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The hands-on equations system and its effect on students' academic performances and retention levels

ProQuest Dissertations and Theses, 2011
Dissertation
Author: Janice M Brown
Abstract:
Educators seek out alternative ways to educate children in this technology-driven society. Children learning styles range from visual learners to tactile learners. Dr. Henry Borenson, the creator of the Hands-On Equations (HOE) module, developed a hands-on manipulative that can be utilized to teach children how to solve linear equations. The purpose of this research is to determine the relationship between the HOE module and ninth-grade students' academic performances and retention levels. The samples for this research are students from a senior high school in rural South Carolina. A quantitative research methodology was utilized to compare the variables. The results of the research indicate that there is not a statistical significant relationship between the variables; however, there does exist a positive correlation between the variables. The HOE module is a hands-on manipulative that offers teachers an alternative approach to teaching children how to solve linear equations.

v Table of Contents Acknowledgments iv List of Tables viii List of Figures ix CHAPTER 1. INTRODUCTION 1 Introduction to the Problem 1 Background of the Study 2 Statement of the Problem 6 Purpose of the Study 6 Rationale 7 Hypotheses 8 Significance of the Study 10 Definition of Terms 12 Assumptions 13 Limitations 14 Nature of the Study 15 Organization of the Remainder of the Study 16 CHAPTER 2. LITERATURE REVIEW 18 Introduction 18 Constructivism Theory 20 Technology as a Learning Enhancer 22 Using Hand Manipulatives to Increase Academic Performance 27

vi Hands-On Equations Learning System and Its Effect on Student Learning 29 No Child Left Behind and Its Effect on Student Learning 30 No Child Left Behind Act History 31 Standardized Assessments and Their Effect on Student Learning 32 Advantages of Standardized Assessments 32 Disadvantages of Standardized Assessments 33 Accountability 34 Best Practices in Teaching Mathematics 40 Teacher-Centered Instruction Versus Student-Centered Instruction 44 Summary 47 CHAPTER 3. METHODOLOGY 51 Introduction 51 Statement of the Problem 51 Hypotheses 52 Research Methodology 53 Research Design 54 Population and Sampling Procedure 55 Instrumentation 56 Validity 58 Reliability 59 Data Collection Procedures 61 Data Analysis Procedures 62

vii Ethical Considerations 64 Summary 65 CHAPTER 4. DATA COLLECTION AND ANALYSIS 67 Introduction 67 Descriptive Data 70 Data Analysis 70 Results 72 Summary 100 CHAPTER 5. RESULTS, CONCLUSIONS, RECOMMENDATIONS 104 Introduction 104 Summary of the Study 105 Summary of Findings and Conclusion 108 Recommendations 112 Implications 116 REFERENCES 119 APPENDIX A. HANDS-ON EQUATIONS PRETEST 124 APPENDIX B. HANDS-ON EQUATIONS POSTTEST 126 APPENDIX C. HANDS-ON EQUATIONS RETENTION LEVEL TEST 128

viii List of Tables Table 1. Descriptive Data for Student Participants 71 Table 2. Regular Education Students’ Descriptive Summarization 75 Table 3. Regular Education Student Nonparticipants’ Pretest, Posttest, and Retention Data 77 Table 4. Regular Education Students’ Posttest/Retention Descriptive Summarization 79 Table 5. Special Education Students’ Pretest, Posttest, and Retention Level Data 81 Table 6. Special Education Nonparticipants’ Pretest, Posttest, and Retention Level 83 Table 7. Special Education Students’ Descriptive Summarization Posttest/Retention Test 85 Table 8. Gifted and Talented Students’ Pretest, Posttest, and Retention Level Data 88 Table 9. Nonparticipant Gifted and Talented Students’ Pretest, Posttest, and Retention Data 90 Table 10. Honor Students’ Pretest, Posttest, and Retention Data 94 Table 11. Nonparticipant Honor Students’ Pretest, Posttest, and Retention Data 96 Table 12. Honor Students’ Descriptive Summarization for Posttest/Retention Test 98

