# Elementary mathematics instruction: Traditional versus inquiry

v Table of Contents Acknowledgments iv List of Tables viii CHAPTER 1. INTRODUCTION 1 Introduction to the Problem 1 Background of the Study 2 Statement of the Problem 4 Purpose of the Study 5 Theoretical Framework 5 Hypothesis and Research Questions 6 Significance of the Study 7 Definition of Terms 9 Assumptions 10 Limitations 11 Nature of the Study 11 Organization of the Remainder of the Study 14 CHAPTER 2. LITERATURE REVIEW 15 Introduction 15 Constructivist Learning Theory 16 Historical Perspective of Constructivist Learning Theory 20 Characteristics of Traditional Instruction 23 The Achievement Gap 24 Math Reform Efforts and Impact on Mathematics Classroom 26

vi The National Council of Teachers of Mathematics Standards 31 Elementary Mathematics Instruction 35 Conclusion 39 CHAPTER 3. METHODOLOGY 42 Introduction 42 Hypothesis and Research Questions 42 Research Design 43 Qualitative Research 44 Quantitative Research 46 Setting of Study 48 Sampling Procedures 49 Instrumentation 51 Data Collection Procedures 55 Ethical Issues 58 Data Analysis Procedures 59 Summary 61 CHAPTER 4. DATA COLLECTION AND ANALYSIS 62 Introduction 62 Purpose of the Study 62 Presentation of Results 63 Qualitative Data Presentation 65 Quantitative Data Presentation 86 Analysis of Results 88

vii Summary 92 CHAPTER 5. RESULTS, CONCLUSIONS, AND RECOMMENDATIONS 93 Introduction 93 Summary of the Study 93 Summary of the Findings and Conclusions 96 Limitations 102 Recommendations 102 REFERENCES 108 APPENDIX A. TEACHER INTERVIEW PROTOCOL AND QUESTIONS 115 APPENDIX B. BIOGRAPHY OF EXPERT PANEL 116

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List of Tables Table 1 - Interviewed Teacher Years of Service and Certification 64

Table 2 - Teacher Response to Interview Question 1 on 78 Resource Choices

Table 3 - Teacher Response to Interview Question 2 on 79 Resource Choices

Table 4 - Teacher Response to Interview Question 3 on 80 Resource Choices

Table 5 - Teacher Response to Interview Question 1 on 81 Instructional Choices

Table 6 - Teacher Response to Interview Question 2 on 82 Instructional Choices

Table 7 - Teacher Response to Interview Question 4 on 83 Instructional Choices

Table 8 - Teacher Response to Interview Question 5 on 83 Instructional Choices

Table 9 - Key Themes from Interviews 84

Table 10 - Mean Scores of Traditional and Inquiry-based Instructional Groups on the 4Sight Benchmark Assessment 86

Table 11 - Comparison of Inquiry-based and Traditional Instructional Groups in the Open-ended Tasks 87

Table 12 - Open-ended, Multiple-choice, and Total Average Comparisons Between Traditional and Inquiry-based Instructional Groups 88

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CHAPTER 1. INTRODUCTION

Introduction to the Problem

There have been substantial changes in what is being taught and the way it is taught in mathematics in the last twenty or so years. The standards published by the National Council of Teachers of Mathematics (NCTM) have placed emphasis on more depth of understanding and less on basic skill knowledge and memorization of facts (NCTM, 2000). The standards have helped to define the difference between teaching mathematics and facilitating the learning of mathematics. Thus, NCTM continues to support and promote reform-based mathematics through their publications and teaching standards for mathematics. Inquiry-based instruction, also called reform mathematics, portrays the teacher as the facilitator of learning. The inquiry-based teacher plans experiences for students that enable them to construct meaning and make connections. Jamar and Pitts (2005) comment on the changing roles of math teachers and state, “math classes should be viewed as places where students rediscover and re-invent concepts and methods of solutions” (p. 128), which emphasizes the construction of knowledge. When students are challenged and actively engaged in the learning experience, student achievement rises dramatically. Students will learn and retain knowledge that has meaning to them or that they can connect with their own life experiences. If a teacher can provide students with a way to make these connections, students will experience greater success with mathematics.

