Differentiated Instruction

THE EFFECTIVENESS OF DIFFERENTIATED INSTRUCTION IN THE
ELEMENTARY MATHEMATICS CLASSROOM

A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE DOCTOR OF EDUCATION

BY
BRIAN E. SCOTT
APPROVED BY:

_______________________________________________
Dr. Patricia Clark, Committee Chairperson

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Date

_______________________________________________
Dr. Nancy Melser, Committee Member

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Date

_______________________________________________
Dr. Nancy Brooks, Committee Member

____________
Date

_______________________________________________
Dr. William Sharp, Committee Member

____________
Date

_______________________________________________
Dr. Johnathan Forbey, Committee Member

____________
Date

THE EFFECTIVENESS OF DIFFERENTIATED INSTRUCTION IN THE
ELEMENTARY MATHEMATICS CLASSROOM

A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE
DOCTOR OF ELEMENTARY EDUCTION
BY
BRIAN SCOTT
DISSERTATION ADVISOR: DR. PATRICIA CLARK

BALL STATE UNIVERSITY
MUNCIE, INDIANA
MARCH 2012

Copyright © Brian E. Scott 2012
All Rights Reserved

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Table of Contents
Abstract…………………………………………………………………………..VI
Acknowledgements………………………………………………………….….VII
Chapter 1: Introduction……………………………………………………………1
Area of Concern…………………………………………………………...1
Statement of the Problem.............................................................................6
Research Questions…………………….………….………………………9
Definition of Important Terms………………………..……………….....10
Significance of the Problem……………………………………………...12
Basic Assumptions……………………………………………………….16
Basic Limitations………………………………………………………...16
Summary…………………………………………………………………17
Chapter 2: Review of the Literature……………………………………………...18
Historical Background…………………………………………………...18
Theory Relevant to Research Question…………………...……………...24
Current Literature………………………………………………………...25
Summary…………………………………………………………………36
Chapter 3: Methodology…………………………………………………………37
Restatement of Purpose…………………………………………………..37
Description of Participants and Setting…………………………………..38
Procedure………………………………...………………………………39
Description of Instrumentation/Measurement Procedures……………….39
Math Assessment……...………………………………..……………40
InView Aptitude Assessment………………….…………………….41
Pre-Treatment Observations…………………………………………41
Training…………………………………………………………...….41
Post Training Observations………………………………………..…42
Research Design………………………….…………………...………….42
Description of Procedures………………..…………………..…………..44
Data Analysis……………………..…………………………..………….45
Chapter 4: Results and Discussion……………………………………………….49
Restatement of the Research Question…………………………………..49
Summary of the Results……………………………………………….…57
Chapter 5: Conclusions and Discussion…………..……………………………...59
Introduction………………...……………………………………………59
Summary of the Study…………………………………………………...60
Discussion of the Results......…...……..…………………………………60

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Summary of Results……………………………….……………..………60
Limitations of the Study…………………………………………………63
Recommendation for Future Research…………………………………...64
Implications for Practice…………………………………………………67
Observations Before Implementation of Differentiate
Instruction………………………………………………………..68
Observations During Implementation of Differentiate
Instruction………………………………………………………..69
Final Summary…………………………………………………………...73
References…………………………………………………………………..……77
List of Tables
Table 1 Comparison of Pretest and Posttest Change
Before and After Treatment………………………………………...…...50
Table 2 Mean Change of Pretreatment and Treatment and
Cumulative Test (Overall)……..………………………………………...51
Table 3 Statistical Comparisons Before and After Treatment of
Unit Paired Tests and Cumulative Test………………..………………...51
Table 4 Mean Change of Pretreatment and Treatment and
Cumulative Test (Gender)………………………...……………...………52
Table 5 Mean Change of Pretreatment and Treatment and
Cumulative Test (Aptitude)……………………………...………….…...55
Appendices
Appendix A Proposal…………………………………………………….85
Appendix B Parent Cover Letter…………………………………………89
Appendix C Introduction to the Study and Consent Form (Parent)……...90
Appendix D Introduction to the Study and Consent Form (Teacher)……93
Appendix E Timeline for Study………………………………………….95
Appendix F Differentiated Instruction Observation Sheet………………96

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ABSTRACT
DISSERATION/THESIS/RESEARCH PAPER/CREATIVE PROJECT: The
Effectiveness of Differentiated Instruction in the Elementary Mathematics Classroom

STUDENT: Brian E.
Scott
DEGREE: Doctor of Education
COLLEGE: Teachers College
DATE: March 2012
PAGES: 96
This study was conducted to determine if differentiated instruction improved student growth. The overall effectiveness was studied as well as that of gender and the aptitude of average and above average students. The study was that of a quasiexperimental design using student subjects in the classrooms of three second-grade teachers. The school in the study was located in an affluent suburb outside of a major city in the Midwest. This quantitative study concluded that differentiated instruction did not have an overall effectiveness at a significant level. Students with a higher academic ability benefited significantly with opportunity to be challenged at a higher level while students of average ability did not. There was no significant difference between the achievement of males and females.

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ACKNOWLDEGEMENTS
I want to first thank Dr. Patricia Clark for her guidance, encouragement, and resourcefulness in directing me towards faculty that would be the most helpful in achieving my goal. I appreciate the time given by Dr. Michael Putman as this quantitative research design was implemented. I appreciate the input and support of
Dr.
Nancy Brooks for helping me see the many cultural perspectives of curriculum development. Dr. William Sharp should be recognized for constructive and practical approach when preparing aspiring administrators for leadership. I would like to thank
Dr. Nancy Melser for her feedback and willingness to share ideas when considering my topic. I would like to finally thank Dr. Johnathan Forbey for his ability to step back from the entire doctoral process and provide the advice needed.
Dr. Timothy Ogle, superintendent of the Avon Community School Corporation, maintained the inspiration for allowing me the opportunity to work in collaboration with him for my administrative internship and allowing me the time needed.
I am grateful to Kelly Jackson, principal of Geist Elementary School, and the three teachers and their classes. Their willingness to be a part of the research and the dedication of their time should be applauded.
Finally, I want to thank my family. My wife, Jeanine, and my daughters, Lauren,
Kelsey, and Carly, put up with my multi-tasking when attending their sporting events and practices, activities, and appointments. They have grown up knowing that Dad always has research materials or laptop computer in tow. As I tell my daughters, and I want to be living proof: ―You can do anything if you put your mind to it.‖

CHAPTER I: INTRODUCTION

Area of Concern
Differentiated instruction has been a ―buzz phrase‖ in American education for many years. Much of what has been written in support of the practice was created in the
1990‘s. Bearne (1996) defines differentiation as an approach to teaching in which teachers proactively modify curricula, teaching methods, resources, learning activities, and student products to address the diverse needs of individual students and small groups of students to maximize the learning opportunity for each student in a classroom.
Differentiated instruction is responsive teaching. Differentiation is a modification of teaching and learning routines and can address a broad range of learners‘ readiness levels, interests, and modes of learning (Tomlinson, 2001). It stems from a teacher‘s solid and growing understanding of how teaching and learning occur, and it responds to varied learners‘ needs for more structure or independence, more practice or greater challenge, and more active or less active approaches to learning.
Supporting the practice are four guiding principles that relate to differentiating classroom practices: (a) a focus on essential ideas and skills in each content area, (b) responsiveness to individual student differences, (c) integration of assessment and instruction, and (d) an ongoing adjustment of content, process, and products to meet individual students‘ levels of prior knowledge, critical thinking, and expression styles
(Tieso, 2003; Tomlinson, 1999). Lending further credence to the approach are seven basic beliefs (Tomlinson, 2000b): (a) same-age students can differ greatly in their life

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circumstances, past experiences, and readiness to learn; (b) these differences have a significant impact on the content and pace of instruction; (c) student learning is improved when they receive support from the teacher that challenges them to work slightly above their comfort level; (d) student learning is enhanced when what they are learning in school is connected to real-life experiences; (e) student learning is also strengthened by authentic learning opportunities; (f) student learning is boosted when they feel they are respected and valued within the context of the school and community; and (g) the overarching goal of schooling is to recognize and promote the abilities of each student.
Lawrence-Brown (2004) reported that differentiated instruction has a great importance for students who struggle in the mastery the grade level curriculum. Two goals are achieved as a result of differentiated instruction. First, and foremost, a high level of achieving the grade level standards for all students is paramount. It is important for teachers to scaffold the instruction as necessary for struggling learners. The second goal is to make curricular adaptations for those students who need it.
Teachers who differentiate are quite aware of the scope and sequence of curriculum prescribed by their state, district, and school. They are also aware of the students in their classrooms who begin each school year spread out along a continuum of understanding and skill. The teacher‘s goal is to maximize the capacity of each learner by teaching in ways that help all learners bridge gaps in understanding and skill and help each learner grow as much as quickly as he or she can (Tomlinson and Eidson, 2003).
Tomlinson (2001) writes that a differentiated classroom provides different paths for