ix List of Figures Figure 1. Regular education students’ pretest/posttest 76 Figure 2. Regular education participants’/nonparticipants’ pretest mean and posttest mean 78 Figure 3. Regular education students’ posttest/retention test scores 79 Figure 4. Regular education participants’/nonparticipants’ posttest and retention test means 80 Figure 5. Special education pretest/posttest scores 82 Figure 6. Special education participants’/nonparticipants’ pretest and posttest means 84 Figure 7. Special education posttest/retention test scores 86 Figure 8. Special education participants’/nonparticipants’ posttest and retention test means 87 Figure 9. Gifted and talented pretest/posttest scores 89 Figure 10. Gifted and talented pretest and posttest level scores for participants and nonparticipants 90 Figure 11. Gifted and talented posttest/retention level scores 92 Figure 12. Gifted and talented participants’ and nonparticipants’ posttest/retention level means 93 Figure 13. Honor students’ pretest/posttest 95 Figure 14. Honor student participants and nonparticipants pretest/posttest 96 Figure 15. Honor students’ posttest/retention test scores 98 Figure 16. Honor student participants’ and nonparticipants’ posttest/retention level test means 99

1

CHAPTER 1. INTRODUCTION

Introduction to the Problem Educators have explored the different avenues of what content to teach and what instructional strategies should be implemented to teach the content. Educators faced the challenge of educating the students of the 21st century. Educators noticed a downwar d trend in the mathematical knowledge of American students as measured by standardized assessments. Since the establishment of the No Child Left Behind Act of 2001 (NCLB) , educators searched for alternative ways to teach all groups of individuals math conc epts, especially abstract concepts. As standards increased, expectations increased. Educators have tested new methodologies to determine which math innovations result in higher academic performances and retention. The Hands-On Equations (HOE) module is a program that uses a hands- on approach to present algebraic concepts to grade school and middle school students

(Barber & Borenson, 2008). Dr. Henry Borenson (as cited in Barber & Borenson, 2006) , the inventor of the HOE module, developed a system of instructions that uses pawns and number cubes to enable grade school students to be successful with basic algebra. This program was designed to teach children in Grades 3–9 to solve equations using their thinking and understanding of basic principles, not to memorize the steps. The intent of this research, as it relates to hands-on manipulatives and mathematics achievement, was

2 to determine the relationship between Borenson’s HOE learning m odule and the academic performance and retention level of rural ninth grade rs with regards to solving linear equations. Ultimately, this research provided educators with valuable information on whether the HOE module helped ninth graders learn how to solve complex linear equations problem. If this module leads to an improvement in academic performances concerning linear equations, then this module can also lead to an academic improvement in other mathematical concepts. In conclusion, this research provided concise ideas on techniques that educators can utilize in the classroom to help their students reach their maximum potential, improve their academic performances, employ their critical thinking skills, and become problem solvers.

Background of the Study In an attempt to close the achievement gaps between Americans and other countries, educators in the United States have implemented new instructional strategies that are utilized in other countries. Allen (2007) concluded that, ― based on research from several countries, manipulative materials in teaching mathematics to students hol d the promise that manipulatives will help students understand the material being taught‖ (p. 7). Globally, as compared to China, India, and other countries, American children have fallen behind, especially in the subject areas of English language arts (ELA) and mathematics (Education Alliance, 2006). Educators have discussed two different aspects of the education: (a) what curriculum is being taught, and (b) how the curriculum is being taught. As students’ interests changed, the way students receive and process information

3 changed. Educators searched for alternative teaching methods to help students reach their maximum potential. The reformation of mathematics can be traced back to the mid-1980s. The reform movement in mathematics was a response to the failure of traditional teaching methods.

The reform movement was a response to the failure of traditional teaching methods, and the emergence of new approaches to the scientific study of how mathematics is learned

(Education Alliance, 2006). Under the mathematical reform move ment, educators scrutinized the curriculum that is currently being taught. This new concept was called the standards-based approach. The standards-based approach, developed by the National Council of Teachers of Mathematics (NCTM, 2000), outlined six principles that should be used to guide instruction: relation to equity, curriculum, teaching, learning, assessment, and technology. The relation to equity is having access to a high quality education, regardless of location or race. This document also noted exactly what content students should know and be able to do. As a result of the mathematics reformation, the new standards- based teaching approach has been created. Mathematics reform documented six basic assumptions about teaching and schooling practices (Education Alliance, 2006): 1. All students must have an opportunity to learn new mathematics. 2. All students have the capacity to learn more mathematics than we traditionally assured. 3. New applications and changes in technology have changed the in structional importance of some mathematical concepts.

