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COURSEWORK ON BASE NUMERATION SYSTEMS AND ITS INFLUENCE ON PRESERVICE ELEMENTARY TEACHERS’ UNDERSTANDING OF PLACE VALUE CONCEPTS By DOROTHY JANICE RADIN Bachelor of Science Minot State University 1971 Master of Arts DePaul University 1992 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF EDUCATION July, 2007 ii COURSEWORK ON BASE NUMERATION SYSTEMS AND ITS INFLUENCE ON PRESERVICE ELEMENTARY TEACHERS’ UNDERSTANDING OF PLACE VALUE CONCEPTS BY DOROTHY J. RADIN Dissertation Approved: ______Patricia L. Jordan________________ Dissertation Advisor ______Margaret M. Scott_______________ ______Caroline Beller__________________ _______Laura L. B. Barnes______________ _______A. Gordon Emslie_______________ Dean of the Graduate College iii ACKNOWLEDGMENTS As this ten year odyssey concludes, I would like to thank my committee members for their contributions and support. To my committee chairperson, Dr. Patricia Jordan, my thanks for handling all my emails and questions with grace and candor. A special thanks to Dr. Laura Barnes for her contributions with the IRB process and for her expertise in the statistical analysis. Thanks to Dr. Margaret Scott for her input into my proposal and for working beyond retirement to see me through. Lastly, I appreciate Dr. Caroline Beller’s commitment to my committee work. All my department colleagues at Oral Roberts University have been a positive influence throughout the years. Chair Ken Preston has urged me on at every step. My friend and coworker Teresa Williams has been a great sounding board for my frustrations and joys. A particular thank you to Mary Lou Miller for helping with my statistics and for understanding my journey. A simple thank you is not enough for my friend and colleague Martha Parrott. Without her cooperation this study would not have been possible. She has been patient, pleasant, and always encouraging as I have strived to complete the work. Finally, my greatest thanks to my family. My children Jeff and Cindy have always been encouraging and have been flexible with their schedules when I needed time for class or for studying. And last, but certainly never least, to my husband Marc for his constant love and support. He has been there through it all, always the cheerleader and my biggest advocate. Thanks honey, and yes, I owe you! iv TABLE OF CONTENTS Chapter Page I. INTRODUCTION…………………………………………………………………..…1 Statement of the Problem ………………………………………………………...2 Context and Need for the Study…………………...……………………..……....3 Purpose…………………………………………………………………………...5 Research Questions ……………………………………………………………....6 Elements of the Study…………………………………………………………....6 Definition of Terms ……………………………………………………………....7 Limitations of the Study……………………………………………………….....8 Conclusion………………………………………………………………………..9 Summary of the Chapters ……………………………………..…………………10 II. LITERATURE REVIEW ……………………………………………...……………..12 Historical Perspective………………………………………………………..….14 PreService Elementary Teachers’ Knowledge of Mathematics ……………..…19 Teacher Knowledge and Student Learning…………………………………..….22 Place Value and Elementary Student Knowledge ……………………..………..25 Numeration Systems ………………………………………………………….....28 Conclusion …………………………………………………………..….…….…34 III. Methodology……………………………………………………………….…..……37 Participants ……………………………………………………………….……..38 Comparison of Classes ………………………………………………….……....39 Data Collection ……………………………………………………….………....46 Instrument ………………………………………………………………….........47 Data Analysis ………………………………………………………..………..…51 Conclusion …………………………………………………………………...….52 IV. FINDINGS …………………………………………………………………….……54 Demographic Information …………………………………...……………...…..55 Rater Reliability…………………………………………………...……...….…56 Question One…………………………………………………………………...58 Question Two ……………………………………………………………….…..63 Question Three……………………………………………………………….…70 Question Four………………………………………………………….………..74 Conclusion ……………………………………………………………..………..75 v Chapter Page V. CONCLUSION………………………………………………….. ………………..77 Summary………………………………………………………...……………77 Conclusions …………………………………………………...………………82 Limitations ……………………………………………………………………87 Recommendations …………………………………………………………….88 BIBLIOGRAPHY…………………………………………………………………….95 APPENDIXES ……………………………………………………………………..…106 APPENDIX A  ASSESSMENT OF PLACE VALUE UNDERSTANDING………………...…..………..107 APPENDIX B – APVU CELL CONTENT……………………….………….114 APPENDIX C – RUBRICS …………………………….………….…………115 APPENDIX D – RUBRIC/TASK CORRELATION…………………...........122 APPENDIX E– INSTITUTIONAL REVIEW BOARD LETTER ……..........124 vi LIST OF TABLES Table Page 1. Comparison of Course Content …………………………………………………40 2. Detail Numeration System Content ……………………………………………..41 3. Matrix of Explicit Place Value Understanding ……………………………….…49 4.. ACT Mean Scores…………………………………………………………….…56 5. Demographic Information ………………………………………………………57 6. APVU Sample Scoring………………………………………………………….57 7. All Tasks Pretest Item Mean Scores …………………………………….………58 8. Pretest Cell Means ……………………………………………………………….60 9. Pretest Rubric Means ………………………………………………………….....62 10. All Tasks Posttest Item Mean Scores ………………………….………….…….63 11. Posttest Cell Means ……………………………………………………………..65 12. Posttest Rubric Means ………………………………………………………….68 13. Without Instruction Rubric Correlations ……………………………………….69 14. With Instruction Rubric Correlations …………………………………………..70 15. Change in Understanding from Pretest to Posttest ……………………………...71 16. Change in Understanding from Pretest to Posttest by Matrix Cell ……………..72 17. Change in Understanding from Pretest to Posttest by Rubric…………..……....73 vii LIST OF FIGURES Figure Page 1. Pretest Total Mean Scores ……………………………………………………..59 2. Posttest Total Mean Scores ……………………………………………….……64 3. Familiar Systematic Cell Posttest Mean Scores ………………………………..66 4. Familiar NonSystematic Cell Posttest Mean Scores ………………………..…66 5. Unfamiliar Systematic Cell Posttest Cell Scores ……………………………..…67 6. Unfamiliar NonUnfamiliar Cell Posttest Mean Scores ……………………..….67 1 CHAPTER 1 INTRODUCTION Degree plans for elementary education majors require coursework in many disciplines. Although there is controversy about what mathematics content courses should be included in the program, most colleges require at least twelve credit hours or a fourcourse equivalent. In these mathematics classes students learn about problem solving, fractions, decimals, geometry, integers, and other topics. Since many students struggle with the content, they often come to class not only afraid of mathematics, but also with deficiencies in their mathematics backgrounds. These students learn basic operations and are usually able to do simple computations, but they lack a depth of understanding. Studies acknowledge that elementary teachers are not adequately prepared with an explicit understanding of mathematics. “It appears that prospective teachers may have mastered basic skills, but they lack the deeper conceptual understanding that is necessary when responding to student questions and extending lessons beyond the basics” (Center for Study of Teaching and Policy, 2001, p. 9). The National Research Council report in 2002 states elementary teachers’ understanding of the mathematics they teach is inadequate. This lack of subject matter depth can lead to students’ dependence on rote memorization of mathematical algorithms to solve problems rather than a development of reasoning skills. As college students they often try to generalize and memorize their way through required mathematics courses, rather than make connections among ideas 2 (National Research Council, 2002). In fact many equate understanding in mathematics with the ability to complete steps in an algorithm (Folk, 2006). Too many future elementary teachers begin college with mathematics deficiencies, both in content and in understanding. Their college mathematics courses focus mainly on computational skills rather than the development of reasoning skills and mathematical ideas. To illustrate these points the Conference Board of Mathematical Sciences uses the place value structure which states that elementary teachers need an explicit understanding of place value to aid children in building a strong numeric foundation (CBMS, 2001). Understanding place value as part of number and number sense is crucial to arithmetic success. Place value is the basis for the representation of whole numbers. Ideas of place value are used to order numbers and to aid in estimation and approximation (CBMS, 2001). Statement of the Problem In the college mathematics classroom, place value is often studied in connection with numeration systems, but elementary preservice teachers question the need to learn different numeration systems. The study of numeration systems includes practice with numberbase problems as well as ancient numeration systems like the Mayan or Babylonian. Number base work typically includes learning to count in different bases, converting back and forth between base ten and other bases, and performing number base computations. With the limited number of mathematics courses required for elementary preservice teachers, every course must provide meaningful instruction and content. Mathematics courses need to adequately prepare these prospective elementary teachers to 3 effectively educate their own future students; therefore, the curriculum for each course is important. Does the study of base numeration systems contribute to elementary preservice teachers’ understanding of place value? Context and Need for the Study Textbooks for mathematics courses for elementary preservice teachers have traditionally included the study of bases and different place value systems in their curricula (Bassarear, 2005; Billstein, Libeskind, & Lott, 2007). Why? Bassarear says the preservice elementary teacher should work with other bases “so that your understanding of the fundamental ideas of the base and place value become deeper and deeper” (2005, p. 115). Many elementary school textbooks explore place value numeration systems. Greenes et al. (2005) includes work with base two and base five in the grade four Houghton Mifflin text. Fennell and Altreri (1998) study the Babylonian system in their edition of the Silver Burdett grade three book. Clements, Jones, Moseley, and Schulman (1998) invite students to construct a place value system in a McGrawHill third grade textbook. Children’s literature also reflects an interest in place value systems. D.A. Adler’s book uses nickels and pennies to show Base Five (Adler, 1975). How to Count Like a Martian (St. John, 1975) has chapters on systems from the imaginary Martian to the ancient Mayan. Luce’s book Counting Systems, The Familiar and the Unusual puts different number systems into practice, such as base nine for baseball teams and base two for lights on and off (Luce, 1969). 4 Are there other reasons to study different place value systems? Casebeer researched the study of numeration systems in school mathematics in the 1960s. He hypothesized three reasons for studying this topic: 1) as foundational information, 2) for greater understanding of our HinduArabic system, and 3) to gain empathy for children’s experiences in learning place value for base ten (Casebeer, 1967). Writing in the 1970s, LeBlanc reiterated some of the same reasons as Casebeer for supporting the study of numeration systems. He suggested that prospective elementary teachers study other base systems to understand the characteristics of our base ten system. He wrote that work with other bases helps with the understanding of groupings and place value and provides elementary preservice teachers with experiences that children may have in learning the HinduArabic system. He drew a clear connection between teacher understanding of place value and children’s development of numeric concepts (LeBlanc, 1976). Research involving place value has primarily focused on elementary students and practicing elementary teachers. Jones’ and Thornton’s review of literature (1993) of the 1980s and early 1990s identified many studies showing children’s deficiencies in place value understanding. Fuson’s research (1990a, 1990b) used manipulatives, such as cubes and baseten blocks, to solve addition and subtraction multidigit problems to help students learn and apply place value concepts. She found that children have “inadequate understanding of the baseten place value system of written multidigit numbers, and consequently longterm errors in multidigit calculation procedures” (Fuson et al., 1997, p. 130). 5 Zazkis and Khoury studied place value, elementary preservice teachers, and their understanding of noninteger numbers in bases otherthanten. They concluded that their preservice students did not fully understand our number system structure. The results suggested that these college students’ partial understanding allowed them to accurately complete problems, but might interfere with conveying place value concepts to children (Zazkis & Khoury, 1994). In 2003, McClain studied elementary preservice teachers’ understandings of place value and multidigit addition and subtraction. In the course of this study she used base eight materials. One conclusion of her research was that the future teachers realized that conceptual understanding was necessary for teaching and explaining place value to others. Her students shifted their goals from “a focus on correct procedures to an emphasis on students’ [children’s] understanding” (McClain, 2003, p. 304). Two recent studies conducted at the University of Texas investigated instructional strategies, elementary preservice teachers, and place value (Hannigan, 1998; Rusch, 1997). FarroLynd’s work (2003) focused on elementary preservice teachers’ understanding and misconceptions of place value. There appears to be no recent research connecting preservice teachers, numeration systems, and place value. The focus of this project will be the teaching of base numeration systems and its impact on elementary preservice teachers’ place value understanding. Purpose This study grew out of classroom practice. The researcher found that her elementary preservice teachers encountered difficulties while studying numeration systems. These students did not see any application for the concepts and usually 6 struggled to understand the content. This project questions the value of teaching numeration systems to elementary preservice teachers. Are the reasons given by textbook authors and researchers valid? In particular, does the study of otherthanten bases increase elementary education students’ understanding of our base ten system? Does place value numeration systems instruction help elementary preservice teachers generalize place value concepts? These are the questions to be researched in this study. Research Questions Specifically, four questions are addressed in this study: 1. What is the level of explicit place value understanding that the elementary preservice teachers participating in the study bring with them to their required mathematics content course? 2. What is the level of explicit place value understanding that the elementary preservice teachers participating in the study have at the end of their required mathematics content course? 3. Is there any significant change in the participants’ explicit place value understanding from the beginning of the semester to the end of the semester? 4. Is there a significant difference in gains made in explicit place value understanding between the class that has numeration system instruction (experimental group) and the class that does not have the numeration system instruction (control group)? Elements of the Study This research was a quantitative quasiexperimental/control group design study. The objective was to compare gains of explicit place value understanding possessed by two groups of elementary preservice teachers taking a required mathematics course. The 7 course curriculum included a section on base numeration systems. The independent variable in the study was the instruction, or withholding of instruction, of base numeration systems. The dependent variable was growth in explicit place value understanding as measured by the Assessment of Place Value Understanding instrument. Two groups of preservice elementary teachers participated in this study, a sample of convenience. One group of 35 was enrolled in a required mathematics content course at a medium size regional state university. A second group of 25 was enrolled in a similar course at a small private university. Both groups’ curriculum contained a section which addressed place value concepts and numeration systems. The instrument used in this research is the Assessment of Place Value Understanding. This pretest and posttest assessment was developed by two PhD. candidates at the University of Texas in 1997. In a review of literature for their separate dissertations, they found “no previously validated assessment instruments that would provide insight into participants’ explicit understanding of place value” (Rusch, 1997, p. 12). They also found existing instruments were directed at children and not appropriate for assessing the knowledge of adults. A description of the Assessment of Place Value Understanding and the validation procedures are included in Chapter 3. The Appendix contains copies of the thirteen item pretest (only the pretest is used in this research), along with information regarding the objectives, rationale, and scoring strategies used. Definition of Terms Numeration System  “A numeration system is a collection of properties and symbols agreed upon to represent numbers systematically” (Billstein et al., 2004, p. 135). 8 Place Value  “Place value assigns a value of a digit depending on its placement in a numeral. To find the value of a digit in a whole number, we multiply the place value of the digit by its face value, where the face value is a digit” (Billstein et al., p. 136). Base Numeration System  A true place value system has a base and a set of symbols, including a symbol for zero. The HinduArabic system used in the United States is an example of a base ten numeration system. Explicit Understanding  “The level of mathematical understanding that is characterized by a precise understanding of the concept being addressed which can be clearly articulated and convincingly justified” (Rusch, 1997, p. 16). Procedural Understanding  “Knowledge of the formal language of mathematics, that is, symbols and syntax; and the rules, algorithms, or procedures used to solve mathematical tasks” (Post & Cramer, 1989, p. 222.). Preservice Teachers  “Person studying to teach mathematics as one of several subjects, future elementary school teachers” (Graeber, 1999, p. 191). Limitations of the Study This research represents a preliminary exploration into the relationship between understanding place value and understanding base numeration systems. This study updates earlier reports, reexamines, and investigates only one small construct of teacher knowledge. With its narrow focus, this study is meant to inform and raise questions regarding course content for elementary preservice teachers in their undergraduate program. The researcher’s hope is that others will take up the charge to investigate this and other related topics about curriculum and expertise in subject matter knowledge. 9 The Assessment of Place Value Understanding was administered during this study. Since this is a relatively new instrument to measure explicit understanding, there may be some hidden limitations on its use. Other assessments were researched for this study, but tests for this population of adult college students were normally of general arithmetic concepts, including topics such as geometry, number theory, fractions, sets, and the four basic operations on the set of whole numbers. None of the instruments focused solely on place value. Due to time constraints, control of participant sampling was not possible. All classes used were chosen by convenience. Although the classes were at different colleges with different instructors, efforts were made to at least select classes with similar students in a comparable college program. All students in the two classes were invited to participate. Even so, the samples lead to major questions regarding generalizabilty. Although teaching methodology was not considered in this research, the teachers have similar pedagogy. Both instructors have many years of university teaching experience, and both are friends and students in the same doctoral program. Texts and curricula for the classes were different, but content pertinent to this research was similar. Content comparisons are discussed in Chapter 3. There was also no way to control for the participant’s concentration and effort while taking the assessments. Although both instructors encouraged the participants, there was no sufficient incentive to ensure their best work. Conclusion Place value is a major component of the HinduArabic numeration system. Although many students are able to gain some expertise in computation, many more find 10 mathematics problems difficult to comprehend. “Knowledge of place value has great implications for success in arithmetic tasks…. Therefore, care should be taken that students develop a meaningful understanding of numbers” (Cathcart, Pothier, Vance, & Bezuk, 2000, p. 106). One major reason for studying base numeration systems in the elementary education major curriculum is the connection to place value. This study examined the relationship: Does studying base numeration systems lead elementary preservice teachers to a higher level of understanding of our baseten system? Summary of the Chapters Chapter One contains the introduction and impetus for the problem. The introduction to the study in Chapter One defends the importance of the topic, states the purpose of the study and identifies the questions to be answered. Included in Chapter One are definitions of terms and an outline of the remainder of this dissertation. Chapter Two begins with the progress of education during the past twenty years. The importance of teachers’ content knowledge is discussed as well as the connection between teaching and student understanding. Research on elementary preservice teachers’ mathematical skills is then addressed. Finally the importance of place value and how children learn is presented. Chapter Three describes the methodology used in this study. Development and piloting information about the instrument used – the Assessment of Place Value Understanding (APVU) – is detailed as well as a matrix of place value grouping schemes used in the instrument. The sample of students is identified and information about the instructors and course content. The five level scale of mathematical understanding 11 employed in the assessment rubrics is defined. Scoring of the tests is discussed and the statistical tests used to analyze the data Chapter Four details the findings of this research. Demographic information that was collected on the participants is summarized. Sample scoring data to check for rater reliability is given. Each questions in the study is answered by analyzing data in total, in each grouping, and in each rubric. Tables show data details and results. Chapter Five summarizes this study and offers conclusions about the research. Finally, recommendations are offered for future work. 12 CHAPTER 2 LITERATURE REVIEW Over the past thirty years national attention has focused on quality education in the United States. Federal legislation and government research set rigorous standards for curriculum, teacher quality, and assessment. The desire for highly qualified teachers became a priority. The National Council of Teachers of Mathematics (NCTM) began publishing standards for the mathematics community in the 1980s with An Agenda for Action (NCTM, 1980). In numerous publications this organization identified content appropriate for students by grade level and defined mathematics and pedagogical content to enhance teacher preparation. The NCTM Professional Standards (1991) emphasized the importance of teachers’ knowledge and their understanding of the subject matter. Since research shows a strong connection between teacher knowledge and student achievement, teachers need to know more content than the subject matter they teach. In order to present mathematics clearly and effectively, teachers must have explicit understanding of the subject. Elementary preservice teachers should have this explicit understanding for all segments of mathematics they will be required to teach. For elementary teachers, place value is a foundation concept. Yet many teachers, and thus their students, do not have an explicit understanding of this topic. As part of the college mathematics curriculum, base numeration systems are often taught to help these preservice teachers understand our base ten system. This research examines the 13 effectiveness of teaching base numeration systems and their effectiveness in improving place value understanding. Specifically, four questions are addressed in this study: 1. What is the level of explicit place value understanding that the elementary preservice teachers participating in the study bring with them to their required mathematics content course? 2. What is the level of explicit place value understanding that the elementary preservice teachers participating in the study have at the end of their required mathematics content course? 3. Is there any significant change in the participants’ explicit place value understanding from the beginning of the semester to the end of the semester? 4. Is there a significant difference in gains made in explicit place value understanding between the class that has numeration system instruction (experimental group) and the class that does not have the numeration system instruction (control group)? This chapter begins with a historical perspective on America’s focus on education and the importance of mathematics. The public call for “highly qualified” teachers and the search for a clear definition of this new term causes all teacher educators to reexamine their instruction. Section two of this chapter sights literature assessing the subject matter qualifications of today’s teachers and preservice teachers. Many teachers and preservice teachers can calculate properly but do not understand the logic behind the algorithms. In section three the researcher makes connections between teacher preparation and children’s achievement in the classroom. Teachers need to have a positive attitude and an understanding of the mathematics they teach. Section four 14 discusses elementary students’ knowledge of place value and the essential nature of this concept. Research with elementary students shows that they do not have a mastery understanding of the place value system. The focus of this study, numeration systems, is discussed in section five. A number of studies are outlined dealing with elementary preservice teachers. Yet there appears to be a void in the literature connecting place value understanding, elementary preservice teachers, and numeration systems. This study is meant to begin the probe into rigorous place value course content for the elementary preservice teacher. Historical Perspective Heading into the 21st century, Americans renewed their focus on quality education. The ‘80s and ‘90s were inundated with multiple reports and studies on schools, student achievement, and teacher effectiveness. Among the reports issued were A Nation At Risk (1983), completed by the National Commission on Excellence in Education, A Nation Prepared: Teaching for the 21st Century (1986) written by the Carnegie Forum on Education, Action for Excellence (1983), authored by the Task Force on Education and the Economy, Education Commission of the States, and A Call for Change (1985), reported by the National Commission for Excellence in Teacher Education. In A Nation At Risk, concern was expressed about “the widespread public perception that something is seriously remiss in our educational system” (National Commission on Excellence in Education, 1983, p 1). A call was issued to reform the Nation’s schools and reverse this declining trend. Recommendations for teachers included higher standards for certification and a competence in an academic discipline. 