# Increasing math success with differentiated instruction

TABLE OF CONTENTS

LIST OF TABLES………………………………………………………………………v

SECTION 1: INTRODUCTION TO THE STUDY .....................................................…1 Problem Statement ............................................................................................................4 Background of Problem ....................................................................................................5 Nature of the Study ..........................................................................................................7 Research Questions ...........................................................................................................8 Quantitative .......................................................................................................................8 Null Hypothesis ................................................................................................................9 Alternative Hypothesis......................................................................................................9 Purpose Statement ...........................................................................................................10 Theoretical Framework ...................................................................................................12 Definition of Terms.........................................................................................................14 Assumptions ....................................................................................................................16 Limitations ......................................................................................................................16 Scope and Delimitations .................................................................................................17 Significance of the Study ................................................................................................18 Implications for Social Change .......................................................................................19 Summary and Transition .................................................................................................21

SECTION 2: LITERATURE REVIEW .........................................................................23 Differentiated Instruction ................................................................................................25 Understanding by Design ................................................................................................34 Mathematics Achievement..............................................................................................36 Mathematical Pedagogy ..................................................................................................40 No Child Left Behind......................................................................................................47 Classroom Assessments ..................................................................................................53 Summary and Transition .................................................................................................56

SECTION 3: RESEARCH METHODOLOGY..............................................................58 Research Design and Approach ......................................................................................60 Setting and Sample .........................................................................................................64 Treatment ........................................................................................................................67 Instrumentation and Materials ........................................................................................75 Data Collection ...............................................................................................................77 Data Analysis ..................................................................................................................78

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Measures to Ensure Participants’ Rights ........................................................................82 Summary and Transition .................................................................................................83

SECTION 4: RESULTS .................................................................................................85 Research Tools ................................................................................................................87 Data Analysis ..................................................................................................................90 Outcomes of Testing Procedures ....................................................................................92 Summary .........................................................................................................................99

SECTION 5: SUMMARY, CONCLUSION, RECOMMENDATIONS .....................102 Research Overview .......................................................................................................102 Results from the Study ..................................................................................................107 Social Implications........................................................................................................110 Recommendation for Action .........................................................................................111 Recommendation for Further Study..............................................................................113 Conclusion ....................................................................................................................115

REFERENCES .............................................................................................................117

APPENDIX A: Pretest and Posttest Scores ..................................................................132 APPENDIX B: Data Use Agreement ...........................................................................133 CURRICULUM VITAE ...............................................................................................136

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LIST OF TABLES Table 1. ABC Middle School CRCT scores, 2009 .........................................................66

Table 2. ABC Middle School CRCT scores ...................................................................66

Table 3. Mathematics domains on seventh grade CRCT ................................................68

Table 4. Average correct/Average percentage correct ....................................................69

Table 5. Independent samples t test ................................................................................93

Table 6. One way analysis of variance summary for posttest .........................................94

Table 7. One-way analysis of covariance summary for pretest group ............................95

Table 8. Estimates: dependent variable: posttest ............................................................95

Table 9. Mean CRCT scores ...........................................................................................98

SECTION 1: INTRODUCTION TO THE STUDY The emphasis in education today is on overall achievement and performance scores, resulting in new standards and curriculum. The No Child Left Behind (NCLB) Act of 2001 (U.S. Department of Education, 2009) holds states, districts, and county schools accountable for student performance and achievement. NCLB relies on benchmark assessments that show progress and identify weaknesses in core academic subjects (U.S. Department of Education, 2005). In an attempt to reduce the achievement deficits experienced by students, school environments have become more complex and diverse, a place where all children are expected to learn and where high learning standards set the vision of educational success for all students (Bennis, 2006). One facet that is adding to the complexity of the school environment is differentiated instruction. Differentiated instruction requires a systematic approach to planning curriculum and instruction for academically diverse learners. It is a way of thinking about the classroom with the goal of honoring each student’s learning needs and maximizing each student’s learning capacity (Tomlinson, 2003, p. 5). Engaging students in learning and encouraging co learning has been recognized as a crucial factor in the construction of knowledge. Schools are faced with the challenge of implementing new state standards which define the skills and topics that students must master (The partnership for 21 st century skills, 2007). Standards are an attempt to answer an essential educational question: What knowledge and skills do we want our children to learn? Standards drive the critical elements of the American educational system; the

