# The effects of graphic organizers on solving linear equations and inequalities

T ABLE OF CONTENTS DEDICATION iii

ACKNOWLEDGEMENTS iv

ABSTRACT vi

LIST OF TABLES x

LIST OF FIGURES xi

CHAPTER 1 1

Background of the Problem 1

Students at Risk for Failure in Mathematics 2

Need for the Study 4

Statement of the Problem 5

Purpose of the Study 5

Research Questions 6

Definition of Terms 7

CHAPTER 2 9

Cognitive Load Theory 10

Schema Theory 13

Affordance Theory 14

Recommended Practices for Teaching Mathematics 16

Literature Review in Terms of Theory and Practice 18

Summary 44

CHAPTER 3 45

Setting 45

Participants 46

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J ustification of Research Design 52

Independent Variables 54

Dependent Variables 59

Data Collection and Scoring 61

Inter-scorers Reliability 62

Data Analysis 63

Fidelity of Implementation and Inter-observer Reliability 64

Instructional Routine 66

CHAPTER 4 70

Results of each Research Question 71

Inter-observer and Inter-rater Reliability 81

CHAPTER 5 83

Discussion 83

Conclusions 83

Difference between Students with Disabilities and General Education Students 88

Relation of Findings to Theory and Previous Research 89

Limitations 90

Directions for Future Research 92

Summary 93

APPENDICES 94

Appendix A: Graphic Organizer A 94

Appendix B: Graphic Organizer B 95

Appendix C: Pretest 1 of Linear Equations and Inequalities 96

Appendix D: Pretest 2 of Linear Equations and Inequalities 97

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A ppendix E: Posttest of Linear Equations and Inequalities 98

Appendix F: Far Generalization Test Items 99

Appendix G: Student Survey for Experimental Groups 101

Appendix H: Student Survey for Control Group 102

Appendix I: Three Point Scoring Rubric for Test of Linear Equations 103

Appendix J: Fidelity of Implementation Measure 104

Appendix K: Example Lesson Plan 106

Appendix M: Institutional Review Board Documents 109

REFERENCES 117

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L ist of Tables Table 1 Summary of Research in Current Review

25

Table 2 Population of Students with Disabilities

45

Table 3 Scale Score Ranges and Corresponding Performance Levels

47

Table 4 Characteristics of Participant Groups

48

Table 5 Student Demographics for Experimental Group E-1

49

Table 6 Student Demographics for Experimental Group E-2

50

Table 7 Student Demographics for Control Group

51

Table 8 Experimental Group E-1 Pretest, Posttest, and Percent Relative Gain

72

Table 9 Experimental Group E-2 Pretest, Posttest, and Percent Relative Gain

73

Table 10 Control Group Pretest, Posttest, and Percent Relative Gain

74

Table 11 Means and Standard Deviations of Relative Gains and Far Generalization Test by Disability Status

79

Table 12 Inter-observer Reliability Coefficients for Fidelity of Implementation

81

Table 13 Inter-rater Reliability Coefficients and Percent Agreement 82

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L ist of Figures

Figure 1 Research Design for non-equivalent groups

54

Figure 2 Linear equation solved using graphic organizer A

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Figure 3 Linear equation solved using graphic organizer B

59

Figure 4 Description of sequential lesson plans

68

Figure 5 Scores earned on pretest and posttest by students in control group

75

Figure 6 Scores earned on pretest and posttest by students in experimental group E-1

76

Figure 7 Scores earned on pretest and posttest by students in experimental group E-2

77

Figure 8 Mean rating for student social validity survey

80

Figure 9 Pseudo graphic organizer used by student in control group 87

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C hapter 1 Exiting high school with successful completion of a comprehensive, challenging curriculum has been linked to increased opportunities in higher education and employment that may otherwise be out of reach (Dunn, Chambers, & Rabren, 2004; Foegen, 2008; Sergio & William, 2008). Included in this demanding curriculum is the expectation that students who graduate from high school will be mathematically literate and competent. The past decade has seen attempts at reform in mathematics and mathematics achievement as a nationwide priority (Goals 2000; National Council of Teachers of Mathematics, 1989, 2000), yet an alarming gap in mathematics achievement and literacy remains between American citizens as compared with to that of other nations (Mullis, Martin & Foy, 2008). As our economy becomes ever more global, proficiency in algebraic and geometric reasoning and problem solving is critical. As recently as two decades ago, high school students were able to opt out of demanding academic courses and instead focus on vocational or business courses. Despite the continuing need for some or all of these preparation courses, many states now require several years of academic studies, including algebra, in order to graduate. Twenty- one states require four years of challenging mathematics, including at least Algebra II (Achieve, 2009). By 2009, twenty-three states had adopted curriculum standards aligned to college and career-ready standards (Achieve, 2009). Additionally, the Common Core Standard Initiative, led by the National Governors’ Association of Best Practices, seeks to coordinate learning standards nationwide, building upon the strengths and lessons of current state standards. Students who are successful in high school Algebra I are more likely to attend college. Additionally, graduating high school with credit in Algebra II makes students twice as likely to graduate from college (National Mathematics Advisory Panel, 2008). Despite these college-ready graduation