4 4. New instructional environments can be created through the use of technological tools. Meaningful mathematics learning is a product of purposeful engagement and interaction that builds on prior experience. Since the establishment of NCLB , the need for new instructional strategies is of significance. NCLB called for increased standardized assessments, improved academic performances, especially in ELA and mathematics, and more teacher accountability ( U.S. Government Accountability Office [GAO], 2005). Standardized assessments are measuring tools for schools to mark student’ s progress, teacher quality, and school quality. According to Honey, Culp, and Spielvogel (2005), ― Efforts to integrate technology into the schools and classroom practices must not only acknowledge but also provide evidence that technology assists in meeting these accountability demands‖ (p. 11). Educators linked teaching methods to students’ academic performances. Throughout the 1980s and early 1990s, teachers used the basic lecture and demonstration technique to teach mathematics (Education Alliance, 2006). The old mathematics content focused on basic skills and basic mathematical comprehension. In an effort to encompass both the new and old mathematical curriculum, the NCTM developed six math strands in 2000: Algebra, Numbers and Operations, Data Analysis, Probability, Geometry, and Measurement. As evident by past national assessments, students’ performances in mathematics are declining (Education Alliance, 2006). In a more recent study by the Institute of Educational Science (IES, 2008) and Texas Instruments (TI), research suggested effective instructional strategies and their

5 impact on students’ mathematical performances. In a compilation of the nation’ s students’ math scores, the IES indicated that 30% of eighth graders are proficient. The IES documented further the need for new instructional strategies as evident from the data in 2007. The IES concluded the following: 1. The average math score in 2007 was higher than any other previous year. 2. Score point increase from 1990 to 2007 ranged from a 13- point gain in numbers and operations to a 24-point gain in algebra. 3. All of the 38 states that participated in both 1990 and 2007 showed incr eases in average mathematics scores. The Education Alliance (2006) stated, ― The poor performance of U.S. students in math can be traced to the method used to teach math at the elementary level. The focus is on specific problems and not on foundations neces sary for understanding higher level math‖ (p. 5). In an effort to increase scores, educators relied on technology and hands- on manipulatives. TI conducted research regarding technology usage in classrooms. TI (n.d.) has consistently conducted research rega rding the usage of graphing calculators in the classroom. TI suggested that effective teaching with graphing calculators can help students develop a better understanding of mathematical concepts, use higher level approaches to solving math problems, and score higher on performances measures. There is little research regarding hands-on manipulatives and their effect on middle school students and beyond. However, Barber and Borenson have conducted

research on the effects of using the HOE module to teach solvi ng linear equations to elementary and middle school age students. Research has already been conducted on fourth and fifth graders in rural areas, sixth graders in regular education classes, fourth-

6 and fifth-grade honor students, and fourth-, fifth-, and sixth- grade special education students (Barber & Borenson, 2006). Research has been conducted on using technology in the math classes to teach math skills to early learners. So me technology equipment included using TI instruments, promethean boards, and other handheld devices.

Statement of the Problem It is not known how or to what extent the HOE learning module affects ninth

graders’ academic performances and retention levels with regards to solving linear equations, in comparison to other instructional methods. One of the main problems in the education world wa s to determine which type of instructional methods work best for the students of the 21st century. This problem existed due to the fact that students learn in different capacities, and the needs of one student differ from the needs of others. The ultimate goal of the education society wa s to determine which instructional methods will help students reach mastery levels, especially in the content areas of m athematics and ELA. The problem of this research tested whether the HOE module had a significant impact on ninth graders’ academic performances and retention levels with regards to solving linear equations, in comparison to other instructional methods. Children have changed, and society seeks alternative ways to educate them. Children are no longer geared towards learning in the old traditional form of using pencil and paper.

