very small sector of the population and is often outdated. A suggested future focus could be to research how high performing students differ from the gifted or highly able students and ways in which teachers might have a greater impact on those students. Conclusion Use of the gap analysis framework (Clark & Estes, 2008) has enabled identification of solutions to the problem of why high performing students are not making requisite growth in mathematics across the year as measured by MAP. Although many of the identified solutions highlighted are specific to KIDS context, these solutions can also be adapted and applied to other settings. Many schools could use their current learning community structures to target professional development in areas of providing feedback to faculty on their instructional methods, utilizing the Effective Mathematical Teaching Practices (NCTM, 2014) collecting, analysing and using student data to inform instruction and collaboratively planning and teaching differentiated lessons. The evaluation framework presented involves Math Growth in High Achieving Elementary Schools 100 related on-the-job continuous professional development and discussion and is designed to produce the desired outcome of greater levels of high performing students’ demonstrating growth and higher achievement levels in math. The vision of the Kampong International Day School (KIDS) is to be “a model of global excellence where learners excel and are challenged to become innovative thinkers and productive, compassionate citizens ready to lead tomorrow’s world” (KIDS 3 System Focus, 2015). The overarching goal of the school related to students’ achievement is that learners have talents and will excel (KIDS System Focus, 2015). In a competitive market, where a schools’ success depends on its ability to deliver what it says it can deliver, it is imperative to be able to confidently respond to parents of high performance students that the needs of their children are being met. This research study demonstrates ways in which a world-renowned school with great expectations of success for all students can become even more effective at meeting the needs of its high performing students. 3 The actual URL not provided in order to protect the identity of the organization. Math Growth in High Achieving Elementary Schools 101 References Assouline, S. G., & Lupowski-Shoplik, A. (2011). Developing math talent: A comprehensive guide to math education for gifted students in elementary and middle school (2 nd ed.). Waco, TX: Prufrock Press. Attard, C. 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Wang, S., McCall, M., Jiao, H., & Harris, G. (2013). Construct validity and measurement invariance of computerized adaptive testing: Application to Measures of Academic Progress (MAP) using confirmatory factor analysis. Journal of Educational and Developmental Psychology, (3)1. Wiliam, D. (2006). Assessment for learning: Why, What, and How? Orbit, 36(2), 2-6. Math Growth in High Achieving Elementary Schools 112 Appendix A: Email Requesting Permission to Conduct Research Hello ______________, Moving from Great to Greater: Math in High Achieving Elementary Schools – A Gap Analysis You are invited to participate in a research study conducted by Susan Shaw at the University of Southern California. Please read through this form and ask any questions you might have before deciding whether or not you want to participate. PURPOSE OF THE STUDY This research study aims to understand how to ensure greater levels of growth in high performing students in mathematics PARTICIPANT INVOLVEMENT If you agree to take part in this study, you will be asked to participate in one 50-minute interview. You do not have to answer any questions you don’t want to. In addition, your math class will be observed for one 25-minute session during the school day. In the interests of confidentiality all interviews will be conducted by an independent researcher who is not your supervisor. All interview responses will be cleaned to protect your anonymity. Please note: 1. You are receiving this email on my behalf (Susan Shaw) and all correspondence will take place between the independent researcher and yourself to continue to protect your anonymity. 2. Participation in this study is entirely voluntary and there are no consequences should you choose not to participate. Your permission to conduct this study will be greatly appreciated. You can indicate your willingness to participate by replying to this email before Thursday, 28 September 2017. Thank you, Researcher (on behalf of Susan Shaw) Math Growth in High Achieving Elementary Schools 113 Appendix B: Interview Protocol Thank you for agreeing to take part in this interview today. I want you to know that your participation today and the answers you give are confidential. Your responses will be shared with the researcher, Susan Shaw, with no names attached so she can continue with her research into this topic. The purpose of this interview is to gain a better understanding of the ways in which you use goal setting, the Common Core State Standards in Mathematics, and Learning Communities (LCs) to inform instruction for high performing students in math. For this interview, high performing students are defined as those students in your classroom who attend the Math Enrichment class, and those who you currently teach math enrichment to in the classroom. There are no right or wrong answers and I know Susan greatly appreciates you taking the time to participate. Do you have any questions before we begin? Interview Questions Warmup #1: What, if anything, do you enjoy about teaching math to high performing students at KIDS? Warmup #2: What, if anything, do you find challenging about teaching math to high performing students at KIDS? 1. Please describe how you use goal setting with your students to achieve the Common Core State Standards in Mathematics. 2. In what ways do you use test data to inform your math instruction for high performing students? If MAP data is not mentioned within their answer, ask: a) In what ways, if any, do you use MAP data to inform math instruction? b) If they say they do not use MAP data to inform math instruction, ask: Why do you not use MAP data to inform math instruction? 3. How well do you know the CCSSM? Probe: What changes have you made to your math instruction as a result of teaching the CCSSM? 4. Please explain how you use students’ prior knowledge to teach the CCSSM. 5. How do you feel about your ability to teach CCSSM to students? (For example, do you feel confident in your ability to teach the CCSSM to students? Are there any areas you think you could improve in?) 6. Do you believe there is a correlation between your understanding of the Common Core and your students achievement in math? (If their answer is yes, ask them): How do you know? Please give some examples. (If their answer is no, ask them): Why not? Math Growth in High Achieving Elementary Schools 114 7. Do you receive fair, accurate, specific and timely feedback about your performance teaching the CCSSM? If their answer is yes, ask them: Who provides this feedback? 8. Please describe the features of an effective LC meeting in your grade level. Probe: What is your role during an effective LC meeting? 9. In what ways, if any, do you utilize the knowledge of your LC members to enhance or modify your math instruction? a) Please describe how you’ve modified a math lesson based on feedback or discussion from a PLC meeting. 10. In what ways, if any, do you group students for math instruction? (If the teacher uses grouping for instruction, ask the following question): How do you decide which group to put each student in? 11. In your experience, what has been the effect of grouping students in mathematics on their subsequent achievement? 12. In what ways, if any, does your behavior impact student learning in your math classroom? 