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acquiring the content, causes the teacher to support students in processing or the making sense of ideas, and helps students create products so they can learn effectively.
A teacher can differentiate instruction in many ways in various subjects.
Instruction can be differentiated based on a student‘s readiness, learning profile, or interest by varying the content, process, or product (Tomlinson, 2001). The main strategies utilized are compacting, independent projects, interest centers or interest groups, tiered assignments, flexible grouping, learning centers, varying questions, mentorships, anchoring activities, and learning contracts.
Because of this active interest in differentiated instruction for all learners, many quick-fix ideas have been proposed. There are many resources, mainly in the form of books, offering teachers a large number of strategies, but these often lack the philosophy of differentiated instruction. Professional development companies create one-day workshops as a means of giving the participant an instant ―everything-you-need-toknow‖ training on differentiated instruction. District and building leaders hire consultants to train faculties in a one-time session on differentiated instruction. Because differentiated instruction is a complex concept, teachers need to remember that professional development cannot provide a ―one-size-fits-all‖ approach to teaching just one lesson plan and will not meet the needs of all of the students in a class. Few teachers automatically know how to lead a classroom that responds to the overwhelming reality of students‘ varying needs. It takes time. As a teacher becomes more comfortable and competent with differentiation, his/her role changes to more of a director of the classroom

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environment, a coach on the side guiding students, and an improviser adjusting the required instruction at the moment based upon ongoing assessment.
One of the strategies of differentiated instruction is flexible grouping, which may take on different forms in the education literature. Flexible grouping is one type of ability grouping, described in a research study conducted by Slavin (1986) as grouppaced mastery learning. Referencing Bloom (1976), he cites group-paced learning as a form of flexible within-class ability grouping, in that students are grouped after each lesson into "masters" and "nonmasters" groups on the basis of a formative test.
Nonmasters received corrective instruction while masters did enrichment activities.
While grouping may be thought to be a negative practice in education, Kulik and
Kulik (1982) reported the results of fifteen studies on student self-concept. In seven studies, self-concept was higher for students in grouped classrooms; in six studies, selfconcept was higher for students from ungrouped classrooms; and in two studies, selfesteem was equal for the two groups. The average effect size (ES) was .01, a trivial value
(Kulik and Kulik, 1982). While these studies were based more on ability grouping, the results are worthy of consideration as a research finding. This is due to flexible grouping having some similarity to ability grouping. The major difference is that ability grouping is often perceived as being static, whereas flexible grouping changes based on the mastery of skills by the student. In a later meta-analysis in 1992, Kulik concluded that evidence on the noncognitive outcomes of grouping is not as clear. Noncognitive instructional outcomes are not often studied by educational researchers, and only tentative conclusions can be drawn despite their importance. Grouping has been thought

5

to have only small effects on student self-esteem. It is not believed to lead talented students to become self-satisfied or smug, nor does it cause a precipitous drop in selfesteem of lower students (Kulik, 1992).
A qualitative study by Carol Tieso (2001) indicated that students in a differentiated classroom setting can become more engaged, motivated, and excited about learning if the curriculum is authentic and meaningful and if appropriate learning goals are provided.
Because students received the necessary modifications to meet their specific learning needs, either more challenging or layered in order to achieve mastery,
Tieso also found that students did not lose their sense of self-concept or self-efficacy.
Assessment is a key component of flexible grouping. It can take varied forms, but it should occur before, during, and at the conclusion of a unit of study. Witham and
Linehan (1995) reported through survey findings of 109 teachers that slightly more than half of the teachers at that time felt that pretests were worth the effort and should be used more often. Teachers with master‘s degree endorsements had a more positive attitude about pretests than those who did not. This leads to the conclusion that slightly less than half of the those who responded felt they were too time consuming, unnecessary, not applicable to certain subjects, and not practical. The authors further reported that
30 percent of the respondents had never pretested in subjects such as spelling or reading, seventeen percent did not pretest in math, and 36 percent did not pretest in grammar or writing. Approximately fifty percent did not pretest in social studies or science.
To avoid making erroneous decisions regarding individual instructional needs, the teacher needs to be aware that not all students are the same. As a result of a synthesis of

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the literature, Tonya Moon (2005) states, ―This assumption leads to erroneous decisions that may (a) prevent students from mastering the intended learning objectives, (b) disengage students from the process of learning because the materials are not accessible to them, (c) result in negative student behaviors or attitudes, or (d) propagate the achievement gap‖ (p. 229).
As teachers are faced with high levels of accountability, often having to prove that learning has taken place, pretesting becomes a critical baseline. The intent of assessment of instruction is to help the teacher make decisions about the best approach to instruction for students. When a teacher assesses students‘ strengths and weaknesses, the teacher uses that information to make decisions about where instruction should begin, or on which particular learning goals to focus (Moon, 2005).
Differentiation occurs concurrently with assessment and grouping. The way assessment is used to create groupings is unlike using it to create stagnant ability groups.
While assessment helps to determine which students need more challenge, which ones are performing at grade level, and which ones need scaffolding to meet the expectations, the teacher must make a decision as to how to make the lesson engaging and focused.
This would consider such approaches such as brain compatibility, learning styles, and cooperative learning strategies.
Statement of the Problem
Differentiated instruction is viewed in schools as a positive approach to meeting the needs of the wide-range of abilities in the classroom, but most of the research supporting it is of a qualitative nature, especially in a heterogeneously mixed classroom.

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Much has been written about its importance in gifted and talented education as well as special education.
While the strategies of differentiated instruction have been used in classrooms for many years, results of the practice have been limited in quantitative studies. Much has also been written about how complex of a concept differentiation is. Educators should be cautious to not make too much of a drastic change in implementing the practice to the classroom. In view of differentiation being grounded in assessment, often of a numerical nature, it appears to be unusual that more quantitative studies have not been done. More schools, administrators, and teachers desire to embrace the practice often not having a full understanding of the time commitment it takes to fully implement it. Teachers and administrators need more research to support the effectiveness of differentiated instruction. For this study, mathematics is being chosen as the subject. Math is more of a skills-based subject and has the flexibility to use many of the differentiated strategies such as flexible grouping, anchoring activities, and compacting.
The Third International Mathematics and Science Study (TIMSS) in 1995 reported that American fourth graders score better than the average of their counterparts in 26 countries. Seven countries were significantly higher than the U.S. and twelve were significantly lower. Six countries were performing about the same as the United States.
In the 2007 study, fourth graders were better than 23 of the 35 countries and lower than the top 8. These top performing countries were located in Asia and Europe. The 2007 average score was better than in 1995 by 11 points. While the average score improved, the United States is still not at the top over the twelve-year period. This demonstrates the

8

importance of investigating whether differentiating instruction can impact the performance of American children when compared to their same-aged peers on a global level. The report also indicated that the achievement gap in mathematics between males and females is still wide with males achieving at higher levels.
Several qualitative studies have been conducted and indicate positive results
(Tomlinson 1995; George, 2005), but there have been few of a quantitative nature. The
Differentiated School: Making Revolutionary Changes and Learning by Tomlinson, C.,
Brimijoin, K., & Narvaez, L (2008) tells of multi-year studies in an elementary school and a high school indicating positive and sustained achievement gains for students in all segments of the achievement spectrum and in a range of subject areas as a result of differentiated instruction. In the high school, the student dropout rate has also fallen sharply, and student participation in Advanced Placement courses has risen by almost half, with AP exam scores holding steady or rising despite the increased enrollment. In both sites, a school-wide emphasis on differentiation has continued for at least seven years, and achievement gains have continued over that time span. This study seeks to answer the question of the effectiveness of differentiated instruction in a quantitative research design. Very few studies of this approach have been done quantitatively, and the field of education could potentially benefit from the findings. In addition, the study attempts to see if an increase in math achievement is accomplished by implementing differentiated instructional strategies.

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Research Questions
This study poses one main question followed by two other questions disaggregated from this main inquiry. The additional questions are being included for different reasons. The first of the related questions will explore an area sparsely found in the literature review of this particular study regarding gender and differentiated instruction. The second will look for some consistency from the study by Tieso finding that students with a greater aptitude benefited greatly from differentiated instruction.
The three questions are:
Does differentiated instruction impact the growth of student learning in the subject of math?
Does differentiated instruction impact the growth of student learning of a particular gender in the subject of math?
Does differentiated instruction impact the growth of student learning of a particular aptitude in the subject of math?
The hypothesis in this study is that differentiated instruction does improve academic achievement. This study will look at the results, not only in terms of the second grade students as a whole, but also in terms of gender and aptitude.
Quantitatively this can be measured by comparing results of instruction that does not differentiate with instruction that does based on growth data using pre- and posttests.
There are three null hypotheses as follows.
1.

There will be no significant change in student mathematics achievement growth using differentiated instructional techniques.

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2.

There will be no significant change in student mathematics achievement growth using differentiated instructional techniques between males and females. 3.