Purpose of the Study The purpose of this research was to determine the relationship between the HOE

learning module and its effect on rural ninth graders’ academic performances and

7 retention level with regards to solving linear equations , in comparison to other instructional methods. The problem of this research emphasized the fact that there wa s a lack of literature regarding the extent the HOE learning module affects ninth graders’

academic performance with regards to solving linear equation. The researcher examined

two components more extensively. The first component of this research was designed to measure whether or not the HOE module will have a significant impact on the ninth graders’ academic performances in regards to solving linear equations. This research investigated whether ninth- grade students are able to use the HOE module to achieve 7 0% mastery on solving linear equations with variables on one side, on both sides, and multisteps as measured by the first posttest. The second component of this research was designed to measure whether or not the HOE module had a significant impact on the ninth graders’ retention level in regards to solving linear equations. This research determined whether or not, after 1 week of not receiving instructions on solving linear equations, ninth graders remembered how to solve linear equations with 70% mastery level with the HOE module as measured by the retention level test.

Rationale According to national math standards, ninth-grade students are expected to solve linear equations both with and without manipulatives successfully by the end of thei r secondary education (NCTM, 2000). The researcher conducted a study that will answer whether ninth-grade middle school students can solve linear equations more accurately

8 with the HOE module. This study explored solving linear equations with variables on one side, variables on both sides, and solving multistep linear equations. In order to answer this question, a correlational study was done. The study was significant at a 0.05 significance level. A total of three tests were gi ven during the research: one pretest, one posttest, and one retention level test. The pretest was given to determine whether ninth- grade students already have experience with solving linear equations. If the students have already received lessons on how to solve linear equations, then this study serves no purpose. The posttest was given to determine whether after receiving instruction using the HOE

module students can solve linear equations with variables on one side, both sides, and multisteps, normally introduced in the eighth grade. The posttest was given after 1 week

of instruction. The retention level test was used to determine whether ninth- grade students can retain the information presented to them on solving linear equatio ns, and still maintain their score from the posttest. The retention level test was given 2 weeks after the pretest. The results of this study inferred that the HOE modules not only led to improved

academic performances, but also provided a technique that students will remember.

Hypotheses This research tested whether the HOE learning module had a significa nt impact on ninth graders’ academic performances and retention levels with regards to solvin g linear equations, in comparison to other instructional methods. In order for one to fully understand the research problem, the following research hypotheses were developed.

9 H1: There is a significant relationship between the Hands-On Equations module and ninth-grade regular education students’ academic performance and retention level regarding solving linear equations. H0: There is not a significant relationship between the Hands-On Equations

module and ninth-grade regular education students’ academic performance and retention level regarding solving linear equations. H2: There is a significant relationship between the Hands-On Equations module and ninth-grade special education students’ academic performance and retention level regarding solving linear equations. H0: There is not a significant relationship between the Hands-On Equations

module and ninth-grade special education students’ academic performance and rete ntion level regarding solving linear equations. H3: There is a significant relationship between the Hands-On Equations module and ninth-grade gifted and talented students’ academic performance and retention level

regarding solving linear equations. H0: There is not a significant relationship between the Hands-On Equations

module and ninth-grade gifted and talented students’ academic performance and retention level regarding solving linear equations. H4: There is a significant relationship between the Hands-On Equations module and ninth-grade honor students’ academic performance and retention level regarding solving linear equations.

10 H0: There is not a significant relationship between the Hands-On Equations

module and ninth-grade honor students’ academic performance and retention level

regarding solving linear equations.

Significance of the Study Since the establishment of NCLB, the need for ne w instructional strategies was significant. NCLB called for increased standardized assessments, improved academic performances, especially in ELA and m athematics, and more teacher accountability (GAO, 2005). This study was significant, because it provided details on which teaching strategies will help close the achievement gap between in dividuals. Additionally this study helped students retain factual and pertinent information at the middle and high school level. Children have changed, and society have searched for alternative ways to educate children. Children are no longer geared towards learning in the traditional form of using pencil and paper. The ultimate goal of the education society wa s to determine which instructional methods will help the students of today reach mastery level, especially in the content areas of mathematics and ELA. This study examined whether the HOE module

led to significant higher grades and mastery levels. The information in this study could aid administrators, teachers, school board members, and parent s in the development and implementation of innovations that promote and enhance student learning that ultimately led to an increase in students’ academic performances and increase students’ retention level.

11 Researchers blamed low math performances in the United States on teaching methodologies. According to Education Alliance (2006), ―The poor per formance of U.S. students in math can be traced to the method used to teach math at the elementary level.