15 In A Nation Prepared: Teaching for the 21st Century, the Carnegie Forum on Education declared that teachers needed a sense of mathematics if their students were to have mathematical ideas and learn reasoning skills. Teachers’ content knowledge must include breadth and depth. “They [teachers] must be people whose knowledge is wideranging and whose understanding runs deep” (1986, p. 25). After President George H.W. Bush convened an education summit in 1989 of the nation’s governors, Congress passed the Goals 2000: Educate America Act. This set a national education agenda. Based on eight goals involving students, teachers, and communities, Goals 2000 required states to develop clear and rigorous standards. Goal Five, emphasizing mathematics and science, advocated more preservice training to help these prospective teachers gain broader and deeper mathematics understanding (Grinstein & Lipsey, 2001). In his second State of the Union address, President Clinton issued a “ ‘Call to Action ’ that included as a priority improving the quality of teachers in every American classroom” (Lewis et al., 1999, p. iii). The next year a Harris Poll revealed that about 90% of Americans believed “the best way to raise student achievement is to ensure a qualified teacher in every classroom” (CBMS, 2001, p. 3). This interest led to the 1998 amendments to the Higher Education Act of 1965. Part of these provisions included teacher quality enhancement grants to improve the quality of the future teaching force by improving the preparation of prospective teachers (United States Department of Education, 2003). Studies and reports were published examining school mathematics. These included Everybody Counts (1989) by the National Research Council and Educating 16 Americans for the 21st Century (1983) reported by the National Science Board Commission. An Agenda for Action (1980), Priorities in School Mathematics (1981), and Guidelines for the Preparation of Teachers of Mathematics (1981) were written by the National Council of Teachers of Mathematics (NCTM). These works made recommendations for school mathematics curricula, teacher training, and public support of the discipline. The National Research Council wrote that the main objective of the elementary school mathematics curriculum should be the concept of number sense. The National Science Board Commission (1983) stressed the importance of number sense and place value and called for high quality teachers. The NCTM encouraged improved preservice teacher education, focusing on teacher skills necessary to help children learn more than computation. The National Council of Teachers of Mathematics compiled their first Curriculum and Evaluation Standards for School Mathematics in 1989. This publication established high expectations for students of mathematics in kindergarten through grade twelve. The document provided a comprehensive “listing by gradelevel bands of the mathematics that students should know about problem solving, communication, connections, and various content aspects of mathematics relevant to those grade levels” (Usiskin & Dossey, 2004, p. 7). The Curriculum Standards were written in response to the increased national interest in the teaching and learning of mathematics and the publication prompted states to develop their own standards to set the bar for student performance. The K – 4 standard six of number sense and numeration identified understanding place value as a “critical step in the development of children’s comprehension of number concepts” (NCTM, 1989, p. 39). Sample learning activities were included. As part of the 17 evaluation standard eight on mathematical concepts, the importance of place value was connected to calculating subtraction problems. The National Council of Teachers of Mathematics followed up its Standards text with the Professional Standards. This publication was based on the assumption that teachers are key in changing the mathematics curriculum and pedagogy. The Professional Standards spelled out “what teachers need to know to teach toward new goals in mathematics education and how teaching should be evaluated for the purpose of improvement” (1991, p. viii). Professional development standard two contained a section on number systems and number sense which identified place value as important content for teachers to understand. Mathematics understanding has also been studied on an international basis. The Second International Mathematics Assessment was completed in 1986, comparing United States student achievement with sixteen other countries (Grinstein & Lipsey, 2001). The Third International Mathematics and Science Study (now called the Trends in International Mathematics and Science Study or TIMSS) compiled data on United States students and students from other countries in 1995, 1999, and 2003 (National Center for Education Statistics, 2001). These studies “suggested that American practices in mathematics education are not yielding the kind of learning that is both desirable and possible” (Watanabe & Thompson, 2004, p. 11). TheUnited States’ mathematics curriculum emphasized procedural skills and provided little depth in subject matter. As the 21st century began, the focus on education continued. President George W. Bush worked to pass the No Child Left Behind Act (NCLB), enacted in 2001. This act, PL 107110, mandated that all teachers must be deemed highly qualified. Although 18 there is disagreement about what it means to be “well qualified,” part of the NCLB definition of “high quality” included an ability to demonstrate expertise in the subject matter. States were charged to raise standards for teacher certification and monitor practices to prepare, train, and recruit high quality teachers (United States Department of Education, 2001). With the higher level of mathematical sophistication needed in the 21st century, increased mathematical skills and a solid understanding of mathematical concepts were needed by educators at all levels. Media and further publications stressed the importance of the subject of mathematics. “School mathematics instruction and the mathematical preparation of teachers are in the spotlight; because after reading and writing, mathematics is widely viewed as the most important component of K12 education” (CBMS, 2001, p. 3). The Nation’s Report Card: Mathematics 2000 claimed that “the ability to know and use mathematics is a necessity in daily life” (Braswell et al., 2001, p. 1). Even television reinforced the importance of mathematics in one of its programs. The series Numb3rs began “We all use math every day, to predict weather, to tell time, to handle money. Math is more than formulas and equations. It’s logic. It’s rationality. It’s using your mind to solve the biggest mysteries we know” (Heuton, Falacci, & Zucker, 2005). In 2000 the National Council of Teachers of Mathematics published an update of its 1989 Standards entitled Principles and Standards for School Mathematics. It stated The need to understand and be able to use mathematics in everyday life and in the workplace has never been greater and will continue to increase. . . . In this changing world, 19 world, those who understand and can do mathematics will have significantly enhanced opportunities and options for shaping their futures (NCTM, 2000, p. 45). PreService Elementary Teachers’ Knowledge of Mathematics With the call for highly qualified teachers, articles and research from the academic community investigated elementary teachers’ knowledge of the subjects they teach. Collias, Pajak and Rigden (2000) begin their article entitled “One Cannot Teach What One Does Not Know” with the quote, “Graduates of university teacher preparation programs in the United States, though well trained in teaching methods, often have insufficient knowledge of the subject matter they will teach” ( p.1). The authors reported that a 2000 survey found thirtythree percent of United States teachers did not believe that a mastery of subject matter was important to teaching. An article by McDiarmid, Ball, and Anderson (1989) on subjectspecific pedagogy used a literature review to defend the statement, “Unfortunately, considerable evidence suggests that many prospective [elementary] teachers do not understand their subjects in depth” ( p. 199). Subject matter content is taken for granted, and teacher preparation programs focus on pedagogical skills. Evidence of poor elementary teachers’ knowledge of content was referenced in an article by Brown, Cooney, and Jones stating that “preservice elementary teachers do not possess a level of mathematical understanding that is necessary to teach elementary school mathematics” (1990, p. 642). Other research by Post, Harel, Behr, and Lesh (1991) specifically found that twenty to thirty percent of teachers scored less than fifty 20 percent on an instrument measuring conceptual knowledge of rational numbers. In fact almost half the teachers missed very fundamental items like ÷3. In their literature review, Brown and Borko found that “research also suggests that prospective elementary teachers often do not have adequate content knowledge when they begin student teaching” (1992, p. 220). Previous research had postulated that this lack of knowledge was due to a focus on procedural rather than conceptual understanding. Students entering mathematics courses associated mathematics with mechanical and abstract symbols and rules. Coursework seemed to help the elementary preservice teachers develop more conceptual understanding about their own learning but did not enhance their thinking about their students’ learning. Preservice teachers should have “ both knowledge of mathematics and knowledge about mathematics” (p. 212). Schwartz and Riedesel (1994) reported on a number of studies that showed teachers and preservice teachers had only a procedural knowledge of mathematics. Duckworth focused on teachers and long division. These teachers knew the algorithm for long division, but they did not know why the procedures worked. Duckworth concluded that teachers’ understanding of arithmetic varied widely. Her second conclusion stressed the connection of understanding and classroom practice. As teachers’ understanding deepened, their teaching methods changed. Wheeler studied preservice teachers and division problems with zero. Sixtytwo elementary preservice teachers were tested and interviewed on problems dealing with a divisor or dividend of zero. Division by zero problems were most difficult for the future teachers. Only 23.1% made no errors. Even though many could do the problems accurately, “many could not adequately elaborate on the question, ‘What is zero?’” (p. 4). 21 An article by Fosnot (1989) described a classroom scenario common for many education majors. They frequently sat in math classes and practiced procedures, proofs, and calculations. They learned to memorize and regurgitate these procedures and rarely were given concrete materials to make the abstract notation meaningful. The result is that many teachers do not understand the concepts they are expected to teach (p. 71). Post and Cramer (1989) found that the mathematical background of new teachers included a concentration on drill and practice rather than an emphasis on understanding the concepts. These authors stated that this lack of conceptual knowledge “dominates school mathematics curricula at virtually all levels” (p. 222). To illustrate this focus on mechanistic skills, a survey from a Minnesota school district was sighted. Elementary junior high teachers reported over sixty percent of class time was spent working textbook problems and practicing speed and accuracy in computational algorithms. This type of instruction using just the printed page generally does not lead to higher level thinking or understanding. McDiarmid and Wilson conducted teacher interviews and learned that many had mastery of mathematical rules but could not explain the reasons behind the rules. One teacher said, “I don’t know why. My teacher never told me” (1992, p. 101). Ball (1990) also found this deficiency. In her interviews a future teacher said, “I absolutely do it [a mathematical calculation] by the rote process I would have to think about it” (p. 458). 22 The students’ idea of “doing” mathematics was to describe the steps of the algorithm, without any thought as to the meaning or reasons for the procedures. In a 2001 report, The Center for the Study of Teaching and Policy concluded that “prospective elementary teachers had relatively sound procedural, or ruledominated knowledge of basic mathematics, especially in arithmetic but had difficulty when pushed to explain why an algorithm or procedure works” (p. 9). They lacked sound conceptual understanding of the underlying concepts of the mathematics. Teacher Knowledge and Student Learning The quality of teachers is a concern because of the strong connection between teachers’ knowledge and student achievement. In its Professional Standards, the National Council of Teachers of Mathematics stated, Knowledge of both the content and discourse of mathematics is an essential component of teachers’ preparation for the profession. Teachers’ comfort with, and confidence in, their own knowledge of mathematics affects both what they teach and how they teach it. Their conceptions of mathematics shape their choice of worthwhile mathematical tasks, the kinds of learning environments they create, and the discourse in their classrooms (1991, p. 132). Studies (Collias, Pajak, & Rigden, 2000; DarlingHammond, 2000) stressed that teachers’ knowledge is an important influence on student progress. The statistical report of DarlingHammond based on the National Assessment of Educational Progress data concluded that “the proportion of wellqualified teachers is by far the most important determinant of student achievement” (2000, p. 30). The Collias, Pajak, and Rigden 23 (2000) article details the STEP program, an initiative to help university faculty in teacher preparation. The program guides universities to reorganize their teacher preparation programs to involve all faculty in an accountability system for graduates’ knowledge. The authors stated that “What a teacher knows and can do makes the crucial difference in what children learn. In fact teacher expertise is the most important factor in student achievement” (p. 3). Bad teachers can adversely effect student learning and put children at an academic disadvantage. A compilation of research report authored by the Center for the Study of Teaching and Policy found that “several studies showed a positive connection between teachers’ subject matter preparation and … higher student achievement, … particularly in mathematics” (2001, p. 7). The report claimed that good teaching could not take place if teachers lacked conceptual understanding of the subject matter. In an annual report on teacher quality published in 2005, the United States Department of Education stated, “Ensuring that America’s teachers are of the highest quality is an important national priority because they hold the key to student success. Simply put, teachers matter” (USDE & OPE, p. 5). A 2001 government publication, The Nation’s Report Card: Mathematics 2000, stressed the importance of teachers having adequate content knowledge. “To better serve the students they teach, teachers need preparation in the content areas of mathematics that are part of their students’ curriculum” (Braswell, p. 135). Reporting on states’ progress toward meeting the NCLB Act requirement of highly qualified teachers, Walsh and Snyder (2004) concluded that understanding subject matter was a critical skill for teachers. They stated that there was no substitute for 24 subject knowledge. If a teacher did not know a concept, then that concept would not be known to the teacher’s students. Grossman, Wilson, and Shulman (1989) reported on the Knowledge Growth in a Profession Project at Stanford University, investigating the role content knowledge plays in instruction preparation. The Project found that “teachers need to understand their subject matter in ways that promote learning” (p. 24). Subject matter knowledge also influenced content in the elementary classroom and classroom practice. The conclusion of the authors in this report was that beginning teachers are not prepared to transform their own learning into a form appropriate for their students. A solution was to have a subjectspecific knowledge of pedagogy, a methods class. This course would unite pedagogy and content and reinforce teacher knowledge. “Teachers have a better chance of being able to help their pupils develop flexible understandings of subject matter if they understand their subject matter well” (McDiarmid, Ball, & Anderson, 1989, p. 199). McLaughlin and Talbert (1993) suggested that if teachers had only cursory knowledge of their subject matter they would be able to teach only the facts or leave learning up to the students. Liping Ma quoted an old Chinese saying, “Know how, and also know why” (Ma, 1999, p. 108). Teachers must not only know how to manipulate an algorithm but they must know why it makes sense mathematically; therefore, they must have conceptual understanding as well as procedural understanding. This higherlevel understanding can improve teachers’ knowledge of mathematics, thus helping to improve students’ mathematical abilities. According to Ma, how well teachers understand the subject matter they teach directly impacts the level of student learning. 25 In 1989, Ball outlined strategies for change regarding teachers and adequate subject matter knowledge. “Teachers should not just be able to do mathematics; if they are to teach for understanding, they must also have a sense for the mathematical meanings underlying the concepts and procedures” (p. 89). Her article in 1990 on the mathematical understanding of prospective teachers stressed the importance of the depth of knowledge teachers must have to present the subject matter effectively to children. Teachers must not only be able to compute problems accurately, they must be able to explain steps, discuss reasoning, and offer alternative solutions. In later research with Bass, Ball wrote that teachers needed to know more than the subject matter knowledge that students learn “in order to have broad perspective on where their students are heading. Teachers’ own knowledge on the subject affects what they teach and how they teach” (Ball & Bass, 2000, p. 86). Hungerford’s article, based on his classroom experiences with preservice teachers, explained that elementary teachers’ attitude about mathematics influence their students’ attitudes. Teachers who are afraid of mathematics are likely to transfer that feeling to their students. It is unlikely that students will develop an appreciate for mathematics if their teachers have little interest in the subject (1991). Place Value and Elementary Student Knowledge In the first NCTM Standards, a strand in the K4 grade group was number sense and numeration. The section began: Children must understand numbers if they are to make sense of the ways numbers are used in their everyday world. They need to use numbers to quantity, to identify locations, to identify a specific object 26 in a collection, to name, and to measure. Furthermore, an understanding of place value is crucial for later work with numbers and computation. (1989, p. 38). According to Dienes (1960), place value and using base ten is our way to communicate numbers, and it is imperative that students understand this concept. “The notation of the number using place value, with the base of ten is a method of communicating numbers, and it is essential that children should learn the meaning of such communication as effectively as possible” (Dienes, p. 51). Place value concepts create a foundation for student number sense. According to Smith, Lambdin, Lindquist, and Reys, Many young children can count to one hundred or beyond, but actually have little or no sense of what [the] numbers … actually mean. Before students can make sense of addition or subtraction of multidigit numbers, it is essential that they develop a good understanding of place value (2001, p. 48). The Principles and Standards for School Mathematics set goals for thinking and reasoning. The authors recommended that “students develop a deep understanding of important concepts and proficiency with important related skills…This recommendation implied that our students deserve to learn more than the procedures of mathematics – they also need to make sense of mathematics” (Lappan, 2005, p. 1). Tracy and Gibbons tested teaching materials and activities to help elementary students and preservice teachers learn decimals and metrics. They used a number line, decimal squares, videos, the calculator, and the United States Metric Association website 27 to develop classroom lessons. These two teachers found that “the problem lies in that the emphasis on place value tends to be on computation, not conceptual understanding. For students to be successful, they must first understand them [concepts]” (1999, p. 2). Textbook authors Lanier and Taylor created an activities manual for preservice elementary students. In it they stated, “One reason many elementary and middle school students have trouble with mathematics is due in part to only being taught an algorithm with no meaning behind it” (2004, p. 11). These authors suggested that teachers should lay a strong foundation and explain to students why algorithms work. Several research papers have shown that elementary students have difficulty learning place value concepts. Articles by Fuson (1990a, 1990b) began with the premise that children perform poorly on place value tasks. She then explored words used in place value (for example, eleven versus ten one) and proposed new textbook characteristics. Given the problem 527 + 435, more than half of the third graders asked were unable to solve it by “counting on with hundreds, tens, and ones from 527 (527, 627, 727, 827, 927, 937, 947, 957, 958, 959, 960, 961, 962)” (1990a, 368). Fuson indicated that, “Many elementary schoolchildren …. do not fully understand the base ten structure of multidigit number words” (1990a, p. 350). Kamii (1985) conducted tests in classrooms in Chicago for place value understanding on students in grades 1 (n = 13), 4 (n = 35), 6 (n = 48), and 8 (n = 41). These tests were given at the same elementary school and at the junior high school that the students would normally attend afterwards. Children were asked to circle the number of objects that the one represented in the number 16. None of the first graders circled ten, the correct answer. Eighteen out of thirtyfive (51%) fourth graders were correct, 28 compared to twentynine out of fortyeight (60%) sixth graders and thirtytwo out of fortyone (78%) eighth graders. When she discussed these results with two mathematics teachers, “they were not surprised. They said that some children never seem to get place value” (p. 63). Ashlock (1990) found that children had difficulties with computational algorithms because they did not have an adequate understanding of place value. He observed that students could often write numerals correctly but could not explain why they were written that way. Children could identify and name place values, but they could not master the place value system as a whole. Work by Ball included a study with nineteen preservice elementary and secondary teachers. She stated that “Since place value is a fundamental idea and since pupils often find it difficult, it seemed a critical area of prospective teachers’ knowledge to investigate” (1988, p. 90). These preservice teachers were given a number of problems to ‘debug’ or find and explain the error in children’s mathematics problems. Her conclusion was that these prospective teachers did not have an explicit understanding of place value. They were able to compute properly, but they were unable to articulate the underlying concepts of the problems. Numeration Systems Historical numeration systems and base number systems are traditionally part of the elementary education major curriculum. Textbooks provide information and practice for systems such as the Babylonian (base 60), Mayan (modified base 20), and other bases. The rationale for this topic being included in the elementary curriculum is typically that studying these systems help students better understand our base ten. 29 Authors Cathcart, Pothier, Vance, and Bezuk state that we study ancient numeration systems “with a view to developing a better understanding and appreciation of our own system” (2000, p. 124). Research involving numeration systems has taken many forms. Hamilton’s classroom work in the 1960s focused on college student understanding of a new ‘Make Believe Arithmetic.’ This creative number system used inventive symbols for a base six system. He found that the concept of base was difficult for students to understand. The use of Arabic numerals in a number system other than base ten was confusing for the students (1961). Hamilton created a new system of base six using new symbols and new words for the numbers. The class worked on addition and multiplication tables and even constructed a ruler for measurement. With these new symbols students were successful in learning and were able to transfer their understanding to base ten. Sawada and Atkinson (1981) replicated Hamilton’s study with a nondecimal invented base five numeration system. They found that their preservice teachers gained empathy for the difficulties children have in understanding base ten numeration. “Students think they know everything there is to know about the counting numbers, the names they say, and the symbols we write, when …they have little or no insight into the system” (p. 367). These researchers stated that many students learned our number system by rote. They concluded that an intensive study of a single base system using nondigit symbols had value in preservice teacher education. Casebeer examined teaching styles for introducing place value systems to preservice teachers in 1967. Assuming a value to incorporating base numeration system content into the elementary education major mathematics curriculum, Casebeer tested 30 two methods of instruction for two weeks in his mathematics classes for preservice elementary teachers. He developed programmed materials to compare two sequences for introducing base numeration systems. One sequence was based on generalizing our base ten concept to other bases, the more traditional approach. The other sequence was based on sets and grouping of sets. Both sequences were used in all four classes studied, half the students receiving instruction in the sequence one method and then sequence two method and the other half receiving instruction in the sequence two method and then sequence one. He found no differences in student performance between the order of the two teaching techniques. Instruction for this research combined both methods, typically beginning with the traditional approach to establishing a connection to base ten and then using manipulatives to discuss sets. Research conducted by Haukebo (1967) investigated sixtytwo elementary preservice teachers in two mathematics classes and their study of numeration systems. He used three different bases with these future teachers over a three week period. Haukebo’s hypothesis was that no differences in baseten understanding would exist between the group receiving instruction in baseten and the group receiving instruction in base numeration systems as measured by his test of arithmetic understanding. A fiftytwo problem test was constructed to test for arithmetic understanding. Problems involved simple computation, place value problems, and base problems. He concluded that although every group of students increased their arithmetic understanding, differences between the two groups were not significant. In 1972 Skipper studied various teaching methods for the study of numeration systems by preservice teachers. He used three groups of students; one employing two 31 sets of manipulatives in the classroom, one using one set of manipulatives in the classroom, and a lecture only group. Approximately fifty students participated in each group. Tests for knowledge used Skipper’s Test on Numeration System (TONS). This was strictly a base computation test using problems like 0.25 = ______ twelve (p. 158) and 103 five + 244 five = ______ (p. 153). His study was inconclusive on the connection of base numeration systems and better understanding of the decimal system. Even so, as a result of his research, Skipper wrote that he thought preservice elementary teachers could learn base systems and recommended that preservice teachers be exposed to more coursework in numeration systems (1972). Hungerford experimented with base five instruction in his college classroom in 1992. He found that his students were frustrated and skeptical about this newly constructed number system, but he received many positive comments about the lessons at the end of the semester. “The new arithmetic forces students to come to grips with place value. Even though the mechanics look much the same as before, they must think about what’s going on and understand how it really works” (p. 1). Zazkis and Khoury (1994) worked with preservice elementary teachers on place value and nondecimal fractions. Twenty students participated. Students were asked to calculate addition and subtraction problems in other bases and then convert nondecimals to base ten. An example of a problem and its incorrect answer follow: 2.23 four + 3.33 four 21.31 four The correct answer should be 12.22 four (p. 204). 32 Students were interviewed about their answers to analyze their paper and pencil responses. The researchers concluded that “preservice teachers’ constructions of place value number systems are fragile and incomplete” (p. 222). A significant number of the preservice teachers were able to perform addition and subtraction calculations but could not explain their work and identify the proper place value of the digits. In 2003, a research study conducted by McClain was published. She developed a sequence of problems involving a fictional candy factory. Packaging for this factory used base eight mathematics. Candies were packaged eight to a roll and eight rolls to a box. Problems relating to this packaging were given to the class of twentyfour elementary preservice teachers who served as subjects in the study. One problem asked students to determine how many candies were in one box, three rolls, and two pieces. Many preservice teachers initially needed to draw and count individual pieces but later devised more simplified figures as their understanding increased. At the beginning of the experiment, McClain stated that “their [preservice teachers] understandings of both place value and multidigit addition and subtraction were very superficial and grounded in rules for manipulating algorithms” (p. 289). At the end of the project, she reported relative proficiency in the tasks. In FarroLynd’s (2003) research she wrote that “ number base work reinforces many of the concepts and procedures learned in base ten such as place value, regrouping, renaming, and computation algorithms” ( p. 7). Her study involved 104 preservice elementary teachers and place value misconceptions, using a twelve item written test and followup interviews. Students participated in the study while taking the second in a two course mathematics sequence where place value was taught in the first course. The test 33 used a problem mixture of rounding whole numbers, base computations, comparing and ordering numbers, and renaming decimals. Sample problems include “Name a decimal between 0.1 and 0.11 (results were 45% correct), and round 99,721 to the nearest thousand (results were 93% correct)” (p. 124 & p. 126). The three base problem accuracies were 11%, 42%, and 17%. Overall, the 104 students averaged 37% correct on the test. She concluded that these preservice teachers had weak conceptual understanding of place value structure, and she recommended further studies of this population and topic. Research by Rusch (1997) involved 206 preservice elementary teachers throughout the state of Texas. The focus of the study was on the influence of different teaching strategies on place value understanding. Three classes employed a constructivist approach to teaching, and six classes used the direct approach, or lecture. Constructivism employs the idea of students constructing their own learning. Teachers become a ‘guide on the side’ rather than a ‘sage on the stage.’ Rather than lecture, the instructors create situations in which students can discover mathematical concepts. Rusch and Hannigan developed the APVU to determine gains in students’ understanding of place value using the two different teaching approaches. Although gains were made by both groups, no significant difference in understanding between groups was found. Hannigan (1998) followed Rusch’s work with a study of place value understanding and the effectiveness of using writing prompts to enhance that understanding. Four classes of approximately twentyfive students each participated. Two instructors each had a treatment class that received freewriting assignments developed by Hannigan and a control class that received no special assignments. Writing 34 prompts invited students to define an algorithm, describe its logic, and give examples using the algorithm. Again, no difference in gains were found in place value understanding between the treatment group and the control group. Conclusion Preservice elementary teachers are required to take a number of mathematics courses as part of their degree plan. In these courses textbooks have traditionally included bases and different place value systems in their curricula (Bassarear, 2005; Billstein, Libeskind, & Lott, 2007; Cathcart, Pothier,Vance, & Bezuk, 2000). The generally accepted reasons for including these topics are twofold: 1) to acquire an explicit understanding of place value, and 2) to gain perspective into elementary students’ difficulties in learning our number system. Does studying base numeration systems help preservice teachers build on their understanding of place value? Place value is a major component of the HinduArabic numeration system. In the early 1960s Dienes stressed the essential nature of place value in communicating the meaning of numbers. The National Council of Teachers of Mathematics promoted the importance of students’ learning place value concepts in its first Standards (1989) within the strand of number sense and numeration, and it continues to emphasize this topic. Research has shown a connection between place value understanding and mathematical competency. “Knowledge of place value has great implications for success in arithmetic tasks…Therefore, care should be taken that students develop a meaningful understanding of numbers” (Cathcart, Pothier,Vance, & Bezuk, 2000, p. 106). Yet elementary students do not have an adequate understanding of place value. Studies by 35 Fuson (1990a, 1990b), Kamii (1985), and Ashlock (1990) found primary student knowledge of place value to be lacking. Based on the findings of previous studies, is this lack of understanding because elementary teachers have a poor understanding of the concepts themselves? Ball (1988, 1990), McDiarmid and Wilson (1992), and Zazkis and Khoury (1994) concluded that elementary and preservice teachers knew facts and could memorize algorithms, but did not understand the underlying mathematical processes. This lack of place value knowledge disrupted the teaching and learning of number concept development and computation. Research has been conducted on the place value understanding of elementary preservice teachers ( Hannigan, 1998; Rusch, 1997). Research has been conducted focusing on numeration systems and elementary preservice teachers’ understanding (Casebeer, 1967; FarroLynd, 2003; Skipper, 1972). Yet no research has been found that attempted to connect numeration systems and place value understanding of elementary preservice teachers. The researcher found no studies utilizing the APVU other than the work by its two authors. Zazkis (1999) wrote that preservice teacher instruction should “provide them with experiences and challenges that reexamine and enrich their understanding of mathematics and its pedagogy” (p. 650). Currently, numeration systems are an accepted part of the elementary preservice teacher mathematics curriculum. With today’s call for highly qualified educators, all teacher educators should examine their curriculum to assure rigorous content. This study attempts to add to the dialogue on the mathematical preparation and required curriculum content established for elementary preservice 36 teachers by investigating the study of numeration systems and their influence on place value understanding. 37 CHAPTER 3 METHODOLOGY This research was a quantitative quasiexperimental/control group design study. The objective was to compare gains of explicit place value understanding possessed by two classes of elementary preservice teachers taking a required mathematics course. The course curriculum included a section on basenumeration systems. The independent variable in the study was the instruction, or withholding of instruction, of basenumeration systems. The dependent variable was growth in explicit place value understanding as measured by the Assessment of Place Value Understanding (APVU) instrument. The four research questions guiding this study were: 1. What is the level of explicit place value understanding that the elementary preservice teachers participating in the study bring with them to their required mathematics content course? 2. What is the level of explicit place value understanding that the elementary preservice teachers participating in the study have at the end of their required mathematics content course? 3. Is there any significant change in the participants’ explicit place value understanding from the beginning of the semester to the end of the semester? 4. Is there a significant difference in gains made in explicit place value understanding 38 between the class that has numeration system instruction (experimental group) and the class that does not have the numeration system instruction (control group)? Participants All of the participants in the study were undergraduate college students pursuing a career in elementary education. Normally the students taking the courses used in this study are sophomores and juniors. The researcher presumed that all students had some preexisting knowledge about the place value concepts and representation in our baseten numeration system. The two classes used in the study were chosen by convenience. Initially, a number of universities expressed interest in helping with this research. Many schools were eliminated due to the semester scheduling of the mathematics content courses. Other schools did not follow through on their first commitment. Only one instructor other than the researcher agreed to aid in this study. This challenge of research as outlined in an article by Heid et al. (2006) reinforces the concern that linking research and practice is a problem. Even with the need for more schoolbased research, teachers are reluctant to allow their classrooms to be laboratories for experimental, unproven content and changes in pedagogy. Conversely, control groups are difficult to assign if teachers perceive the intervention of the researcher to be impractical, unusable, or intrusive. One class of approximately 35 students was taught by an instructor at a regional, medium sized public university (experimental group). One class of approximately 25 students was taught by the researcher at a small private university (control group). The two colleges are in close geographic proximity to each other in the same suburban area. The instructors were both experienced teachers with many years of college teaching 39 experience, and they were friends and fellow students in the same doctoral program. All students in the two classes were invited to participate in the research project. Comparison of Classes Both mathematics courses were components of the required degree plan at their respective universities and were designed specifically for elementary education majors with enrollments limited to that group of majors. The curriculum in both courses used in the study normally includes sections on numeration systems. At the private university, the course was usually taken as the second or third course in a four course sequence. At the public university, the course could be taken second, third, or fourth in a similar fourcourse required sequence. Neither school had a specific mathematics methods course for elementary education majors. Since two different texts were used in the courses at the two schools, content was compared. Table 1 gives the number of weeks that the major course content at the two schools was presented as well as the order in which the material was taught. Although material in the two courses was not identical, the basic place value topics were similar as well as the focus objectives on numeration systems. Table 2 details the specific numeration system content taught by each instruction during the semester. The instructional style of the professors was not a component of this research. Each instructor was free to use any method of her choice, including lectures, modeling, and videos. However, it was known to the researcher that both instructors used a constructivist approach and employed many manipulatives in their classrooms. Handson activities include work with baseten and multibase blocks. 40 Table 1 Comparison of Course Content__________________________________________ Numeration system instruction Without With (Control group) (Experimental group) Topic Number of Weeks Topic Number of Weeks Problem solving One Problem solving One Sets One Number theory Two Prenumber skills Onehalf Sets One & onehalf Numeration systems Egyptian, Roman, HinduArabic, time One & onehalf Prenumber skills Onehalf 4 basic operations Two Numeration systems Two & onehalf Integers One 4 basic operations Three Number theory One & onehalf Rationals One & onehalf Decimals One Base numeration systems One 41 Table 2 Detail Numeration System Content____________________________________________ Numeration system instruction Without With_________________ HinduArabic, baseten Egyptian Egyptian Roman Roman Chinese Time Greek Time delay for study HinduArabic, baseten Babylonian Babylonian Mayan Mayan Bases (multi – 2, 3, 4, 5, others) Bases (multi – 2, 3, 4, 5, others) Note: Content in bold taught after the posttest for this research. The researcher (control group) delayed teaching the base numeration system content section of the course curriculum while the study was in place. Only simple information about these numeration systems was introduced. In particular, notation of the base systems was introduced so that students’ work on the second test would better reflect their understanding of place value concepts rather than their lack of knowledge about the problem structure. For example, the researcher explained to her students that 234 eight indicated a number in the base eight numeration system, not a HinduArabic base ten number. Base system instruction was given at the end of the semester after the test was administered for the second time and the study had been completed. At that time the researcher taught the concepts of base numeration systems, including addition, 42 subtraction, and multiplication. Sections from the syllabus delayed for the study were taught using textbook material. The instructor (experimental group) taught base numeration systems as usual which involved extending the use of baseten blocks for the HinduArabic system to multibase blocks for the different number bases. Babylonian and Mayan systems were also studied to show the variety of base numeration systems in other contexts. Class notes state: As early numeration systems began to use grouping they became conceptually harder, and at the same time easier to use on a mechanical basis. In our HinduArabic system we group by tens. Children need experience grouping by twos, threes, . . . all the way to tens. The idea of an exchange point is the key to understanding place value. Also, grouping using smaller numbers provides more practice than waiting until ten objects have been collected. (Parrott, 2007, p. 89). The following three pages were taken from the instructor’s class notes for the experimental group to provide some insight as to the method of teaching of these numeration systems and the instructional approaches employed. 43 44 45 (Parrott, 2007, pp. 115, 118,119). 46 Both instructors taught traditional place value concepts, including expanded notation, and the HinduArabic system (our baseten system). “Place value assigns a value to a digit depending on its placement in a numeral” (Billstein, Libeskind, & Lott, 2007, p. 155). An example of a place value question from the control group text would be to give the place value of the underlined numeral: 827, 367 (Billstein, Libeskind, & Lott, 2007, p. 166). An example of a problem from the experimental group class: “Consider the numbers 40 and 400 in the HinduArabic numeration system. What does the 4 represent in each number?” (Angel & Porter, 1997, p. 180). Expanded notation is illustrated by the number 1234 = (1•1000) + (2•100) + (3•10) + (4•1). Both texts ask students to write numerals in expanded form and the reverse. Both instructors used materials such as the base ten blocks, place value charts, chip trading, and Cuisenaire rods to reinforce baseten concept development. Data Collection Data collection consisted of three parts: 1. Collection of demographic information on all participants. 2. Administration of Form A of the Assessment of Place Value Understanding (APVU) instrument at the beginning of the semester before numeration system concepts and place value were taught. 3. Administration of the same form of the APVU near the end of the semester after numeration system concepts and place value were/were not taught in the individual class sections. The APVU instruments were given as part of the normal classroom requirements. Anonymous demographic information was collected, as were the average ACT scores of 47 the two groups which were compared to check for equivalence between the groups. Consent to use the scores was included in consent forms acknowledging subjects’ participation in the project. Students were allowed approximately 45 minutes to take each one of the two assessments at each location. This time allotment was similar to the time given in the test developers’ previous studies. Administration of both the first test and second test was scheduled as close together as possible at the two schools to maintain similar total class time at each location. The first administration of the APVU test was given to each group at about the third week of the semester. This was later than planned due to inclement weather, but no place value concepts had been taught at that point in the semester. The second administration of the APVU test was given by the researcher during week fourteen of the semester, allowing enough time for the base numeration systems content to have been taught to the control group before the end of the semester. The instructor of the experimental group administered the second test during the last week of the semester. Instrument The Assessment of Place Value Understanding (APVU, pretest and parallel posttest) was developed by Rusch and Hannigan in 1997 as part of their Ph.D. work at the University of Texas. For this study only the pretest form of the test was used. This choice was made due to the original authors’ lack of confidence and documentation regarding the comparability of the initial two forms. After its introductory use, Hannigan (1998), one of the APVU authors, expressed concern in her dissertation about some problems which arose indicating that the two forms may not have been truly parallel. 48 One minor change was made in the original test by this researcher. For questions 3, 8, and 13, relating to numberbase problems, the researcher asked the students to be more specific about describing their processes in calculating the solutions. No change was made in the rubrics for these problems. This assessment was based on a matrix of place value grouping schemes shown in Table 3 from Rusch. The APVU was piloted multiple times as the authors continued to revise and refine the questions. According to Rusch, “It was necessary to pilot the assessment several times to ensure clarity in the communication of each question, validity of the questions, and to control for the time required to complete the assessment. Furthermore, rubrics for assessing the responses to the assessment questions were developed and carefully tested for clarity and interrater reliability” (1997, p. 51). The instrument was first piloted orally with sixteen elementary preservice teachers at the University of Texas. Content was analyzed, and a second handwritten test was given to the same sixteen participants to develop a time frame for giving the test in a paperpencil format. A third pilot was conducted with nineteen students at Austin Community College and 265 University of Texas students. A stratified sample of 45 tests was selected for use in the development and analysis of the rubrics. Five outside evaluators scored the test responses to check the reliability of the assessment rubrics. During the course of their research, the authors, Rusch and Hannigan, used the APVU for thirteen classes and 312 students (Rusch, 1997). 49 Table 3 Matrix of Explicit Place Value Understanding___________________________________ Systematic NonSystematic Systematic grouping schemes Nonsystematic grouping are those in which the number schemes are those in which the of items required to create one number of items required to group of the next place value create one group of the next remains constant. place value varies from place to place. Familiar A familiar grouping scheme is The Base 10 System Time one which most preservice seconds, minutes, hours, days, teachers have come across in weeks, . . . both the traditional mathematics curriculum and Metric Measurement Imperial Measurement in their daily living routines. inches, feet, yards, … cups, pints, quarts, gallons, … Unfamiliar An unfamiliar grouping Base n Systems Foreign Coinage scheme is one which uses a place value structure, but (other than base 10) British pence, shilling, which preservice teachers may pound, … not have come across in either Base 2, Base 5, Base 16 their school years or their daily life. (1997, p. 38) 50 The tests used in this study were graded using the Rusch  Hannigan rubrics. Rater reliability was checked by analyzing scoring differences between the two instructors on a small sample of tests. Twenty percent of the first test was scored by the instructors. Results of this analysis are discussed in Chapter 4. The rubrics employ a 1 – 5 scale. The numbers in the scale indicate the level of student understanding based on the descriptions found. 1. Algorithmic: Knowledge of how to manipulate symbols to get an "answer." This level of mathematical knowledge does not necessarily imply understanding of why the algorithmic steps make sense. 2. Tacit: Algorithmic knowledge with some intuitive understanding of the logical foundation from which the concept or algorithm emerges. This understanding is, however, somewhat vague and as a result it is difficult for the individual to articulate the logic which brings meaning to the concept. 3. Explicit: Precise understanding of the concept being addressed which can be clearly articulated and convincingly justified. Explicit understanding includes a clear understanding of the connections between the concept being addressed and related concepts, as well as an ability to articulate the logical development of related algorithmic procedures and generalizations. 4. Pedagogical (Self and Peers): Knowledge of how they (self) and individuals of similar age and experience (peers) build mathematical understanding. Able to thoughtfully choose pedagogical tools and strategies and are able to use language effectively to teach peers. 51 5. Pedagogical(Children): Knowledge of how children build mathematical understanding. Able to thoughtfully choose pedagogical tools and strategies and are able to use language effectively to teach children (Rusch, 1997, p. 34). Data Analysis Using Rusch’s and Hannigan’s work as a guide, the researcher used simple means and standard deviations for a total score to partially answer questions one and two. Question one: What is the level of explicit place value understanding that the elementary preservice teachers participating in the study bring with them to their required mathematics content course? Question two: What is the level of explicit place value understanding that the elementary preservice teachers participating in the study have at the end of their required mathematics content course? These statistics were compared using an independent sample ttest. A univariate analysis of variance was used to compare means in the four cells of the matrix (familiar systematic, familiar nonsystematic, unfamiliar systematic, and unfamiliar nonsystematic). Boxplots were constructed to give more specific information about the data. Correlations were calculated to examine possible connections within the data. To answer questions three and four, again total mean scores were compared – pre versus post means in the four cells, and means for rubric scores were compared. Question three: Is there any significant change in the participants’ explicit place value understanding from the beginning of the semester to the end of the semester? Question four: Is there a significant difference in gains made in explicit place value understanding between the group that had numeration system instruction (experimental group) and the 52 group that did not have the numeration system instruction (control group)? Analysis was done with a splitplot ANOVA for question three. Gains were calculated for question four, and then an independent ttest was utilized to discover significant differences. Of particular interest to the researcher were comparisons in the unfamiliar systematic grouping cell, as this in particular focused on nondecimal base systems, an important facet of this study. A modified Bonferroni correction was used to determine . Normally used at the 0.05 level, due to the large number of dependent variables, was recalculated to 0.05/11 = 0.0045. No qualitative data was collected, although many test questions asked the elementary preservice teachers to explain their answers. An example was question one. This question asked subjects to analyze an elementary student’s error pattern. As part of the question, the elementary preservice teacher is asked to “Briefly describe your assessment of what Caroline does not understand” (Rusch, 1997, p. 124). These comments were helpful as descriptive data to accompany the quantitative results. Conclusion This research was designed to assess place value understanding of preservice elementary teachers. The APVU was developed in 1997 just for this population; thus, it was chosen as the instrument to be used. Due to the lack of teacher commitment, only two classes participated in the project. These classes were compared by teacher, by content, and by student, with the help of a demographic sheet. The researcher used one form of the APVU as a pretest and posttest during the semester. Timing for these pretests and posttests was coordinated between the instructors of both groups. The assessment 53 tasks were scored and analyzed by the researcher in three categories: total, cell or grouping, and rubric. Comparisons were drawn on pretest and posttest means. To validate earlier literature and research, the hypothesis of this study is that the class receiving normal instruction in nonbaseten numeration systems (experimental group) will show a significant gain in scores, first exam to second, when compared to the without instruction class of students (control group). Of particular interest to the researcher are the gains for both groups in the unfamiliar systematic cell which contained the base numeration systems problems. Chapter Four will use a number of statistical tests and graphs to analyze the pretest and posttest scores of the two groups. These results can then be used to answer the research questions. Conclusions and recommendations will be stated in Chapter Five. 54 CHAPTER 4 FINDINGS This study investigated the influence of base numeration instruction on place value understanding for elementary preservice education majors. Two university classes of elementary preservice teachers were used to test the hypothesis that base numeration system instruction would increase the conceptual understanding of place value for the participants. Using a sample of convenience, the researcher and another instructor administered a pretest to each class near the beginning of the semester. During the semester the researcher’s class (control group) did not receive base system instruction. This instruction was incorporated at the end of the semester after the study was completed. During the semester the other instructor taught her class (experimental group) following the usual syllabus, including base system content. At the end of the semester the same test was given to both groups. The instrument used was the Assessment of Place Value Understanding (APVU). This thirteen item test was developed by Rusch (1997) and Hannigan (1998) at the University of Texas to specifically test place value understandingof preservice elementary teachers. The assessment can be found in Appendix A of this study. Four questions guided this research: 1. What is the level of explicit place value understanding that the elementary preservice 55 teachers participating in the study bring with them to their required mathematics content course? 2. What is the level of explicit place value understanding that the elementary preservice teachers participating in the study have at the end of their required mathematics content course? 3. Is there any significant change in the participants’ explicit place value understanding from the beginning of the semester to the end of the semester? 4. Is there a significant difference in gains made in explicit place value understanding between the class that has numeration system instruction (experimental group) and the class that does not have the numeration system instruction (control group)? Demographic Information During the semester of research, two classes of elementary education major students were invited to participate. One, at a private university, was taught by the researcher. One, at a public university, was taught by an instructor known to the researcher. Both universities are located in the same metropolitan area of a southwestern United States city. Each student was required to take the Assessment of Place Value Understanding (APVU) twice, once at the beginning of the semester and again near the end of the semester. Students were then asked by the researcher to use their test results for this study. Only scores for students who completed these three components were used in this research. This reduced the sizes of the samples to 23 for the control group and 32 for the experimental group. 56 These 55 students then completed a demographic information sheet. The main purpose for this additional request was to give the researcher some perspective about the students and to check for similarity of groups. In particular, ACT scores were solicited. This selfreported data is summarized in Table 4 Due to the small sample sizes, equivalency testing for the classes was not attempted. Table 4 ACT Mean Scores_______________________________________________________ Numeration system instruction Without With Number of respondents 8 14 Percent of respondents 35% 44% Mean 22.75 19.71 Other demographic information revealed that all students were female. Only two – at the public university – were parttime students; the other 53 reported fulltime student status. All students were taking their second or third college mathematics course. A summary of other demographic information is show in Table 5. Rater Reliability After data collection, a twenty percent sample of the pretests was scored by the researcher and the instructor. This was implemented to assess interrater reliability and measurement accuracy. Discussion of the scores between the raters then allowed for refinement of rubric meaning. The rubrics established by the authors of the APVU utilize 57 Table 5 Demographic Information__________________________________________________ Numeration system instruction Without With Average age Range 19.6 18  22 29.4 21  49 Average GPA Range 3.34 2  4 3.28 1.8  3.86 a one to five point scale to rate student understanding with : (1) algorithmic understanding; (2) tacit understanding; (3) explicit understanding; (4) self and peer pedagogical understanding; and (5) child pedagogical understanding. The data for this sample is summarized in Table 6. These results paralleled results previously obtained by the authors of the APVU; thus, it was determined that there was sufficient consistency to allow only the researcher to score all tasks and tests. Table 6 APVU Sample Scoring_____________________________________________________ Number of items Percent of items Differ by 0 322 86.6 Differ by 1 37 9.9 Differ by 2 10 2.7 Differ by 3 3 0.8 Differ by 4 0 0 58 Question One The first research question was: What is the level of explicit place value understanding that the elementary preservice teachers participating in the study bring with them to their required mathematics content course? This question was initially analyzed for all tasks for both the without instruction (control group) and instruction students (experimental group). A sample mean and standard deviation was calculated for each group’s scores. This information is summarized in Table 7. Table 7 All Tasks Pretest Item Mean Scores Numeration system instruction Without With Number of participants 23 32 Mean 1.95 1.95 Standard deviation 0.73 0.49 These scores indicated that the students achieved the same baseline results. No significant difference was found in the means. The boxplots in Figure 1 show somewhat similar distributions with similar medians of 1.81 and 1.845. As the standard deviations indicate, the noninstruction students’ scores varied more. A score close to two on the scale of mathematical understanding used in this research indicates some intuitive knowledge of the concept foundation. However, this understanding is somewhat vague, and the student may have difficulty articulating the logic behind their work. 59 Figure 1 To further define scores and place value understanding, the pretest scores were analyzed by cell. Matrix cell definitions were presented in Chapter 3. A table identifying the relationship of cell content and problems from the APVU is found in Appendix B. Pretest cell scores for both sites are presented in Table 8. Means and standard deviations were calculated for each cell and analyzed using a univariate analysis of variance to determine significant differences in the baseline scores of the two groups. Best scores were achieved in the familiar systematic cell. These problems focused on the vocabulary of place value understanding and the ability of students to evaluate elementary student work. Scores approaching three on the rubric scale indicate more precise understanding of the concepts. 