2 Student achievement in mathematics is of great concern because of the level of accountability schools are held to with respect to federal and state educational policies. Inquiry-based instruction is a method used to meet these demands (Speziale, Reese, Landon, & Jesberg, 2007). However, many elementary teachers are reluctant to change instructional approaches for two reasons: their level of content knowledge and the belief that traditional instruction yields higher levels of achievement. It is possible that elementary math teachers would move toward inquiry-based instruction given concrete data that would support such a change. A comparison of 4Sight Benchmark Assessment (Success For All Foundation [SFAF], 2008) scores between students in an inquiry-based learning environment to those students in a traditional setting, within their own district, may provide the necessary evidence. This study classified elementary teachers in an urban school district as traditional, inquiry-based, or other with respect to primary strategy for the delivery of instruction in mathematics classes. Teachers’ perceptions of the impact of their choices in materials and instructional strategies on student achievement were explored. District-administered 4Sight Benchmark Assessment (SFAF, 2008) scores were compared for those students receiving traditional and inquiry-based instruction in an effort to determine if there is any statistically significant difference in achievement between these instructional approaches.

Background of the Study

The interest in mathematics within the kindergarten (K) through grade 12 educational systems in the United States has increased dramatically in the last 10 years and, as a result, so has the research. Many of the research studies in mathematics focus on

3 not only what to teach but how to teach it (Frykholm, 2004). One reason for the escalation of attention on curriculum and instruction in math is the publication of the Trends in International Math and Science Study (TIMSS) report and the No Child Left Behind (NCLB) act. These two events have highlighted the need for an increase in achievement in mathematics for students in the United States. The TIMSS report, very specific to mathematics, indicates that teachers in the United States do not use “rich mathematical problems that focus on concepts and connections” (Stigler & Hiebert, 2004, p.15). Instead, teachers use these types of problems to develop basic computational skills and procedures. Students are not exposed to multiple methods for solving problems, just to an algorithm used to solve that particular type of problem. The standards published by the National Council of Teachers of Mathematics (NCTM) also supports this notion by placing emphasis on more depth of understanding and less on basic skill knowledge and memorization of facts (NCTM, 2000), reinforcing the fact that memorization of algorithms is not conducive to students understanding of mathematics. When students acquire a deep level of understanding of mathematics, it is said that they have conceptual understanding (NCTM, 2000). A student displaying conceptual understanding will be able to explain why a concept is important, when to use it, comprehend the mathematics, and make connections (Kilpatrick, Swafford, & Findell, 2001). Students with conceptual understanding know mathematical algorithms because they have spent time developing them in relation to other mathematical ideas and background knowledge, not because they have memorized facts (Wu, 1999). Facts and methods to solve problems are not isolated but instead are connected to meaningful

4 mathematical understanding possessed by the student with a conceptual framework (Mathematical Sciences Education Board [MSED], 2007). This conceptual framework will allow students to make connections and see mathematics in a variety of contexts. One method of instruction that exposes students to experiences that create an opportunity for in-depth development of mathematical concepts and encourages the use of multiple means for solving problems is inquiry-based instruction (Wu, 1999). Inquiry- based instruction is a method where “teachers orchestrate a learning environment that allows students to develop a deeper understanding of content” (Speziale, Reese, Landon, & Jesberg, 2007, p. 11). A depth of understanding is necessary for students to retain and apply what they have learned. In addition, Speziale et al. indicates that teachers must possess a deep understanding of the math and the skill in pedagogy required to allow students to develop this depth.

Statement of the Problem

The effects of inquiry-based instruction on the academic achievement, as measured by the 4Sight Benchmark Assessment (SFAF, 2008), of elementary mathematics students are unknown. Many traditional elementary teachers are more comfortable with direct instruction in mathematics because it allows them more control over what is explicitly taught, the rate of instructional time per topic, and quantity of content covered (Richardson, 2003). In an effort to help teachers uncover the most effective type of mathematics instruction, 4Sight scores were compared between classrooms with traditional instruction and inquiry-based learning. The results of such a

5 comparison may assist teachers with the choice of the type of instruction necessary to maximize the achievement of their students.

