# The effects of a Concrete, Representational, Abstract (CRA) instructional model on Tier 2 first-grade math students in a Response to Intervention model: Educational implications for number sense and computational fluency

vi TABLE OF CONTENTS

Page ABSTRACT ................................................................................................................. iii ACKNOWLEDGEMENTS ...........................................................................................v LIST OF TABLES ...................................................................................................... viii LIST OF FIGURES ...................................................................................................... ix

CHAPTER 1 INTRODUCTION ...................................................................................1 THE RESEARCH PROBLEM: BACKGROUND AND NEED .............................................................................. 1 RATIONALE ............................................................................................................................................. 3 RESPONSE TO INTERVENTION .................................................................................................................. 3 THEORETICAL FRAMEWORK .................................................................................................................... 6 INSTRUCTIONAL MODEL ......................................................................................................................... 7 STUDY PURPOSE AND DESIGN ................................................................................................................. 8 DEFINITION OF TERMS ........................................................................................................................... 11 LIMITATIONS ......................................................................................................................................... 11 SIGNIFICANCE OF STUDY ....................................................................................................................... 12

CHAPTER 2 LITERATURE REVIEW ....................................................................14 NUMBER SENSE ..................................................................................................................................... 14 DEVELOPMENT OF NUMBER SENSE ....................................................................................................... 15 COMPUTATION ...................................................................................................................................... 21 NUMBER SENSE ASSESSMENT ............................................................................................................... 24 MATHEMATICS DIFFICULTIES ................................................................................................................ 28 ADDITION INTERVENTION ..................................................................................................................... 32 CONCRETE REPRESENTATIONAL ABSTRACT ......................................................................................... 35 RESPONSE TO INTERVENTION ................................................................................................................ 39 SUMMARY ............................................................................................................................................. 45

CHAPTER 3 METHODOLOGY ...............................................................................45 INTRODUCTION ...................................................................................................................................... 47 PARTICIPANTS ....................................................................................................................................... 49 INSTRUMENTATION ............................................................................................................................... 50 Lembke-Foegen Early Numeracy Indicators ..........................................................50 Curriculum-Based Measurement-Mathematics (CBM-M) .....................................52 Test of Early Mathematics Ability (TEMA-3) .........................................................52 PROCEDURES ......................................................................................................................................... 54 Lembke-Foegen .......................................................................................................54 Curriculum Based Assessment ................................................................................54 TEMA ......................................................................................................................55 Expeditions to Numeracy ........................................................................................56 Curriculum Based Assessments ..............................................................................59 DATA ANALYSIS ................................................................................................................................... 59 TEMA ......................................................................................................................59 Curriculum Based Measurement ............................................................................60 Hierarchical Linear Modeling ................................................................................63

vii CHAPTER 4 RESULTS ..............................................................................................65 INTRODUCTION ...................................................................................................................................... 65 TEST OF EARLY MATHEMATICS ABILITY-3 RD EDITION .......................................................................... 69 CURRICULUM BASED ASSESSMENTS ..................................................................................................... 71 Tukey Method ..........................................................................................................76 Percent of Non-Overlapping Data ..........................................................................78 Split Middle Trend Line ..........................................................................................80 LEMBKE-FOEGEN EARLY NUMERACY INDICATORS .............................................................................. 82 GROWTH CURVE MODELING ................................................................................................................. 85 SUMMARY ............................................................................................................................................. 91

CHAPTER 5 DISCUSSION, IMPLICATIONS, AND RECOMMENDATIONS ..............................................................................................93 INTRODUCTION ...................................................................................................................................... 93 SUMMARY OF MAJOR FINDINGS ............................................................................................................ 93 LINKS TO PRIOR RESEARCH ................................................................................................................... 96 LIMITATIONS OF THE STUDY ............................................................................................................... 100 SUMMARY OF CONCLUSIONS ............................................................................................................... 101 EDUCATIONAL IMPLICATIONS ............................................................................................................. 103 RECOMMENDATIONS ........................................................................................................................... 105

REFERENCES ............................................................................................................108