60 Table 8 Pretest Cell Means________________________________________________________ Numeration system instruction Without With Familiar systematic 2.60 (.87) 2.43 (.70) Familiar nonsystematic 2.26 (1.20) 2.40 (1.16) Unfamiliar systematic 1.30 (.75) 1.32 (.52) Unfamiliar nonsystematic 2.29 (1.34) 2.38 (1.00) Note: Standard deviation in parentheses, * p < .0045 The familiar nonsystematic cell problems dealt with time problems. Scores around two again indicated some intuitive knowledge; however, the large standard deviations showed a broad range of scores (and therefore knowledge) among the two groups. Lowest scores were discovered in the unfamiliar systematic cell. Tasks here required the students to apply place value concepts in different base numeration systems. Some problems were typical base computation problems, while others required creativity and sophistication to interpret the task. A score of one on the rubric indicates a basic knowledge of an algorithm, but does not imply further understanding. The unfamiliar nonsystematic cell problems presented students with an unusual setting for place value 61 concepts. Results here were similar to the familiar nonsystematic cell, although higher standard deviations indicate a larger range of values. As with the “all tasks” scores, the cell analyses revealed similar baseline results. All cells produced no significant differences and homogeneity of variance. To further explore the pretest scores, means and standard deviations were calculated for the tasks by rubric. The seven rubrics used are defined in Appendix C. A table showing the correlation of rubrics and problems can be found in Appendix D. Results for this rubric analysis are summarized in Table 9. Highest scores were found with the error reproduction rubric. This rubric was utilized with question numbers 1 and 12 on the test – debugging elementary student work. A score of three, explicit understanding, indicates the students gave an inaccurate reproduction of the error, but there was some evidence of partial recognition of the error pattern. No significant differences were found among the scores on any of the rubrics. Standard deviations were above one (or one level of understanding) for the without instruction students’ Error Reproduction and both sets of students analysis of symbolic representation. This rubric was used with the foreign coinage problems, numbers 9 and 10. A level two, tacit understanding, for this rubric indicates the students used base ten techniques to arrive at an answer, not place value representation. Thus, for question one of this research, students at both schools achieved similar baseline results. The overall averages of 1.95 showed almost a tacit level of understanding. The more specific numbers of the cells and rubrics showed a range of 62 understanding of algorithmic (choice of correct representations) to explicit (error reproduction), with most cells and rubrics in the level 2 tacit range. Table 9 Pretest Rubric Means______________________________________________________ Numeration system instruction Rubric Without With Error reproduction 3.41 (1.19) 3.05 (.92) Depth of analysis 2.00 (.83) 1.89 (.70) Use of descriptive language 1.73 (.68) 1.66 (.67) Choice of correct representations 1.13 (.34) 1.16 (.37) Accurate computation 2.27 (.78) 2.30 (.50) Analysis of comp. method 1.45 (.91) 1.53 (.59) Analysis of sym. representation 2.04 (1.42) 2.31 (1.18) Note: Standard deviation in parentheses, * p < .0045 63 Question Two The second research question was: What is the level of explicit place value understanding that the elementary preservice teachers participating in the study have at the end of their required mathematics content course? This question was initially analyzed for all tasks for the public and private university students. A sample mean and standard deviation was calculated for each school’s student scores. This information is summarized in Table 10 . Table 10 All Tasks Posttest Item Mean Scores __________________________________________ Numeration system instruction Without With Number of participants 23 32 Mean 2.35 2.59 Standard deviation 0.53 0.70 Similar analyses were conducted with the posttest data as with the pretest data. A comparison of the posttest means showed no significant difference (p = 0.16) in the level of explicit understanding of place value concepts. Again boxplots were constructed from the data to show median differences of 2.19 versus 2.61 in Figure 2. Scores were more evenly distributed within the instruction class of students. To look for specific differences or levels of understanding, the posttest scores were analyzed by cell and by rubric, similar to the analyses of the pretest scores. The results are summarized in Tables 11 and 12. 64 Figure 2 Significant differences were found in the unfamiliar nonsystematic cell. This grouping of problem topics included the otherthanbaseten base problems, numbers three, five, six, eight, eleven, and thirteen on the APVU. Some of these problems involved basic computation in the other bases; some involved a higher level of problemsolving as well as base place value knowledge to complete the questions. The difference in scores was anticipated by the researcher. Prior to the posttest, the researcher’s students (control group) received only instruction on base ten problems and place value concepts. The instructor’s students (experimental group) had over two weeks of instruction with different base systems and had classroom experience with base computation. Although analyses showed no significant differences in mean scores in the other three cells, the boxplots in Figures 3  6 show clear differences in distribution. In two 65 cells, the familiar systematic cell dealing with children’s error patterns and the familiar nonsystemic cell of time problems, the without instruction students (control group) had a higher 25th, 50th, and 75th percentile. Table 11 Posttest Cell Means__________________________________________________ Numeration system instruction_____ Without With Familiar systematic 3.00 (.76) 2.80 (.80) Familiar nonsystematic 3.33 ** (1.02) 2.56 ** (1.05) Unfamiliar systematic 1.53 * (.51) 2.47 * (.96) Unfamiliar nonsystematic 2.57 (.81) 2.60 (.88) Note: Standard deviation in parentheses, * p < .0045, ** p < .05 66 Figure 3 Data NonInstruction Instruction 5 4 3 2 1 3 2.75 Familiar Systematic Cell Posttest Mean Scores Figure 4 Data NonInstruction Instruction 5 4 3 2 1 3.25 2.375 Familiar NonSystematic Cell Posttest Mean Scores 67 Figure 5 Data NonInstruction Instruction 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 1.46 2.615 Unfamiliar Systematic Cell Posttest Mean Scores Figure 6 Data NonInstruction Instruction 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 2.67 2.83 Unfamiliar NonSystematic Cell Posttest Mean Scores 68 The analysis by rubric identified no areas of significant difference at the = 0.0045 level. However, differences were apparent at the 0.05 level. These rubrics included: choice of correct representation, accurate computation, and analysis of computation method. Since all of these rubrics were primarily used for the unfamiliar systematic cell problems, it would follow that significance in these cells would bring significance in the rubrics. A table showing the correlation of rubrics and problems can be found in Appendix D. Table 12 Posttest Rubric Means_____________________________________________________ Numeration system instruction Without With Error reproduction 3.87 (1.11) 3.33 (.99) Depth of analysis 2.32 (.71) 2.11 (.64) Use of descriptive language 2.01 (.65) 1.95 (.69) Choice of correct representations 1.04 ** (.21) 1.25 ** (.44) Accurate computation 2.64 ** (.57) 3.10 ** (.86) Analysis of comp. method 2.00 ** (.63) 2.58 ** (.95) 69 Within With Analysis of sym. representation 2.48 (.81) 2.53 (.98) Note: Standard deviation in parentheses, * p < .0045, ** p < .05 Tables 13 and 14 show the correlations of posttest rubrics with each other. Significant correlations showing linear relationships exist between many of the rubrics. These results are consistent with the rubric/task correlation table found in Appendix D. Table 13 Without Instruction Rubric Correlations___________________________________ DA DL AC CM SR CR ER .558 ** .422 * .357 .392 .175 .222 DA .937 ** .530 ** .639 ** .538 ** .210 DL .470 * .589 ** .558 ** .081 AC .888 ** .652 ** .222 CM .581 ** .116 CR .141 Note: * p < .05, ** p < .01 ER = Error Reproduction; DA = Depth of Analysis; DL = Use of Descriptive Language; AC = Accurate Computation; CM = Analysis of Computation Method; CR = Choice of Correct Representation; SR = Analysis of Symbolic Representation 70 Table 14 With Instruction Rubric Correlations______________________________________ DA DL AC CM SR CR ER .730 ** .696** .590** .479** .207 .083 DA .930 ** .531 ** .564 ** .040 .243 DL .563** .571 ** .105 .201 AC .926 ** .531** .055 CM .433* .104 CR .244 Note: * p < .05, ** p < .01 ER = Error Reproduction; DA = Depth of Analysis; DL = Use of Descriptive Language; AC = Accurate Computation; CM = Analysis of Computation Method; CR = Choice of Correct Representation; SR = Analysis of Symbolic Representation Question Three The third research question was: Is there any significant change in the participants’ explicit place value understanding from the beginning of the semester to the end of the semester? To analyze the data, a repeated measure analysis of variance statistic was calculated. As with the previous two questions, data was compared by pretest and posttest across all tasks (in total), by cell, and then by rubric. For the pretest and posttest all task means, no significant differences were 71 found. This can be explained by reexamining the data discussion for question 2. Posttest averages for the without instruction students (control group) were higher in the cell for the time problems. Posttest averages for the instruction students (experimental group) were significantly higher in the cell for the base problems. Thus, these posttest differences contributed to no overall difference between pretest and posttest means. Table 15 summarizes these results. An ANOVA performed on the pretest/posttest scores for each of the two classes did find differences in each class (p = 0.04 and p = 0.00, respectively). This showed there was some growth in the participant’s place value understanding. Table 15 Change in Understanding from Pretest to Posttest_______________________________ Numeration system instruction Without With Pretest 1.95 1.95 Posttest 2.35 2.59 The repeated measures ANOVA indicated significant differences in the unfamiliar systematic cell. This demonstrated the significant effect of instruction on the pretest to posttest change of scores in the base numeration grouping of problems. These results are similar to the posttest analyses. In the unfamiliar cell, the instruction students increased their average scores by more than one level of understanding. Unfortunately, even their ending score of 2.47 indicates less than a precise understanding of the place value concepts. Table 16 provides the summary of analysis of cells. 72 Table 16 Change in Understanding from Pretest to Posttest by Matrix Cell___________________ Numeration system instruction Without With Familiar systematic Pretest Posttest 2.60 3.00 2.43 2.80 Familiar nonsystematic Pretest Posttest 2.26 3.33 2.40 2.56 Unfamiliar systematic Pretest Posttest 1.30 1.53 * 1.32 2.47 * Unfamiliar nonsystematic Pretest Posttest 2.29 2.57 2.38 2.60 Note: * p < .0045, ** p < .05 Analysis by rubric found no significant differences between the two groups. Growth was achieved on tasks in every rubric, but there was not sufficient evidence that instruction had an effect on the change in scores. Only the error reproduction rubric indicates a reasonably sophisticated level of understanding. A Level 3 score shows a partial recognition of the error pattern, while a Level 4 score gives evidence that the error is understood as only a computational error. 73 Correct answers to Problem one – Caroline’s Addition Error – are 7110, 757, and 1112. The error pattern work for Problem one should look like: 3 252 + 585 7110 An example of a level 4 score is shown in “Mary’s” work – 719, 757, and 1112. In this case Mary recognized the pattern correctly and then added incorrectly – 2 + 3 + 5 = 9. An example of a level 3 score is shown in “Cindy’s” work – 739, 757, and 3011. She recognized Caroline’s basic error of adding left to right, but didn’t replicate the second error of carrying the wrong digit. For example: 1 252 + 585 739 Rubric analyses are summarized in Table 17. Table 17 Change in Understanding from Pretest to Posttest by Rubric_______________________ Numeration system instruction Rubric Without With Error reproduction Pretest Posttest 3.41 3.87 3.05 3.33 Depth of analysis Pretest Posttest 2.00 2.32 1.89 2.11 74 Without With Use of descriptive language Pretest Posttest 1.73 2.01 1.66 1.95 Choice of correct representations Pretest Posttest 1.13 1.14 1.16 1.25 Analysis of comp. method Pretest Posttest 1.45 2.00 1.53 2.58 Analysis of sym.representation Pretest Posttest 2.04 2.48 2.31 2.53 Note: * p < .0045, ** p < .05 Question Four The fourth research question was: Is there a significant difference in gains made in explicit place value understanding between the class that has numeration system instruction (experimental group) and the class that does not have the numeration system instruction (control group)? Independent sample t tests were calculated for the gains for each group in total, in cells, and in rubrics. Only the unfamiliar systematic cell revealed a significant difference with a mean gain of 0.2335 for the without instruction students (control group) and a mean gain of 1.1522 for the instruction students (p = 0.00) (experimental group). The scores in 75 this cell reflect knowledge of problems relating to the research focus, base n problems where n 10. Final means in this cell were 1.53 for the without instruction control group students. On the scale of explicit understanding used for this research, this score indicates the students possessed an understanding between algorithmic meaning (process with no meaning) and tacit meaning (intuition of the meaning behind the process). The final mean of 2.47 in the base n cell for the instruction student scores’ (experimental group) indicates an understanding between tacit meaning and explicit meaning (clear understandings of the connections). Conclusion Beginning scores for both groups of students on the APVU indicated a similar baseline level of understanding of place value concepts. The overall average of 1.95 for both groups showed almost a tacit level of understanding – some intuitive knowledge, but vague understanding and poor articulation of the mathematical logic. This similarity of scores carried through the cell analyses and rubric analyses. No significant differences were found between the two groups. Means for the pretest were generally in the level one and level two range. Level one indicates algorithmic understanding, computation without meaning. Both groups showed an increase in scores in total, by cell, and by rubric. However, this increase still left many means in the algorithmic and tacit levels. Overall, both group means were in the two range, 2.35 and 2.59. Cell and rubric means were also generally at the tacit level of understanding. For the without instruction students, only the familiar systematic cell (error pattern 76 debugging) and the familiar nonsystematic cell (time) indicated an explicit level of understanding. For the instruction students, none of the cell means were above the tacit level. Rubric analyses indicated the best understanding for all subjects was in the area of error reproduction. Here all scores were in level 3, explicit understanding. However, on these debugging problems, students were not able to use descriptive language or show a depth of analysis. Analysis of data leads to interpretation of this data and its implications. These numbers must be put in perspective and related to the research questions of this study. Chapter Five will provide a summary of these results along with implications of this research and recommendations for further study. 77 CHAPTER 5 CONCLUSION Summary As the No Child Left Behind Act is revamped and renewed, there is much debate about highly qualified teachers and improved student learning. New studies are investigating the mathematics content knowledge and the skills that teachers must demonstrate to improve student achievement. Questions are being raised about what mathematicians/mathematics educators can do to improve the mathematical preparation of teachers. New work by Thames (2006) reports that many college courses today leave teachers unprepared, lacking sufficient mathematical knowledge needed for competent teaching. This, in turn, affects student learning. Perhaps more “rigorous content relevant to teaching and learning” needs to be explored (p. 2). That search for meaningful curriculum has been the focus of this research. At present, in most mathematics sequences for elementary education majors, the topic of base numeration systems is studied. Authors and researchers give rationale for this content, but is that rationale correct? Does this subject help students acquire an explicit understanding of place value? This study investigated the role of base numeration system instruction as it relates to place value understanding in preservice elementary teachers. To accomplish this task, the Assessment of Place Value Understanding (APVU) was administered to fiftyfive 78 preservice elementary teachers at two universities in the southwest. These students were enrolled in a required mathematics course developed specifically for elementary education majors. The APVU was chosen to ascertain knowledge because it was specifically developed for this population to measure place value understanding. This assessment instrument connected the population of elementary preservice teachers and place value concepts and included a section of problems involving base numeration systems. These three elements were components of this research. This chapter provides a brief summary of the numerical findings of this research and reviews the four guiding questions. Conclusions about preservice elementary teacher place value understanding are discussed using the rubrics and the five levels of understanding associated with the assessment instrument, the APVU. These conclusions are then compared to previous studies. Limitations of the study are listed before improvements and changes lead to recommendations for future research. During the course of a single semester, one version of the APVU was given twice to each of the two groups, once early in the semester (pretest) and once near or at the end of the semester (posttest). During the weeks of class, regular instruction was provided to the students at the public university by an instructor known to the researcher. No changes were made to the instructor’s methods or syllabus. The conventional curriculum covered such topics as base ten place value and base numerations systems as part of the content. This group of subjects was identified as the experimental group. During the same weeks, the private university students received instruction from the researcher in base ten place value and all syllabus topics except base numeration systems. Once the posttest was 79 given to the researcher’s class, the missing topic of base numeration systems was taught at the end the semester. This group was identified as the control group. The choice of the two classes in the sample was constructed by convenience. Although many schools were contacted about participating in the study, only one followed through on the commitment. The researcher used her own class as the noninstruction/ delayed group because no other instructor contacted wanted to deviate from the normal sequence of topics. The pretest scores provided baseline information on the place value understanding that the preservice elementary teachers had prior to the semester of instruction. Question one asked: What is the level of explicit place value understanding that the elementary preservice teachers participating in the study bring with them to their required mathematics content course? Although there was much concern about the sampling used in this study, pretest scores showed no significance differences between the two groups of students. The overall means for both groups was a 1.95 on a scale from 1 to 5 of understanding. Medians were 1.81 for the without instruction group (control group) versus 1.845 for the instruction or experimental group. Boxplots showed similar distributions. Based on this five level scale, these average scores were at less than a tacit level. This score level indicates that students possess the ability to use an algorithm but demonstrate vague understanding of its logic. The thirteenitem test was then analyzed by the four groupings (cells) of similar problems and by rubrics utilized on each problem. No significant differences between the two groups were found in any category. Thus, both groups entered the study with comparable levels of place value understanding. 80 Using the five level scale of understanding , the elementary preservice teachers’ average scores indicated almost a tacit level of understanding. The posttest scores provided information on the place value understanding that the preservice elementary teachers had after the semester of instruction. Question two asked: What is the level of explicit place value understanding that the elementary preservice teachers participating in the study have at the end of their required mathematics content course? Results of the posttest scores showed a slightly higher total mean score for the instruction students – 2.35 for the without instruction (control) group versus the mean score of 2.59 for the instruction (experimental) group. However, this was not enough to be statistically significant. Individual cell analysis found a difference in means in only one of the areas – base problems. The instruction students did significantly better with the base problems, demonstrating a mean of 2.47 compared to the without instruction group (control group) mean of 1.53. This result was expected due to the study design. Thus, both groups of students demonstrated an average tacit level of understanding of place value as the semester ended. Rubric analyses showed no significant differences between the two groups. However three rubrics, used for problems included in the unfamiliar systematic (base) cell, did show marked differences. Choice of correct representation, accurate computation, and analysis of computation method rubrics showed higher means for the instruction (experimental) group. A comparison of scores of the choice of correct representation, rubric used for Problem 11 of the APVU, found that its mean was 1.25 for the instruction group and 1.04 for the without instruction (control) group. The accurate computation rubric score means, used for Problems 2, 3, 5, 6, 7, 8, 9, 10, and13, were 81 3.10 for the instruction group and 2.64 for the without instruction group. Means of 2.58 for the instruction group and 2.00 for the without instruction group were found for the analysis of computation method rubric, used for Problems 2, 3, 5, 6, 7, 8, 9, 10, and 13. Next the pretest and posttest for both groups of study are analyzed. Comparing scores across the groups answered question three. Question three asked: Is there any significant change in the participants’ explicit place value understanding from the beginning of the semester to the end of the semester? Mean scores for the two groups were analyzed for significant changes in understanding from pretest to posttest. Each group, individually, showed significant growth, but the means across the groups were not significant. Only the unfamiliar systematic cell dealing with base numeration problems showed significance at the = 0.0045 level between the two groups. Again, this result was anticipated due to the study design. No rubric analyses produced significant differences. Thus, there were no significant changes in place value understanding across the groups from pretest to posttest. Lastly, question four explored the gains made by the students during the semester. Question four asked: Is there a significant difference in gains made in explicit place value understanding between the class that has numeration system instruction (experimental group) and the class that does not have the numeration system instruction (control group)? The differences in gains for both groups of students were calculated and analyzed. Total mean scores and rubric scores calculated for each of the two groups showed no differences in gains. Only the unfamiliar systematic cell problems indicated a significant difference, a gain on mean scores of 0.23 (1.30 to 1.53) for the noninstruction students and 1.15 (1.32 to 2.47) for the instruction students. This information correlated 82 with other data previously obtained. Thus, there were no significant gains in understanding as measured by the APVU for the students in either group. Conclusions The Assessment of Place Value Understanding (APVU) was developed to indicate levels of mathematical understanding from algorithmic to pedagogical for preservice elementary teachers. A rubric scale of one to five corresponds to the five levels of depth of understanding. Each question on the APVU used two to three rubrics to evaluate the understanding students demonstrated on each problem. The lowest level referred to as algorithmic understanding indicates a student has knowledge of computational procedures, but not necessarily an understanding of why the process works. The second level described as tacit understanding indicates the student has some intuitive reasoning ability, but an inability to articulate the logic of the concept. At level three, explicit understanding, the student demonstrates a precise understanding of the concept with good articulation of the logic and a clear sense of connections among ideas. Level four named pedagogical, (self and peers), denotes students who know how they and their peers build mathematical understanding. Lastly, level five, pedagogical, (children), asserts that students have a level of understanding needed for effective teaching. Students at this level know how others build knowledge and how to choose appropriate tools for teaching. Using Problem 4 from the APVU as a guide for discussion, the following represent student responses that serve as examples of each level of understanding. Analyzing problem 4 required the use of the Depth of Analysis and Use of Descriptive 83 Language rubrics. These rubrics are described in Appendix C. Level 1 – Algorithmic “He thinks he can pick any chip to represent the number he wants.” Level 2  Tacit “He does not fully understand place value. He thinks in terms of units.” Level 3  Explicit “He is not understanding place values, ones and tens. He is counting by ones only.” Level 4  Pedagogical (self/peers) “Bobby isn’t using place value skills. He circled the six (1s) but when Mrs. Jones pointed to the two he just circles 2 not realizing that the 2 had a place value and should have been a 20.” Level 5  Pedagogical (children) “Bobby doesn’t understand the place value that the 2 represents. He is viewing it as a single digit not as the number of tens that the second digit represents. The two represents 84 2 groups of two or twenty, so he is not understanding the value of the position.” In this study it was disappointing to find that both groups of students began the semester with less than a tacit understanding of place value (1.95 mean score). Current research in workshops sponsored by the Mathematical Sciences Research Institute on critical issues in mathematical education promotes the importance of clear explanations, choosing useful examples, and evaluating students’ ideas (Thames, 2006). Students were unable to provide explanations or accurately assess student work on the APVU. Also disappointing in the results was that after a semester of instruction, all the students’ understanding had improved to only a midlevel tacit understanding. These low mean scores and some large standard deviations within the groups (more than one level) may have been the result of varied abilities. These scores also may have been the result of student apathy in answering all the questions. Because there was no incentive for students to do their best work, a number of problems, or parts of problems, were left blank by each group of students. Approximately twenty to twentyfive percent of all students left at least one problem blank. The instruction (experimental) group left an average of 2.125 questions blank on the pretest and 1.67 problems blank on the posttest. The without instruction (control) group left an average of 2 problems blank on the pretest and 1.625 problems blank on the posttest. More blanks occurred for problems 11, 12, and 13, although all problems from 3 to 13 had at least one blank per group. These blanks were scored as a zero and had an affect on the calculation of mean scores and standard deviations. Interpretation of the blanks was difficult for the researcher to assess without followup. No answers to the problems may have indicated 85 confusion about the question, no knowledge of how to answer the question, or no desire to answer the question. Earlier examination of the pretests could have allowed the researcher time to question students about the problems they did not complete. Unfortunately, all tests were examined at the end of the semester to help with the rater reliability regarding scoring. Due to already small sample sizes, it was not possible for the researcher to discard tests with blanks. Analyses indicate that for these 55 students, the treatment of base numeration system instruction did not significantly impact understanding as measured by the APVU. There was, however, growth in each class’ place value understanding. The change in the means from pretest to posttest for the total, in every cell, and in every rubric indicate increases in understanding for both groups of students. These results do suggest that instruction in this content area had a positive influence on the development of understanding in place value and numeration systems. Student responses from Problem 1 show this increase in instruction. Problem 1 on the APVU asked students to replicate and explain Caroline’s addition error pattern. One student on the pretest wrote, “She is adding from left to right instead of right to left.” On the posttest the same student answered, “Caroline doesn’t understand to add from right to left. She is adding as if she is reading a book. She also carries the wrong numbers to the next digits place” The posttest response is still not at a pedagogical level, but the student has begun to include place value language in her response. Authors of the APVU, in their studies ten years ago, had similar findings. Rusch’s research with place value understanding, constructivist, and direct instruction concluded that “the participants’ failure to use place value concepts in even moderately 86 sophisticated ways suggests that, by the end of the course dealing with place value concepts, they had not yet achieved the recommended ‘explicit’ level of understanding” (Rusch, 1997, p. 104). Gains made by the constructivist class of students were not significantly greater than the gains made by the direct instruction students. Hannigan’s study with place value understanding and writing prompts concluded that students did show some growth in their place value understanding. However, understanding was still not at the desired level for teachers, and the treatment of freewriting made no difference in the students’ scores (Hannigan, 1998). The current study’s findings support the results of these two research projects. Recent research in elementary preservice teachers’ place value understanding by FarroLynd found a “lack of foundational knowledge of the structure of place value” (FarroLynd, 2003, p. 104). Her study focused on misconceptions of place value concepts by preservice elementary teachers. Although the current study’s focus was on numeration system instruction and used a different assessment instrument, this researcher found a similar lack of understanding. Skipper’s study in 1972 using manipulatives with numeration systems concluded instruction methods provided no significant difference in place value knowledge. The current research studied understanding, not just knowledge, and found no significant different in place value understanding. Haukebo’s research in 1967 was similar to this project in that the research focused on place value under