13. Do you believe you are able to provide effective instruction that can make a difference to your students’ growth and achievement in mathematics? (If their answer is yes, ask): Please could you give some examples? (If their answer is no, ask): Why not? 14. It is said that working in a high achieving school such as KIDS an atmosphere of competition and fear of failure amongst teachers may be created – do you believe this statement is true or false for your experience? (Pause for their answer then ask): Please explain why or why not Thank you for your responses. Your time is much appreciated and your answers will be strictly confidential. Math Growth in High Achieving Elementary Schools 115 Appendix C: Interview Questions Listed by Critical Areas, Knowledge, Motivation, and Organizational Influences Table 6. Interview Questions Listed by Critical Areas, Knowledge, Motivation, and Organizational Influences Interview Questions Critical Areas and Knowledge, Motivation and Organizational Influences What, if anything, do you enjoy about teaching math to high ability students at KIDS? Warmup Question What, if anything, do you find challenging about teaching math to high ability students at KIDS? Warmup Question 1. Please describe how you use goal setting with your students in mathematics. Critical Area 1 2. In what ways do you use test data to inform your math instruction for high performing students? (K#2) If MAP data is not mentioned in the answer to the above question, ask the following question: a) In what ways, if any, do you use MAP data to inform math instruction? (K #2) b) If they say they do not use MAP data to inform math instruction, ask: Why do you not use MAP data to inform math instruction? Knowledge #2 3. How well do you know the CCSSM? Probe: What changes have you made to your math instruction as a result of teaching the CCSSM? Knowledge #1 4. Please explain how you use students’ prior knowledge to teach the CCSSM. Critical Area 2 5. How do you feel about your ability to teach CCSSM to students? (For example, do you feel confident in your ability to teach the CCSSM to students? Are there any areas you think you could improve in?) (M#1) Motivation #1 6. Do you believe there is a correlation between your understanding of the Common Core and your students achievement in math? (If their answer is yes, ask them): How do you know and please give examples. (If their answer is no, ask them): Why not? (M#4) Knowledge #3 Motivation #4 7. Do you receive fair, accurate, specific and timely feedback about your performance teaching the CCSSM? (O#1) If their answer is yes, ask them: Who provides this feedback? (O#1). Organization #1 8. Please describe the features of an effective LC meeting in your grade level. Probe: What is your role during an effective LC meeting? Critical Area 3 9. In what ways, if any, do you utilize the knowledge of your LC members to enhance or modify your math instruction? a) Please describe how you’ve modified a math lesson based on feedback or discussion from a LC meeting. Critical Area 3 Math Growth in High Achieving Elementary Schools 116 Other Knowledge, Motivation and Organizational Barriers KNOWLEDGE BARRIERS: • Teachers are knowledgeable about the effectiveness and impact of grouping on student achievement and can group all students accordingly. (K#4) 10. In what ways, if any, do you group students for math instruction? (If the teacher uses grouping for instruction, ask the following question): How do you decide which group to put each student in? 11. In your experience, what has been the effect of grouping students in mathematics on their subsequent achievement? • Teachers understand how their behavior impacts student learning. (K#5) 12. In what ways, if any, does your behavior impact student learning in your math classroom? Knowledge #4 Knowledge #5 MOTIVATIONAL BARRIERS: • Teachers believe they are capable of effective instruction that can make a difference to their students’ growth and achievement in mathematics. M#2 • Successful teachers believe students should demonstrate growth in math achievement. M#3 13. Do you believe you are able to provide effective instruction that can make a difference to your students’ growth and achievement in mathematics? (If their answer is yes, ask): Please could you give some examples? (If their answer is no, ask): Why not? Motivation #2 Motivation #3 ORGANIZATIONAL BARRIER: Working in a high achieving school may create atmosphere of competition and fear of failure amongst teachers. (O#2) 14. It is said that working in a high achieving school such as KIDS an atmosphere of competition and fear of failure amongst teachers may be created – do you believe this statement is true or false for your experience? (Pause for their answer then ask): Please explain why or why not. Organization #2 Thank you for your responses. Your time is much appreciated and your answers will be strictly confidential. Math Growth in High Achieving Elementary Schools 117 Appendix D: Observation Protocol The purpose of the observation as far as practicable is to provide evidence that teachers: • Know the Common Core State Standards in Mathematics (CCSSM) • Know how to use math test data to inform instruction for students • Know the effectiveness and impact of grouping on student achievement and can group all students accordingly. • Understand how their behavior impacts student learning. • Are confident in their ability to teach CCSSM to students The observation should take place without disrupting the teacher or students. Take note of the following where possible (adapted from Merriam & Tisdell, 2016, p. 141): 7. The physical setting: What is the physical setting like for this lesson? How is space allocated? What objects, resources, technologies are in the setting? 8. The participants: Describe who is in the scene, how many people and their roles. What are the ways in which the participants organize themselves? 9. Activities and interactions: What is going on? Is there a definable sequence of activities? How do the students interact with the activity and one another? What norms or rules structure the activities and interactions? When did the activity begin? How long does it last? Is it a typical activity or unusual? 10. Conversation: what is the content of the conversations in this setting? Who speaks to whom? Who listens? Quote directly, paraphrase and summarize conversations. 11. Your own behavior: How is your role affecting the scene you are observing? What thoughts are you having about what is going on? Record these in the observer comments section. 12. After the observation take time to record as many field notes as possible about the observation. Date: Time: Setting: Class A The physical setting: What is the physical setting like for this lesson? How is space allocated? What objects, resources, technologies are in the setting? The participants: Describe who is in the scene, how many people and their roles. What are the ways in which the participants organize themselves? Math Growth in High Achieving Elementary Schools 118 Activities and interactions: What is going on? Is there a definable sequence of activities? How do the students interact with the activity and one another? What norms or rules structure the activities and interactions? When did the activity begin? How long does it last? Is it a typical activity or unusual? Conversation: what is the content of the conversations in this setting? Who speaks to whom? Who listens? Quote directly, paraphrase and summarize conversations. Time: Observer Notes Activities and interactions: What is going on? Is there a definable sequence of activities? How do the students interact with the activity and one another? What norms or rules structure the activities and interactions? When did the activity begin? How long does it last? Is it a typical activity or unusual? Conversation: what is the content of the conversations in this setting? Who speaks to whom? Who listens? Quote directly, paraphrase and summarize conversations. Time: Observer Notes