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r equirements, there is a growing need for remedial math instruction in basic algebra skills at the post-secondary level (National Center for Educational Statistics, 2003), indicating that even for students graduating and going on to college, proficiency in algebra remains a concern. Students At-risk for Failure in Mathematics With the passage of No Child Left Behind (NCLB, 2002), accountability in mathematics and reading became a priority. Once excluded from accountability measures, students with disabilities were now expected to have access to the general curriculum, as well as demonstrate proficiency in rigorous high-stakes exit exams. These exams can present a challenge to many students, including those who have historically done well in school. To those students who have intellectual and emotional difficulties, the task can be intimidating and demoralizing. In 2007, 41% more students with disabilities scored below the basic level of performance in mathematics than students without disabilities (Maccini, Strickland, Gagnon, & Malmgren, 2008). Many s tudents with learning disabilities and emotional disabilities have demonstrated difficulties in hi gher-level mathematics, including algebraic problem solving (Maccini, McNaughton, & Ruhl, 1999). Additionally, students with disabilities may exhibit difficulty with self-regulation skills during problem solving, deficits in computational skills, and problems with higher-level mathematics that require reasoning skills (Gagnon & Maccini, 2001). These students often struggle when organizing and processing information and tend to commit errors when attempting to solve problems involving multiple procedures (Maccini, Mulcahy, & Wilson, 2007). Data from the National Center for Educational Statistics (NCES) indicate current trends in grades four and eight suggest improvement in math scores at the elementary level, yet accountability reports, including that of the National Assessment of Educational Progress

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( NAEP), indicate a decline in mathematics achievement overall (NCES, 2010). Assessments at the eighth grade level report 32% of students scoring at or above the proficient level, yet by the end of grade 12, only 23% of students remain at or above the level of proficiency (National Mathematics Advisory Panel, 2008). New York State requires students to pass six exams, including one in either Algebra or Geometry, in order to earn a Regent’s diploma. Although 84% of general education students in New York State earned a Regent’s diploma in 2008 by demonstrating proficiency in four main content areas, only 48% of students with disabilities were able to graduate with the same credentials. A considerable amount of research has been conducted in the area of mathematics learning at the elementary and middle school levels. In 2001, the National Research Council convened a panel of experts to synthesize the research in order to provide recommendations to guide instruction, curriculum development and improve student learning. Like the recommendation made by the NCTM (1989, 2000), the report suggested using more open-ended problems throughout instruction. The report added that there is a need for more teacher- mediated instruction and increased explicit instruction, a direct contradiction of the proposed practices put forth by the NCTM, and argued for a blend of both approaches. Although the National Research Council syntheses failed to include any studies that focused on students who were having difficulties in math, nor did they address how instructional methods are best modified to meet the needs of diverse populations, they did indicate that the suggested practices would likely benefit students who were struggling in math. This research study using graphic organizers to guide students in solving linear equations is based on the integration of three distinct, but interrelated cognitive learning theories: Cognitive Load Theory, Schema Theory, and Affordance Theory. A common characteristic of