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curriculum that schools follow, the textbooks students read, and the tests they take (The partnership for 21 st century skills, 2006). Standardized test scores, specifically in Georgia, have indicated that there has been a decline in the comprehension of basic mathematical fundamentals among middle school students (Georgia Department of Education, 2008). The purpose of this quantitative quasi-experimental casual-comparative study was to implement differentiated instruction in an 8th grade mathematics classroom and to examine the potential relationship that differentiated instruction has on the achievement and comprehension of mathematical fundamentals, possibly resulting in increased mathematics comprehension and standardized test scores. Schools are paying much more attention to the alignment of curriculum and instruction (Jennings & Rentner, 2006). Over the past two decades, state agencies and educational groups have paid considerable attention to describing what students need to know but without addressing the more complex thinking and technical skills that will govern 21 st century life. Today’s standards privilege fact-based learning and core areas of study (The partnership for 21 st

century skills, 2006). The schools, state, and government are also analyzing test score data more closely today than they have in the past. It is important to understand that while standards and tests can help define priorities and measure progress, they cannot turn schools around on their own. ―Test scores are information for an assessment system — they are not the system itself‖ (Darling-Hammond, 2002). Teachers have the critical role of providing suitable experiences that will facilitate learning (Palmer, 2005).

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The first major paradigm shift in learning theories began in the mid 1960s in the field of early childhood education based on the work of Piaget (1963). Piaget fundamentally altered the views of the child and the nature of mental development (Singer & Revenson, 1996). The central theme of this paradigm is the view of the child as active in constructing not only knowledge but intelligence itself. Knowledge is what you already know. It is fact or condition of knowing something with familiarity and is often gained through experience or association. Intelligence is the ability to learn or understand or deal with new or trying situations. In other words, it's the ability to successfully apply your knowledge. Researchers today have shifted their focus from how knowledge is acquired to how it is constructed and how the child can maintain and build upon that knowledge. The current Georgia mathematic curriculum is cummulative; meaning each year students add to or build upon knowledge gained in the previous academic year. Changes in the mathematic curriculum over the past 4 years in the state of Georgia have increased the difficulty level and the number of mathematical concepts learned, while reducing the review time in the upcoming year (Georiga Department of Education, 2008). Lecturing, memorizing, and drilling on mathematical concepts do not give the students time to take in information, make sense of ideas, and express what they have learned. The current drop in standardized test scores and classroom assessments have prompted new applications, resources, and methods of classroom instruction (Arhar, 2003). Educational researchers have investigated differentiated instruction on a number of levels and from

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various perspectives (Arhar, 2003). Many schools are initiating the implementation of differentiated instruction into the classroom (Georgia Department of Education, 2008). The middle-grade years constitute the greatest learner variability of any segment in school, with students of the same age varying in intellectual development, as they do in physical, emotional, and moral development.Tomlinson (2005) investigated how differentiated instruction develops the unique capacities of learners and developed theories regarding the role of the teacher in the differentiated classroom. Schools that aspire to be successful in educating adolescents must attend to their varying needs, interests, and readiness levels (National Middle School Association, 2005). Problem Statement The problem addressed in this study is that in ABC Middle School, there has been a decline in the comprehension of basic mathematical fundamentals among the 8th grade students. According to the National Council of Teachers of Mathematics ([NCTM], 2006), ―Excellence in mathematics education requires high expectations and strong support for all students‖ (p.11). Educators need to hold their students to high expectations and provide the necessary resources and support for the students to accomplish success. NCTM noted, ―Equity requires accommodating differences to help everyone learn mathematics‖ (p. 13). Accommodating differences can be defined as differentiated instruction. Teachers must see to it that ―mathematics can and will be learned by all students‖ (p.13). All students can learn, but not all students learn the same way.