The focus is on specific problems and not on foundations necessary for understanding higher level math‖ (p. 5). In an effort to increase scores, educators relied on technology and hands-on manipulatives. Many studies have been conducted regarding technology usage in the classroom and increased academic performances. For example, TI has conducted research on graphing calculators in the United Kingdom, Fr ance, and in the United States. There is a gap in research regarding hands-on manipulatives and their

effect on middle school students and beyond. This study helped developed and find techniques to incorporate in the classroom to help middle school student s increase their academic performances. Educators also faced the challenge of integrating t echniques in the class that helped students to retain knowledge not just memorize facts for a short moment. This research provided information on what kind of instru ctional techniques will help students to retain knowledge over a period of time. The findings of this study benefited the educational society, especially educators and stud ents at the middle school level and beyond. In conclusion, the results of this research opened up new avenues for educators to consider when developing instructional strategies to incorporate into the classroom to help students reach and exceed their maximum potential.

12 Definition of Terms There are a number of terms that are important to this study. As such the following terms were operationally defined. Blue pawns. The student game pieces that are used to represent the variable X

(Barber & Borenson, 2008). Flat laminated balance. A representation of a balance scale printed on paper, laminated for protection (Barber & Borenson, 2008). Hands-On Equations (HOE) module. A program that uses a hands- on approach to presenting algebraic concepts to grade school and middle school students ( Barber & Borenson, 2008). Honor students. Can be defined a s those students, who display exemplar academic performances and moral, along with high achievement (Stanlick, 2006). Linear equation. An equation in which the exponent of the variable is one (Larson, Boswell, Kanold, & Stiff, 2001). Red numbered cubes. The student game pieces used to represent the positive constant (Barber & Borenson, 2008). Retention test. U sed to determine the extent to which the students were able to solve equations using pictorial notation after 3 weeks of no HOE instruction ( Barber & Borenson, 2008). Scatterplot. A pictorial image of two sets of scores for participants on a graph. Scatter plots are used to plot data points on a horizontal and a vertical axis in the attempt to show how much one variable is affected by another (Creswell, 2005).

13 Teacher’s balance scale and game pieces. A stationary balance scale and game pieces used by the instructor in the front of the room to illustrate the equations ( Barber & Borenson, 2008). Variable. A letter used to represent a range of numbers, as it refers to the letter X , in Borenson’s HOE system. The variable can be changed in order to change the characteristics or value of an expression or equation (Larson et al., 2001).

Assumptions The following assumptions were presented in this study: 1. The ninth- grade students have no prior knowledge of how to solve linear equations. 2. The ninth-grade students have no attendance issues. 3. The class is organized and structured. 4. The teacher has prior knowledge of how to use the HOE module. 5. The ninth graders who took th e pretest will also take the posttest and the retention level test. 6. The ninth graders will receive no additional support from any other individual during the pretest or posttests. 7. The ninth-grade students have sufficient time to complete the tests. 8. The ninth-grade students did not retake any tests.

14 Limitations The following limitations were used in this research study: 1. This study is not intended to study the HOE instruction effect on teacher work satisfaction. 2. This study is not intended to compare student’ s attendance rates while using the HOE module. 3. This study is not intended to show students’ retention level after a 2- month period of time. 4. This study is not intended to apply to any grade level except for ninth grade. 5. This study is not to learn all char acteristics of students or teachers or to measure what they can or can not do in all math concepts. 6. This study is not intended to learn all things about the effect of th e HOE method of instruction on ninth-grade students. The primary interest is to learn about the effect on student’s ability to solve algebraic linear equations. 7. This study is not intended to suggest that the HOE module is better than using technology as an instructional method. 8. This study is not intended to replace textbooks usage. 9. This study is not intended to suggest using the HOE module as the only

standardized assessment. 10. This study is not intended to suggest the HOE module will change students’

interest regarding mathematics. 11. This study is not intended to replace any existing curriculum.