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Title  Coursework on Base Numeration Systems and Its Influence on Preservice Elementary Teachers' Understanding of Place Value Understanding 
Date  20070701 
Author  Radin, Dorothy Janice 
Keywords  mathematics;numeration systems, place value understanding; elementary preservice teachers 
Department  Education (all programs) 
Document Type  
Full Text Type  Open Access 
Abstract  The purpose of this study was to examine the effectiveness of base numeration system instruction on place value understanding for college elementary education majors. Participants in the study were students in two university mathematics classes, one class at a private college and one class at a public college. Both colleges were in the same southwestern United States city metropolitan area. Each participant completed the Assessment of Place Value Understanding twice, pretest and posttest. Ttests, analysis of variance, and repeated measures ANOVA were used on means of total scores, cell scores, and rubric scores to test the four research questions. Only the mean scores of the cell of base n problems, n not equal to 10, showed signigicance differences between the two classes (p = 0.00). This was expected due to the research design and treatment of base numeration system instruction. All averages by total, cell, and rubric showed gains from pretest to posttest, meaning students' understanding of place value concepts improved. However, final understanding was at a less than sophisticated level. Instruction led to higher scores in the base problems, but did not improve overall place value understanding. More research is needed to examine the connection between elementary teachers' classroom instruction and the foundational concept of place value understanding. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  COURSEWORK ON BASE NUMERATION SYSTEMS AND ITS INFLUENCE ON PRESERVICE ELEMENTARY TEACHERS’ UNDERSTANDING OF PLACE VALUE CONCEPTS By DOROTHY JANICE RADIN Bachelor of Science Minot State University 1971 Master of Arts DePaul University 1992 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF EDUCATION July, 2007 ii COURSEWORK ON BASE NUMERATION SYSTEMS AND ITS INFLUENCE ON PRESERVICE ELEMENTARY TEACHERS’ UNDERSTANDING OF PLACE VALUE CONCEPTS BY DOROTHY J. RADIN Dissertation Approved: ______Patricia L. Jordan________________ Dissertation Advisor ______Margaret M. Scott_______________ ______Caroline Beller__________________ _______Laura L. B. Barnes______________ _______A. Gordon Emslie_______________ Dean of the Graduate College iii ACKNOWLEDGMENTS As this ten year odyssey concludes, I would like to thank my committee members for their contributions and support. To my committee chairperson, Dr. Patricia Jordan, my thanks for handling all my emails and questions with grace and candor. A special thanks to Dr. Laura Barnes for her contributions with the IRB process and for her expertise in the statistical analysis. Thanks to Dr. Margaret Scott for her input into my proposal and for working beyond retirement to see me through. Lastly, I appreciate Dr. Caroline Beller’s commitment to my committee work. All my department colleagues at Oral Roberts University have been a positive influence throughout the years. Chair Ken Preston has urged me on at every step. My friend and coworker Teresa Williams has been a great sounding board for my frustrations and joys. A particular thank you to Mary Lou Miller for helping with my statistics and for understanding my journey. A simple thank you is not enough for my friend and colleague Martha Parrott. Without her cooperation this study would not have been possible. She has been patient, pleasant, and always encouraging as I have strived to complete the work. Finally, my greatest thanks to my family. My children Jeff and Cindy have always been encouraging and have been flexible with their schedules when I needed time for class or for studying. And last, but certainly never least, to my husband Marc for his constant love and support. He has been there through it all, always the cheerleader and my biggest advocate. Thanks honey, and yes, I owe you! iv TABLE OF CONTENTS Chapter Page I. INTRODUCTION…………………………………………………………………..…1 Statement of the Problem ………………………………………………………...2 Context and Need for the Study…………………...……………………..……....3 Purpose…………………………………………………………………………...5 Research Questions ……………………………………………………………....6 Elements of the Study…………………………………………………………....6 Definition of Terms ……………………………………………………………....7 Limitations of the Study……………………………………………………….....8 Conclusion………………………………………………………………………..9 Summary of the Chapters ……………………………………..…………………10 II. LITERATURE REVIEW ……………………………………………...……………..12 Historical Perspective………………………………………………………..….14 PreService Elementary Teachers’ Knowledge of Mathematics ……………..…19 Teacher Knowledge and Student Learning…………………………………..….22 Place Value and Elementary Student Knowledge ……………………..………..25 Numeration Systems ………………………………………………………….....28 Conclusion …………………………………………………………..….…….…34 III. Methodology……………………………………………………………….…..……37 Participants ……………………………………………………………….……..38 Comparison of Classes ………………………………………………….……....39 Data Collection ……………………………………………………….………....46 Instrument ………………………………………………………………….........47 Data Analysis ………………………………………………………..………..…51 Conclusion …………………………………………………………………...….52 IV. FINDINGS …………………………………………………………………….……54 Demographic Information …………………………………...……………...…..55 Rater Reliability…………………………………………………...……...….…56 Question One…………………………………………………………………...58 Question Two ……………………………………………………………….…..63 Question Three……………………………………………………………….…70 Question Four………………………………………………………….………..74 Conclusion ……………………………………………………………..………..75 v Chapter Page V. CONCLUSION………………………………………………….. ………………..77 Summary………………………………………………………...……………77 Conclusions …………………………………………………...………………82 Limitations ……………………………………………………………………87 Recommendations …………………………………………………………….88 BIBLIOGRAPHY…………………………………………………………………….95 APPENDIXES ……………………………………………………………………..…106 APPENDIX A  ASSESSMENT OF PLACE VALUE UNDERSTANDING………………...…..………..107 APPENDIX B – APVU CELL CONTENT……………………….………….114 APPENDIX C – RUBRICS …………………………….………….…………115 APPENDIX D – RUBRIC/TASK CORRELATION…………………...........122 APPENDIX E– INSTITUTIONAL REVIEW BOARD LETTER ……..........124 vi LIST OF TABLES Table Page 1. Comparison of Course Content …………………………………………………40 2. Detail Numeration System Content ……………………………………………..41 3. Matrix of Explicit Place Value Understanding ……………………………….…49 4.. ACT Mean Scores…………………………………………………………….…56 5. Demographic Information ………………………………………………………57 6. APVU Sample Scoring………………………………………………………….57 7. All Tasks Pretest Item Mean Scores …………………………………….………58 8. Pretest Cell Means ……………………………………………………………….60 9. Pretest Rubric Means ………………………………………………………….....62 10. All Tasks Posttest Item Mean Scores ………………………….………….…….63 11. Posttest Cell Means ……………………………………………………………..65 12. Posttest Rubric Means ………………………………………………………….68 13. Without Instruction Rubric Correlations ……………………………………….69 14. With Instruction Rubric Correlations …………………………………………..70 15. Change in Understanding from Pretest to Posttest ……………………………...71 16. Change in Understanding from Pretest to Posttest by Matrix Cell ……………..72 17. Change in Understanding from Pretest to Posttest by Rubric…………..……....73 vii LIST OF FIGURES Figure Page 1. Pretest Total Mean Scores ……………………………………………………..59 2. Posttest Total Mean Scores ……………………………………………….……64 3. Familiar Systematic Cell Posttest Mean Scores ………………………………..66 4. Familiar NonSystematic Cell Posttest Mean Scores ………………………..…66 5. Unfamiliar Systematic Cell Posttest Cell Scores ……………………………..…67 6. Unfamiliar NonUnfamiliar Cell Posttest Mean Scores ……………………..….67 1 CHAPTER 1 INTRODUCTION Degree plans for elementary education majors require coursework in many disciplines. Although there is controversy about what mathematics content courses should be included in the program, most colleges require at least twelve credit hours or a fourcourse equivalent. In these mathematics classes students learn about problem solving, fractions, decimals, geometry, integers, and other topics. Since many students struggle with the content, they often come to class not only afraid of mathematics, but also with deficiencies in their mathematics backgrounds. These students learn basic operations and are usually able to do simple computations, but they lack a depth of understanding. Studies acknowledge that elementary teachers are not adequately prepared with an explicit understanding of mathematics. “It appears that prospective teachers may have mastered basic skills, but they lack the deeper conceptual understanding that is necessary when responding to student questions and extending lessons beyond the basics” (Center for Study of Teaching and Policy, 2001, p. 9). The National Research Council report in 2002 states elementary teachers’ understanding of the mathematics they teach is inadequate. This lack of subject matter depth can lead to students’ dependence on rote memorization of mathematical algorithms to solve problems rather than a development of reasoning skills. As college students they often try to generalize and memorize their way through required mathematics courses, rather than make connections among ideas 2 (National Research Council, 2002). In fact many equate understanding in mathematics with the ability to complete steps in an algorithm (Folk, 2006). Too many future elementary teachers begin college with mathematics deficiencies, both in content and in understanding. Their college mathematics courses focus mainly on computational skills rather than the development of reasoning skills and mathematical ideas. To illustrate these points the Conference Board of Mathematical Sciences uses the place value structure which states that elementary teachers need an explicit understanding of place value to aid children in building a strong numeric foundation (CBMS, 2001). Understanding place value as part of number and number sense is crucial to arithmetic success. Place value is the basis for the representation of whole numbers. Ideas of place value are used to order numbers and to aid in estimation and approximation (CBMS, 2001). Statement of the Problem In the college mathematics classroom, place value is often studied in connection with numeration systems, but elementary preservice teachers question the need to learn different numeration systems. The study of numeration systems includes practice with numberbase problems as well as ancient numeration systems like the Mayan or Babylonian. Number base work typically includes learning to count in different bases, converting back and forth between base ten and other bases, and performing number base computations. With the limited number of mathematics courses required for elementary preservice teachers, every course must provide meaningful instruction and content. Mathematics courses need to adequately prepare these prospective elementary teachers to 3 effectively educate their own future students; therefore, the curriculum for each course is important. Does the study of base numeration systems contribute to elementary preservice teachers’ understanding of place value? Context and Need for the Study Textbooks for mathematics courses for elementary preservice teachers have traditionally included the study of bases and different place value systems in their curricula (Bassarear, 2005; Billstein, Libeskind, & Lott, 2007). Why? Bassarear says the preservice elementary teacher should work with other bases “so that your understanding of the fundamental ideas of the base and place value become deeper and deeper” (2005, p. 115). Many elementary school textbooks explore place value numeration systems. Greenes et al. (2005) includes work with base two and base five in the grade four Houghton Mifflin text. Fennell and Altreri (1998) study the Babylonian system in their edition of the Silver Burdett grade three book. Clements, Jones, Moseley, and Schulman (1998) invite students to construct a place value system in a McGrawHill third grade textbook. Children’s literature also reflects an interest in place value systems. D.A. Adler’s book uses nickels and pennies to show Base Five (Adler, 1975). How to Count Like a Martian (St. John, 1975) has chapters on systems from the imaginary Martian to the ancient Mayan. Luce’s book Counting Systems, The Familiar and the Unusual puts different number systems into practice, such as base nine for baseball teams and base two for lights on and off (Luce, 1969). 4 Are there other reasons to study different place value systems? Casebeer researched the study of numeration systems in school mathematics in the 1960s. He hypothesized three reasons for studying this topic: 1) as foundational information, 2) for greater understanding of our HinduArabic system, and 3) to gain empathy for children’s experiences in learning place value for base ten (Casebeer, 1967). Writing in the 1970s, LeBlanc reiterated some of the same reasons as Casebeer for supporting the study of numeration systems. He suggested that prospective elementary teachers study other base systems to understand the characteristics of our base ten system. He wrote that work with other bases helps with the understanding of groupings and place value and provides elementary preservice teachers with experiences that children may have in learning the HinduArabic system. He drew a clear connection between teacher understanding of place value and children’s development of numeric concepts (LeBlanc, 1976). Research involving place value has primarily focused on elementary students and practicing elementary teachers. Jones’ and Thornton’s review of literature (1993) of the 1980s and early 1990s identified many studies showing children’s deficiencies in place value understanding. Fuson’s research (1990a, 1990b) used manipulatives, such as cubes and baseten blocks, to solve addition and subtraction multidigit problems to help students learn and apply place value concepts. She found that children have “inadequate understanding of the baseten place value system of written multidigit numbers, and consequently longterm errors in multidigit calculation procedures” (Fuson et al., 1997, p. 130). 5 Zazkis and Khoury studied place value, elementary preservice teachers, and their understanding of noninteger numbers in bases otherthanten. They concluded that their preservice students did not fully understand our number system structure. The results suggested that these college students’ partial understanding allowed them to accurately complete problems, but might interfere with conveying place value concepts to children (Zazkis & Khoury, 1994). In 2003, McClain studied elementary preservice teachers’ understandings of place value and multidigit addition and subtraction. In the course of this study she used base eight materials. One conclusion of her research was that the future teachers realized that conceptual understanding was necessary for teaching and explaining place value to others. Her students shifted their goals from “a focus on correct procedures to an emphasis on students’ [children’s] understanding” (McClain, 2003, p. 304). Two recent studies conducted at the University of Texas investigated instructional strategies, elementary preservice teachers, and place value (Hannigan, 1998; Rusch, 1997). FarroLynd’s work (2003) focused on elementary preservice teachers’ understanding and misconceptions of place value. There appears to be no recent research connecting preservice teachers, numeration systems, and place value. The focus of this project will be the teaching of base numeration systems and its impact on elementary preservice teachers’ place value understanding. Purpose This study grew out of classroom practice. The researcher found that her elementary preservice teachers encountered difficulties while studying numeration systems. These students did not see any application for the concepts and usually 6 struggled to understand the content. This project questions the value of teaching numeration systems to elementary preservice teachers. Are the reasons given by textbook authors and researchers valid? In particular, does the study of otherthanten bases increase elementary education students’ understanding of our base ten system? Does place value numeration systems instruction help elementary preservice teachers generalize place value concepts? These are the questions to be researched in this study. Research Questions Specifically, four questions are addressed in this study: 1. What is the level of explicit place value understanding that the elementary preservice teachers participating in the study bring with them to their required mathematics content course? 2. What is the level of explicit place value understanding that the elementary preservice teachers participating in the study have at the end of their required mathematics content course? 3. Is there any significant change in the participants’ explicit place value understanding from the beginning of the semester to the end of the semester? 4. Is there a significant difference in gains made in explicit place value understanding between the class that has numeration system instruction (experimental group) and the class that does not have the numeration system instruction (control group)? Elements of the Study This research was a quantitative quasiexperimental/control group design study. The objective was to compare gains of explicit place value understanding possessed by two groups of elementary preservice teachers taking a required mathematics course. The 7 course curriculum included a section on base numeration systems. The independent variable in the study was the instruction, or withholding of instruction, of base numeration systems. The dependent variable was growth in explicit place value understanding as measured by the Assessment of Place Value Understanding instrument. Two groups of preservice elementary teachers participated in this study, a sample of convenience. One group of 35 was enrolled in a required mathematics content course at a medium size regional state university. A second group of 25 was enrolled in a similar course at a small private university. Both groups’ curriculum contained a section which addressed place value concepts and numeration systems. The instrument used in this research is the Assessment of Place Value Understanding. This pretest and posttest assessment was developed by two PhD. candidates at the University of Texas in 1997. In a review of literature for their separate dissertations, they found “no previously validated assessment instruments that would provide insight into participants’ explicit understanding of place value” (Rusch, 1997, p. 12). They also found existing instruments were directed at children and not appropriate for assessing the knowledge of adults. A description of the Assessment of Place Value Understanding and the validation procedures are included in Chapter 3. The Appendix contains copies of the thirteen item pretest (only the pretest is used in this research), along with information regarding the objectives, rationale, and scoring strategies used. Definition of Terms Numeration System  “A numeration system is a collection of properties and symbols agreed upon to represent numbers systematically” (Billstein et al., 2004, p. 135). 8 Place Value  “Place value assigns a value of a digit depending on its placement in a numeral. To find the value of a digit in a whole number, we multiply the place value of the digit by its face value, where the face value is a digit” (Billstein et al., p. 136). Base Numeration System  A true place value system has a base and a set of symbols, including a symbol for zero. The HinduArabic system used in the United States is an example of a base ten numeration system. Explicit Understanding  “The level of mathematical understanding that is characterized by a precise understanding of the concept being addressed which can be clearly articulated and convincingly justified” (Rusch, 1997, p. 16). Procedural Understanding  “Knowledge of the formal language of mathematics, that is, symbols and syntax; and the rules, algorithms, or procedures used to solve mathematical tasks” (Post & Cramer, 1989, p. 222.). Preservice Teachers  “Person studying to teach mathematics as one of several subjects, future elementary school teachers” (Graeber, 1999, p. 191). Limitations of the Study This research represents a preliminary exploration into the relationship between understanding place value and understanding base numeration systems. This study updates earlier reports, reexamines, and investigates only one small construct of teacher knowledge. With its narrow focus, this study is meant to inform and raise questions regarding course content for elementary preservice teachers in their undergraduate program. The researcher’s hope is that others will take up the charge to investigate this and other related topics about curriculum and expertise in subject matter knowledge. 9 The Assessment of Place Value Understanding was administered during this study. Since this is a relatively new instrument to measure explicit understanding, there may be some hidden limitations on its use. Other assessments were researched for this study, but tests for this population of adult college students were normally of general arithmetic concepts, including topics such as geometry, number theory, fractions, sets, and the four basic operations on the set of whole numbers. None of the instruments focused solely on place value. Due to time constraints, control of participant sampling was not possible. All classes used were chosen by convenience. Although the classes were at different colleges with different instructors, efforts were made to at least select classes with similar students in a comparable college program. All students in the two classes were invited to participate. Even so, the samples lead to major questions regarding generalizabilty. Although teaching methodology was not considered in this research, the teachers have similar pedagogy. Both instructors have many years of university teaching experience, and both are friends and students in the same doctoral program. Texts and curricula for the classes were different, but content pertinent to this research was similar. Content comparisons are discussed in Chapter 3. There was also no way to control for the participant’s concentration and effort while taking the assessments. Although both instructors encouraged the participants, there was no sufficient incentive to ensure their best work. Conclusion Place value is a major component of the HinduArabic numeration system. Although many students are able to gain some expertise in computation, many more find 10 mathematics problems difficult to comprehend. “Knowledge of place value has great implications for success in arithmetic tasks…. Therefore, care should be taken that students develop a meaningful understanding of numbers” (Cathcart, Pothier, Vance, & Bezuk, 2000, p. 106). One major reason for studying base numeration systems in the elementary education major curriculum is the connection to place value. This study examined the relationship: Does studying base numeration systems lead elementary preservice teachers to a higher level of understanding of our baseten system? Summary of the Chapters Chapter One contains the introduction and impetus for the problem. The introduction to the study in Chapter One defends the importance of the topic, states the purpose of the study and identifies the questions to be answered. Included in Chapter One are definitions of terms and an outline of the remainder of this dissertation. Chapter Two begins with the progress of education during the past twenty years. The importance of teachers’ content knowledge is discussed as well as the connection between teaching and student understanding. Research on elementary preservice teachers’ mathematical skills is then addressed. Finally the importance of place value and how children learn is presented. Chapter Three describes the methodology used in this study. Development and piloting information about the instrument used – the Assessment of Place Value Understanding (APVU) – is detailed as well as a matrix of place value grouping schemes used in the instrument. The sample of students is identified and information about the instructors and course content. The five level scale of mathematical understanding 11 employed in the assessment rubrics is defined. Scoring of the tests is discussed and the statistical tests used to analyze the data Chapter Four details the findings of this research. Demographic information that was collected on the participants is summarized. Sample scoring data to check for rater reliability is given. Each questions in the study is answered by analyzing data in total, in each grouping, and in each rubric. Tables show data details and results. Chapter Five summarizes this study and offers conclusions about the research. Finally, recommendations are offered for future work. 12 CHAPTER 2 LITERATURE REVIEW Over the past thirty years national attention has focused on quality education in the United States. Federal legislation and government research set rigorous standards for curriculum, teacher quality, and assessment. The desire for highly qualified teachers became a priority. The National Council of Teachers of Mathematics (NCTM) began publishing standards for the mathematics community in the 1980s with An Agenda for Action (NCTM, 1980). In numerous publications this organization identified content appropriate for students by grade level and defined mathematics and pedagogical content to enhance teacher preparation. The NCTM Professional Standards (1991) emphasized the importance of teachers’ knowledge and their understanding of the subject matter. Since research shows a strong connection between teacher knowledge and student achievement, teachers need to know more content than the subject matter they teach. In order to present mathematics clearly and effectively, teachers must have explicit understanding of the subject. Elementary preservice teachers should have this explicit understanding for all segments of mathematics they will be required to teach. For elementary teachers, place value is a foundation concept. Yet many teachers, and thus their students, do not have an explicit understanding of this topic. As part of the college mathematics curriculum, base numeration systems are often taught to help these preservice teachers understand our base ten system. This research examines the 13 effectiveness of teaching base numeration systems and their effectiveness in improving place value understanding. Specifically, four questions are addressed in this study: 1. What is the level of explicit place value understanding that the elementary preservice teachers participating in the study bring with them to their required mathematics content course? 2. What is the level of explicit place value understanding that the elementary preservice teachers participating in the study have at the end of their required mathematics content course? 3. Is there any significant change in the participants’ explicit place value understanding from the beginning of the semester to the end of the semester? 4. Is there a significant difference in gains made in explicit place value understanding between the class that has numeration system instruction (experimental group) and the class that does not have the numeration system instruction (control group)? This chapter begins with a historical perspective on America’s focus on education and the importance of mathematics. The public call for “highly qualified” teachers and the search for a clear definition of this new term causes all teacher educators to reexamine their instruction. Section two of this chapter sights literature assessing the subject matter qualifications of today’s teachers and preservice teachers. Many teachers and preservice teachers can calculate properly but do not understand the logic behind the algorithms. In section three the researcher makes connections between teacher preparation and children’s achievement in the classroom. Teachers need to have a positive attitude and an understanding of the mathematics they teach. Section four 14 discusses elementary students’ knowledge of place value and the essential nature of this concept. Research with elementary students shows that they do not have a mastery understanding of the place value system. The focus of this study, numeration systems, is discussed in section five. A number of studies are outlined dealing with elementary preservice teachers. Yet there appears to be a void in the literature connecting place value understanding, elementary preservice teachers, and numeration systems. This study is meant to begin the probe into rigorous place value course content for the elementary preservice teacher. Historical Perspective Heading into the 21st century, Americans renewed their focus on quality education. The ‘80s and ‘90s were inundated with multiple reports and studies on schools, student achievement, and teacher effectiveness. Among the reports issued were A Nation At Risk (1983), completed by the National Commission on Excellence in Education, A Nation Prepared: Teaching for the 21st Century (1986) written by the Carnegie Forum on Education, Action for Excellence (1983), authored by the Task Force on Education and the Economy, Education Commission of the States, and A Call for Change (1985), reported by the National Commission for Excellence in Teacher Education. In A Nation At Risk, concern was expressed about “the widespread public perception that something is seriously remiss in our educational system” (National Commission on Excellence in Education, 1983, p 1). A call was issued to reform the Nation’s schools and reverse this declining trend. Recommendations for teachers included higher standards for certification and a competence in an academic discipline. 