Purpose of the Study

The purpose of this study was to explore teacher perceptions of the impact of their instructional and resource choices on student achievement in elementary mathematics. The study sought to determine what impact traditional and inquiry-based instruction have on the achievement of elementary students in mathematics through a comparison of 4Sight Benchmark Assessment (SFAF, 2008) scores. Furthermore, if a difference in achievement does exist, the study sought to identify the most significant area: basic skills and procedures or the application of learning through problem solving. The 4Sight Benchmark Assessment (SFAF, 2008) is administered to all elementary students three to four times each school year. It allows teachers to predict how well their students will do on the state assessments and it assures that district curriculum maps are being followed consistently throughout the district.

Theoretical Framework

The current reality of instruction in most elementary math classrooms is a teacher- centered environment where the teacher is viewed as a dispenser of knowledge. This type of instruction is based on the behaviorist principles of learning that were particularly popular in the 1950s and 1960s (Cuban, 2001; Ormrod, 2004). Behaviorist learning theory examines the observable behavior of the students and focuses on the acquisition of factual knowledge (Orlich et al, 2007). The cognitive processes are not the focus, and students enter each new learning experience with little or no prior knowledge. An

6 increase in brain research and conceptual knowledge development has highlighted the need to move away from this type of learning environment and toward a classroom that is more student-centered (Glen, 2002). Thus, it is inherently obvious why behaviorist learning principles are mostly contradictory to the NCTM (1991 & 2000), which has consistently supported instruction that engages students in mathematics with a focus on developing conceptual understanding. Constructivism is an educational philosophy that supports the importance of students developing conceptual understanding in mathematics through inquiry-based learning (Alsup, 2005; Brooks & Brooks, 1999). Inquiry-based learning asserts that students must first make sense of something by experimentation and then construct or reconstruct their knowledge system based on this new experience. Piaget emphasizes that children construct knowledge by making meaning or connections at their level to their world (Kuschner & Forman, 1998). Piaget’s work was the primary force driving the shift to constructivism as a central learning theory in mathematics from behaviorist learning theory. The shift to constructivist theory in mathematics education is often titled “mathematics reform.” It is described as a move from an emphasis on memorization of facts and algorithms to the development of an in-depth understanding of mathematical concepts (Widmaier, 2004).

Hypothesis and Research Questions

This study was designed to address a hypothesis and several questions that are related to, and possibly lead to, an understanding of the research problem.

7 Hypothesis The type of instruction chosen by teachers, based on a self-assessment categorization of traditional or inquiry, has a significant impact on the achievement of elementary students in mathematics. Research Question One What are the perceptions of elementary teachers regarding the impact of traditional versus inquiry-based instructional strategies on student achievement in mathematics? Research Question Two Is there a significant relationship between the teachers’ choices of specific instructional strategies and materials and their beliefs in choosing the most effective instructional approach?

Significance of the Study

The topic of instructional strategies in mathematics in the elementary setting is very important for many reasons. Elementary teachers struggle with mathematics instruction because their own conceptual understanding is often superficial (Baturo, 2004). It is for this reason that most elementary classrooms are very teacher-centered and provide students with ways to memorize material instead of allowing them to develop conceptual understanding. Many elementary teachers do not see the need for movement toward teaching for conceptual understanding because their own lack of content knowledge does not allow them to see the need for mathematical depth (Weiss, Banilower, McMahon, & Smith, 2001).