There will be no significant change in student mathematics achievement growth using differentiated instructional techniques among ability levels.
Definition of Important Terms

Anchoring activity Students automatically move when they complete an assigned task during a daily lesson or unit of study. It is important in maintaining a productive work environment and ensures wise use of a student‘s time (Tomlinson, 2001).
Examples of anchoring activities are reading, journal writing, portfolio management, and practicing skills through manipulatives or sorting activities. Anchoring activities may take the form of learning centers. It is possible that some students will participate in these activities more than others.
Aptitude This is the capacity for learning a student demonstrates. It can be considered synonymous with ability. Some in the field of education would erroneously consider this to be equivalent to a student‘s intelligence quotient (IQ). Aptitude is measured using an assessment tool designed to measure a student‘s ability often in a classroom setting to a group of students at the same time. The Otis-Lennon School
Abilities Test (OLSAT) by Pearson Assessment and the Cognitive Abilities Test
(CogAT) published by Riverside Publishing are examples. An IQ is best measured using a tool given by a trained psychologist in a one-on-one situation.

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Learning Centers Learning centers can be ―stations‖ or collections of materials learners use to explore topics or practice skills. Teachers can adjust learning center tasks to readiness levels or learning styles (Tomlinson, 2001).
Compacting This is a 3-step process that (1) assesses what a student knows about material to be studied and what the student still needs to master, (2) plans for learning what is not known and excuses the student from what is known, and (3) plans for freedup time to be spent in enriched or accelerated study (Tomlinson, 2001).
Tiered assignments In a heterogeneous classroom, a teacher uses varied levels of activities to ensure that students explore ideas at a level that builds on their prior knowledge and prompts continued growth. Student groups use varied approaches to explore essential ideas (Tomlinson, 2001). At times these may look similar to varying questions. Varying questions In class discussions and on tests, teachers vary the sorts of questions posed to learners based on their readiness, interests, and learning styles
(Tomlinson, 2001). These types of questions may also vary on tiered assignments.
Flexible grouping Students are part of many different groups—and also work alone—based on the match of the task to student readiness for instruction, interest, or learning style. Teachers may create skills-based groups that are heterogeneous or homogeneous in readiness level. Sometimes students select work groups, and sometimes teachers select them. Sometimes student group assignments are purposeful and sometimes random (Tomlinson, 2001).

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Contracts Contracts take a number of forms that begin with an agreement between student and teacher. The teacher grants certain freedoms and choices about how a student will complete tasks, and the student agrees to use the freedoms appropriately in designing and completing work according to specifications (Tomlinson, 2001).
Pretesting Pretesting is a technique used to find out what a student knows and does not know prior to formal instruction. This form of assessment is often the driving force in determining how, why, and what type of differentiated instruction will take place. Significance of the Problem
While there is a great amount of support for the strategies related to differentiated instruction, there have not been a large number of research studies indicating that differentiated instruction helps students grow in their academic endeavors. Kronberg,
York-Barr, Arnold, Gombos, Truex, Vallejo, and Stevenson (1997) report that classrooms are becoming increasingly heterogeneous, and teachers frequently work amidst complex and sometimes unpredictable situations. Jenkins, Jewell, Leicester,
O‘Connor, Jenkins, & Troutner (1990), indicated that heterogeneity is represented by students with diverse cultural, racial, religious, and linguistic backgrounds; family structures; socioeconomic status; and ability levels. Our public schools have a vast majority of teachers who are currently or soon will be expected to teach students with markedly diverse backgrounds and abilities. It has been estimated that the range of instructional levels among students (those not receiving any special services) in many

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general education classrooms is an average of 5.4 grade equivalents (Jenkins, Jewell,
Leicester, O‘Connor, Jenkins, and Trouter, 1990).
Students with diverse backgrounds and abilities pose new and different challenges as teachers seek to meaningfully include and effectively educate all students (Kronberg,
York-Barr, Arnold, Gombos, Truex, Vallejo, & Stevenson, 1997). Most certainly, as diversity among students increases so must the differentiation of teaching and learning
(Kronberg, York-Barr, Arnold, Gombos, Truex, Vallejo, & Stevenson, 1997). The problem in classrooms for academically diverse student populations is likely systemic.
Literature has proven in many forms that the patterns of inattention to student variance are evident and related specifically to numerous learner exceptionalities, such as giftedness, special education, second language acquisition, multicultural learners, and students from low economic backgrounds. The problems lie in beliefs and practices related to teaching, learning, and the nature of people as learners, the best ways to teach, and how students should learn. It is critical that educators understand and address the systemic issues. Contemporary schools with their diverse populations must consider the learning needs to be shaped by readiness, interest, or learning profile. Teachers have to address these varied academic needs and investigate and address the persistent and longheld teacher beliefs. While this study does not address diversity beyond gender and aptitude, the impact of this awareness and the curriculum and instruction teachers plan and deliver to diverse learners is critical. Such an approach and consideration of the big issues may well be a precursor to addressing effectively the particular learning needs of

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specific learners and populations of learners (Tomlinson, Brighton, Hertberg, Callahan,
Moon, Brimijoin, Conover, and Reynolds, 2003, p. 125).
A substantial amount of the research regarding differentiated instruction is at a superficial level. As Tomlinson et al explain, the responsive classroom to meet academically diverse needs is a new concept. Knowing how to create an atmosphere of this responsiveness is the basis that might ultimately support such classrooms. For example, we do not know the particular range of students for whom differentiated, heterogeneous classrooms might be effective (Tomlinson, Brighton, Hertberg, Callahan,
Moon, Brimijoin, Conover, and Reynolds, 2003). We do not know which of a variety of potential models of teaching and learning might best serve the learning needs of students who differ as learners. Quantitatively we know little regarding the relative impact of differentiated instruction based on learner readiness needs versus interests, versus learning profiles—nor whether it is important to address all of those elements simultaneously. Likewise, we need to investigate the impact of such classroom elements as learning environment and the effect on achievement of diverse populations
(Tomlinson, Brighton, Hertberg, Callahan, Moon, Brimijoin, Conover, and Reynolds,
2003).
In a paper presented before the Association for Supervision and Curriculum
Development, Ronis (1999) shared that The National Council of Teachers of
Mathematics (NCTM) recommends through its Principles and Standards that instruction needs to be more than mastery of facts and routine skills but should result in all students gaining understanding and the ability to apply mathematical concepts in new situations.

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Ronis goes on to describe six pillars of performance tasks that she states are braincompatible, another way to differentiate instruction. These elements are to establish clear goals using the content standards, employ authentic tasks, emphasize critical levels and performance standards, provide models of excellence, teach strategies explicitly, and provide ongoing assessments with timely feedback. The sixth pillar aligns well with differentiated instruction. Assessment is a critical component as indicated by it driving the planning, guiding, and evaluating of instruction. She contends that teachers must move away from teach, test/grade, and move on, to adopt the practice of teach, assess, adjust, and assess.
Small group instruction, the opportunity to have students focusing on skills of the greatest need, extension for students who have mastery of the core curriculum of skills, and the provision of individualized instruction make good sense. A teacher can differentiate instruction in many ways in various subjects. Instruction can be differentiated based on a student‘s readiness, learning profile, or interest by varying the content, process, or product (Tomlinson, 2001a). The main strategies utilized are compacting, independent products, interest centers or interest groups, tiered assignments, flexible grouping, learning centers, varying questions, mentorships, anchoring activities, and contracts. The strategies of flexible grouping, anchoring activities, and tiered assignments will be utilized. In this study, it should be noted that a teacher may start out focusing on one or two strategies and will end up incorporating others.

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Basic Assumptions
One of the basic assumptions in this study would be that teachers administer any and all assessments with fidelity. Teachers will receive two full days of training prior to implementation of the experiment. The training will include an overview of differentiated teaching strategies including video demonstrations of teachers using these techniques. Teachers in the study will receive training to better use differentiated instruction through improved understanding of assessment data, classroom management, and creating multiple pathways for learning. Because the teachers involved are not new to the profession, one would believe that long-standing instructional strategies these teachers have developed in the past could potentially be used again. Each teacher will be observed for two 30-minute periods prior to the training and for two 30-minute periods after the training during the implementation phase. These observations serve as a baseline of typical instructional practices before the training and to ensure fidelity of the application of training sessions during the implementation of deliberate differentiated instruction. Basic Limitations
Three second grade teachers with classroom sizes of approximately 26 students each will participate in the study. Two of these are special education inclusion classrooms with four students each identified with some sort of minor learning disability such as attention deficit hyperactivity disorder or difficulty with reading and language development. All three classes will be in the same school, which could potentially help with the collaboration and fidelity of the instructional strategies employed.

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While a control and experimental design would have been preferred, the quasiexperimental design was chosen due to the teachers‘ willingness to participate. The smaller number of participants was also a factor to validate the results more accurately.
The study might have a different result with a larger sample of teacher participants and classrooms of students.
The subject school is a public school in an affluent suburb northeast of
Indianapolis, Indiana. It could be challenging to generalize the results, especially if they are positive, to other schools of more diverse demographics due to the perceived notion that differentiated instruction will only work in this certain group. For this study, the convenient population will limit this study to gender and aptitude. The small sample of special education students, the lack of racial diversity, and few students of lower socioeconomic status in this particular school would not permit a valid or deep analysis of these subgroups.
Summary
Because there is little quantitative research on differentiated instruction, this study could add some very important information to the approach and its effectiveness. There is little doubt as to the necessity to address so many avenues for learning due to the wide range of students. As the review of the literature will imply, differentiated instruction has been provided for some students for quite some time. Due to its wide acceptance in the fields of gifted education and special education, two seemingly polar opposites, it is critical that more research be done in heterogeneous classrooms.