Appendices A. LEMBKE-FOEGEN INSTRUMENT ...................................................................... 114 B. EXPEDITIONS TO NUMERACY CURRICULUM .......................................... 115 C. DIGI-BLOCKS .................................................................................................................. 117 D. STUDENT PROGRESS RECORD SHEET ........................................................... 118 E. EXPEDITIONS TO NUMERACY FIDELITY SELF-CHECK ..................... 119 F. FUCHS AND FUCHS CURRICULUM BASED ASSESSMENT .................. 121 G. IRB CONSENT .................................................................................................................. 122 H. STUDENT ASSENT ........................................................................................................ 123

LIST OF TABLES

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Table 2.1 Ten Frame from Expeditions to Numeracy .............................................. 16 Table 2.2 Core Mastery Goals .................................................................................. 18 Table 3.1 Tema-3 Reliability Coefficients ................................................................ 53 Table 3.2 Lembke-Foegen Cut Scores ...................................................................... 54 Table 3.3 Guide to Interpreting TEMA-3 Math Ability Scores ................................ 60 Table 4.1 TEMA-3 Scores ........................................................................................ 68 Table 4.2 TEMA-3 Selected Student Scores ............................................................ 70 Table 4.3 Computation Probes Median Scores ......................................................... 71 Table 4.4 Lembke-Foegen Early Numeracy Scores ................................................. 82 Table 4.5 Emily’s Lembke-Foegen Scores ............................................................... 83 Table 4.6 Matt’s Lembke-Foegen Scores ................................................................. 83 Table 4.7 Estimates of Effects from Individual Growth Models .............................. 87 Table 4.8 Explanation of Growth Curve Data .......................................................... 99

ix LIST OF FIGURES

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Figure 4.1 Emily’s Weekly CBM............................................................................... 82 Figure 4.2 Matt’s Weekly CBM ................................................................................. 83 Figure 4.3 Emily’s Tukey Analysis ............................................................................ 85 Figure 4.4 Matt’s Tukey Analysis .............................................................................. 86 Figure 4.5 Emily’s Percent of Non-Overlapping Data ............................................... 87 Figure 4.6 Matt’s Percent of Non-Overlapping Data ................................................. 88 Figure 4.7 Emily’s Split Trend Line Analysis ........................................................... 89 Figure 4.8 Matt’s Split Trend Line Analysis .............................................................. 90 Figure 4.9 Fitted Growth Trajectories ........................................................................ 97

CHAPTER 1 INTRODUCTION

The Research Problem: Background and Need

Instructional models which assist students who are struggling to gain access to mathematics are crucial to the development of an effective educational system. Meeting the needs of such students has long been the focus of research in education. However, many believe that educators have demonstrated little success in improving the American student’s education. No Child Left Behind (NCLB, 2001), signed into law by President George Bush in 2002, brought a dramatic increase in reporting and accountability measurements for K-12 schools in the United States. NCLB established four pillars to help educators deliver a more valuable education. The first pillar provides stronger accountability for student achievement. States must close the achievement gap for all students, including those who are disadvantaged. Annual reports to local communities in each state must show adequate yearly gains. If insufficient progress is attained, then supplemental services, such as tutoring and after-school programs, must be provided to help students make significant progress. The second pillar of NCLB provides more local control and decision making for states and communities. School districts may use their federal funding with less regulation, allowing them to direct funds to areas where needs are most critical, including teacher salaries, professional development, or recruitment. The third pillar emphasizes the importance of using curriculum derived from solid scientific research base. Federal funding for school districts is provided to develop and support such programs. The final pillar of NCLB provides choices for parents in low

2 performing schools. Students attending a school that has not made adequate yearly progress over two consecutive years may transfer to better achieving schools in their district. If a school fails to meet state standards for three years, students may be eligible for free supplemental services (US Department of Education, 2008). In addition, the reauthorization of the Individuals with Disabilities Education Act (IDEA) was signed into law in December 2004. This reauthorization shared alignment with NCLB and facilitated a change in the identification of students with specific learning disabilities. Effective July 2005, states could no longer require school districts to use a discrepancy model for the identification of students with a specific learning disability. The discrepancy model required a severe difference between a student’s intellectual ability (as measured on an intelligence quotient test) and educational achievement when determining if the child needed special education services. Educators waited until there were sufficient gaps to provide students with special services. This model placed a disproportionate number of minority students into special education classrooms (wrightlaw.com). IDEA 2004 permits the use of a process that relies on scientific, researched-based interventions to determine whether a child qualifies for special education services. In fact, IDEA 2004 requires that students be provided with appropriate instruction, delivered by a qualified teacher who formally documented the student’s progress at reasonable intervals of time, prior to any student being referred for an evaluation. Qualified teachers must be deemed as such through the NCLB teacher certification regulations (IDEA, 2004).