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s tudents demonstrating a weakness or disability in math centers on the inability to solve mathematical problems with multiple steps (Pedrotty, Bryant, & Hammill, 2000). Pedrotty and colleagues (2000) also noted students impulsively jumped into problems and performed various operations with no set procedure of determining the correct operation. Pedrotty and colleagues found students did not know where to begin their calculations and failed to recognize operant symbols. The graphic organizer in the current study was designed specifically for student interaction and affords the students both a starting point and a way to determine the correct operation. Unlike most graphic organizers that assist students in comprehending content by organizing and categorizing units of information, this graphic organizer guides the students to perform specific mathematical procedures in order to attain a goal. Affordances in the design of the graphic organizer lead to a progression of specific actions. Need for the Study Many students with learning disabilities and emotional disabilities have dem onstrated difficulties in higher level mathematics, including algebraic problem solving (Maccini, McNaughton, & Ruhl, 1999). Additionally, students with disabilities may exhibit difficulty with self-regulation skills during problem solving, deficits in computational skills, and problems with higher level mathematics that require reasoning skills (Gagnon & Maccini, 2001). These students often struggle organizing and processing information and tend to commit errors when attempting to solve problems involving multiple procedures (Maccini, Mulcahy, & Wilson, 2007). In their analysis of empirical research on teaching mathematics to low-achieving students, Baker, Gersten, & Lee (2002) identified three practices having significant effect on student achievement in math: (a) progress monitoring and reporting to students, teachers, and parents, (b) peer-assisted learning, and (c) direct/explicit instruction. Only three of the 15 studies

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i n the analysis by Baker, Gersten, & Lee (2002) included students with disabilities at the secondary level, and of those only 16% included topics in algebra or geometry, despite the National Advisory Panel’s Final Report listing linear equations as the second major topic recommended to be consistently included in all school algebra curriculum (National Mathematics Advisory Panel, 2008). Statement of the Problem Although numerous studies have been conducted to determine the most effective way to teach solving linear equations to typically achieving students, (Jimenez, Browder, & Courtade, 2008; Pirie & Martin, 1997; Rojano & Martinez, 2009), few studies examine higher order math skills for students with disabilities or students at risk for failure. Many of the studies conducted thus far involve computational mathematics and lower level math skills such as fractions, and simple word problems. It is critical to identify interventions that will better enable students with disabilities or at risk for failure in mathematics to achieve at higher levels. Purpose of the Study Research has clearly shown the benefits of using direct and explicit instruction to teach complex mathematics concepts. Additionally, strategy instruction in the form of graphic organizers and mnemonic devices has proven effective for students with disabilities and low- achieving students. Indeed, analyses of effect sizes from various studies indicate a combination of both approaches to be the most beneficial. The purpose of this study was to determine the effect of using direct instruction (DI) and strategy instruction (SI) with a topic specific graphic organizer to teach students who are at risk for failure in mathematics and students with disabilities how to solve linear equations and inequalities.

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R esearch Questions The following research questions were addressed in this study: 1. What effects will combined direct/strategy instruction have on the performance of at-risk students and students with disabilities on solving equations and inequalities? 2. What effects will combined direct/strategy instruction which incorporates a graphic organizer have on the performance of at-risk students and students with disabilities on solving equations and inequalities? 3. What are the differences between the performance of students who use two different graphic organizers to solve linear equations and inequalities? 4. What are the differences between students who were taught to solve linear equations and inequalities with direct/strategy instruction versus those who were taught with direct/strategy instruction using the graphic organizers? 5. What will be the effect of instruction on students’ ability to transfer skills to novel problems found on a standardized state math assessment? 6. How will students rate the social validity of using the graphic organizers to solve linear equations and inequalities? Limitations, Assumptions, and Design Controls This research endeavor took place in a real school with typical teachers and students. Due to the lack of artificial constraints, often a main component of purely empirical research, the findings should be authentic and generalizable to similar settings. Having said this, the authentic setting also provides realistic concerns and constraints. The major design limitation is the possibility of having to exclude some of the student participants due to attendance issues.

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A ttendance at the participating school has historically been of great concern. Some students were not present for some or all of the instruction and consequently were excluded in the final analysis. Further possible limitations will be discussed in chapter three and again in the chapter five. Definition of Key Terms Included throughout this dissertation is the discussion and presentation of data and literature related to this study. The following list includes key terms and their definitions. Student with disabilities: A child evaluated in accordance with §§300.304 through 300.311 as having mental retardation, a hearing impairment (including deafness), a speech or language impairment, a visual impairment (including blindness), a serious emotional disturbance (referred to in this part as “emotional disturbance”), an orthopedic impairment, autism, traumatic brain injury, and other health impairment, a specific learning disability, deaf-blindness, or multiple disabilities, and who, by reason thereof, needs special education and related services (NICHCY, 2010). At-Risk Students: Students who have educational needs not associated with a disability and who have one or more risk factors including poor performance in school, low economic- status, member of minority group, poor school attendance, attending school cited for failure to make Adequate Yearly Progress as measured on the state and federal school report card. Direct instruction: A systematic, explicit form of instruction that includes faced-paced drills, repetition, and includes the following components: Clear presentation of objective, teacher modeling, guided practice, independent practice, and corrective feedback Strategy instruction: Teaching and using specific learning strategies to enable students to learn and retrieve information and includes mnemonics, verbalization of steps (think-alouds).