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Background of the Problem Tomlinson (2005) stated, ―Developing academically responsive classrooms is important for a country built on the twin values of equity and excellence‖ (p. 34). The teacher must be able to teach students with a wide range of readiness levels, varying interests, and varying learning profiles (Tomlinson, 2005). As stated by the Alliance for Excellent Education (2008): As globalization has progressed, American educational progress has stagnated. Today, the United States’ high school graduation rate ranks near the bottom among developed nations belonging to the Organization for Economic Co- operation and Development (OECD). And on virtually every international assessment of academic proficiency, American secondary school students’ performance varies from mediocre to poor. Given that human capital is a prerequisite for success in the global economy, U.S. economic competitiveness is unsustainable with poorly prepared students feeding into the workforce. (p. 1)

Unless the United States begins to focus on preparing all students for college and the modern workplace, this downward trend will only get worse (Alliance for excellent education, 2008). The following details how 15-year-old students from the United States compare with 15-year-olds in other OECD member countries in the Program for International Student Achievement (PISA) measures of academic proficiency. According to the OECD (2007), the following information is for the subject of mathematics literacy: The United States ranks 25th of 30 OECD countries in mathematics literacy, and the average score of 474 fell well below the OECD average of 498. Scores have not measurably changed since 2003, when the United States ranked 24th of 29 countries. Over one quarter (28.1 percent) of American fifteen-year-olds performed below the baseline level of mathematics proficiency at which students begin to demonstrate the kind of skills that enable them to use mathematics actively in daily life. (p. 7)

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Currently a computer based math remediation program called Classworks is offered at ABC Middle School. This program has gone from class room computers only to web-based, allowing the program to be accessed at home. Access from home provides students with the opportunity for remediation or acceleration outside of school. In addition to web access, the students can also check out a CD from the media center to take home and use if the student does not have Internet access. The XYZ school district has mandated that 95% of target students, the students right on the border of passing (bubble students), the students who just barely passed (high impact), and Title I students, must get an additional 45 minutes of instruction a week on either language arts or math. The CRCT scores from 2008 are in the Classworks system, which allows the program to analyze and build a program and assign units based on the student’s individual weaknesses. In addition to the Classworks program, ABC Middle School has built in morning activity periods on Tuesday and Thursday mornings that will focus on additional math and language arts remediation or acceleration for the students (ABC Middle School documentation, 2009). Despite these interventions, improvements have not been realized. The intent of this study was to address integration of differentiated instruction in the mathematics classroom and the design and implementation of research-based instructional strategies to improve student achievement in mathematics. This study should contribute to the body of knowledge needed to address and promote social change by providing guidance and resources regarding the steps that educators and school systems can take to be highly

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effective in increasing students’ mathematics achievement in the classroom and on high- stakes tests. Nature of the Study This casual-comparative quantitative research study focused on the achievement and comprehension of mathematical fundamentals among 8th grade mathematics students when differentiated instructional strategies are implemented in the classroom. Mathematics achievement scores were compared between a group of 8th grade students who participated in a mathematics classroom taught by an educator whose teaching methodology was differentiated instruction and a group of 8th grade students who received traditional instruction from an educator whose teaching methodology was traditional textbook pedagogy. The study compared the type of instruction to the level of mathematics achievement of the students. Every child comes into the classroom with varying levels of knowledge, experiences, and interests in understanding mathematical concepts. It is the critical responsibility of the teachers to provide learning opportunities that motivate and challenge students, no matter what their learning styles. To change teaching models, educators have to be willing to address how brains learn and immerse students in interactive, real-life, complex experiences out of which one can process new ideas (Mitchell & Hobson, 2005). To produce effective change in schools a new teaching model has to come from educators themselves.