Abstract (if available)

Abstract This study examined the performance gap experienced by high achieving grade four and five students at the Kallang International Day School (KIDS) who were not demonstrating growth in math results from fall to spring as measured by the Measures of Academic Progress (MAP) test (NWEA, 2017). The purpose of this study was to apply the gap analysis problem-solving framework (Clark & Estes, 2008) to identify root causes of the knowledge, motivation and organizational factors that prevented teachers from effectively teaching 100% of the math content required for greater numbers of high performing students to demonstrate growth in the MAP test. The subjects of this study were teachers of grade four and five at the Kallang International Day School. Interview and observational data were collected and analyzed and used to validate and inform possible solutions for the knowledge, motivation and organization influences. Research-based solutions were recommended to close the knowledge, motivation and organization gaps, and included providing teachers with frequent, accurate, specific and timely feedback regarding their use of differentiated mathematics instruction for student groupings and using learning community time to collaboratively plan specific conceptual and skills based math lessons differentiated for high performing students. Critical behaviors of the stakeholders were also examined and recommendations included generating student math goals based on fall MAP results, teaching with the Effective Mathematical Teaching Practices, (NCTM, 2014) and using them to create plans for modifying classroom instruction to enhance high performing students’ progress. The outcomes of this study may be used by this school and others to improve the growth in math achievement for high performing students.

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Creator Shaw, Susan Joy (author)

Core Title Moving from great to greater: Math growth in high achieving elementary schools - A gap analysis

School Rossier School of Education

Degree Doctor of Education

Degree Program Education (Leadership)

Publication Date 07/26/2018

Defense Date 05/16/2018

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

Tag differentiated mathematics instruction,effective mathematics teaching practices,elementary school,feedback,high achieving students,high performing students,instructional groupings,learning communities,MAP testing,math growth,OAI-PMH Harvest,student math goals

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