3 Rationale IDEA and NCLB have heightened awareness regarding the need for reform in the way students are identified for special education. Historically, there has been a disproportionate number of minority students placed in special education, including classes for specific learning disabilities (Zhang & Katsiyannis, 2002). African Americans account for 14.8% of the general population; however, they represent 20% of special education classifications (Losen & Orfield, 2002). Blanchett (2006) asserts that placements are usually made by school personnel using subjective referral and eligibility criteria, despite the original the intent of special education to provide objective evaluation, instruction, and assessment for students (Blanchett & Shealey, 2005). When African American students are placed into special education classes they are typically segregated from the regular education population, have higher dropout rates, lower standardized assessment scores, and watered-down curriculum (Ferri & Connor, 2005). The National Research Council (2002) asserts that improving the regular education instruction students receive can reduce the placement of minority students in special education, and increase the likelihood of success for all students. Response to Intervention The mandates of IDEA and NCLB have led some states to adopt a new model for identifying students with specific learning disabilities. This model is called Response to Intervention (RtI). RtI is divided into three tiers and requires educators to screen all students, not just those students that a teacher may notice having difficulty in the curriculum. The National Research Center on Learning Disabilities describes RtI in the following way:

4 RtI is an assessment and intervention process for systematically monitoring student progress and making decisions about the need for instructional modifications or increasingly intensified services using progress monitoring data. The following is the fundamental question of RtI procedures: Under what conditions will a student successfully demonstrate a response to the curriculum? Thus interventions are selected and implemented under rigorous conditions to determine what will work for the student (Retrieved August 28, 2008).

The first tier of RtI calls for universal screening of all students. At this tier, all students are receiving standard research-based curriculum prescribed by the district as part of their comprehensive education. Research-based curricula are curricula that have been tested through experimental measures and whose results have been published in a peer reviewed journal (Kovaleski, 2007). Many advocates of RtI recommend that all students have a one-time screening to determine eligibility. However, short-term continuous curriculum-based assessment is the ideal model for identifying students with potential learning challenges since some students may “recover” or catch up to their peers on their own (Fuchs, Fuchs & Hollenbeck, 2007). This provides many data points to determine eligibility for more specialized services. Those students who are labeled as non-responders in the regular curriculum progress to Tier 2 instruction. At this tier students are taught in small groups, ideally for one-half hour, three times a week, over an eight-week period (Fuchs, Compton, Fuchs, Paulsen, Bryant, & Hamlett, 2005). These students are provided with a researched-based program that has been proven successful with learners of similar need (wrightslaw.com). The program is delivered by a qualified individual, such as a teacher or instructional aide, and weekly progress monitoring tracks student progress. Best practice for evaluation at

5 the end of the cycle is determined by adequate progress based on either criterion- referenced or norm-referenced estimates of weekly improvement (Fuchs et al., 2005). Children who are identified as responders in Tier 2 are placed again in Tier 1 with progress monitoring to ensure continued success. However, if a student is still determined to be a non-responder, not meeting the goals of progress monitoring then the child is sent to Tier 3. In Tier 3 students are either tested for a specific learning disability and/or given even more intensive interventions in groups of one or two to help them overcome their learning challenges, thus increasing student success in their academic endeavors. In recent years, classrooms have become more inclusive, including students with diverse needs. Meeting the needs of all students has become a challenge for classroom teachers, as some research indicates the need for direct instruction and other literature indicates the need for creative problem-solving ability. The National Council of Teachers of Mathematics (NCTM), in their document Principles and Standards for School Mathematics (2000), called for a mathematics education that permits all students access to an education which enables them to solve problems creatively. Such education will assist students in gaining access to careers previously not available to those who do not have an understanding of mathematics. According to the Principles and Standards (2000) students must learn mathematics with understanding, actively building upon prior mathematical knowledge. Teachers need to provide mathematical instruction that allows students to construct knowledge based on prior skills and understandings. The current research in the area of diverse learning needs indicates that teachers need to focus on explicit teaching (Hudson, Miller, & Butler, 2006). Hudson et al. suggest that instruction

6 for struggling learners should contain explicit instruction, and have students actively constructing their mathematical knowledge.