8 In many instances, elementary teachers do not recognize the need for continuously improving mathematics instruction because, from their perspective, teacher- centered instruction is successful. Inquiry-based learning fosters collaboration which increases interest for teachers in improving their own practices (Altun, & Buyukduman, 2007). It is through collaboration that teachers are exposed to a multitude of instructional strategies that offer them insight into how students learn. They are exposed to a variety of strategies that can be used to keep students motivated and engaged in order for conceptual understanding to develop. Finally, the elementary students with no conceptual understanding move on to middle and high school with no foundation for numbers; hence, they struggle with mathematics for many years thereafter. There is a great deal of research on the underachievement of American students, especially in high school, as compared to their European counterparts. One of the main differences is the lack of conceptual understanding students have of mathematics. Although the amount of research supporting inquiry-based instruction is overwhelming, teachers are not willing to alter their methods because of uncertainty of the achievement results of their students. If teachers have the opportunity to receive achievement data of students in their own district, comparing traditional and inquiry-based instructional approaches, then perhaps they will make informed decisions about the instruction they offer their students. This study is significant because it draws a comparison between student-centered and teacher-centered instruction that will possibly guide elementary teachers in making instructional decisions most appropriate for their students. It highlights the importance of instructional choices in mathematics and the impact on student achievement. Principals

9 and teacher leaders may be able to use this study to guide the professional development offered to teachers. In addition, improving elementary math instruction by focusing on conceptual development will assist students in becoming more confident and successful in mathematics throughout their education. Furthermore, in the research community, this study could showcase the need for further research with a variety of student populations with respect to mathematics instruction.

Definition of Terms

There are many terms that were used in this study that can be interpreted in multiple ways and thus need clarification. Algorithm. In mathematics, an algorithm refers to a sequence of mathematical calculations or steps that will lead to a solution or partial solution to a problem. Students can often memorize a sequence of steps without any understanding of what is happening mathematically. Conceptual understanding. In this study, the phrase refers to students’ depth of knowledge surrounding a mathematical concept. This means that students have developed a level of knowledge that allows them to understand the reason why something such as a formula or rule works, instead of memorizing an algorithm. Constructivist Theory. This is a theory that is defined as “a dynamic and interactive conception of learning in which all knowledge is constructed; it is a product of the cognitive acts of the individual” (Greene, 1995) that are based on prior knowledge and experiences.

10 Inquiry-based instruction. This is a type of student-centered instruction where students are given multiple means to gather and interpret information. The student makes connections that are personal and meaningful, thus achieving a deeper development of a concept. This process is accomplished through learning experiences that are student- centered and involve problem solving and questioning. Metacognition. This is the insight into one’s own process of learning. Student-centered instruction. This is an approach to instruction that involves students taking an active role in the learning process through the use of discovery, open- ended types of problems, writing, questioning, sharing and justifying with peers, and reflection. This list is not meant to be exhaustive but, instead, demonstrates some examples of student-centered instruction. Students tend to have more motivation, the level and amount of knowledge is more substantial, and it creates more interest in the content being taught. Teacher-centered instruction. In this instructional approach, teachers control what is taught, how it is taught and when it is taught. The mode of instruction is through lecture, where teachers transmit the knowledge to the students who are usually seated in rows facing the instructor (Cuban, 2006). The role of the student is primarily to receive knowledge and not to question or problem solve.

Assumptions

There are several assumptions related to this study. 1. The scores of elementary students on 4Sight Benchmark Assessment (SFAF, 2008) were used to demonstrate achievement, and it is assumed

11 that any growth occurring is a result of the instructional technique utilized. 2. It was also assumed that the student scores on these assessments accurately represent their level of mathematical knowledge. 3. In addition, it was assumed that the multiple-choice and open-ended questions accurately distinguish between skills and procedures, and problem-solving and reasoning abilities of students. 4. Teachers completed a survey about the type of techniques they use for instruction in math. It was assumed that the teachers completed the survey in a manner that accurately reflects the instructional approach that was utilized in their classroom for mathematics. 5. The National Staff Development Council (NSDC) audit team categorized the teachers as traditional or inquiry-based from the results of the survey. It was assumed that the NSDC audit team members accurately categorized the teachers’ instructional approach. 6. The survey is a modification of the 2000 National Survey of Mathematics and Science Education from Horizon Research, Inc. (Weiss, Banilower, McMahon, & Smith, 2001). It was assumed the survey has the capacity to categorize teachers as purveyors of either traditional or inquiry-based instruction with respect to the definitions used within this study. 7. It was assumed that the level of teacher competence is randomly assigned to both groups.