CHAPTER II: REVIEW OF THE LITERATURE

Historical Background
Basing her ideas on the work of Vygotsky (1986); Csikszentmihalyi (1997); and
Sternberg, Torff, and Grigorenko (1998), Tomlinson writes:
There is ample evidence that students are more successful in school and find it more satisfying if they are taught in ways that are responsive to their readiness levels (e.g. Vygotsky, 1986), interests (e.g., Csikszentmihalyi, 1997), and learning profiles (e.g., Sternberg, Torrf, & Grigorenko). Another reason for differentiating instruction relates to teacher professionalism. Expert teachers are attentive to students ' varied learning needs (Danielson, 1996); to differentiate instruction, then, is to become a more competent, creative, and professional educator. (2000a, p. 3)
It is common to find references in articles related to differentiated instruction tying the approach to the work of cognitive psychologist, Jerome Bruner. Bruner in the
1960‘s and 70‘s believed that a student could learn just about anything if the structure of the lesson was designed appropriately. According to Bell (2004):
In The Process of Education, Bruner offers the foundational conjecture that ―it is the underlying premise of laboratory exercises that doing something helps one understand it.‖ Derived from the individualistic Piagetian view of the active construction of knowledge through inquiry that was emerging at the time— which fueled the central arguments of Process—it helped launch the subsequent

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‗learning through discovery‘ movement and ultimately the recognition that discovery was being taken as an end rather than a means for knowledge construction. (p. 8)
As Bruner (1985) indicates, there is no one way or one kind of learning.
Conflicting thoughts on the best approaches to teaching and how students learn best are at odds with the instructional practices of a preceding generation. It is not appropriate to think that one learning theory would eventuate into one winning over the others. Any learner has a host of learning strategies to command. Perhaps the best choice is not a choice of one, but an appreciation of the variety possible. The appreciation of that variety is what makes the practice of education something more than a scripted exercise in cultural rigidity and sameness.
Bruner, in collaboration with Kenney, made a connection to the ―doing‖ students need to experience in their learning with mathematics. The study was based on Piaget‘s stages of learning development and the concern for children who need concrete examples and are not ready for formal operations. The study included four eight-year-olds, two boys and two girls. Each had an IQ between 120 and 130. They were from professional, middle-class families, and attended third grade in a private school. A well-respected mathematics researcher and a psychology assistant provided instruction with a focus on the human thought processes. The instruction emphasized concrete construction. Bruner and Kenney found that mathematics should be viewed as a microcosm of intellectual development. Engagement is a critical component for building the foundation of

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understanding. Once the learner has a firm grasp on the concrete, the abstract and symbolic can be achieved.
Slavin has conducted and written about grouping and cooperative learning. In the
1980‘s he wrote about the effects of ability grouping on students, both negative and positive. Slavin (1987) found that above average students benefitted from being a part of a static group of similar achievers. In a synthesis of research of ability grouping he defined one type of ability grouping, and he found the following regarding the regrouping of students for reading and mathematics instruction:
Often, students are assigned to heterogeneous homeroom classes for part or most of the day, but are "regrouped" according to achievement level for one or more subjects. In the elementary grades, regrouping is often done for reading (and occasionally mathematics), where all students at a particular grade level have reading scheduled at the same time and are re-sorted from their heterogeneous homerooms into classes that are relatively homogeneous in reading level (p.295).
Slavin recounts the research findings of Provus (1960) regarding regrouping in mathematics. The results were positive countered by a smaller study where the results were not quite as positive. Students in 11 classes in a suburb of Chicago were regrouped from their heterogeneous homerooms into relatively homogeneous mathematics classes at the same grade level. The original study conducted by Provus during the 1957-58 school year garnered the following results: 1) children were familiar with more arithmetic concepts grouped as to ability than children not so grouped; 2) children who were grouped were just as proficient as those children who were not grouped, and some

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children grouped as to ability were more familiar with the fundamentals normally taught at their grade level than children who were not grouped as to ability; 3) a comparison of three different levels of students, indicated that bright children, benefitted most, average children may have profited slightly, and the slow learners may have profited no more from ability grouping than they would have from a heterogeneous class; 4) children in the upper elementary grades achieved at a substantially higher level than the children in these grades during the year before the experiment; 5) student interest in the subject matter was noticed in the control or experimental groups and assumed to be associated to ability grouping; 6) teachers largely supported the program, and most expressed a desire to teach homogeneous classes the following year. Based on these findings, ability grouping for arithmetic instruction was extended to the third through eighth grades in the Homewood schools for the 1958-59 academic year (Provus, 1960).
Slavin (1987) counterbalances the Provus findings with research conducted in
1963 by Davis and Tracy in the now discontinued National Teacher of Mathematics
Council‘s journal, Arithmetic Teacher, regarding the harm of ability grouping. Slavin writes, ―In contrast, Davis and Tracy (1963) found that regrouping for mathematics was detrimental to the achievement of students in a rural North Carolina town. However, this study compared only two schools, and there were substantial achievement differences at pretest‖ (p. 310). Slavin goes on to point out that there was no attempt made to include differentiated materials for students. All classes used the same grade level textbook.
Tomlinson, in a 1995 edition of Gifted Child Quarterly, found several challenges regarding a change to a differentiated instructional approach. The school in the study

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was located in an affluent community in the Midwest. The school experienced high levels of achievement as indicated by the statewide testing instrument. Tomlinson found initial teacher opposition toward modifying instruction to suit varied needs of the learners. In addition, the initiative to implement a more proactive approach to instruction came from the administrative level, thus making an impact on the teacher‘s sense of selfefficacy. Other obstacles included teachers‘ perceptions that differentiated instruction would be a passing fad, something experienced by other professional development initiatives. There were also concerns over time allocated to prepare for differentiated lesson, unease over student assessments and preparation for testing, disquiet regarding classroom management and perceived teacher insecurity over a change in their role
(Tomlinson, 1995). Teachers who adopted the use of differentiated techniques demonstrated that attitude for the need for the changes was more prevalent than age or years of experience. Teachers who experienced early successes with differentiation were more likely to persist (Tomlinson, 1995). Tomlinson concluded that schools need to do more than explore at a superficial level when it comes to addressing academic diversity.
Teachers at the elementary school level often have great flexibility but might lack some in-depth subject area knowledge often due to having elementary classroom background, and secondary teachers have deep subject area capability but lack a variety of instructional approaches. Teachers were quite concerned that classrooms would become out of control. By allowing and encouraging collaboration, Tomlinson found that the classroom management issues expressed by those who more resistant had the potential to subside. 23

Tomlinson, Moon, and Callahan (1995) conducted an investigation by surveying administrators and teachers about instructional practice in middle school populations.
The findings were that few teachers take into account student cultural differences, learning profile, or interests when creating lesson plans. To address diversity, instructional design was prevalently teaching teams. Building principals and teachers also indicated that middle school students were more socially focused than academic, required basic skill instruction, were concrete thinkers, were extrinsically motivated, and indicated that varying the curriculum and offering students choice of tasks the least important approach with this age group of students. Teachers also indicated that cooperative learning strategies best addressed the varying academic needs of the students in the classroom including those with lower academic as well as those students who would be considered gifted. The authors suggest that different models of professional development be reviewed to support teachers better for the implementation of differentiated strategies.
People with strong ties to gifted and talented education paved the way for highly able students to experience a differentiated approach to learning. Experts in the field supported a separate curriculum and approach for teaching high ability students. As time progressed, teachers saw other teaching strategies such as learning styles, Multiple
Intelligences, and brain-compatible techniques, as paths to differentiating instruction.
With the implementation of No Child Left Behind (NCLB) in 2001, schools and districts are held accountable through benchmark assessment results. Educators began to look at differentiated instruction as an instructional practice in which all students could achieve.