Theoretical Framework This study will draw on Jerome Bruner’s theory of Interactional Cognitive Development. Bruner believes that knowledge is developed through three modes of representation that allows learners to construct an understanding of their world. The first mode is the Enactive Representation. This mode refers to “representing past events through appropriate motor response” (Driscoll, 2005). Students learning at this level need to have concrete examples that they can physically manipulate through their own actions. The use of manipulatives in mathematics education is important in the development of student’s enactive representations necessary to develop number sense, which can then subsequentially be applied to more advanced mathematical skills. The Iconic Representation mode refers to the development of skills through the use of images. This would include diagrams and pictures of concepts that allow the learner to identify the important features of a concept. Once the student has shown a mastery of this level of representation, he/she is ready to move to the third mode called Symbolic Representation. At this phase of development learners are able to use symbols, such as language and mathematical notations, to represent concepts being developed. Bruner cautioned that pushing learners to the symbolic level quickly, or skipping the enactive and iconic modes all together, can impede the development of a concept (Driscoll, 2005). According to Bruner (1966) culture plays a significant role in the development of knowledge. Bruner believed that learners progressed through various stages of

7 development, but these stages were not age dependent. Any student could learn a concept, at any age or developmental level, as long as he/she was given the proper prerequisite knowledge and experiences. Bruner believes that any idea can be presented to students via a discovery process where children are led through experiences that allow them to use their prior knowledge to gain an understanding, facilitated by the use of active dialogue. K-12 learning should be an active, engaging process not haphazard in nature, but rather the result of careful planning. Bruner believes schools should equip students with the skills and tools necessary for the culture in which they live. Bruner believes that if the education system did a better job helping students transfer their skills to their culture, schools would be more successful (Driscoll, 2005). Instructional Model The Concrete-Representational-Abstract (CRA) approach offers students explicit instruction in mathematics and assists the learner in constructing his/her knowledge through the use of multiple representations of a concept (Hudson et al., 2006). This approach contains three stages of instruction, similar to Bruners stages of knowledge development. The first stage is the Concrete Stage, where the teacher models the concept with concrete materials such as base ten blocks, digi-blocks, or fraction bars. The use of manipulatives has been proven useful in assisting struggling learners to acquire and maintain various mathematical skills (Cass, Cates, Smith, & Jackson, 2003). In the Representation Stage, the teacher changes the concrete representation to a semi-concrete level that may include drawings or tallies. This stage prevents the distortion of concepts by connecting the pictorial with the concrete, which has been proven instrumental for

8 students with learning challenges in mathematics (Witzel, Mercer & Miller, 2003). In the final stage, called the Symbolic Stage, a model is used which includes numerals and/or symbols as the sole representation of a number sentence or concept. Witzel et al. (2003) believe that the value of the use of symbols in mathematics is to work beyond what one can see or touch and establish the connections to other situations. Meaningful connections between these representations are created for students to assist them in the development of a mathematical concept (Access Center, n.d.). This meaningful development of concepts is crucial for young children. Research indicates that 66% of the variance in first-grade student math achievement can be attributed to a student’s number sense (Jordan, Kaplan, Locuniak, & Ramineni, 2007). Number Sense is defined as “abilities related to counting, number patterns, magnitude comparisons, estimating and number transformations” (Berch, as quoted in Jordan et al., 2007). Research conducted by Jordan et al. (2007) indicates that number sense is a predictor of mathematics achievement by the end of first-grade. Their study also reveals a strong and significant correlation between number combination retrieval and story problems, suggesting that these skills are fundamental to learning conventional mathematics. It is therefore crucial that children experiencing early difficulty in mathematics be given explicit instruction in number sense in order to increase their success in mathematics throughout later grades. Study Purpose and Design The purpose of this study is to examine the effects of explicit instruction on a child’s ability to develop number sense and computational fluency. The following research questions were addressed:

9 1. What are the effects of a concrete-representational-abstract instructional model on students’ achievement as measured on a mathematical achievement test?