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Limitations

There are several limitations in this study. 1. This study could not control what instruction the students received or did not receive at home or outside of the school environment. 2. The 4Sight Benchmark Assessment (SFAF, 2008) that was used to measure student achievement is comprised of mainly multiple-choice questions that, in many instances, cannot completely predict the depth of student understanding, only procedural knowledge and skills. 3. There was no control over teacher effectiveness relative to their chosen instructional delivery in mathematics. 4. The limited sample size of 10 inquiry-based teachers and 10 traditional teachers may prohibit the results from being generalized to other populations.

Nature of the Study

This mixed-methods study attempted to determine the significance of traditional or inquiry-based instruction on student achievement in elementary mathematics. A causal-comparative design was utilized to identify which type of learning environment produces the highest level of achievement in mathematics. The population is elementary students in grades three and four who received instruction in math from a teacher using either traditional or inquiry-based teaching techniques. The quantitative portion of the study utilized student achievement data gathered from the 4Sight Benchmark Assessment (SFAF, 2008). Permission was obtained for use

13 of these scores and the comparisons from the district superintendent. However, all student names were removed, as well as the teachers’ identity, for publishing purposes. A grounded theory design was used for the qualitative portion of the study, which includes categorizing the instructional approach of teachers and determining their perceptions about instructional approach choices. All teachers considered for the study completed an instructional survey. In addition, many of the participants were observed by a NSDC team member, which assisted in identifying their teaching style and perceptions about the impact of their instructional choices on student achievement. The researcher conducted individual interviews with each teacher participating in the research study to further develop knowledge of teacher perceptions of instructional and resource choices. The data gathered for the placement of the teachers into the categories of traditional or inquiry-based instruction were validated through the use of two sources. This was accomplished with the teacher survey, possible classroom observations, and interviews by the researcher. Descriptions about teacher habits, planning, and teaching techniques were gathered from the interviews conducted by the researcher and reported as supportive evidence of belonging in either the traditional or inquiry-based category for instructional design. The instructional categories teachers were placed in were not communicated to the participating teachers. The results of this study are intended to guide the selection of math programs, professional development offerings, and instructional practices. It is the explicit intention of this study to use the data to influence teachers in their instructional decision making in an effort to maximize student achievement in mathematics.

14 Organization of the Remainder of the Study

Chapter 1 gives the overall background and purpose of the research to be investigated. The remainder of this paper is divided into four additional chapters. Chapter 2 explores the literature that is related to inquiry-based learning and traditional instruction. Chapter 3 expands on the research methodology utilized in this study. Chapter 4 displays and evaluates the data collected using the research methodology described in Chapter 3. The conclusion, Chapter 5, is a summary of the results and any implications that can be drawn from the data in Chapter 4. Chapter 5 also elaborates on possible recommendations for extensions of this study and future research.

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CHAPTER 2. LITERATURE REVIEW

Introduction

This chapter develops a historical perspective of constructivism and a theoretical framework for a student-centered learning environment in elementary mathematics. The literature review includes an appraisal and an analysis of research related to inquiry-based learning, with a focus on the elementary grades. It is divided into seven main sections comprised of: introduction to constructivist theory; a historical perspective of constructivist theory; traditional instruction; the achievement gap; the impact of mathematics reform on inquiry-based learning; the NCTM and the philosophy represented with respect to mathematics instruction; and the current state of elementary mathematics instruction. Researchers help to support the validity of some educational theories through findings revealed in their studies. The first two sections of the literature review focus primarily on the work of Vygotsky (1978), Piaget (1979), and Von Glasersfeld (1995) in an effort to develop a clear picture of the constructivist learning theory. In addition, a summary is given on the debate between traditional instruction and what is known as reform mathematics. The focus of the review is literature relating to elementary math instructional practices and their relationship to student achievement.