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Rather than emphasizing the same teaching strategy that may have been appropriate for some, schools began to discover that the same strategy may not be effective for the most at-risk students. Even in the presence of high-quality curriculum and instruction, we will fall woefully short of the goal of helping each learner build a good life through the power of education unless we build bridges between the learner and learning (Tomlinson, 2001).
Theory Relevant to Research Question
Multiple theories can be related to differentiated instruction. These are not limited to but include Howard Gardner‘s Multiple Intelligence, brain-compatible learning, learning styles, and cooperative learning. Two dominant theories are Lev
Vygotsky‘s theory of zone of proximal development (ZPD) and Robert Sternberg‘s three cognitive styles of learning.
Vygotsky was very concerned with the role of socialization in development.
(Parker, 1979) In an explanation of the ZPD, Vygostsky wrote in 1978:
Learning and development are interrelated from the child‘s very first day of life. A well known and empirically established fact is that learning should be matched in some manner with the child‘s developmental level (Vygotsky, p. 82).
A child‘s ZPD is that point in a learning experience that is slightly more challenging than what he or she can do alone. By students working within their zone, they are not interacting with work that is too easy or too difficult for them. The learning is meaningful because it appropriately meets the child‘s readiness level (Hall, 2002).
Vygotsky claims the ZPD furnishes psychologists and educators with a tool through which the internal course of development can be understood (Vygotsky, 1978). He

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continues to say that an essential feature of learning is that it creates the zone of proximal development. Learning stimulates various internal developmental processes that are able to operate only when the child is interacting with people in his environment and in cooperation with his peers. Once these processes become automatic, they become part of the child‘s independent developmental achievement. It is important to remember the notion that developmental processes do not coincide with learning processes. Instead, the developmental process falls behind the learning process; this sequence then results in zones of proximal development (Vygotsky, 1978).
Sternberg, along with research associate, Elena Grigorenko, looked at three major thinking styles. These are analytical, creative, and practical. In two studies, one conducted in 1995 and one in 1997, thinking styles were shown to impact how much a student learned. The 1995 study found that students were more positively evaluated by and received better grades from teachers who matched their styles than those who did not
(Sternberg and Grigorenko, 1997). In another study conducted in 1997 of 199 high school students at a Yale University Summer Psychology Program, Sternberg and
Grigorenko found consistent positive relations between preferred style and performance.
When these abilities were used to predict school achievement, and then styles were added in through a hierarchical regression, styles made a significant incremental contribution to the prediction equation (Sternberg and Grigorenko, 1997).
Current Literature
The current research includes information published after 2001. Several themes were clear in the review of the literature. These were references to the previous theories

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and research of a central group of experts, effect on a diverse population of students, and chief instructional strategies and arrangements. Other themes worth noting are the need for teacher training to bring change in classroom instruction and managing the classroom environment. Many references in the literature include easily found research studies conducted by Joe Renzulli, Sally Reis, and Robert Slavin. Many of these go back as far as the early
1970‘s. Other reviews of the research cite Carol Tomlinson and Lev Vygotsky‘s theory of zone of proximal development as well as the aforementioned researchers.
Renzulli created the Schoolwide Enrichment Model (SEM) as a means of meeting the varied academic needs of students. The SEM is an organizational format that gives both enrichment and accelerated options through a continuum of high ability and integration. It allows for the highest potential growth for students and enables them to escalate their experience through options that might be available through other service delivery components. This can be in the form of general enrichment, highly individualized curriculum modifications for advanced learners, and first-hand investigative opportunities. The model also includes a broad array of specific grouping arrangements based on commonalities in abilities, interests, learning styles, and preferences for various modes of expression (Renzulli and Reis, 2002). Acceleration options include grade skipping, enrollment in college classes, and numerous supplementary program options that provide opportunities for talent development in specialized areas, such as Math League, Invention Convention, and National History Day
Competition. Other components of the model include performance-based assessment of

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student strengths, individual and group counseling, and various special placement options
(within and outside the school) based on high degrees of proficiency and potential
(Renzulli and Reis, 2002).
Tomlinson would be considered to be the most contemporary authority on differentiated instruction. She has written several books, has authored or co-authored many articles in educational journals, and is a featured speaker at education conferences.
Others have published books using her ideas and applying the concepts into practical classrooms practices and strategies.
In her master‘s research project, Melinda Good found that most of the literature regarding differentiated instruction has been produced by Carol Ann Tomlinson.
Although her experience and expertise are undeniable, the fact remains that this single viewpoint is the basis of much of the available differentiation literature. Clearly, the field would be expanded by the input of additional researchers (Good, 2006, p. 31).
As noted in Rock, Gregg, Ellis, and Gable (2008), Stamps (2004), Hall (2002), and Tieso (2005), experimental research studies on differentiated instruction are scarce.
Although it seems logical to assume that greater scholastic gains would be made by students by teachers creating learning experiences tailored to their needs, controlled studies have not been performed to verify this idea. Particularly in this era of emphasis on quantifiable, verifiable data, having results that demonstrate the effectiveness of differentiation would likely increase its widespread acceptance as an educational strategy
(Good, 2006).

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Baumgartner, Lipowski, and Rush (2003) in their dissertation researched differentiated approaches that included flexible grouping in reading, student choice of various tasks, increased self-selected reading time, and access to various reading materials. The researchers found improvements in students‘ instructional reading levels.
They also found the number of comprehension strategies used, mastery of phonemic and decoding skills, and attitudes toward reading also improved.
For her doctoral dissertation, Schlag (2009) examined the effects of flexible grouping on reading instruction at the fifth grade level. The study was of a quasiexperimental design measured by a pretest and posttest. Students moved from different reading groups as the readiness level of the student changed. The dependent variable was reading achievement scores measured by the Standardized Test of Achievement in
Reading (STAR) test. Students took the computerized test every two weeks for the eight weeks of the study and were grouped based on the results. The teacher then instructed the students at their reading level. Schlag found that implementing flexible grouping made a significant gain in reading achievement.
Interestingly the studies that have been done indicate promise for all students.
Students with learning disabilities who have low achievement, or are considered gifted benefit the most. Differentiated instruction has its roots in gifted education, but the strategies have also been embraced as a means to support students with learning problems. As Tieso (2005) cited in Rock, Gregg, Ellis, and Gable (2008) reported, many of these studies are qualitative in nature indicating positive emotional outcomes in terms of motivation, task commitment, and excitement about learning.

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Riener and Willingham, a psychologist and neuroscientist, respectively, at the
University of Virginia, believe that adapting classroom instruction to attend to learning styles is not beneficial. They base this position on at least two arguments. Their research indicates that the brain does not work the way learning style theory suggests. Riener and
Willingam (2010) say that if people used auditory strengths to store what they learned in the brain, the result would be the awareness of sound differences, not meaning about math or a novel. They emphasize that students remember meaning, not auditory qualities. They also make an argument against using kinesthetic or visual instructional techniques. Reiner and Willingham have noted that findings that occur in a lab setting with individual learners do not necessarily transfer into classroom practice where there are many variables at work in the learning process; however, Riener and Willingham recognize that people clearly learn in different ways.
Although more stagnant than flexible grouping, Preckel and Brull (2008) found that the self-esteem of gifted females who were ability grouped was lower. The opposite was found to be true with the males who were ability grouped. In the literature review of their study, similar findings were shared by Catsambis, S., Mulkey, L. M., & Crain, R. L.
(1999, 2011).
In a review of the literature, Subban included the findings of Hodge (1997) stating an improvement in mathematics achievement but not in reading for students using differentiated techniques for preparing for tests leading Hodge to question whether more traditional approaches are more appropriate during reading instruction. Blozowich
(2005) in Subban (2006) found teachers created lessons for differentiating instruction

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looked similar to tracking lessons even though teachers used a variety of techniques concluding that continuous professional development along with strong consultation and conversation was necessary. Again, Subban cites research conducted by McAdamis published in the Journal of Staff Development in 2001 indicating a significant improvement in test scores in a previously low-scoring district in Missouri. It was reported that teachers noticed that students were more enthusiastic and motivated. It also found that sustaining differentiation would require professional development, mentoring, and time for planning over a five-year period with all levels of administration and teachers involved and dedicated to the change. Subban shares one more finding by
Affholder in an unpublished doctoral thesis in 2003. Affholder‘s investigation concluded that students gained more responsibility, and teachers gained more and more selfconfidence as they experienced more success with implementing the techniques. As a result, teachers were willing to try an increased number of strategies. This was especially true of teachers with more experience, knew the curriculum well, and received more extensive training with the methods.
Much of the literature emphasizes the amount of diversity in the elementary classroom. In an example of a mixed-ability classroom, one can find students with learning disabilities, gifted students, students who are socio-economically disadvantaged, children with language needs, and others from a wide range of cultures. Castle, Deniz, and Tortora (2005) contend that differentiated instruction is necessary to meet the varied needs of all students in the classroom. Their study indicated student achievement

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improved after experiencing differentiated instruction over several years, but no conclusion could be made regarding the impact of the idea in the classroom.
George (2005) supports differentiated instruction linked with public education and the mixed-ability classrooms in today‘s schools. The mixed-ability classroom is a reflection of the variety in American society. He goes on to argue that gifted and talented students will not be challenged and will not reach their potential or will become behavior problems due to boredom in the classroom. As students prepare for standardized tests,
Tieso (2004) believes interests, abilities, and strengths are in conflict with a one way approach of teaching. As legislation requires programs for the gifted to be implemented, budget restraints place classroom teachers in a position of meeting the needs of these students in a mixed-ability classroom of students.
Beecher and Sweeny (2008) summarize one school‘s approach to reducing the achievement gap among culturally diverse groups of students through differentiated curriculum and school-wide enrichment teaching and learning. They tell of a school that transformed itself from a plan of remediation to one that enriched students. The approach resulted in improved student achievement and a reduction of the achievement gap between rich and poor and across the various ethnic groups. The school accomplished this by analyzing the strengths and weaknesses in all areas. This resulted in a change in the mission statement, a broad-based instructional strategic plan, specific learning objectives, and detailed action plans. Considering the students‘ interests and choices, it made sense to the teaching faculty to provide learning experiences that were responsive to the learning characteristics of a diverse student population. The school determined