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Research Questions This study analyzed the achievement and comprehension of mathematical fundamentals among 8th grade general education students when differentiated instructional strategies were used while delivering instruction. The question that guided this research study is: Does differentiated instruction have a positive effect on 8th grade mathematics achievement? Quantitative Methods The quantitative data of the study included a pre and post test, along with Georgia CRCT scores, to evaluate the impact of the type of instruction and student achievement using traditional and differentiated instruction. The independent variable was defined as the type of instruction, and the dependent variable was the mathematics achievement as measured by mathematics test scores. The population for this study consisted of 60 students, 30 in the differentiated instruction classroom and 30 in the traditional classroom. The hypotheses will be tested using independent t-tests to examine the differential impact of instruction and to analyze the data. It was hypotheszied that students participating in the differentiated instruction classroom would achieve steady academic improvement in mathematics and improve their scores on the standardized tests. I believed that implementing the ideas and philosophies related to differentiated instruction would promote a climate of success and expectations in which all students could be successful. The data were collected, recorded, and analyzed to answer the following quantitative questions:

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1. Is there a significant difference between students in the traditional instruction classroom and the differentiated instruction classroom on the 8th grade mathematics posttest when controlling for preexisting differences on- the mathematics pretest? Null Hypothesis: There is no significant difference when teaching using text book pedagogy and teaching using differentiated instruction among middle school students and the achievement and comprehension of mathematical fundamentals on the 8th grade mathematics posttest while controlling for the mathematics pretest. Alternative Hypothesis: There is a significant difference when teaching using text book pedagogy and teaching using differentiated instruction among middle school students and the achievement and comprehension of mathematical fundamentals on the 8th grade mathematics posttest while controlling for the mathematics pretest. 2. What are the differences in student achievement level on the Georgia CRCT between students taught using whole-class instruction and students taught with differentiated instruction? Null Hypothesis: There will not be a statistically significant difference in math achievement levels on the Georgia CRCT mathematics test between students taught using whole-class instruction and differentiated instruction. Alternative Hypothesis: There will be a statistically significant difference in math achievement levels on the Georgia CRCT mathematics test between students taught using whole-class instruction and differentiated instruction.

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Differentiated instruction is a teaching method designed to meet the needs of students in a mixed ability classroom (Edwards et al., 2006). Teachers set the same goals and teach the same skills to all students, but student learning styles and academic abilities are also considered, therefore making it a practical option to vary activities according to the diverse levels of learners (Sternberg & Zhang, 2005). I implemented differentiated instruction including specifically tiered activities. Pre and Posttest scores along with CRCT Mathematics test scores determined if differentiated instruction increased the achievement and comprehension of mathematical fundamentals of 8th grade students. Purpose Statement The purpose of this casual-comparative quantitative study was to determine the impact of differentiated instructional strategies that would encourage learning and co learning, possibly resulting in increased mathematics comprehension and standardized test scores. The study compared the type of mathematics instruction to the level of mathematics achievement of 8th grade students at the ABC School who participated in this study. The independent variable was defined as the type of instruction, differentiated instruction, or traditional instruction, and the dependent variable was defined as the mathematics achievement measured by one 8th grade mathematics pre and posttest and one CRCT test. The control variable will be the pretest and CRCT tests. An independent samples t-test, ANOVA, and ANCOVA will be used to analyze the data. Implementing differentiated instruction techniques provides the students of multiple learning styles opportunities to comprehend and retain mathematical fundamentals in a variety of ways (Wing & Beal, 2004). Success on standardized tests