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Graphic Organizer: Graphic organizers are communication devices that show the or ganization or structure of concepts as well as relationships between concepts.

Inclusion Classroom: A classroom that consists of both disabled and non-disabled students being taught the same curriculum by a general education teacher in a general education setting.

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C hapter 2 The No Child Left Behind Act of 2002, (NCLB, 2002) required all public schools to assess the mathematics and reading skills of all students in grades three through eight . Additionally, schools were required to implement research-based instructional programs. As a result of reform efforts including the re-authorization of the Individuals with Disabilities Education Act and NCLB, teachers are increasingly seeking out methods to reconcile instructional methods that challenge general education students and at the same time meet the needs of students with disabilities (Karp & Voltz, 2000). Collectively, these legislative mandates informed the current practices which include increased inclusionary education at the secondary level as well as requirements of all students, including those with disabilities, to participate in high stakes assessment and exit exams. Because of the difficulties students with learning disabilities and low-achieving students often exhibit in mathematics, combined with the rigorous curricular demands and high stakes testing, it is imperative to document teaching and learning strategies that will better enable students with disabilities to meet these higher standards with success. An instructional intervention that may have merit is the use of graphic organizers to assist students in solving higher level mathematics problems. For decades, researchers have examined the effectiveness of graphic organizers in assisting students with content-area knowledge, reading comprehension and written expression (Doyle, 1999; Gaustello, Beasley, & Sinatra, 2000; Horton, Lovitt, & Bergerud, 1990). To date, little research has examined the efficacy of utilizing similar principles of graphic organizers applied to solving higher-level secondary mathematics, such as linear equations and inequalities.

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To provide a framework for the proposed study, this chapter will present (a) the theoretical framework on which the intervention is grounded, (b) the recommended practices in general education and special education, and (c) the research findings related to math interventions for at-risk students. Theoretical Framework Instructional design and delivery, regardless of the content, should be shaped by the theoretical assumptions regarding cognition and learning. It is from this conception that the theoretical structure of this research arises. Three main learning theories are proposed as the scaffolds for this research endeavor: Cognitive Load Theory, Schema Theory, and Affordance Theory. These three theories are especially relevant to the research questions being asked in this study. The study seeks to determine the effectiveness of two distinct graphic organizers in assisting students to learn how to solve linear equations. The manner by which the information is presented to the learner can be manipulated to optimize the working memory of the learner, directing the learner’s attention to relevant stimuli, and develop automaticity and schema development. Cognitive Load Theory Cognitive load theory (CLT) is concerned with interactions of information processing structures in the working and long-term memory with the cognitive design of the human brain. Namely, the instructional implications of CLT are of great importance, especially when considered in light of cognitive challenges of students with disabilities. In general, cognitive load is the amount of mental resources necessary to process a given problem (Adcock, 2000). CLT theorists have identified two sources of cognitive load as intrinsic and extraneous cognitive load.

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A third category of cognitive load described by Ayers, (2006) includes germane cognitive load. Intrinsic cognitive load refers to the level of complexity of a task as measured by the number of interacting elements (Sweller, 1994). If elements of a task can be learned in isolation, there is less interactivity, and consequently, a lower intrinsic cognitive load. Extraneous and germane cognitive load, in contrast, is the amount of interaction of elements afforded by the instructional design of materials. Extraneous cognitive load may be taxed if excessive elements require more resources in the working memory, thereby inhibiting effective learning. Theoretically, working memory can only deal with limited amounts of information at a given time. Cognitive load theorists stipulate the typical person can work with no more than seven mental elements or cognitive units at any given time (Sweller, van Merrienboer, & Paas, (1988). If those elements are novel to the learner, the number can be as low as two or three (Paas, Renkl & Sweller, 2003). Pass et al. distinguish between individual cognitive units in the working, or short-term memory, from schemas that are stored in the long-term memory. As students gain understanding of conceptual units, they are organized into a schema, or larger picture, which is then linked to existing knowledge through relationships and generalities. The germane cognitive load, resources required for schema development, is related to effective instructional techniques that lessen the extraneous load, and as such, is closely tied with leaner characteristics. Intrinsic and extraneous constructs are believed to be additive in nature and can have a negative influence on working memory. Intrinsic load is determined by the material’s level of element interactivity (Paas, Renkl, & Sweller, 2003), and cannot be modified by instructional design. For example, performing an operation on any component of an equation has consequences for the entire equation, resulting in the need to attend to several elements simultaneously (high element interactivity). This is true regardless of the type of instruction. Extraneous load is that is