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what would be implemented simultaneously, as well as what would be implemented over a series of years as determined by the strategic plan. Teachers rewrote the curriculum for reading, writing, mathematics, and social studies to include enrichment experiences and differentiated instruction. Although there were enrichment opportunities built into the day, the learning environment extended to an afterschool program inspired by
Enrichment Clusters. Staff development was essential to the success of each new initiative, and a significant amount of time was devoted to teacher training. Teachers were provided with training, modeling, coaching, and planning time to integrate the new ideas and skills into their lessons (Beecher and Sweeny, 2008).
While it is not an empirical research study, data were gathered over a period of eight years in which information was drawn from the school‘s meeting agendas, strategic plan, professional development, and from specific areas of curriculum. The school documented its success through students‘ positive attitudes about school, increased engagement of learning, and improved achievement on district and state assessments
(Beecher and Sweeny, 2008). Graphic data indicated that over a seven year period Asian students improved by over 60%, African-American by over 20%, and White and
Hispanic students by over 5%. Students qualifying for free or reduced lunch improved by nearly 30% on state achievement tests.
Tomlinson, Brimijoin, and Narvaez (2008) report on the experiences of two schools on the differentiated instruction journey. Their book indicates the structure of differentiation in each of the schools and survey results in support of differentiation.

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Tomlinson (2009) presented results that one of the schools improved steadily in all subjects tested over the three-year period studied.
Through her research on differentiated instruction for her dissertation, Bosier
(2007) investigated what research studies have been done on the topic of differentiated instruction in math. The purpose of her research was to 1) review the perceptions of differentiated instruction of upper elementary math teachers as an effective and instructional tool, 2) develop a link between mathematic student achievement and teacher commitment of implementing differentiated instruction in the classroom, and 3) determine teacher perceptions of the advantages and disadvantages of differentiated mathematics instruction. This was a mixed methods study. Bosier compared beginning and ending achievement data in the fall and spring and drew conclusions from the teachers‘ perceptions.
In a review of studies regarding direct instruction, Gujjar (2007) found students receiving direct instruction in a small group setting performed better in reading, math, and social studies than those in whole group arrangements. Because the groupings are flexible and change as needed, ongoing assessment becomes necessary. Pre-assessment can also be in the form of teacher or textbook created assessments, interest inventories, learning style inventories, and other non-academic instruments.
In her quantitative study published in 2005, Tieso found student mathematics achievement improved based on the effects of three grouping plans on fourth and fifth grade students. In this experimental study she used a design of flexible grouping within the classroom, ability grouping, and the Joplin Plan (grouping based on readiness across

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three adjacent grade levels) to investigate the effectiveness. The study was a result of grouping research for gifted students conducted by Pascow (1962), Kulik and Kulik
(1982), Slavin (1987), Rogers (1991), and Mills, Ablard, and Gustin (1994). Tieso‘s study added a new level of complexity with a differentiated curriculum for students who demonstrated mastery of the grade level skills. The comparison group used lessons from the textbook without additional resources. The revision group experienced higher level abstract questions and lessons that crossed into other subject areas. This was a result of the removal of unchallenging and repetitive curriculum. The differentiated group experienced lessons using the principles of a differentiated curriculum. Tieso hypothesized students receiving the adjusted lessons and differentiated curriculum would demonstrate positive gains. Tieso used random assignment of classroom treatment groups and then subdivided into three equal member groups of high, middle, and low using a curriculum based pretest. Teachers who participated in the study were provided with lessons created by the researcher. Posttests followed as the measure of success. The posttests were re-administered three weeks later and indicated that the new learning had positive long-term effects. Other validity concerns were students taking the same forms of the pretest and posttest, sharing ideas among teachers in spite of receiving direction to avoid the temptation, managing groups of students, losing time in students transitioning from one room to another, and teaching materials not being equally distributed.
Tieso‘s results indicated that flexible grouping combined with a differentiated or revised curriculum positively impacted achievement for gifted students. The findings suggest that teachers need to maintain an active involvement of curriculum, especially for general

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education and gifted students. The current curriculum and enrichment opportunities should be meaningful, stretch students‘ thinking, and build on concepts that are longlasting.
Walker (2007) includes a positive experience with Dynamic Pedagogy, a curriculum, instruction, and assessment mathematics intervention, seeking to improve students‘ skills from a conceptual understanding, computational fluency, problem solving, and procedures. In describing the program‘s five-pronged approach, she references Vygotsky‘s sociocultural perspective of cognitive development and
Sternberg‘s Triarchic Theory of Intelligence resulting in mixed results as reported by the
University of Connecticut‘s National Research Center on the Gifted and Talented due to clear limitations to the study, including non-random sample and significant interaction between the covariate and grouping variables.
Ongoing assessment and flexibly grouping students based on readiness, interest, learning profile, and prior knowledge are also repeated themes in the literature.
Pretesting students prior to starting a unit was a common practice. This type of assessment could be formal or informal. The pre-assessment information provided a path for teachers to follow (Tomlinson, 2001). Students who scored at a mastery level would have their unit compacted. Often the teacher is better able to adjust the pacing, common content, and removal of unnecessary review of skills when pretesting is done (Renzulli &
Reis, 2002). This would allow some students to do alternate activities based on interest.
Students indicating weak mastery of skills would receive intensive instruction in a small group setting.

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Summary
Current understandings on differentiated instruction have primarily been through the work of one expert, Carol Tomlinson; however, the review of the literature includes the work, thoughts, and research studies of this educational approach in combination with others in the field. Considering the review of the literature, a quantitative research study on effectiveness of differentiated instruction is needed. In this continuing age of teacher accountability and quantitative evidence of student achievement, it is necessary to harvest evidence in a similar mode to support a practice to which many teachers in the classroom aspire. CHAPTER III: METHODOLOGY

Restatement of Purpose
The purpose of this study was to determine the effects of differentiated instruction on student achievement in mathematics. Of the main differentiated instructional strategies, compacting, flexible grouping, and anchoring activities were examined. Most of the studies on this topic have been of a qualitative nature with positive results. This study was quantitative and addressed the following question: Does differentiated instruction impact the growth of student learning in the subject of math?
H0 = There will be no significant change in student mathematics achievement using differentiated instructional techniques.
The study will also examine if using differentiated instructional techniques impact the growth of student learning in mathematics achievement based on gender differences and ability.
H1 = There will be no significant change in student mathematics achievement using differentiated instructional techniques between males and females.
H2 = There will be no significant change in student mathematics achievement using differentiated instructional techniques among ability levels.
This was a study sample of convenience due to the school being accessible to the researcher. It was conducted over four units of study spanning approximately 13 weeks.
The four units at the time the study were counting money, money applications, area and perimeter, and fraction concepts. The repeated measures quasi-experimental research

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design allowed the researcher to quantitatively investigate the relationship of differentiated instruction on mathematical achievement as a means of increasing growth in mathematics achievement. Additional attention to demographic variables, often thought to influence in mathematics performance, strengthened the design by allowing a deeper analysis of the differentiated instructional activities.
Description of Participants and Setting
Participants were three female teachers. Teacher A has been a second or third grade teacher for fifteen years and holds a Master‘s degree. She was a former school level Teacher of the Year. Her class of 24 students had 12 boys and 12 girls with four requiring a special education plan in an inclusion setting. Teacher B has 25 years of experience, has been nominated as the school‘s Teacher of the Year twice and represented the district at the state level. She has always taught primary grades. Her class consisted of 26 students consisting of 12 boys and 14 girls. One of her students is considered an English Language Learner. Teacher C has been teaching for eight years in second grade. Her classroom has 13 boys and 12girls. Four of the students had an individualized education plan in the classroom setting. There were two other teachers in the grade level who did not participate in the study.
The school was in an affluent area of a suburb northeast of Indianapolis, Indiana, with 95.7% of the students‘ families able to pay for school lunch. During the 2009-10 school year the racial demographics of the school were 84.7% White, 2.3% Hispanic,
3.2% Black, 5.2% Multi-racial, and 4.3% Asian. There were 695 students in grades kindergarten through fourth grade. Third and fourth graders took the Indiana Statewide