15 Nature of the Study This research study involved analyzing statistical data to determine the relationship between the variables, HOE module, rural ninth- grade academic performances, and rural ninth graders’ retention level. This study was analyzed using a correlational research study design. The samples in this study were rural ninth- grade students who were chosen using both the stratified sampling technique and the random simple sampling technique. The data for this research were collected using the rural ninth

graders’ pretest, posttest, and retention level test. The GraphPad software was used to

calculate and display the t-test results. This study was considered significant at the al pha level of 0.500. The research methodology utilized in this research study was the quantitative design. The actual design incorporated was the correlational research design. This methodology and design were utilized because the purpose of the research wa s to determine the effects of the HOE module on rural ninth-grade honor students, gifted and talented students, regular education students, and special education students in comparison to students who did not receive the HOE training. In order to determine how effective the HOE module was on ninth graders’ academic performanc es and retention level a statistical comparison was needed. This type of study can only be answered by using a quantitative study as reflected by past similar research studies. This type of study allowed the researcher to analyze how effective the HOE module is on ninth graders’

academic performances and retention level. The quantitative design allowed the researcher to visually recognize the relation that will be displayed on a scatterplot.

16 The samples for this research were ninth-grade rural special educa tion students, ninth-grade rural honor students, ninth-grade rural gifted and talented students, and ninth - grade rural regular education students. The research samples were chosen using a stratified sampling technique. Once the groups were chosen, then a simple r andom sampling technique was utilized to further simplify the population. The data for this research were gathered from observations, pretest, posttest, and retention level test. The HOE module has a positive effect on teaching third, fourth, fifth, and sixth graders on how to solve linear equations, a concept normally taught in ninth grade or higher, as denoted by past studies. The purpose of this research wa s to determine the relationship between the HOE learning module and its effect on rural ninth graders’

academic performance and retention level s with regards to solving linear equations, in comparison to other instructional methods.

Organization of the Remainder of the Study This research examined the effects of the HOE module on rural ninth graders’

academic performances and retention levels reg arding solving linear equations, in comparison to other instructional methods. Th e following linear equations that were explored are as follows: variables on one side and on both sides, and solving linear

equations with multisteps. This research studied special education students, regular education students, gifted and talented students, and honor students, in the ninth grade, over a 2-week period. The remainder of this study was divided up as follows. Cha pter 2 presents a literature review of the effects of hands- on manipulative and technology usage in the classroom on students’ academic performances and retention level. This chapter

17 summarized other similar research and their findings, conclusions, and im plications for further study. Chapter 3 discusses the research methodology a nd research design that was utilized to carry out this research. This chapter provided details of which research methodology is chosen and how it is selected. Chapter 3 discusses the res earch design, which incorporated the sampling techniques. Chapter 4 outlines the data collection techniques and data analysis. This chapter also explained how the data were collected and from whom. Chapter 5 provides a summary of the results, conclus ions, and recommendations. This chapter provides suggestions for further research, discusses

limitations, and provides suggestions for improvement. The estimated completion time for this research project was 2–4 months from the approval date of the proposal.

18

CHAPTER 2. LITERATURE REVIEW

Introduction Since the establishment of NCLB , educators seek out new alternatives, in contrast to the traditional paper and pencil technique, to increase academic performances in the classroom. The educational society is increasing educational demands and standards. As society changes into a technological society, students’ interests also change. Since the 1990s educators realize the growing trend of school dropouts and retention level. In 1990, the IES noted that only 52% of American students in eighth grade scored basic on the standardized assessment, 15% at the proficient level, and 2% at the advanced or above level. Educators realize that the United States is falling behind in education globally, especially in mathematics. Since the decline of standardized test scores educato rs closely look at how math is taught and what is taught. Teachers seek out methods to utilize in the classroom to help students reach their potential and meet states standardized assessments.

Full document contains 140 pages
Abstract: Educators seek out alternative ways to educate children in this technology-driven society. Children learning styles range from visual learners to tactile learners. Dr. Henry Borenson, the creator of the Hands-On Equations (HOE) module, developed a hands-on manipulative that can be utilized to teach children how to solve linear equations. The purpose of this research is to determine the relationship between the HOE module and ninth-grade students' academic performances and retention levels. The samples for this research are students from a senior high school in rural South Carolina. A quantitative research methodology was utilized to compare the variables. The results of the research indicate that there is not a statistical significant relationship between the variables; however, there does exist a positive correlation between the variables. The HOE module is a hands-on manipulative that offers teachers an alternative approach to teaching children how to solve linear equations.