2. What are the effects of a concrete-representational- abstract instructional model on Tier 2 students’ development of computational fluency?

3. What are the differences in student growth, measured via a progress monitoring tool, when students receive concrete-representational-abstract instruction?

These questions were investigated in a single subject design study, as is common in many RtI studies due to the small number of participants qualifying for Tier 2 interventions. This method has a behaviorist background that allows the researcher to observe behavioral changes over time, where each participant acts as his/her own control. All first-grade students at the study sites were screened using: 1.) the Early Numeracy Indicators, a curriculum based assessment developed by Lembke and Foegen (2005), 2.) Fuchs and Fuchs Curriculum Based Assessments (CBM) and 3.) teacher informal assessments. Winter norms developed by Lembke and Foegen were used to identify students performing at or below the 35 th percentile. The Lembke and Foegen instrument is based on research studies indicating that items assessing number magnitude judgments, reading numerals, and quantity discrimination are early predictors of mathematical achievement (Baker, Gersten, Flojo, Katz, Chard, & Clark, 2002; Clarke & Shinn, 2004; Mazzocco & Thompson, 2005). Curriculum based assessment has been used for more than 25 years in special education as a mechanism to both monitor students’ progress toward IEP goals, and identify students at-risk (Deno, 2003; Shinn, 2007; Stecker, Fuchs, & Fuchs, 2005). Stecker, Fuchs and Fuchs (2005) highlight three distinguishing features of CBM. First,

10 the CBM must be used to assess students’ progress toward long term goals that are general, rather than specific in nature. The second feature is frequent monitoring of student’s progress at least once per week, as it is a formative assessment that reflects performance over time. To be used effectively, data obtained on the assessments must be used to drive the curriculum by making instructional decisions based upon these assessment results (Stecker, et al., 2005). The third crucial feature of CBM is the use of a sound instrument which is technically adequate. The instruments typically include a set of standardized and validated assessments in mathematics computation, mathematics applications, and other reading skills that take between one and four minutes to administer (Shinn, 2007). There are many documented studies that support the use of progress monitoring as part of the RtI model of evaluating student’s response to an intervention (Deno, 2003; Fuchs & Fuchs, 1986; Fuchs & Vaughan, 2005). During each week of the small group instruction all first-grade students were assessed using Monitoring Basic Skills Progress (MBSP). Students who are identified as at-risk in both the regular curriculum by teacher observation and the Lembke-Foegen instrument, or the CBM were given the Test of Early Mathematic Abilities (TEMA-3) assessment to measure their mathematics achievement of concepts and skills. It provides norm-referenced scores, including scale and standard scores, and can be used to report progress and analyze growth. If a student was labeled at-risk by these initial screenings and the classroom teacher, he/she was identified as a Tier 2 student. These students will receive thirty instructional sessions using the Expeditions to Numeracy program, which has a CRA-based approach

11 embedded into its lessons. At the conclusion of the intervention, the Lembke-Foegen and TEMA-3 assessments were re-administered again to assess student growth. Definition of Terms In order to clearly understand the research questions, the following definitions are provided to ensure that the researcher and reader are clear about the concepts to be studied: Number sense is defined as the ability to understand the base-ten number system, estimate, make sense of numbers, and recognize the relative and absolute magnitude of numbers (NCTM, 2000, p.32) Place value is defined as the value of a digit in the base-ten number system. This value is based upon its location in relation to other digits in the number. Computational fluency is a student’s ability to have efficient, flexible and accurate methods for computing (NCTM, 2000). Explicit instruction systematically breaks concepts down into small sub-skills and provides experiences for students to develop conceptual understanding through guided discovery. Response to Intervention (RtI) is an assessment and intervention model which systematically monitors student progress to make instructional decisions which allow for increasingly intensified services as student need indicates (National Research Center on Learning Disabilities, 2008). Limitations This study has several limitations. The first limitation is the use of a single subject design. Although this is the most common model used to study special education