15 In A Nation Prepared: Teaching for the 21st Century, the Carnegie Forum on Education declared that teachers needed a sense of mathematics if their students were to have mathematical ideas and learn reasoning skills. Teachers’ content knowledge must include breadth and depth. “They [teachers] must be people whose knowledge is wideranging and whose understanding runs deep” (1986, p. 25). After President George H.W. Bush convened an education summit in 1989 of the nation’s governors, Congress passed the Goals 2000: Educate America Act. This set a national education agenda. Based on eight goals involving students, teachers, and communities, Goals 2000 required states to develop clear and rigorous standards. Goal Five, emphasizing mathematics and science, advocated more preservice training to help these prospective teachers gain broader and deeper mathematics understanding (Grinstein & Lipsey, 2001). In his second State of the Union address, President Clinton issued a “ ‘Call to Action ’ that included as a priority improving the quality of teachers in every American classroom” (Lewis et al., 1999, p. iii). The next year a Harris Poll revealed that about 90% of Americans believed “the best way to raise student achievement is to ensure a qualified teacher in every classroom” (CBMS, 2001, p. 3). This interest led to the 1998 amendments to the Higher Education Act of 1965. Part of these provisions included teacher quality enhancement grants to improve the quality of the future teaching force by improving the preparation of prospective teachers (United States Department of Education, 2003). Studies and reports were published examining school mathematics. These included Everybody Counts (1989) by the National Research Council and Educating 16 Americans for the 21st Century (1983) reported by the National Science Board Commission. An Agenda for Action (1980), Priorities in School Mathematics (1981), and Guidelines for the Preparation of Teachers of Mathematics (1981) were written by the National Council of Teachers of Mathematics (NCTM). These works made recommendations for school mathematics curricula, teacher training, and public support of the discipline. The National Research Council wrote that the main objective of the elementary school mathematics curriculum should be the concept of number sense. The National Science Board Commission (1983) stressed the importance of number sense and place value and called for high quality teachers. The NCTM encouraged improved preservice teacher education, focusing on teacher skills necessary to help children learn more than computation. The National Council of Teachers of Mathematics compiled their first Curriculum and Evaluation Standards for School Mathematics in 1989. This publication established high expectations for students of mathematics in kindergarten through grade twelve. The document provided a comprehensive “listing by gradelevel bands of the mathematics that students should know about problem solving, communication, connections, and various content aspects of mathematics relevant to those grade levels” (Usiskin & Dossey, 2004, p. 7). The Curriculum Standards were written in response to the increased national interest in the teaching and learning of mathematics and the publication prompted states to develop their own standards to set the bar for student performance. The K – 4 standard six of number sense and numeration identified understanding place value as a “critical step in the development of children’s comprehension of number concepts” (NCTM, 1989, p. 39). Sample learning activities were included. As part of the 17 evaluation standard eight on mathematical concepts, the importance of place value was connected to calculating subtraction problems. The National Council of Teachers of Mathematics followed up its Standards text with the Professional Standards. This publication was based on the assumption that teachers are key in changing the mathematics curriculum and pedagogy. The Professional Standards spelled out “what teachers need to know to teach toward new goals in mathematics education and how teaching should be evaluated for the purpose of improvement” (1991, p. viii). Professional development standard two contained a section on number systems and number sense which identified place value as important content for teachers to understand. Mathematics understanding has also been studied on an international basis. The Second International Mathematics Assessment was completed in 1986, comparing United States student achievement with sixteen other countries (Grinstein & Lipsey, 2001). The Third International Mathematics and Science Study (now called the Trends in International Mathematics and Science Study or TIMSS) compiled data on United States students and students from other countries in 1995, 1999, and 2003 (National Center for Education Statistics, 2001). These studies “suggested that American practices in mathematics education are not yielding the kind of learning that is both desirable and possible” (Watanabe & Thompson, 2004, p. 11). TheUnited States’ mathematics curriculum emphasized procedural skills and provided little depth in subject matter. As the 21st century began, the focus on education continued. President George W. Bush worked to pass the No Child Left Behind Act (NCLB), enacted in 2001. This act, PL 107110, mandated that all teachers must be deemed highly qualified. Although 18 there is disagreement about what it means to be “well qualified,” part of the NCLB definition of “high quality” included an ability to demonstrate expertise in the subject matter. States were charged to raise standards for teacher certification and monitor practices to prepare, train, and recruit high quality teachers (United States Department of Education, 2001). With the higher level of mathematical sophistication needed in the 21st century, increased mathematical skills and a solid understanding of mathematical concepts were needed by educators at all levels. Media and further publications stressed the importance of the subject of mathematics. “School mathematics instruction and the mathematical preparation of teachers are in the spotlight; because after reading and writing, mathematics is widely viewed as the most important component of K12 education” (CBMS, 2001, p. 3). The Nation’s Report Card: Mathematics 2000 claimed that “the ability to know and use mathematics is a necessity in daily life” (Braswell et al., 2001, p. 1). Even television reinforced the importance of mathematics in one of its programs. The series Numb3rs began “We all use math every day, to predict weather, to tell time, to handle money. Math is more than formulas and equations. It’s logic. It’s rationality. It’s using your mind to solve the biggest mysteries we know” (Heuton, Falacci, & Zucker, 2005). In 2000 the National Council of Teachers of Mathematics published an update of its 1989 Standards entitled Principles and Standards for School Mathematics. It stated The need to understand and be able to use mathematics in everyday life and in the workplace has never been greater and will continue to increase. . . . In this changing world, 19 world, those who understand and can do mathematics will have significantly enhanced opportunities and options for shaping their futures (NCTM, 2000, p. 45). PreService Elementary Teachers’ Knowledge of Mathematics With the call for highly qualified teachers, articles and research from the academic community investigated elementary teachers’ knowledge of the subjects they teach. Collias, Pajak and Rigden (2000) begin their article entitled “One Cannot Teach What One Does Not Know” with the quote, “Graduates of university teacher preparation programs in the United States, though well trained in teaching methods, often have insufficient knowledge of the subject matter they will teach” ( p.1). The authors reported that a 2000 survey found thirtythree percent of United States teachers did not believe that a mastery of subject matter was important to teaching. An article by McDiarmid, Ball, and Anderson (1989) on subjectspecific pedagogy used a literature review to defend the statement, “Unfortunately, considerable evidence suggests that many prospective [elementary] teachers do not understand their subjects in depth” ( p. 199). Subject matter content is taken for granted, and teacher preparation programs focus on pedagogical skills. Evidence of poor elementary teachers’ knowledge of content was referenced in an article by Brown, Cooney, and Jones stating that “preservice elementary teachers do not possess a level of mathematical understanding that is necessary to teach elementary school mathematics” (1990, p. 642). Other research by Post, Harel, Behr, and Lesh (1991) specifically found that twenty to thirty percent of teachers scored less than fifty 20 percent on an instrument measuring conceptual knowledge of rational numbers. In fact almost half the teachers missed very fundamental items like ÷3. In their literature review, Brown and Borko found that “research also suggests that prospective elementary teachers often do not have adequate content knowledge when they begin student teaching” (1992, p. 220). Previous research had postulated that this lack of knowledge was due to a focus on procedural rather than conceptual understanding. Students entering mathematics courses associated mathematics with mechanical and abstract symbols and rules. Coursework seemed to help the elementary preservice teachers develop more conceptual understanding about their own learning but did not enhance their thinking about their students’ learning. Preservice teachers should have “ both knowledge of mathematics and knowledge about mathematics” (p. 212). Schwartz and Riedesel (1994) reported on a number of studies that showed teachers and preservice teachers had only a procedural knowledge of mathematics. Duckworth focused on teachers and long division. These teachers knew the algorithm for long division, but they did not know why the procedures worked. Duckworth concluded that teachers’ understanding of arithmetic varied widely. Her second conclusion stressed the connection of understanding and classroom practice. As teachers’ understanding deepened, their teaching methods changed. Wheeler studied preservice teachers and division problems with zero. Sixtytwo elementary preservice teachers were tested and interviewed on problems dealing with a divisor or dividend of zero. Division by zero problems were most difficult for the future teachers. Only 23.1% made no errors. Even though many could do the problems accurately, “many could not adequately elaborate on the question, ‘What is zero?’” (p. 4). 21 An article by Fosnot (1989) described a classroom scenario common for many education majors. They frequently sat in math classes and practiced procedures, proofs, and calculations. They learned to memorize and regurgitate these procedures and rarely were given concrete materials to make the abstract notation meaningful. The result is that many teachers do not understand the concepts they are expected to teach (p. 71). Post and Cramer (1989) found that the mathematical background of new teachers included a concentration on drill and practice rather than an emphasis on understanding the concepts. These authors stated that this lack of conceptual knowledge “dominates school mathematics curricula at virtually all levels” (p. 222). To illustrate this focus on mechanistic skills, a survey from a Minnesota school district was sighted. Elementary junior high teachers reported over sixty percent of class time was spent working textbook problems and practicing speed and accuracy in computational algorithms. This type of instruction using just the printed page generally does not lead to higher level thinking or understanding. McDiarmid and Wilson conducted teacher interviews and learned that many had mastery of mathematical rules but could not explain the reasons behind the rules. One teacher said, “I don’t know why. My teacher never told me” (1992, p. 101). Ball (1990) also found this deficiency. In her interviews a future teacher said, “I absolutely do it [a mathematical calculation] by the rote process I would have to think about it” (p. 458). 22 The students’ idea of “doing” mathematics was to describe the steps of the algorithm, without any thought as to the meaning or reasons for the procedures. In a 2001 report, The Center for the Study of Teaching and Policy concluded that “prospective elementary teachers had relatively sound procedural, or ruledominated knowledge of basic mathematics, especially in arithmetic but had difficulty when pushed to explain why an algorithm or procedure works” (p. 9). They lacked sound conceptual understanding of the underlying concepts of the mathematics. Teacher Knowledge and Student Learning The quality of teachers is a concern because of the strong connection between teachers’ knowledge and student achievement. In its Professional Standards, the National Council of Teachers of Mathematics stated, Knowledge of both the content and discourse of mathematics is an essential component of teachers’ preparation for the profession. Teachers’ comfort with, and confidence in, their own knowledge of mathematics affects both what they teach and how they teach it. Their conceptions of mathematics shape their choice of worthwhile mathematical tasks, the kinds of learning environments they create, and the discourse in their classrooms (1991, p. 132). Studies (Collias, Pajak, & Rigden, 2000; DarlingHammond, 2000) stressed that teachers’ knowledge is an important influence on student progress. The statistical report of DarlingHammond based on the National Assessment of Educational Progress data concluded that “the proportion of wellqualified teachers is by far the most important determinant of student achievement” (2000, p. 30). The Collias, Pajak, and Rigden 23 (2000) article details the STEP program, an initiative to help university faculty in teacher preparation. The program guides universities to reorganize their teacher preparation programs to involve all faculty in an accountability system for graduates’ knowledge. The authors stated that “What a teacher knows and can do makes the crucial difference in what children learn. In fact teacher expertise is the most important factor in student achievement” (p. 3). Bad teachers can adversely effect student learning and put children at an academic disadvantage. A compilation of research report authored by the Center for the Study of Teaching and Policy found that “several studies showed a positive connection between teachers’ subject matter preparation and … higher student achievement, … particularly in mathematics” (2001, p. 7). The report claimed that good teaching could not take place if teachers lacked conceptual understanding of the subject matter. In an annual report on teacher quality published in 2005, the United States Department of Education stated, “Ensuring that America’s teachers are of the highest quality is an important national priority because they hold the key to student success. Simply put, teachers matter” (USDE & OPE, p. 5). A 2001 government publication, The Nation’s Report Card: Mathematics 2000, stressed the importance of teachers having adequate content knowledge. “To better serve the students they teach, teachers need preparation in the content areas of mathematics that are part of their students’ curriculum” (Braswell, p. 135). Reporting on states’ progress toward meeting the NCLB Act requirement of highly qualified teachers, Walsh and Snyder (2004) concluded that understanding subject matter was a critical skill for teachers. They stated that there was no substitute for 24 subject knowledge. If a teacher did not know a concept, then that concept would not be known to the teacher’s students. Grossman, Wilson, and Shulman (1989) reported on the Knowledge Growth in a Profession Project at Stanford University, investigating the role content knowledge plays in instruction preparation. The Project found that “teachers need to understand their subject matter in ways that promote learning” (p. 24). Subject matter knowledge also influenced content in the elementary classroom and classroom practice. The conclusion of the authors in this report was that beginning teachers are not prepared to transform their own learning into a form appropriate for their students. A solution was to have a subjectspecific knowledge of pedagogy, a methods class. This course would unite pedagogy and content and reinforce teacher knowledge. “Teachers have a better chance of being able to help their pupils develop flexible understandings of subject matter if they understand their subject matter well” (McDiarmid, Ball, & Anderson, 1989, p. 199). McLaughlin and Talbert (1993) suggested that if teachers had only cursory knowledge of their subject matter they would be able to teach only the facts or leave learning up to the students. Liping Ma quoted an old Chinese saying, “Know how, and also know why” (Ma, 1999, p. 108). Teachers must not only know how to manipulate an algorithm but they must know why it makes sense mathematically; therefore, they must have conceptual understanding as well as procedural understanding. This higherlevel understanding can improve teachers’ knowledge of mathematics, thus helping to improve students’ mathematical abilities. According to Ma, how well teachers understand the subject matter they teach directly impacts the level of student learning. 25 In 1989, Ball outlined strategies for change regarding teachers and adequate subject matter knowledge. “Teachers should not just be able to do mathematics; if they are to teach for understanding, they must also have a sense for the mathematical meanings underlying the concepts and procedures” (p. 89). Her article in 1990 on the mathematical understanding of prospective teachers stressed the importance of the depth of knowledge teachers must have to present the subject matter effectively to children. Teachers must not only be able to compute problems accurately, they must be able to explain steps, discuss reasoning, and offer alternative solutions. In later research with Bass, Ball wrote that teachers needed to know more than the subject matter knowledge that students learn “in order to have broad perspective on where their students are heading. Teachers’ own knowledge on the subject affects what they teach and how they teach” (Ball & Bass, 2000, p. 86). Hungerford’s article, based on his classroom experiences with preservice teachers, explained that elementary teachers’ attitude about mathematics influence their students’ attitudes. Teachers who are afraid of mathematics are likely to transfer that feeling to their students. It is unlikely that students will develop an appreciate for mathematics if their teachers have little interest in the subject (1991). Place Value and Elementary Student Knowledge In the first NCTM Standards, a strand in the K4 grade group was number sense and numeration. The section began: Children must understand numbers if they are to make sense of the ways numbers are used in their everyday world. They need to use numbers to quantity, to identify locations, to identify a specific object 26 in a collection, to name, and to measure. Furthermore, an understanding of place value is crucial for later work with numbers and computation. (1989, p. 38). According to Dienes (1960), place value and using base ten is our way to communicate numbers, and it is imperative that students understand this concept. “The notation of the number using place value, with the base of ten is a method of communicating numbers, and it is essential that children should learn the meaning of such communication as effectively as possible” (Dienes, p. 51). Place value concepts create a foundation for student number sense. According to Smith, Lambdin, Lindquist, and Reys, Many young children can count to one hundred or beyond, but actually have little or no sense of what [the] numbers … actually mean. Before students can make sense of addition or subtraction of multidigit numbers, it is essential that they develop a good understanding of place value (2001, p. 48). The Principles and Standards for School Mathematics set goals for thinking and reasoning. The authors recommended that “students develop a deep understanding of important concepts and proficiency with important related skills…This recommendation implied that our students deserve to learn more than the procedures of mathematics – they also need to make sense of mathematics” (Lappan, 2005, p. 1). Tracy and Gibbons tested teaching materials and activities to help elementary students and preservice teachers learn decimals and metrics. They used a number line, decimal squares, videos, the calculator, and the United States Metric Association website 27 to develop classroom lessons. These two teachers found that “the problem lies in that the emphasis on place value tends to be on computation, not conceptual understanding. For students to be successful, they must first understand them [concepts]” (1999, p. 2). Textbook authors Lanier and Taylor created an activities manual for preservice elementary students. In it they stated, “One reason many elementary and middle school students have trouble with mathematics is due in part to only being taught an algorithm with no meaning behind it” (2004, p. 11). These authors suggested that teachers should lay a strong foundation and explain to students why algorithms work. Several research papers have shown that elementary students have difficulty learning place value concepts. Articles by Fuson (1990a, 1990b) began with the premise that children perform poorly on place value tasks. She then explored words used in place value (for example, eleven versus ten one) and proposed new textbook characteristics. Given the problem 527 + 435, more than half of the third graders asked were unable to solve it by “counting on with hundreds, tens, and ones from 527 (527, 627, 727, 827, 927, 937, 947, 957, 958, 959, 960, 961, 962)” (1990a, 368). Fuson indicated that, “Many elementary schoolchildren …. do not fully understand the base ten structure of multidigit number words” (1990a, p. 350). Kamii (1985) conducted tests in classrooms in Chicago for place value understanding on students in grades 1 (n = 13), 4 (n = 35), 6 (n = 48), and 8 (n = 41). These tests were given at the same elementary school and at the junior high school that the students would normally attend afterwards. Children were asked to circle the number of objects that the one represented in the number 16. None of the first graders circled ten, the correct answer. Eighteen out of thirtyfive (51%) fourth graders were correct, 28 compared to twentynine out of fortyeight (60%) sixth graders and thirtytwo out of fortyone (78%) eighth graders. When she discussed these results with two mathematics teachers, “they were not surprised. They said that some children never seem to get place value” (p. 63). Ashlock (1990) found that children had difficulties with computational algorithms because they did not have an adequate understanding of place value. He observed that students could often write numerals correctly but could not explain why they were written that way. Children could identify and name place values, but they could not master the place value system as a whole. Work by Ball included a study with nineteen preservice elementary and secondary teachers. She stated that “Since place value is a fundamental idea and since pupils often find it difficult, it seemed a critical area of prospective teachers’ knowledge to investigate” (1988, p. 90). These preservice teachers were given a number of problems to ‘debug’ or find and explain the error in children’s mathematics problems. Her conclusion was that these prospective teachers did not have an explicit understanding of place value. They were able to compute properly, but they were unable to articulate the underlying concepts of the problems. Numeration Systems Historical numeration systems and base number systems are traditionally part of the elementary education major curriculum. Textbooks provide information and practice for systems such as the Babylonian (base 60), Mayan (modified base 20), and other bases. The rationale for this topic being included in the elementary curriculum is typically that studying these systems help students better understand our base ten. 29 Authors Cathcart, Pothier, Vance, and Bezuk state that we study ancient numeration systems “with a view to developing a better understanding and appreciation of our own system” (2000, p. 124). Research involving numeration systems has taken many forms. Hamilton’s classroom work in the 1960s focused on college student understanding of a new ‘Make Believe Arithmetic.’ This creative number system used inventive symbols for a base six system. He found that the concept of base was difficult for students to understand. The use of Arabic numerals in a number system other than base ten was confusing for the students (1961). Hamilton created a new system of base six using new symbols and new words for the numbers. The class worked on addition and multiplication tables and even constructed a ruler for measurement. With these new symbols students were successful in learning and were able to transfer their understanding to base ten. Sawada and Atkinson (1981) replicated Hamilton’s study with a nondecimal invented base five numeration system. They found that their preservice teachers gained empathy for the difficulties children have in understanding base ten numeration. “Students think they know everything there is to know about the counting numbers, the names they say, and the symbols we write, when …they have little or no insight into the system” (p. 367). These researchers stated that many students learned our number system by rote. They concluded that an intensive study of a single base system using nondigit symbols had value in preservice teacher education. Casebeer examined teaching styles for introducing place value systems to preservice teachers in 1967. Assuming a value to incorporating base numeration system content into the elementary education major mathematics curriculum, Casebeer tested 30 two methods of instruction for two weeks in his mathematics classes for preservice elementary teachers. He developed programmed materials to compare two sequences for introducing base numeration systems. One sequence was based on generalizing our base ten concept to other bases, the more traditional approach. The other sequence was based on sets and grouping of sets. Both sequences were used in all four classes studied, half the students receiving instruction in the sequence one method and then sequence two method and the other half receiving instruction in the sequence two method and then sequence one. He found no differences in student performance between the order of the two teaching techniques. Instruction for this research combined both methods, typically beginning with the traditional approach to establishing a connection to base ten and then using manipulatives to discuss sets. Research conducted by Haukebo (1967) investigated sixtytwo elementary preservice teachers in two mathematics classes and their study of numeration systems. He used three different bases with these future teachers over a three week period. Haukebo’s hypothesis was that no differences in baseten understanding would exist between the group receiving instruction in baseten and the group receiving instruction in base numeration systems as measured by his test of arithmetic understanding. A fiftytwo problem test was constructed to test for arithmetic understanding. Problems involved simple computation, place value problems, and base problems. He concluded that although every group of students increased their arithmetic understanding, differences between the two groups were not significant. In 1972 Skipper studied various teaching methods for the study of numeration systems by preservice teachers. He used three groups of students; one employing two 31 sets of manipulatives in the classroom, one using one set of manipulatives in the classroom, and a lecture only group. Approximately fifty students participated in each group. Tests for knowledge used Skipper’s Test on Numeration System (TONS). This was strictly a base computation test using problems like 0.25 = ______ twelve (p. 158) and 103 five + 244 five = ______ (p. 153). His study was inconclusive on the connection of base numeration systems and better understanding of the decimal system. Even so, as a result of his research, Skipper wrote that he thought preservice elementary teachers could learn base systems and recommended that preservice teachers be exposed to more coursework in numeration systems (1972). Hungerford experimented with base five instruction in his college classroom in 1992. He found that his students were frustrated and skeptical about this newly constructed number system, but he received many positive comments about the lessons at the end of the semester. “The new arithmetic forces students to come to grips with place value. Even though the mechanics look much the same as before, they must think about what’s going on and understand how it really works” (p. 1). Zazkis and Khoury (1994) worked with preservice elementary teachers on place value and nondecimal fractions. Twenty students participated. Students were asked to calculate addition and subtraction problems in other bases and then convert nondecimals to base ten. An example of a problem and its incorrect answer follow: 2.23 four + 3.33 four 21.31 four The correct answer should be 12.22 four (p. 204). 