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i mposed by the instructional format and can have a negative effect on learning. Consider the example of a table used to represent statistical findings from a school-wide survey, with corresponding labels and keys physically separated from the table. To begin to understand the table and formulate implications, a student must first integrate the textual labels with the corresponding table components. According to cognitive load theorists, if intrinsic load is low, then a high extraneous load likely would not have an effect on learning. Conversely, the combination of high element interactivity and high extraneous load could potentially overload the working memory and negatively affect learning. Germane cognitive load, on the other hand, is concerned with effective instructional design that contributes to schema construction and automation. The cognitive load needed to solve a simple step linear equation such as 2x = 6 may seem miniscule at first glance. Most students have likely learned a general algorithm for solving this and similar problems, namely applying the inverse property to maintain equivalence and isolate the variable. But upon a deeper inspection of the conceptual knowledge related to solving the problem, the chain of cognitive units consists of several arithmetical operations and equalities (the intrinsic cognitive load), with more than one way to show them. Extraneous cognitive load for this problem is associated with the manner in which it is presented. For example,using a geometric approach to solving this equation would require learners to transform 2x = 6 into y = 2x and y = 6. Students must then be able to construct a graph and identify the x coordinate of the intersection of the lines as the solution. This method may help deepen the conceptual understanding of the linear equation, but the cognitive load requires many more resources than the arithmetic model (Tossavainen, 2009).

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The principals of cognitive load theory can be applied to instructional design to affect how written, spoken, and diagrammatic information can best be presented in order to increase learning. Research in the area of cognitive load theory is vibrant and preliminary studies indicated the significance of accounting for student characteristics, especially prior knowledge, in manipulating extraneous cognitive load in instruction. For at risk students and students with disabilities, decreasing element interactivity in conjunction with increasing germane cognitive load is likely to lead to conditions under which student learning will occur. Schema Theory A schema is a knowledge structure similar to a framework with which to consolidate and assimilate new learning into existing understandings. Schemata provide the context with which new experiences are interpreted and are the sum of a person’s knowledge (Winn & Snider, 1996). According to Dye (2000), the use of graphic organizers as a learning tool has its roots in schema theory. Graphic organizers provide a link for the learner to create a connection between new concepts and existing knowledge. Strategies that exploit schema theory include sequencing instruction from simple to advanced, providing pre-instruction through advanced organizers, and using brainstorming and mapping. Schema-based instruction has been shown to support word problem solving skills in students with learning disabilities and students at risk for failure in mathematics (Hutchinson, 1993). A key feature of schema-based instruction is that it uses diagrams to note important information and highlight key relationships among ideas (Jitendra, DiPipi, & Perron-Jones, 2002) In the current study, the teacher used the graphic organizer to focus students’ attention to the mathematical structure of the problem as a function of successive operations. As students gained domain specific knowledge and competence of simple to complex equations, they added

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t o their existing schemas. Because this intervention was conducted at the end of grade nine, the students already had a schema for the concept of equations, including operations, symbols, equality, and balance. This particular strategic intervention was designed to assist students in creating and manipulating partial schemas, then combining those with their existing schema. For example, in the previous equation, 2x = 6, students would first be directed to focus on the term ‘2x’. The concept of two as the multiplying coefficient was taught to mastery, with many repetitions and examples. Students with little schema acquisition have a greater difficulty in learning complex skills due to the fact of the high cognitive demand of the task (Ayres, 2006). Students developed automaticity and understanding of transposing elements of the equation, they were able to incorporate them into their current understandings. As problems progressed in difficulty, such as 2x + 5 = 11, the student incorporated the additional step into their existing schema. To relate this to cognitive load theory, as students are more and more proficient at solving specific types of equations, they can combine those sub skills into one schema. When learning more complex mathematical concepts or equations, the new, richer schema will be considered as a ‘whole’ unit of information. The germane load then is reduced and students will be able to more effectively use their cognitive resources. Affordance/Perceptual Learning Theory Affordance theory (Gibson, 1979) is an attempt to explain human perception and knowledge in terms not only of experience, but also of expectations. Affordances are described as inherent uses of the artifacts that humans interact with in the environment and are typically considered in terms of an ecological perspective. The design of an affordance can offer clues on how it may or may not be used. Examples of affordances are a door to be opened, a button to be