16 Constructivist Learning Theory

The constructivist theory can be traced back to well before the 18 th century in the works of many psychologists, sociologists, and philosophers. In the last 100 years, constructivism has influenced educational learning theories in very meaningful ways. The work of Von Glasersfeld (1987), Jean Piaget (1979), and L. S. Vygotsky (1978) has made a substantial impact on translating the theories of constructivism to effective educational teaching practices within the classroom. Constructivism is a philosophical perspective used to describe a learning theory that can best be seen on a continuum consisting of varying degrees, such as radical, social, and modern constructivism. Along this continuum, the work of many theorists emerges, evolves, and blends to fully develop the Constructivist Learning Theory. Jean Piaget, a developmental psychologist, provides a substantial part of the foundation on which constructivism has been built. Piaget is well known in education for his work in genetic epistemology. Piaget believed that knowledge or conceptual understanding stems from interaction with a concept. The interaction allows the student to work a concept into a schema that is in alignment with the student’s current knowledge structures (Von Glasersfeld, 1997). His theories on the developmental stages of logical thinking in children are very important to the learning of mathematics. According to Piagets’ theories, students construct rules through understanding of patterns of outcomes resulting from actions performed with mathematical material (Piaget, 1972). The results of such actions are viewed by the student as feedback, which affirm or refute their thinking. It is through this type of feedback that students further develop their conceptual understanding of a mathematical

17 idea. Students independently progress through this entire process with no social interaction. Thus, the possible value of communication is completely disregarded in Piaget’s theory of learning. One aspect of Piaget’s work that is not supported by the educational community is his portrayal of the process of learning as individualistic, removing the social and cultural influence (Jaworski, 1999). Piaget goes as far as indicating that teaching a student without first allowing him or her the opportunity for self-discovery deprives the student of developing conceptual understanding. Von Glasersfeld (1987) labels Piaget’s theories as being “radical constructivist,” which he defines as two distinct but interdependent principles. The first principle defines acquisition of knowledge as an active process where students construct their own knowledge. The second principle stresses that the learner is flexible, constantly taking in new information and organizing it. This allows the learner to fit the newly acquired information into his or her perception of reality. Neither of Von Glasersfeld’s principles, which classify the work of Piaget, account for social or cultural interaction. Von Glasersfeld labels his work as radical constructivist, although many aspects of his work can be labeled social constructivist as well. Social constructivism does consider the impact of language and social activity in the learning environment. However, social constructivists differ slightly from radical constructivist in their views of social interaction. Social constructivists claim that communication and social interaction are a “direct means for the sharing of knowledge” (Von Glasersfeld, 1995, p. 191), whereas radical constructivists view it as a means for acquiring and refining knowledge.

18 Von Glasersfeld’s (1987) work does indicate the importance of communication and social interaction in an educational setting and specifically in problem solving. He stresses the importance of allowing students to express their thinking and include it as part of a whole class discussion. This allows the students to continue to construct conceptual understanding; however it is the role of the teacher to guide and direct this process. Communication, a component of radical and social constructivism, is a very important aspect of constructivist theory because it is through the process of talking and sharing ideas that individuals develop understanding (Jaworski, 1999). In the work of Von Glasersfeld, it is apparent that communication and social interaction are vital in the learning process. Specifically, “talking about the situation is conducive to reflection” and upon reflection Von Glasersfeld (1995) states that “in order to describe verbally what we are perceiving, doing or thinking, we have to distinguish and characterize the items and relations we are using” (p. 188). This allows students to focus on the characteristics of a problem or concept and begin to analyze the process through which the solution is attained. This reflective process also affords students the opportunity to determine possible errors in their thinking without being directly told if a solution is correct or incorrect. Vygotsky (1997) stipulates that if a student develops a deep understanding of a concept, it is because appropriate and thought-provoking investigative opportunities have been provided along with the tools necessary to accomplish the task. If the learning path does not involve obstacles, or knowledge is handed to the student verbally or otherwise, little learning will occur. When one person is the dispenser of knowledge, this limits communication and social interaction to a single direction: from teacher to student. This is challenging as