12 students, this design limits the ability to extrapolate the results to other populations. However, it does provide the basis for further research. Another limitation is the small sample size. Approximately 10 to 15% of the student population should qualify for Tier 2 services. If the population size is greater in a Tier 2 intervention, the instruction offered to all regular education students needs to be examined. The small sample size lowers the power of the study. Finally, the population of students participating in the study is not representative of the general population of the United States, as they are predominately upper-middle class children. Significance of Study The research on RtI and Tier 2 interventions is limited and in its infancy. While many schools are implementing many reading programs for Tier 2 students, mathematics interventions have not been fully studied or implemented. Current research indicates the rigor of mathematics courses students enroll in are powerful predictors of school mathematics achievement in later education (Tate, 2005). In order for all students to reach their full potential, educators must provide students with a variety of opportunities to be successful. Berry (2008) indicates that the affective connection that teachers make with their students influences their academic outcomes. The use of small group instruction can facilitate this connection. This study looked at a CRA instructional model that offers explicit instruction in computational fluency. This model of instruction provided the students with lessons which were broken down into small sub-skills, but still involved teacher directed discovery of the concepts underlying the development of addition and subtraction strategies. Through rich discourse and conceptual activities in the classroom, students developed strategies to increase their computational fluency and

13 over-all mathematics achievement. Selection of students for this study was based on research indicating that young students’ ability to identify numbers, make quantity comparisons, and identify the missing number in patterns are predictors of later success in mathematics. Designing and implementing targeted mathematics programs which include small group instruction, should enable students to learn prerequisite skills, which will contribute to success in future grades.

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CHAPTER 2 LITERATURE REVIEW

Number Sense

This work is grounded in the developmental stages of number sense, a topic discussed widely in the field of mathematics education. Berch (2005) identified thirty different definitions. According to his research the definition can encompass “awareness, intuition, recognition, knowledge, skill, ability, desire, feel, expectation, process conceptual structure, or mental number line” (p. 333). Berch further indicates that number sense can be attributed to natural ability or acquired skills. The National Council of Teachers of Mathematics (NCTM), in their Principles and Standards for School Mathematics (2000) defines number sense as: “The ability to decompose numbers naturally, use particular numbers like 100 or ½ as referents, use of the relationship among arithmetic operations to solve problems, understand the base ten number system, estimate, make sense of numbers, and recognize the relative and absolute magnitude of numbers.” (p.32)

The NCTM states that number sense is the cornerstone of elementary mathematics and should be developed between grades kindergarten and two. Certainly performing number combinations such as single digit addition facts is essential for success in mathematics. However, developing computational fluency must go hand-in-hand with an understanding of the meaning of the operation (NCTM, 2000). Berch and Mazzocco (2007) claim that number sense cannot be taught; rather it is an innate ability. Engaging in mathematical games and thinking can develop these skills (Berch et al., 2007). In contrast, this nativist position is refuted by researchers who

15 believe number sense is linked to the cerebral part of the brain, which under normal circumstances can be developed spontaneously. Other researchers such as Gersten, Jordan, and Flojo (2005) have discovered that informal knowledge of numbers is linked to number sense in young children. These researchers believe that the link between mathematics relationships, principles and procedures can be enhanced by gaining informal knowledge before a child enters school. Kroesbergen, Van Luit, Van Lieshout, Van Loosbroek, and Van de Rijt (2009) studied the relationship between domain-general and domain-specific factors and early numeracy. The researchers utilized 115 randomly chosen students in the Netherlands to study executive functions, fluid intelligence, subitizing, and language. Their results indicate 45% of the variance in early numeracy scores can be explained by measures which assess planning, updating, and inhibition, with updating showing the highest correlation. A second finding indicates that subitizing accounts for 22% of the variance in counting skills. These results indicate the need for the development of number sense skills through the use of flexible strategies. Development of Number Sense Landmarks such as the Ten Frame (see Figure 2.1) are important tools in the development of a child’s early number sense (Fosnot, 2001). Without these landmarks children rely on strategies such as “counting on” that can become problematic as the child needs to rework the strategy of counting from the beginning. For example, a student using a rudimentary “counting on” strategy when adding 3+5 would have to count three times. First, they count three objects, then five objects and finally start from the beginning to re-count all eight items. In order for the child to discontinue the use of this