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Testing for Educational Progress-Plus (ISTEP+) assessment in the spring of 2010.
Ninety-seven and four-tenths percents of the third graders and 94.9% of the fourth graders received a passing score in mathematics.
Student participants in the study aligned closely with the demographic composition described above. They had to be second graders between the ages of seven and nine. Students were heterogeneously mixed, including eight who received special education services in an inclusionary model. Children of advanced ability were included.
The students‘ pretest and posttest results for each unit, as well as the aptitude test data and gender were collected and included in the descriptive analyses of the sample. Only data from student subjects who were present at the beginning and end of the research study, and who had parental agreement to participate, were included in order to make valid comparisons.
Procedure
Description of Instrumentation/Measurement Procedures
The effects of differentiated instruction were measured through a repeated quasiexperimental research design. Teachers needed to access not only the pretest and posttest results, but they also needed each of the student‘s aptitude test results. These were used as a means to explore whether there was a difference among low ability, average ability, and high ability students. The researcher met with the teachers in the study three weeks prior to beginning to review what grade level mathematics topics would be taught. A proposed time line was also presented (Appendix E). Teachers also provided insight as to what units of study would be of similar difficulty for students based on their

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experience. The pre- and posttest procedures were modeled for teachers in the form of a pre-assessment of differentiated instructional techniques during the twelve hours of training teachers received. Two separate observations were made before the training as well as during the implementation. The purpose of the first two observations was to determine to what extent teachers understood differentiation. Two thirty-minute observations by the researcher were conducted during math instruction as a means of determining which strategies were implemented. The tally of strategies implemented and observations allowed the researcher to compare these to the change in the pre- and posttest results (Appendix F). A fifth test was given three weeks after the conclusion of two non-treatment units, and a sixth test was given three weeks after the last posttest to the treatment units to determine retention of the skills learned.
Math Assessment
Students were administered four pre- and posttests as well as two cumulative review assessments three weeks after the second non-treatment unit and three weeks after the second unit in which differentiated instruction was implemented. For each lesson or objective of the unit, four or five questions were developed for students to answer. Each unit contained five to six objectives, thus creating a unit assessment of between 21 and 25 questions. Mastery of the skill after instruction was defined to be at 75% accuracy or better. Four or five questions were used to allow for the student to miss one question and still maintain this threshold. Students‘ mathematics achievement was measured in the form of a researcher created pretest before instruction and repeated again after the full unit instruction had concluded. The cumulative tests had two questions per skill. Unlike

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the previous assessments, these tests were summative in nature and served as a tool for determining if the learning experienced was retained. The textbook provided assessments were the primary resource for the questions.
InView Aptitude Assessment
Students were given the InView Cognitive Abilities Test published by
CTB/McGraw-Hill. The standard deviation for this assessment is 16. For example a score of 100, which considered average, would have a range of 84 to 116. This is a test given annually to second graders in this school and was used as a means of determining the students‘ aptitude.
Pre-Treatment Observations
Each teacher was observed for two 30-minute periods before and after training on implementing differentiated instruction. The purpose of the first two observations was to determine to what extent teachers understood differentiation. A tally of differentiated instructional strategies observed was recorded.
Training
The training sessions on differentiated instruction were facilitated by the researcher and took place over a two-day period. The first day focused on the theory behind differentiated instruction and a demonstration of the of the various strategies of differentiated instruction and the foundation including Vygotsky‘s zone of proximal development and an overview of Sternberg cognitive styles, a second underpinning of differentiation. The strategies were viewed through an Association for Supervision and
Curriculum Development training module.

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The second day emphasized the strategies of disaggregating assessment data, compacting, flexible grouping, and anchoring activities. Managing a differentiated classroom and the lessons to be used for instruction were included. The lessons were designed and agreed upon by the three teacher participants to reduce the variability of the instruction. A menu of anchoring activities for enrichment was also created. Teachers then planned how to use any help students received from the instructional assistants, including the high school cadet teacher, resources available for anchoring activities to provide additional practice as well as extend the learning, and reviewed for consistency in interpretation of the skills to be taught for the differentiated units.
Post-Training Observations
The units of study for differentiation were area and perimeter and fraction concepts. Two observations of approximately 30 minutes each occurred. The two observations during the implementation were done to ensure that strategies presented during the two-day training were being used. One was during the unit on area and perimeter, and the other was during the unit on fractions. A tally of strategies observed was kept.
Research Design
The research design was inspired by a quantitative study published by Tieso
(2005). The week before beginning the unit, teachers administered the pretest. After it was scored, teachers had the opportunity to see the results including the responses of the students. This was done twice before the teachers received the twelve hours of training on differentiated instruction over a two-day period. An emphasis of the training included

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compacting, flexible grouping, and anchoring activities. Teachers then developed instruction implementing the strategies learned for the next two units of study. Students were grouped based on the InView intervals as a means of determining the level of student who achieved the best in a differentiated classroom. Due to only having participants of average and above average ability in this study, a comparison was made with and without differentiated instruction between these two aptitude ranges. Names were not used and were coded instead. Only students, who were present at the beginning and end of the study and who had completed all four of the pre- and posttests, as well as a minimum of 75% of the instructional experiences, were tracked. The dependent variable was the measure of the pre-and posttest. The independent variable was the differentiated instruction. Internal validity was determined by the comparison of the non-treatment and treatment of the instruction. External validity was determined by the consistency of the results between the two treatments within the same classroom and between the three classrooms. A threat to the construct validity was the level of use of the strategies implemented, the commitment to use the strategies, the researcher‘s attitude when conducting the training and potentially influencing the teacher attitude, and prior knowledge and experience of the teachers in using differentiated instructional strategies.
The process of this experimental design was as follows.

Pretest

Instruct as Usual

Posttest

After the twelve hours of differentiated instruction training occurred, the second step was

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altered and the process changed as follows.

Pretest

Differentiated
Instruction

Posttest

Description of Procedures
This research study took place over a time period of thirteen weeks through four units of study in mathematics. Approximately one week before the beginning of each of the first two units, a pretest was administered. Teachers were provided with the results in a summative manner and were given the actual responses for each of the questions for each student. Two units were used in order to increase the reliability and validity of the study. Teachers then engaged in twelve hours of training in differentiated instructional design. The first six hours included an overview of differentiated instruction and the different techniques, including Vygotsky‘s zone of proximal development and an overview of Sternberg cognitive styles, a second foundational underpinning of differentiation. The second six hours emphasized the strategies of disaggregating assessment data, compacting, flexible grouping, and anchoring activities. Managing a differentiated classroom and the lessons to be used for instruction were included. The lessons were designed and agreed upon by the three teacher participants to reduce the variability of the instruction. A menu of anchoring activities for enrichment was also created. 45

Teachers administered pretests of four to five questions per skill. Teachers then reviewed the results as a means of flexibly grouping the students for instruction.
Students needing direct instruction as indicated by achieving less than 100% on each objective on the pretest met with the teacher. As teachers learned during the 12-hour training, students at this level may be dismissed early from the lesson if the student demonstrated understanding. Teachers implemented the researcher-facilitated designed lessons. Students not in the lesson experienced the enriching anchoring activities. A posttest was given to all students to measure growth. This assessed the impact of the challenging anchoring activities and alternative assignments. The researcher also made two observations of each teacher during the period of differentiated instruction. The tally of strategies implemented and observations allowed the researcher to compare these to the change in the pre- and posttest results.
Data Analysis
The study was implemented with a repeated measures design. The change between pretests and posttests of four mathematics units in three second grade classrooms was compared. The first two units did not have a deliberate attempt to implement strategies associated with differentiated instruction; the second two did. A comparison was made of the following.
Comparison One: The growth (or lack thereof) in mathematics learning in students after two units of non-treatment compared to two units of treatment with differentiated instruction, based on the change from the pretests and posttests using a dependent t-test.

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Comparison Two: The growth (or lack thereof) in mathematics learning in students by gender after two units of non-treatment compared to two units of treatment with differentiated instruction, based on the change from the pretests and posttests using a dependent t-test during differentiated instruction.
Comparison Three: The growth (or lack thereof) in mathematics learning in students with differing ability after two units of non-treatment compared to two units of treatment with differentiated instruction. This will be based on the change from the pretests and posttests using a dependent t-test of two ranges of scores from the InView
Cognitive Abilities Test during differentiated instruction. The average range is between
86-114, and the above average range will be any score greater than 114.
A comparison by special education, English Language Learners, poverty level, or ethnic or racial diversity was not done due to the school not having a large enough number of students in any of these subgroups.
A finding was considered significant at a 0.05 level for all three comparisons.
These analyses needed to be conducted at the classroom level due to the lack of random assignment of the students to classrooms since students with special needs were clustered together for ease in providing services.
To determine the significance of growth from non-treatment to treatment, the following formula was applied: m equals the average growth of each pre- and posttest combination. M equals the mean of each variable before and after treatment.

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M Before = (m Unit 1 + m Unit 2)/2
M Treatment = (m Unit 3 + m Unit 4)/2
M After = (m Retention 1,2 + m Retention 3,4)/2
To accommodate for the small sample of three classroom teachers, an effect size analysis was done. The effect size was calculated as follows.
Mean Difference = (Mean Treatment – Mean Before)
SD pooled = (SD before n before) + (SD treatment n treatment) / (n before + n treatment)
Cohen‘s d = M difference / SD pooled
A second effect size was determined for retention of skills learned and was calculated as follows.
Mean Difference = (Mean Retention 3,4 – Mean 1,2)
SD pooled = (SD Retention 1,2 n Retention 1,2) + (SD Retention 3,4 n Retention 3,4) / (n Retention 1,2 + n Retention 3,4)
Cohen‘s d = M difference / SD pooled
The value of Cohen‘s d effect size fell into one of the following levels of effectiveness. < 0.2 = not effective
0.2-0.8 = moderately effective
> 0.8 = highly effective
The test of retention was a comparison of the average of the two tests, one cumulative after the first two units without differentiation and the second after the treatment of differentiated instruction included. The pretest and posttest change as a

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result of the training and the implementation of differentiated instuction was simply reported as the mean percentage increase or decrease.