32 Students were interviewed about their answers to analyze their paper and pencil responses. The researchers concluded that “preservice teachers’ constructions of place value number systems are fragile and incomplete” (p. 222). A significant number of the preservice teachers were able to perform addition and subtraction calculations but could not explain their work and identify the proper place value of the digits. In 2003, a research study conducted by McClain was published. She developed a sequence of problems involving a fictional candy factory. Packaging for this factory used base eight mathematics. Candies were packaged eight to a roll and eight rolls to a box. Problems relating to this packaging were given to the class of twentyfour elementary preservice teachers who served as subjects in the study. One problem asked students to determine how many candies were in one box, three rolls, and two pieces. Many preservice teachers initially needed to draw and count individual pieces but later devised more simplified figures as their understanding increased. At the beginning of the experiment, McClain stated that “their [preservice teachers] understandings of both place value and multidigit addition and subtraction were very superficial and grounded in rules for manipulating algorithms” (p. 289). At the end of the project, she reported relative proficiency in the tasks. In FarroLynd’s (2003) research she wrote that “ number base work reinforces many of the concepts and procedures learned in base ten such as place value, regrouping, renaming, and computation algorithms” ( p. 7). Her study involved 104 preservice elementary teachers and place value misconceptions, using a twelve item written test and followup interviews. Students participated in the study while taking the second in a two course mathematics sequence where place value was taught in the first course. The test 33 used a problem mixture of rounding whole numbers, base computations, comparing and ordering numbers, and renaming decimals. Sample problems include “Name a decimal between 0.1 and 0.11 (results were 45% correct), and round 99,721 to the nearest thousand (results were 93% correct)” (p. 124 & p. 126). The three base problem accuracies were 11%, 42%, and 17%. Overall, the 104 students averaged 37% correct on the test. She concluded that these preservice teachers had weak conceptual understanding of place value structure, and she recommended further studies of this population and topic. Research by Rusch (1997) involved 206 preservice elementary teachers throughout the state of Texas. The focus of the study was on the influence of different teaching strategies on place value understanding. Three classes employed a constructivist approach to teaching, and six classes used the direct approach, or lecture. Constructivism employs the idea of students constructing their own learning. Teachers become a ‘guide on the side’ rather than a ‘sage on the stage.’ Rather than lecture, the instructors create situations in which students can discover mathematical concepts. Rusch and Hannigan developed the APVU to determine gains in students’ understanding of place value using the two different teaching approaches. Although gains were made by both groups, no significant difference in understanding between groups was found. Hannigan (1998) followed Rusch’s work with a study of place value understanding and the effectiveness of using writing prompts to enhance that understanding. Four classes of approximately twentyfive students each participated. Two instructors each had a treatment class that received freewriting assignments developed by Hannigan and a control class that received no special assignments. Writing 34 prompts invited students to define an algorithm, describe its logic, and give examples using the algorithm. Again, no difference in gains were found in place value understanding between the treatment group and the control group. Conclusion Preservice elementary teachers are required to take a number of mathematics courses as part of their degree plan. In these courses textbooks have traditionally included bases and different place value systems in their curricula (Bassarear, 2005; Billstein, Libeskind, & Lott, 2007; Cathcart, Pothier,Vance, & Bezuk, 2000). The generally accepted reasons for including these topics are twofold: 1) to acquire an explicit understanding of place value, and 2) to gain perspective into elementary students’ difficulties in learning our number system. Does studying base numeration systems help preservice teachers build on their understanding of place value? Place value is a major component of the HinduArabic numeration system. In the early 1960s Dienes stressed the essential nature of place value in communicating the meaning of numbers. The National Council of Teachers of Mathematics promoted the importance of students’ learning place value concepts in its first Standards (1989) within the strand of number sense and numeration, and it continues to emphasize this topic. Research has shown a connection between place value understanding and mathematical competency. “Knowledge of place value has great implications for success in arithmetic tasks…Therefore, care should be taken that students develop a meaningful understanding of numbers” (Cathcart, Pothier,Vance, & Bezuk, 2000, p. 106). Yet elementary students do not have an adequate understanding of place value. Studies by 35 Fuson (1990a, 1990b), Kamii (1985), and Ashlock (1990) found primary student knowledge of place value to be lacking. Based on the findings of previous studies, is this lack of understanding because elementary teachers have a poor understanding of the concepts themselves? Ball (1988, 1990), McDiarmid and Wilson (1992), and Zazkis and Khoury (1994) concluded that elementary and preservice teachers knew facts and could memorize algorithms, but did not understand the underlying mathematical processes. This lack of place value knowledge disrupted the teaching and learning of number concept development and computation. Research has been conducted on the place value understanding of elementary preservice teachers ( Hannigan, 1998; Rusch, 1997). Research has been conducted focusing on numeration systems and elementary preservice teachers’ understanding (Casebeer, 1967; FarroLynd, 2003; Skipper, 1972). Yet no research has been found that attempted to connect numeration systems and place value understanding of elementary preservice teachers. The researcher found no studies utilizing the APVU other than the work by its two authors. Zazkis (1999) wrote that preservice teacher instruction should “provide them with experiences and challenges that reexamine and enrich their understanding of mathematics and its pedagogy” (p. 650). Currently, numeration systems are an accepted part of the elementary preservice teacher mathematics curriculum. With today’s call for highly qualified educators, all teacher educators should examine their curriculum to assure rigorous content. This study attempts to add to the dialogue on the mathematical preparation and required curriculum content established for elementary preservice 36 teachers by investigating the study of numeration systems and their influence on place value understanding. 37 CHAPTER 3 METHODOLOGY This research was a quantitative quasiexperimental/control group design study. The objective was to compare gains of explicit place value understanding possessed by two classes of elementary preservice teachers taking a required mathematics course. The course curriculum included a section on basenumeration systems. The independent variable in the study was the instruction, or withholding of instruction, of basenumeration systems. The dependent variable was growth in explicit place value understanding as measured by the Assessment of Place Value Understanding (APVU) instrument. The four research questions guiding this study were: 1. What is the level of explicit place value understanding that the elementary preservice teachers participating in the study bring with them to their required mathematics content course? 2. What is the level of explicit place value understanding that the elementary preservice teachers participating in the study have at the end of their required mathematics content course? 3. Is there any significant change in the participants’ explicit place value understanding from the beginning of the semester to the end of the semester? 4. Is there a significant difference in gains made in explicit place value understanding 38 between the class that has numeration system instruction (experimental group) and the class that does not have the numeration system instruction (control group)? Participants All of the participants in the study were undergraduate college students pursuing a career in elementary education. Normally the students taking the courses used in this study are sophomores and juniors. The researcher presumed that all students had some preexisting knowledge about the place value concepts and representation in our baseten numeration system. The two classes used in the study were chosen by convenience. Initially, a number of universities expressed interest in helping with this research. Many schools were eliminated due to the semester scheduling of the mathematics content courses. Other schools did not follow through on their first commitment. Only one instructor other than the researcher agreed to aid in this study. This challenge of research as outlined in an article by Heid et al. (2006) reinforces the concern that linking research and practice is a problem. Even with the need for more schoolbased research, teachers are reluctant to allow their classrooms to be laboratories for experimental, unproven content and changes in pedagogy. Conversely, control groups are difficult to assign if teachers perceive the intervention of the researcher to be impractical, unusable, or intrusive. One class of approximately 35 students was taught by an instructor at a regional, medium sized public university (experimental group). One class of approximately 25 students was taught by the researcher at a small private university (control group). The two colleges are in close geographic proximity to each other in the same suburban area. The instructors were both experienced teachers with many years of college teaching 39 experience, and they were friends and fellow students in the same doctoral program. All students in the two classes were invited to participate in the research project. Comparison of Classes Both mathematics courses were components of the required degree plan at their respective universities and were designed specifically for elementary education majors with enrollments limited to that group of majors. The curriculum in both courses used in the study normally includes sections on numeration systems. At the private university, the course was usually taken as the second or third course in a four course sequence. At the public university, the course could be taken second, third, or fourth in a similar fourcourse required sequence. Neither school had a specific mathematics methods course for elementary education majors. Since two different texts were used in the courses at the two schools, content was compared. Table 1 gives the number of weeks that the major course content at the two schools was presented as well as the order in which the material was taught. Although material in the two courses was not identical, the basic place value topics were similar as well as the focus objectives on numeration systems. Table 2 details the specific numeration system content taught by each instruction during the semester. The instructional style of the professors was not a component of this research. Each instructor was free to use any method of her choice, including lectures, modeling, and videos. However, it was known to the researcher that both instructors used a constructivist approach and employed many manipulatives in their classrooms. Handson activities include work with baseten and multibase blocks. 40 Table 1 Comparison of Course Content__________________________________________ Numeration system instruction Without With (Control group) (Experimental group) Topic Number of Weeks Topic Number of Weeks Problem solving One Problem solving One Sets One Number theory Two Prenumber skills Onehalf Sets One & onehalf Numeration systems Egyptian, Roman, HinduArabic, time One & onehalf Prenumber skills Onehalf 4 basic operations Two Numeration systems Two & onehalf Integers One 4 basic operations Three Number theory One & onehalf Rationals One & onehalf Decimals One Base numeration systems One 41 Table 2 Detail Numeration System Content____________________________________________ Numeration system instruction Without With_________________ HinduArabic, baseten Egyptian Egyptian Roman Roman Chinese Time Greek Time delay for study HinduArabic, baseten Babylonian Babylonian Mayan Mayan Bases (multi – 2, 3, 4, 5, others) Bases (multi – 2, 3, 4, 5, others) Note: Content in bold taught after the posttest for this research. The researcher (control group) delayed teaching the base numeration system content section of the course curriculum while the study was in place. Only simple information about these numeration systems was introduced. In particular, notation of the base systems was introduced so that students’ work on the second test would better reflect their understanding of place value concepts rather than their lack of knowledge about the problem structure. For example, the researcher explained to her students that 234 eight indicated a number in the base eight numeration system, not a HinduArabic base ten number. Base system instruction was given at the end of the semester after the test was administered for the second time and the study had been completed. At that time the researcher taught the concepts of base numeration systems, including addition, 42 subtraction, and multiplication. Sections from the syllabus delayed for the study were taught using textbook material. The instructor (experimental group) taught base numeration systems as usual which involved extending the use of baseten blocks for the HinduArabic system to multibase blocks for the different number bases. Babylonian and Mayan systems were also studied to show the variety of base numeration systems in other contexts. Class notes state: As early numeration systems began to use grouping they became conceptually harder, and at the same time easier to use on a mechanical basis. In our HinduArabic system we group by tens. Children need experience grouping by twos, threes, . . . all the way to tens. The idea of an exchange point is the key to understanding place value. Also, grouping using smaller numbers provides more practice than waiting until ten objects have been collected. (Parrott, 2007, p. 89). The following three pages were taken from the instructor’s class notes for the experimental group to provide some insight as to the method of teaching of these numeration systems and the instructional approaches employed. 43 44 45 (Parrott, 2007, pp. 115, 118,119). 46 Both instructors taught traditional place value concepts, including expanded notation, and the HinduArabic system (our baseten system). “Place value assigns a value to a digit depending on its placement in a numeral” (Billstein, Libeskind, & Lott, 2007, p. 155). An example of a place value question from the control group text would be to give the place value of the underlined numeral: 827, 367 (Billstein, Libeskind, & Lott, 2007, p. 166). An example of a problem from the experimental group class: “Consider the numbers 40 and 400 in the HinduArabic numeration system. What does the 4 represent in each number?” (Angel & Porter, 1997, p. 180). Expanded notation is illustrated by the number 1234 = (1•1000) + (2•100) + (3•10) + (4•1). Both texts ask students to write numerals in expanded form and the reverse. Both instructors used materials such as the base ten blocks, place value charts, chip trading, and Cuisenaire rods to reinforce baseten concept development. Data Collection Data collection consisted of three parts: 1. Collection of demographic information on all participants. 2. Administration of Form A of the Assessment of Place Value Understanding (APVU) instrument at the beginning of the semester before numeration system concepts and place value were taught. 3. Administration of the same form of the APVU near the end of the semester after numeration system concepts and place value were/were not taught in the individual class sections. The APVU instruments were given as part of the normal classroom requirements. Anonymous demographic information was collected, as were the average ACT scores of 47 the two groups which were compared to check for equivalence between the groups. Consent to use the scores was included in consent forms acknowledging subjects’ participation in the project. Students were allowed approximately 45 minutes to take each one of the two assessments at each location. This time allotment was similar to the time given in the test developers’ previous studies. Administration of both the first test and second test was scheduled as close together as possible at the two schools to maintain similar total class time at each location. The first administration of the APVU test was given to each group at about the third week of the semester. This was later than planned due to inclement weather, but no place value concepts had been taught at that point in the semester. The second administration of the APVU test was given by the researcher during week fourteen of the semester, allowing enough time for the base numeration systems content to have been taught to the control group before the end of the semester. The instructor of the experimental group administered the second test during the last week of the semester. Instrument The Assessment of Place Value Understanding (APVU, pretest and parallel posttest) was developed by Rusch and Hannigan in 1997 as part of their Ph.D. work at the University of Texas. For this study only the pretest form of the test was used. This choice was made due to the original authors’ lack of confidence and documentation regarding the comparability of the initial two forms. After its introductory use, Hannigan (1998), one of the APVU authors, expressed concern in her dissertation about some problems which arose indicating that the two forms may not have been truly parallel. 48 One minor change was made in the original test by this researcher. For questions 3, 8, and 13, relating to numberbase problems, the researcher asked the students to be more specific about describing their processes in calculating the solutions. No change was made in the rubrics for these problems. This assessment was based on a matrix of place value grouping schemes shown in Table 3 from Rusch. The APVU was piloted multiple times as the authors continued to revise and refine the questions. According to Rusch, “It was necessary to pilot the assessment several times to ensure clarity in the communication of each question, validity of the questions, and to control for the time required to complete the assessment. Furthermore, rubrics for assessing the responses to the assessment questions were developed and carefully tested for clarity and interrater reliability” (1997, p. 51). The instrument was first piloted orally with sixteen elementary preservice teachers at the University of Texas. Content was analyzed, and a second handwritten test was given to the same sixteen participants to develop a time frame for giving the test in a paperpencil format. A third pilot was conducted with nineteen students at Austin Community College and 265 University of Texas students. A stratified sample of 45 tests was selected for use in the development and analysis of the rubrics. Five outside evaluators scored the test responses to check the reliability of the assessment rubrics. During the course of their research, the authors, Rusch and Hannigan, used the APVU for thirteen classes and 312 students (Rusch, 1997). 49 Table 3 Matrix of Explicit Place Value Understanding___________________________________ Systematic NonSystematic Systematic grouping schemes Nonsystematic grouping are those in which the number schemes are those in which the of items required to create one number of items required to group of the next place value create one group of the next remains constant. place value varies from place to place. Familiar A familiar grouping scheme is The Base 10 System Time one which most preservice seconds, minutes, hours, days, teachers have come across in weeks, . . . both the traditional mathematics curriculum and Metric Measurement Imperial Measurement in their daily living routines. inches, feet, yards, … cups, pints, quarts, gallons, … Unfamiliar An unfamiliar grouping Base n Systems Foreign Coinage scheme is one which uses a place value structure, but (other than base 10) British pence, shilling, which preservice teachers may pound, … not have come across in either Base 2, Base 5, Base 16 their school years or their daily life. (1997, p. 38) 50 The tests used in this study were graded using the Rusch  Hannigan rubrics. Rater reliability was checked by analyzing scoring differences between the two instructors on a small sample of tests. Twenty percent of the first test was scored by the instructors. Results of this analysis are discussed in Chapter 4. The rubrics employ a 1 – 5 scale. The numbers in the scale indicate the level of student understanding based on the descriptions found. 1. Algorithmic: Knowledge of how to manipulate symbols to get an "answer." This level of mathematical knowledge does not necessarily imply understanding of why the algorithmic steps make sense. 2. Tacit: Algorithmic knowledge with some intuitive understanding of the logical foundation from which the concept or algorithm emerges. This understanding is, however, somewhat vague and as a result it is difficult for the individual to articulate the logic which brings meaning to the concept. 3. Explicit: Precise understanding of the concept being addressed which can be clearly articulated and convincingly justified. Explicit understanding includes a clear understanding of the connections between the concept being addressed and related concepts, as well as an ability to articulate the logical development of related algorithmic procedures and generalizations. 4. Pedagogical (Self and Peers): Knowledge of how they (self) and individuals of similar age and experience (peers) build mathematical understanding. Able to thoughtfully choose pedagogical tools and strategies and are able to use language effectively to teach peers. 51 5. Pedagogical(Children): Knowledge of how children build mathematical understanding. Able to thoughtfully choose pedagogical tools and strategies and are able to use language effectively to teach children (Rusch, 1997, p. 34). Data Analysis Using Rusch’s and Hannigan’s work as a guide, the researcher used simple means and standard deviations for a total score to partially answer questions one and two. Question one: What is the level of explicit place value understanding that the elementary preservice teachers participating in the study bring with them to their required mathematics content course? Question two: What is the level of explicit place value understanding that the elementary preservice teachers participating in the study have at the end of their required mathematics content course? These statistics were compared using an independent sample ttest. A univariate analysis of variance was used to compare means in the four cells of the matrix (familiar systematic, familiar nonsystematic, unfamiliar systematic, and unfamiliar nonsystematic). Boxplots were constructed to give more specific information about the data. Correlations were calculated to examine possible connections within the data. To answer questions three and four, again total mean scores were compared – pre versus post means in the four cells, and means for rubric scores were compared. Question three: Is there any significant change in the participants’ explicit place value understanding from the beginning of the semester to the end of the semester? Question four: Is there a significant difference in gains made in explicit place value understanding between the group that had numeration system instruction (experimental group) and the 52 group that did not have the numeration system instruction (control group)? Analysis was done with a splitplot ANOVA for question three. Gains were calculated for question four, and then an independent ttest was utilized to discover significant differences. Of particular interest to the researcher were comparisons in the unfamiliar systematic grouping cell, as this in particular focused on nondecimal base systems, an important facet of this study. A modified Bonferroni correction was used to determine . Normally used at the 0.05 level, due to the large number of dependent variables, was recalculated to 0.05/11 = 0.0045. No qualitative data was collected, although many test questions asked the elementary preservice teachers to explain their answers. An example was question one. This question asked subjects to analyze an elementary student’s error pattern. As part of the question, the elementary preservice teacher is asked to “Briefly describe your assessment of what Caroline does not understand” (Rusch, 1997, p. 124). These comments were helpful as descriptive data to accompany the quantitative results. Conclusion This research was designed to assess place value understanding of preservice elementary teachers. The APVU was developed in 1997 just for this population; thus, it was chosen as the instrument to be used. Due to the lack of teacher commitment, only two classes participated in the project. These classes were compared by teacher, by content, and by student, with the help of a demographic sheet. The researcher used one form of the APVU as a pretest and posttest during the semester. Timing for these pretests and posttests was coordinated between the instructors of both groups. The assessment 53 tasks were scored and analyzed by the researcher in three categories: total, cell or grouping, and rubric. Comparisons were drawn on pretest and posttest means. To validate earlier literature and research, the hypothesis of this study is that the class receiving normal instruction in nonbaseten numeration systems (experimental group) will show a significant gain in scores, first exam to second, when compared to the without instruction class of students (control group). Of particular interest to the researcher are the gains for both groups in the unfamiliar systematic cell which contained the base numeration systems problems. Chapter Four will use a number of statistical tests and graphs to analyze the pretest and posttest scores of the two groups. These results can then be used to answer the research questions. Conclusions and recommendations will be stated in Chapter Five. 54 CHAPTER 4 FINDINGS This study investigated the influence of base numeration instruction on place value understanding for elementary preservice education majors. Two university classes of elementary preservice teachers were used to test the hypothesis that base numeration system instruction would increase the conceptual understanding of place value for the participants. Using a sample of convenience, the researcher and another instructor administered a pretest to each class near the beginning of the semester. During the semester the researcher’s class (control group) did not receive base system instruction. This instruction was incorporated at the end of the semester after the study was completed. During the semester the other instructor taught her class (experimental group) following the usual syllabus, including base system content. At the end of the semester the same test was given to both groups. The instrument used was the Assessment of Place Value Understanding (APVU). This thirteen item test was developed by Rusch (1997) and Hannigan (1998) at the University of Texas to specifically test place value understandingof preservice elementary teachers. The assessment can be found in Appendix A of this study. Four questions guided this research: 1. What is the level of explicit place value understanding that the elementary preservice 55 teachers participating in the study bring with them to their required mathematics content course? 2. What is the level of explicit place value understanding that the elementary preservice teachers participating in the study have at the end of their required mathematics content course? 3. Is there any significant change in the participants’ explicit place value understanding from the beginning of the semester to the end of the semester? 4. Is there a significant difference in gains made in explicit place value understanding between the class that has numeration system instruction (experimental group) and the class that does not have the numeration system instruction (control group)? Demographic Information During the semester of research, two classes of elementary education major students were invited to participate. One, at a private university, was taught by the researcher. One, at a public university, was taught by an instructor known to the researcher. Both universities are located in the same metropolitan area of a southwestern United States city. Each student was required to take the Assessment of Place Value Understanding (APVU) twice, once at the beginning of the semester and again near the end of the semester. Students were then asked by the researcher to use their test results for this study. Only scores for students who completed these three components were used in this research. This reduced the sizes of the samples to 23 for the control group and 32 for the experimental group. 