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determines the student’s ability levels, identifies weaknesses, and is required for promotion. New mathematic standards implemented in the state of Georgia have increased the complexity of the math curriculum in addition to raising the minimum score required to pass the CRCT (Georgia Department of Education, 2009). The challenge of the study was to determine if implementing differentiated instruction affects the mathematics achievement of 8th grade students as compared to 8th grade students taught in a traditional instruction classroom. The study was a design that involved two preexisting classes of mathematics students. The population was comprised of 49 8th grade mathematics students who had received general education mathematics instruction for the 2008-2009 school year. The students are required to have test scores on record for the sixth and seventh grade CRCT and the 8th grade mathematics pretest. The sample, comprised of 49 students, included 23 students receiving differentiated instruction, and 27 students receiving traditional instruction. One of the main reasons for conducting this quantitative casual-comparative study was to see if differentiated instruction resulted in better test scores than traditional instruction. New standards for 8th grade mathematics were implemented during the 2007- 2008 school year (Georgia Department of Education, 2008). Over the past 4 years, the state of Georgia changed the state standards for mathematics from the Quality Core Curriculum (QCC) to the current Georgia Performance Standards (GPS). The changes implemented a more rigorous curriculum in grades 6-8 with a more defined outline of standards necessary for the students to master. A drop in the average standardized test

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score was expected as a result of the new standards, as have been seen in years past when implementing new standards (Georgia Department of Education, 2008). A more rigorous curriculum, new testing standards, and score requirements contributed to the expected drop in the average standardized test scores. The excessive drop in the average score among middle school students has ignited an investigation into new differentiated instruction techniques in the mathematics classroom. Theoretical Framework After conducting a preliminary literature review, I was able to determine that adequate research was present and available on differentiated instruction, high-stakes testing, and mathematics. The theoretical framework that guided this research consisted of the utilization of differentiated instruction to increase test scores. Differentiated instruction influences educational thinking about curriculum and instruction. Tomlinson (2008) described differentiated instruction as a learning theory and strategy that hypothesizes that students differ in their learning profiles and that all students learn best when the curriculum and instruction are adjusted to be compatible with individual learner profiles. Subban’s (2007) study on differentiation emphasized the integrated curriculum in which students study a topic from multiple perspectives, offering an assortment of learning options designed to appeal to student’s various readiness levels. Situations should be structured such that learners become actively involved with content through manipulation of materials and social interaction (Schunk, 2004). Kelly (2005) stated, Since the 1970s a series of assessments of U.S. student’s performance has revealed an overall level of mathematical proficiency well below what s desired

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and needed…..[the International Mathematics and Science Study rated U.S.] students among the worst in the world. (p. 18)

This indicated that students in the United States are experiencing major difficulties attaining mathematics proficiency. According to Lawrence-Brown (2004), there are five classroom elements that teachers can differentiate, or modify, to increase the likelihood that each student will learn as much as possible, as efficiently as possible: content, process, products, affect, and learning environment. In addition, there are three student characteristics that teachers can respond to as they craft curriculum: readiness, interest, and learning profile (Lawrence-Brown, 2004). Differentiated instruction is a strategy that recognizes and supports a classroom that comprises students from different cultures, with varying learning styles, and different levels of social and emotional maturity (Lawrence- Brown, 2004). Content is what students should know, understand, and be able to master as a result of a segment of study. Content is derived from national, state, and local standards and defined by local curriculum, standards, and textbooks (Georgia Department of Education, 2008). The line between content and process is small; but, traditionally, the process is when the teacher asks the students to stop listening or reading and to begin making sense out of information, ideas, and skills they have accessed (Tomlinson, 2003). Finally, a product is a means by which students demonstrate what they have come to know, understand, and be able to do (Tomlinson, 2003) Students come to school with common affective needs (Tomlinson & Edison, 2003). Students need to feel safe, secure, that they belong to the group, and are important to it. Students need to share a common ground with their peers, feel challenged, and that

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they can succeed. Educators need to create a flexible learning environment, which is a hallmark of a differentiated classroom (Tomlinson, 2003). A more detailed discussion of the literature on differentiated instruction, mathematical misconceptions, and math intervention for students is in section 2. Definitions of Terms Affect: How students link thought and feeling in the classroom (Tomlinson & Edison, 2003) Content: What is taught and how it gives students access to the information and ideas that matter (Tomlinson & Edison, 2003). Criterion-Referenced Competency Test (CRCT): CRCT is a criterion-referenced test that was designed to measure how well students acquire, learn, and accomplish the knowledge and skills set forth in a specific curriculum or unit of instruction (Georgia Department of Education, 2006). Differentiated Instruction: A teaching approach in which teachers adapt their instruction to student differences. Rather than developing a curriculum aimed at the common student, teachers modify their instruction to meeting individual students’ readiness levels, preferences, and interests (Tomlinson, 2003). In differentiated instruction, it is recognized that there is a diverse student population, and this recognition enables educators to teach students of varying abilities in one class. Interest: What a student enjoys learning about, thinking about, and doing (Tomlinson & Edison, 2003).