CHAPTER IV: RESULTS

Restatement of the Research Questions
The hypothesis in this study was that differentiated instruction does improve the growth of academic achievement. This study looks at the results, not only in terms of the second grade students as a whole, but also gender and ability. Quantitatively this was measured by comparing instruction that does not differentiate based on growth data with instruction that does, using pre- and posttests. The following three null hypotheses were stated as:
1.

There will be no significant change in student mathematics achievement using differentiated instructional techniques.

2.

There will be no significant change in student mathematics achievement using differentiated instructional techniques between males and females.

3.

There will be no significant change in student mathematics achievement using differentiated instructional techniques among ability levels.

Does differentiated instruction impact the growth of student learning in the subject of math?
In order to determine whether or not differentiated instruction impacted student learning in mathematics, the mean difference in pretest and posttest scores was calculated for the two units implemented without and with differentiated instruction (Table 1).

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Table 1 Comparison of Pretest and Posttest Change Before and After Treatment

Condition
OVERALL
Female
Male
Avg. Aptitude
Above Avg.
Aptitude

n
61
31
30
36
25

Percent
Diff. of
Unit 12
Pre- and
Post
41.77
47.86
41.87
41.19
32.92

Percent
Diff. of
Unit 13
Pre- and
Post
34.63
35.06
27.27
42.67
32.10

Avg. Pct.
Chg. of
Units 12 and 13
38.20
41.46
34.83
41.93
32.51

Percent
Diff. of
Unit 25
Pre- and
Post
41.80
41.65
39.76
37.30
46.55

Percent
Diff. of
Unit 26
Pre- and
Post
36.54
37.15
38.08
34.78
42.59

Avg. Pct.
Chg of
Units 25 and 26
39.17
39.40
38.92
36.04
44.57

Avg. Pct.
Chg of
Units 25 and 26 and Units
12 and
13
+0.93
-2.06
+4.09
-5.89
+12.06

A relationship of the mean of the two units of study prior to implementation compared with the two after implementation showed a 0.93% increase from the average pre-treatment change to the treatment change when viewing the last column of Table 1.
Students also took an assessment approximately three weeks after the conclusion of the second unit (Unit 13). The same process occurred after Unit 26, the second unit of the treatment. The results are shown in Table 2. The purpose was to determine the retention of the material learned. There was a positive change of 3.17% from Cumulative
1 to Cumulative 2.
A dependent sample t-test was conducted to determine if there was a significant difference in the average changes in scores between the pretreatment units (12 and 13) and the two units after training in differentiated instruction was conducted. The results of the change between pretreatment and treatment instruction were not significant at the
0.05 level as demonstrated in Table 2; (mean = 39.17, SD = 1.68; t = .33, p > .05).

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Table 2 Mean Changes of Pretreatment and Treatment and Cumulative Tests (Overall)
Change After
Differentiated
Instruction
M2
SD2
39.17
1.68

Change Before
Differentiated
Instruction
M1
SD1
38.20
1.36

Results of t Tests
Condition
n t df
p.05
Overall Chg of
61
0.33
60
< .74
25 and 26 from
12 and 13
Overall Cum.
61
87.05
1.34
83.88
1.37
1.81
60
< .08
Test 2 from
Cum Test 1
Notes: 1 reflects pretreatment units 12 and 13 or CUM 1; 2 reflects treatment units 25 and 26 or CUM 2

A dependent t-test was conducted to determine if there was a significant difference in the scores on the two cumulative tests which was administered approximately three weeks after instruction. Two questions per skill taught in the units were on the assessment. The results of the change between pretreatment and treatment were not significant at the 0.05 level as demonstrated in Table 2; (mean = 87.05, SD =
1.34; t = 1.81, p > 0.05).
When comparing the pretreatment units to the treatment units, the implementation of differentiated instruction was found to be relatively ineffective, ES=0.04. The cumulative assessments indicated a low effectiveness, ES=0.23. Table 3 reflects this.
Table 3 Statistical Comparisons Before and After Treatment of Unit Paired Tests and
Cumulative Tests
SD1

SD2

Cohen‘s d

61

Percent
Change
+0.93

1.36

1.69

.04

61

+3.17

1.37

1.35

.23

Condition

N

Difference between mean of two non-treatment units and the two treatment units
Difference between mean of cumulative tests of non-treatment and treatment

Notes: 1 reflects pretreatment units 12 and 13 or CUM 1; 2 reflects treatment units 25 and 26 or CUM 2

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Based on the low effect size and the result of the comparison of the unit test pairs, and the cumulative tests, differentiated instruction did not have an impact on the learning of students in general, and the null hypothesis is accepted; (t = 0.33, p > .05).
Does differentiated instruction impact the growth of student learning of a particular gender in the subject of math?
Thirty-one females and thirty males participated in the study. In order to determine whether or not differentiated instruction impacted student learning based on gender in mathematics, the mean difference in pretest and posttest scores was again calculated for the two units implemented without differentiated instruction and for the two units implemented after differentiated instruction (Table 4).
Table 4 Mean Changes of Pretreatment and Treatment and Cumulative Tests (Gender)
Change After
Differentiated
Instruction
M2
SD2
39.40
1.69

Change Before
Differentiated
Instruction
M1
SD1
41.46
1.30

Results of t Tests
Condition
n t df
p.05
Female Chg
31
-0.48
30
< .63 of 25 and 26 from 12 and
13
Male Chg of
30
38.92
1.71
34.83
1.36
1.02
29
< .32
25 and 26 from 12 and
13
Female Cum
31
84.84
1.70
81.86
1.52
1.59
30
< .12
2 Test and
Cum 1 Test
Average
Male Cum 2
30
89.33
0.81
85.97
1.18
1.59
29
< .12
Test from
Cum 1 Test
Notes: 1 reflects pretreatment units 12 and 13 or CUM 1; 2 reflects treatment units 25 and 26 or CUM 2

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In viewing Table 4, there was a slight decrease in female test scores from before to after treatment. Male change performance increased 4.09% after differentiated instructional strategies were implemented in the mathematics instruction.
A dependent t-test to determine if there was a significant difference in the average changes in scores between the pretreatment units (12 and 13) and the two units (25 and
26) after training in differentiated instruction was conducted based on gender. The result of the change between pretreatment and treatment for females was not significant at the
0.05 level as demonstrated in Table 4; (mean = 39.40, SD = 1.69; t = -0.48, p > 0.05). In fact, differentiated instruction appeared to be detrimental; however, for males, it was not harmful, but the results of the positive change for males between pretreatment and treatment were not significant at the 0.05 level as demonstrated in Table 2; (mean =
38.92, SD = 1.71; t = 1.02 p > .05).
A dependent t-test to determine if there was a significant difference in the scores on the two cumulative tests that were administered approximately three weeks after instruction for the purpose of analyzing the effect of differentiated instruction in mathematics on gender. Two questions per skill taught in the units were on the assessment. The results of the change between pretreatment and treatment were not significant at the 0.05 level for either males or females.
The comparison of female results as indicated in Table 4 was consistent when reviewing the cumulative tests and comparing them to the end-of-unit tests; (mean =
84.84, SD = 1.70; t = 1.59, p > .05).

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The comparison for males proved to be more positive when reviewing the change of unit averages and the cumulative averages and comparing them to the end-of-unit tests averages; (mean = 89.33, SD = 0.81; t = 1.59, p > 0.05).
Neither males nor females had significant growth when comparing the cumulative test change to the paired unit tests when differentiated instruction was implemented. When looking at the results of both cumulative assessment results, the null hypothesis is accepted indicating that differentiated instruction does not benefit one gender over the other. Similar results were found with either analyzing the paired unit assessments or the cumulative assessments.
Does differentiated instruction impact the growth in mathematics learning of students with differing aptitudes?
In order to determine whether or not differentiated instruction impacted student learning in mathematics based on ability, the mean difference in pretest and posttest scores was calculated for the two units implemented without differentiated instruction and for the two units implemented after differentiated instruction (Table 5).

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Table 5 Mean Changes of Pretreatment and Treatment and Cumulative Tests (Aptitude)
Change After
Differentiated
Instruction
M2
SD2
36.04
1.68

Change Before
Differentiated
Instruction
M1
SD1
41.93
1.31

Results of t Tests
Condition
n t df
p.05
Avg.
36
-1.76
35
< .09
Aptitude
Chg. of 25 and 26 from
12 and 13
Above Avg.
25
44.57
1.65
32.51
1.32
2.40
24