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Learning Environment: This refers to the way the classroom feels and functions (Tomlinson, 2003). The differentiated classroom should include areas in which students can work quietly, as well as collaborate with others and materials that reflect diverse ability levels and learning style (Tomlinson, 2003). Learning Profile: A student’s preferred mode of learning (Tomlinson & Edison, 2003). Process: How students come to understand and own the knowledge, understanding, and skills essential to a topic (Tomlinson & Edison, 2003). Product: How a student demonstrates what he or she has come to know, understand, and be able to do as a result of a segment of study (Tomlinson & Edison, 2003). Proficient: Georgia CRCT proficient scores for mathematics are 800-849 which meet standard and represent proficient student achievement level in accordance with NCLB (Georgia Department of Education, 2008). Performance level: The CRCT has three performance levels: Exceeds the Standard, Meets the Standards, and Does Not Meet the Standard (Georgia Department of Education, 2008). Readiness: The current knowledge, understanding, and skill level a student has related to a particular sequence of learning (Tomlinson & Edison, 2003). Scale score: A mathematical raw score that provides a uniform metric for interpreting and comparing scores (Georgia Department of Education, 2008).

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No Child Left Behind Act of 2001 (NCLB): The NCLB is a landmark in education reform designed to improve student achievement and change the culture of America’s schools. Under NCLB, each state must measure every public school student’s progress in reading and math in each of Grades 3 through 8 and at least once during Grades 10 through 12 (U.S. Department of Education, 2005). Assumptions Assumptions in this study included, but were not limited to, the roles of the teacher and the students. I assumed participating teachers and students would provide accurate, honest responses when completing assessments. It can be assumed that students are trying their best when completing assessments and have been equally distributed according to mathematical ability levels. In addition, it can be assumed that participants actively engaged in the study of mathematics and the students had the necessary support at home that encouraged them to attend school every day, complete homework as assigned, and study rigorously. Limitations There were several limitations to this study. First, it was conducted in a single geographical area. Second, attendance, reading levels, home life, and previous knowledge have the ability to hinder the performance of the students. Third, the study was limited to 8th-grade students. The male to female ratio in the class room could be a potential weakness of the study. These factors limited the generalization of the study to school districts in other regions with other populations. This study did not consist of true randomization because I was not in charge of the selection process for students who

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would be enrolled in the classes. Fourth, the study took into account the student’s performance from homework assignments and grades on teacher-constructed tests. Inclusion of this information could have a mitigating effect on findings. Scope and Delimitations The purpose of this casual-comparative quantitative study was to successfully implement differentiated instruction in the 8th grade mathematics classroom and to investigate if there was a positive effect on the mathematics achievement and comprehension of the 8th grade students on state and classroom assessments. The goal of this research project was to compare methodology intended on increasing math fundamentals for standardized test taking. Documentation of assessments determined if differentiated instruction increased math success. Results will be shared with faculty, administration, and county level administrators when research is complete. The independent variable in this study was the type of classroom instruction, traditional textbook instruction or differentiated instruction. The experimental group had differentiated instruction as the intervention, and the control group had a traditional instruction classroom. The two groups were compared on the dependent variable, which was defined as mathematics achievement measured by classroom assessments and standardized tests. This study initially confined itself to the 8th grade mathematics classes at the target middle school. Analysis was performed using ANOVA, an independent samples t test, and an ANCOVA. The findings indicated that implementation of differentiated instruction does positively impact student achievement. The students experienced success