56 These 55 students then completed a demographic information sheet. The main purpose for this additional request was to give the researcher some perspective about the students and to check for similarity of groups. In particular, ACT scores were solicited. This selfreported data is summarized in Table 4 Due to the small sample sizes, equivalency testing for the classes was not attempted. Table 4 ACT Mean Scores_______________________________________________________ Numeration system instruction Without With Number of respondents 8 14 Percent of respondents 35% 44% Mean 22.75 19.71 Other demographic information revealed that all students were female. Only two – at the public university – were parttime students; the other 53 reported fulltime student status. All students were taking their second or third college mathematics course. A summary of other demographic information is show in Table 5. Rater Reliability After data collection, a twenty percent sample of the pretests was scored by the researcher and the instructor. This was implemented to assess interrater reliability and measurement accuracy. Discussion of the scores between the raters then allowed for refinement of rubric meaning. The rubrics established by the authors of the APVU utilize 57 Table 5 Demographic Information__________________________________________________ Numeration system instruction Without With Average age Range 19.6 18  22 29.4 21  49 Average GPA Range 3.34 2  4 3.28 1.8  3.86 a one to five point scale to rate student understanding with : (1) algorithmic understanding; (2) tacit understanding; (3) explicit understanding; (4) self and peer pedagogical understanding; and (5) child pedagogical understanding. The data for this sample is summarized in Table 6. These results paralleled results previously obtained by the authors of the APVU; thus, it was determined that there was sufficient consistency to allow only the researcher to score all tasks and tests. Table 6 APVU Sample Scoring_____________________________________________________ Number of items Percent of items Differ by 0 322 86.6 Differ by 1 37 9.9 Differ by 2 10 2.7 Differ by 3 3 0.8 Differ by 4 0 0 58 Question One The first research question was: What is the level of explicit place value understanding that the elementary preservice teachers participating in the study bring with them to their required mathematics content course? This question was initially analyzed for all tasks for both the without instruction (control group) and instruction students (experimental group). A sample mean and standard deviation was calculated for each group’s scores. This information is summarized in Table 7. Table 7 All Tasks Pretest Item Mean Scores Numeration system instruction Without With Number of participants 23 32 Mean 1.95 1.95 Standard deviation 0.73 0.49 These scores indicated that the students achieved the same baseline results. No significant difference was found in the means. The boxplots in Figure 1 show somewhat similar distributions with similar medians of 1.81 and 1.845. As the standard deviations indicate, the noninstruction students’ scores varied more. A score close to two on the scale of mathematical understanding used in this research indicates some intuitive knowledge of the concept foundation. However, this understanding is somewhat vague, and the student may have difficulty articulating the logic behind their work. 59 Figure 1 To further define scores and place value understanding, the pretest scores were analyzed by cell. Matrix cell definitions were presented in Chapter 3. A table identifying the relationship of cell content and problems from the APVU is found in Appendix B. Pretest cell scores for both sites are presented in Table 8. Means and standard deviations were calculated for each cell and analyzed using a univariate analysis of variance to determine significant differences in the baseline scores of the two groups. Best scores were achieved in the familiar systematic cell. These problems focused on the vocabulary of place value understanding and the ability of students to evaluate elementary student work. Scores approaching three on the rubric scale indicate more precise understanding of the concepts. 60 Table 8 Pretest Cell Means________________________________________________________ Numeration system instruction Without With Familiar systematic 2.60 (.87) 2.43 (.70) Familiar nonsystematic 2.26 (1.20) 2.40 (1.16) Unfamiliar systematic 1.30 (.75) 1.32 (.52) Unfamiliar nonsystematic 2.29 (1.34) 2.38 (1.00) Note: Standard deviation in parentheses, * p < .0045 The familiar nonsystematic cell problems dealt with time problems. Scores around two again indicated some intuitive knowledge; however, the large standard deviations showed a broad range of scores (and therefore knowledge) among the two groups. Lowest scores were discovered in the unfamiliar systematic cell. Tasks here required the students to apply place value concepts in different base numeration systems. Some problems were typical base computation problems, while others required creativity and sophistication to interpret the task. A score of one on the rubric indicates a basic knowledge of an algorithm, but does not imply further understanding. The unfamiliar nonsystematic cell problems presented students with an unusual setting for place value 61 concepts. Results here were similar to the familiar nonsystematic cell, although higher standard deviations indicate a larger range of values. As with the “all tasks” scores, the cell analyses revealed similar baseline results. All cells produced no significant differences and homogeneity of variance. To further explore the pretest scores, means and standard deviations were calculated for the tasks by rubric. The seven rubrics used are defined in Appendix C. A table showing the correlation of rubrics and problems can be found in Appendix D. Results for this rubric analysis are summarized in Table 9. Highest scores were found with the error reproduction rubric. This rubric was utilized with question numbers 1 and 12 on the test – debugging elementary student work. A score of three, explicit understanding, indicates the students gave an inaccurate reproduction of the error, but there was some evidence of partial recognition of the error pattern. No significant differences were found among the scores on any of the rubrics. Standard deviations were above one (or one level of understanding) for the without instruction students’ Error Reproduction and both sets of students analysis of symbolic representation. This rubric was used with the foreign coinage problems, numbers 9 and 10. A level two, tacit understanding, for this rubric indicates the students used base ten techniques to arrive at an answer, not place value representation. Thus, for question one of this research, students at both schools achieved similar baseline results. The overall averages of 1.95 showed almost a tacit level of understanding. The more specific numbers of the cells and rubrics showed a range of 62 understanding of algorithmic (choice of correct representations) to explicit (error reproduction), with most cells and rubrics in the level 2 tacit range. Table 9 Pretest Rubric Means______________________________________________________ Numeration system instruction Rubric Without With Error reproduction 3.41 (1.19) 3.05 (.92) Depth of analysis 2.00 (.83) 1.89 (.70) Use of descriptive language 1.73 (.68) 1.66 (.67) Choice of correct representations 1.13 (.34) 1.16 (.37) Accurate computation 2.27 (.78) 2.30 (.50) Analysis of comp. method 1.45 (.91) 1.53 (.59) Analysis of sym. representation 2.04 (1.42) 2.31 (1.18) Note: Standard deviation in parentheses, * p < .0045 63 Question Two The second research question was: What is the level of explicit place value understanding that the elementary preservice teachers participating in the study have at the end of their required mathematics content course? This question was initially analyzed for all tasks for the public and private university students. A sample mean and standard deviation was calculated for each school’s student scores. This information is summarized in Table 10 . Table 10 All Tasks Posttest Item Mean Scores __________________________________________ Numeration system instruction Without With Number of participants 23 32 Mean 2.35 2.59 Standard deviation 0.53 0.70 Similar analyses were conducted with the posttest data as with the pretest data. A comparison of the posttest means showed no significant difference (p = 0.16) in the level of explicit understanding of place value concepts. Again boxplots were constructed from the data to show median differences of 2.19 versus 2.61 in Figure 2. Scores were more evenly distributed within the instruction class of students. To look for specific differences or levels of understanding, the posttest scores were analyzed by cell and by rubric, similar to the analyses of the pretest scores. The results are summarized in Tables 11 and 12. 64 Figure 2 Significant differences were found in the unfamiliar nonsystematic cell. This grouping of problem topics included the otherthanbaseten base problems, numbers three, five, six, eight, eleven, and thirteen on the APVU. Some of these problems involved basic computation in the other bases; some involved a higher level of problemsolving as well as base place value knowledge to complete the questions. The difference in scores was anticipated by the researcher. Prior to the posttest, the researcher’s students (control group) received only instruction on base ten problems and place value concepts. The instructor’s students (experimental group) had over two weeks of instruction with different base systems and had classroom experience with base computation. Although analyses showed no significant differences in mean scores in the other three cells, the boxplots in Figures 3  6 show clear differences in distribution. In two 65 cells, the familiar systematic cell dealing with children’s error patterns and the familiar nonsystemic cell of time problems, the without instruction students (control group) had a higher 25th, 50th, and 75th percentile. Table 11 Posttest Cell Means__________________________________________________ Numeration system instruction_____ Without With Familiar systematic 3.00 (.76) 2.80 (.80) Familiar nonsystematic 3.33 ** (1.02) 2.56 ** (1.05) Unfamiliar systematic 1.53 * (.51) 2.47 * (.96) Unfamiliar nonsystematic 2.57 (.81) 2.60 (.88) Note: Standard deviation in parentheses, * p < .0045, ** p < .05 66 Figure 3 Data NonInstruction Instruction 5 4 3 2 1 3 2.75 Familiar Systematic Cell Posttest Mean Scores Figure 4 Data NonInstruction Instruction 5 4 3 2 1 3.25 2.375 Familiar NonSystematic Cell Posttest Mean Scores 67 Figure 5 Data NonInstruction Instruction 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 1.46 2.615 Unfamiliar Systematic Cell Posttest Mean Scores Figure 6 Data NonInstruction Instruction 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 2.67 2.83 Unfamiliar NonSystematic Cell Posttest Mean Scores 68 The analysis by rubric identified no areas of significant difference at the = 0.0045 level. However, differences were apparent at the 0.05 level. These rubrics included: choice of correct representation, accurate computation, and analysis of computation method. Since all of these rubrics were primarily used for the unfamiliar systematic cell problems, it would follow that significance in these cells would bring significance in the rubrics. A table showing the correlation of rubrics and problems can be found in Appendix D. Table 12 Posttest Rubric Means_____________________________________________________ Numeration system instruction Without With Error reproduction 3.87 (1.11) 3.33 (.99) Depth of analysis 2.32 (.71) 2.11 (.64) Use of descriptive language 2.01 (.65) 1.95 (.69) Choice of correct representations 1.04 ** (.21) 1.25 ** (.44) Accurate computation 2.64 ** (.57) 3.10 ** (.86) Analysis of comp. method 2.00 ** (.63) 2.58 ** (.95) 69 Within With Analysis of sym. representation 2.48 (.81) 2.53 (.98) Note: Standard deviation in parentheses, * p < .0045, ** p < .05 Tables 13 and 14 show the correlations of posttest rubrics with each other. Significant correlations showing linear relationships exist between many of the rubrics. These results are consistent with the rubric/task correlation table found in Appendix D. Table 13 Without Instruction Rubric Correlations___________________________________ DA DL AC CM SR CR ER .558 ** .422 * .357 .392 .175 .222 DA .937 ** .530 ** .639 ** .538 ** .210 DL .470 * .589 ** .558 ** .081 AC .888 ** .652 ** .222 CM .581 ** .116 CR .141 Note: * p < .05, ** p < .01 ER = Error Reproduction; DA = Depth of Analysis; DL = Use of Descriptive Language; AC = Accurate Computation; CM = Analysis of Computation Method; CR = Choice of Correct Representation; SR = Analysis of Symbolic Representation 70 Table 14 With Instruction Rubric Correlations______________________________________ DA DL AC CM SR CR ER .730 ** .696** .590** .479** .207 .083 DA .930 ** .531 ** .564 ** .040 .243 DL .563** .571 ** .105 .201 AC .926 ** .531** .055 CM .433* .104 CR .244 Note: * p < .05, ** p < .01 ER = Error Reproduction; DA = Depth of Analysis; DL = Use of Descriptive Language; AC = Accurate Computation; CM = Analysis of Computation Method; CR = Choice of Correct Representation; SR = Analysis of Symbolic Representation Question Three The third research question was: Is there any significant change in the participants’ explicit place value understanding from the beginning of the semester to the end of the semester? To analyze the data, a repeated measure analysis of variance statistic was calculated. As with the previous two questions, data was compared by pretest and posttest across all tasks (in total), by cell, and then by rubric. For the pretest and posttest all task means, no significant differences were 71 found. This can be explained by reexamining the data discussion for question 2. Posttest averages for the without instruction students (control group) were higher in the cell for the time problems. Posttest averages for the instruction students (experimental group) were significantly higher in the cell for the base problems. Thus, these posttest differences contributed to no overall difference between pretest and posttest means. Table 15 summarizes these results. An ANOVA performed on the pretest/posttest scores for each of the two classes did find differences in each class (p = 0.04 and p = 0.00, respectively). This showed there was some growth in the participant’s place value understanding. Table 15 Change in Understanding from Pretest to Posttest_______________________________ Numeration system instruction Without With Pretest 1.95 1.95 Posttest 2.35 2.59 The repeated measures ANOVA indicated significant differences in the unfamiliar systematic cell. This demonstrated the significant effect of instruction on the pretest to posttest change of scores in the base numeration grouping of problems. These results are similar to the posttest analyses. In the unfamiliar cell, the instruction students increased their average scores by more than one level of understanding. Unfortunately, even their ending score of 2.47 indicates less than a precise understanding of the place value concepts. Table 16 provides the summary of analysis of cells. 72 Table 16 Change in Understanding from Pretest to Posttest by Matrix Cell___________________ Numeration system instruction Without With Familiar systematic Pretest Posttest 2.60 3.00 2.43 2.80 Familiar nonsystematic Pretest Posttest 2.26 3.33 2.40 2.56 Unfamiliar systematic Pretest Posttest 1.30 1.53 * 1.32 2.47 * Unfamiliar nonsystematic Pretest Posttest 2.29 2.57 2.38 2.60 Note: * p < .0045, ** p < .05 Analysis by rubric found no significant differences between the two groups. Growth was achieved on tasks in every rubric, but there was not sufficient evidence that instruction had an effect on the change in scores. Only the error reproduction rubric indicates a reasonably sophisticated level of understanding. A Level 3 score shows a partial recognition of the error pattern, while a Level 4 score gives evidence that the error is understood as only a computational error. 73 Correct answers to Problem one – Caroline’s Addition Error – are 7110, 757, and 1112. The error pattern work for Problem one should look like: 3 252 + 585 7110 An example of a level 4 score is shown in “Mary’s” work – 719, 757, and 1112. In this case Mary recognized the pattern correctly and then added incorrectly – 2 + 3 + 5 = 9. An example of a level 3 score is shown in “Cindy’s” work – 739, 757, and 3011. She recognized Caroline’s basic error of adding left to right, but didn’t replicate the second error of carrying the wrong digit. For example: 1 252 + 585 739 Rubric analyses are summarized in Table 17. Table 17 Change in Understanding from Pretest to Posttest by Rubric_______________________ Numeration system instruction Rubric Without With Error reproduction Pretest Posttest 3.41 3.87 3.05 3.33 Depth of analysis Pretest Posttest 2.00 2.32 1.89 2.11 74 Without With Use of descriptive language Pretest Posttest 1.73 2.01 1.66 1.95 Choice of correct representations Pretest Posttest 1.13 1.14 1.16 1.25 Analysis of comp. method Pretest Posttest 1.45 2.00 1.53 2.58 Analysis of sym.representation Pretest Posttest 2.04 2.48 2.31 2.53 Note: * p < .0045, ** p < .05 Question Four The fourth research question was: Is there a significant difference in gains made in explicit place value understanding between the class that has numeration system instruction (experimental group) and the class that does not have the numeration system instruction (control group)? Independent sample t tests were calculated for the gains for each group in total, in cells, and in rubrics. Only the unfamiliar systematic cell revealed a significant difference with a mean gain of 0.2335 for the without instruction students (control group) and a mean gain of 1.1522 for the instruction students (p = 0.00) (experimental group). The scores in 75 this cell reflect knowledge of problems relating to the research focus, base n problems where n 10. Final means in this cell were 1.53 for the without instruction control group students. On the scale of explicit understanding used for this research, this score indicates the students possessed an understanding between algorithmic meaning (process with no meaning) and tacit meaning (intuition of the meaning behind the process). The final mean of 2.47 in the base n cell for the instruction student scores’ (experimental group) indicates an understanding between tacit meaning and explicit meaning (clear understandings of the connections). Conclusion Beginning scores for both groups of students on the APVU indicated a similar baseline level of understanding of place value concepts. The overall average of 1.95 for both groups showed almost a tacit level of understanding – some intuitive knowledge, but vague understanding and poor articulation of the mathematical logic. This similarity of scores carried through the cell analyses and rubric analyses. No significant differences were found between the two groups. Means for the pretest were generally in the level one and level two range. Level one indicates algorithmic understanding, computation without meaning. Both groups showed an increase in scores in total, by cell, and by rubric. However, this increase still left many means in the algorithmic and tacit levels. Overall, both group means were in the two range, 2.35 and 2.59. Cell and rubric means were also generally at the tacit level of understanding. For the without instruction students, only the familiar systematic cell (error pattern 76 debugging) and the familiar nonsystematic cell (time) indicated an explicit level of understanding. For the instruction students, none of the cell means were above the tacit level. Rubric analyses indicated the best understanding for all subjects was in the area of error reproduction. Here all scores were in level 3, explicit understanding. However, on these debugging problems, students were not able to use descriptive language or show a depth of analysis. Analysis of data leads to interpretation of this data and its implications. These numbers must be put in perspective and related to the research questions of this study. Chapter Five will provide a summary of these results along with implications of this research and recommendations for further study. 77 CHAPTER 5 CONCLUSION Summary As the No Child Left Behind Act is revamped and renewed, there is much debate about highly qualified teachers and improved student learning. New studies are investigating the mathematics content knowledge and the skills that teachers must demonstrate to improve student achievement. Questions are being raised about what mathematicians/mathematics educators can do to improve the mathematical preparation of teachers. New work by Thames (2006) reports that many college courses today leave teachers unprepared, lacking sufficient mathematical knowledge needed for competent teaching. This, in turn, affects student learning. Perhaps more “rigorous content relevant to teaching and learning” needs to be explored (p. 2). That search for meaningful curriculum has been the focus of this research. At present, in most mathematics sequences for elementary education majors, the topic of base numeration systems is studied. Authors and researchers give rationale for this content, but is that rationale correct? Does this subject help students acquire an explicit understanding of place value? This study investigated the role of base numeration system instruction as it relates to place value understanding in preservice elementary teachers. To accomplish this task, the Assessment of Place Value Understanding (APVU) was administered to fiftyfive 78 preservice elementary teachers at two universities in the southwest. These students were enrolled in a required mathematics course developed specifically for elementary education majors. The APVU was chosen to ascertain knowledge because it was specifically developed for this population to measure place value understanding. This assessment instrument connected the population of elementary preservice teachers and place value concepts and included a section of problems involving base numeration systems. These three elements were components of this research. This chapter provides a brief summary of the numerical findings of this research and reviews the four guiding questions. Conclusions about preservice elementary teacher place value understanding are discussed using the rubrics and the five levels of understanding associated with the assessment instrument, the APVU. These conclusions are then compared to previous studies. Limitations of the study are listed before improvements and changes lead to recommendations for future research. During the course of a single semester, one version of the APVU was given twice to each of the two groups, once early in the semester (pretest) and once near or at the end of the semester (posttest). During the weeks of class, regular instruction was provided to the students at the public university by an instructor known to the researcher. No changes were made to the instructor’s methods or syllabus. The conventional curriculum covered such topics as base ten place value and base numerations systems as part of the content. This group of subjects was identified as the experimental group. During the same weeks, the private university students received instruction from the researcher in base ten place value and all syllabus topics except base numeration systems. Once the posttest was 79 given to the researcher’s class, the missing topic of base numeration systems was taught at the end the semester. This group was identified as the control group. The choice of the two classes in the sample was constructed by convenience. Although many schools were contacted about participating in the study, only one followed through on the commitment. The researcher used her own class as the noninstruction/ delayed group because no other instructor contacted wanted to deviate from the normal sequence of topics. The pretest scores provided baseline information on the place value understanding that the preservice elementary teachers had prior to the semester of instruction. Question one asked: What is the level of explicit place value understanding that the elementary preservice teachers participating in the study bring with them to their required mathematics content course? Although there was much concern about the sampling used in this study, pretest scores showed no significance differences between the two groups of students. The overall means for both groups was a 1.95 on a scale from 1 to 5 of understanding. Medians were 1.81 for the without instruction group (control group) versus 1.845 for the instruction or experimental group. Boxplots showed similar distributions. Based on this five level scale, these average scores were at less than a tacit level. This score level indicates that students possess the ability to use an algorithm but demonstrate vague understanding of its logic. The thirteenitem test was then analyzed by the four groupings (cells) of similar problems and by rubrics utilized on each problem. No significant differences between the two groups were found in any category. Thus, both groups entered the study with comparable levels of place value understanding. 80 Using the five level scale of understanding , the elementary preservice teachers’ average scores indicated almost a tacit level of understanding. The posttest scores provided information on the place value understanding that the preservice elementary teachers had after the semester of instruction. Question two asked: What is the level of explicit place value understanding that the elementary preservice teachers participating in the study have at the end of their required mathematics content course? Results of the posttest scores showed a slightly higher total mean score for the instruction students – 2.35 for the without instruction (control) group versus the mean score of 2.59 for the instruction (experimental) group. However, this was not enough to be statistically significant. Individual cell analysis found a difference in means in only one of the areas – base problems. The instruction students did significantly better with the base problems, demonstrating a mean of 2.47 compared to the without instruction group (control group) mean of 1.53. This result was expected due to the study design. Thus, both groups of students demonstrated an average tacit level of understanding of place value as the semester ended. Rubric analyses showed no significant differences between the two groups. However three rubrics, used for problems included in the unfamiliar systematic (base) cell, did show marked differences. Choice of correct representation, accurate computation, and analysis of computation method rubrics showed higher means for the instruction (experimental) group. A comparison of scores of the choice of correct representation, rubric used for Problem 11 of the APVU, found that its mean was 1.25 for the instruction group and 1.04 for the without instruction (control) group. The accurate computation rubric score means, used for Problems 2, 3, 5, 6, 7, 8, 9, 10, and13, were 81 3.10 for the instruction group and 2.64 for the without instruction group. Means of 2.58 for the instruction group and 2.00 for the without instruction group were found for the analysis of computation method rubric, used for Problems 2, 3, 5, 6, 7, 8, 9, 10, and 13. Next the pretest and posttest for both groups of study are analyzed. Comparing scores across the groups answered question three. Question three asked: Is there any significant change in the participants’ explicit place value understanding from the beginning of the semester to the end of the semester? Mean scores for the two groups were analyzed for significant changes in understanding from pretest to posttest. Each group, individually, showed significant growth, but the means across the groups were not significant. Only the unfamiliar systematic cell dealing with base numeration problems showed significance at the = 0.0045 level between the two groups. Again, this result was anticipated due to the study design. No rubric analyses produced significant differences. Thus, there were no significant changes in place value understanding across the groups from pretest to posttest. Lastly, question four explored the gains made by the students during the semester. Question four asked: Is there a significant difference in gains made in explicit place value understanding between the class that has numeration system instruction (experimental group) and the class that does not have the numeration system instruction (control group)? The differences in gains for both groups of students were calculated and analyzed. Total mean scores and rubric scores calculated for each of the two groups showed no differences in gains. Only the unfamiliar systematic cell problems indicated a significant difference, a gain on mean scores of 0.23 (1.30 to 1.53) for the noninstruction students and 1.15 (1.32 to 2.47) for the instruction students. This information correlated 82 with other data previously obtained. Thus, there were no significant gains in understanding as measured by the APVU for the students in either group. Conclusions The Assessment of Place Value Understanding (APVU) was developed to indicate levels of mathematical understanding from algorithmic to pedagogical for preservice elementary teachers. A rubric scale of one to five corresponds to the five levels of depth of understanding. Each question on the APVU used two to three rubrics to evaluate the understanding students demonstrated on each problem. The lowest level referred to as algorithmic understanding indicates a student has knowledge of computational procedures, but not necessarily an understanding of why the process works. The second level described as tacit understanding indicates the student has some intuitive reasoning ability, but an inability to articulate the logic of the concept. At level three, explicit understanding, the student demonstrates a precise understanding of the concept with good articulation of the logic and a clear sense of connections among ideas. Level four named pedagogical, (self and peers), denotes students who know how they and their peers build mathematical understanding. Lastly, level five, pedagogical, (children), asserts that students have a level of understanding needed for effective teaching. Students at this level know how others build knowledge and how to choose appropriate tools for teaching. Using Problem 4 from the APVU as a guide for discussion, the following represent student responses that serve as examples of each level of understanding. Analyzing problem 4 required the use of the Depth of Analysis and Use of Descriptive 83 Language rubrics. These rubrics are described in Appendix C. Level 1 – Algorithmic “He thinks he can pick any chip to represent the number he wants.” Level 2  Tacit “He does not fully understand place value. He thinks in terms of units.” Level 3  Explicit “He is not understanding place values, ones and tens. He is counting by ones only.” Level 4  Pedagogical (self/peers) “Bobby isn’t using place value skills. He circled the six (1s) but when Mrs. Jones pointed to the two he just circles 2 not realizing that the 2 had a place value and should have been a 20.” Level 5  Pedagogical (children) “Bobby doesn’t understand the place value that the 2 represents. He is viewing it as a single digit not as the number of tens that the second digit represents. The two represents 84 2 groups of two or twenty, so he is not understanding the value of the position.” In this study it was disappointing to find that both groups of students began the semester with less than a tacit understanding of place value (1.95 mean score). Current research in workshops sponsored by the Mathematical Sciences Research Institute on critical issues in mathematical education promotes the importance of clear explanations, choosing useful examples, and evaluating students’ ideas (Thames, 2006). Students were unable to provide explanations or accurately assess student work on the APVU. Also disappointing in the results was that after a semester of instruction, all the students’ understanding had improved to only a midlevel tacit understanding. These low mean scores and some large standard deviations within the groups (more than one level) may have been the result of varied abilities. These scores also may have been the result of student apathy in answering all the questions. Because there was no incentive for students to do their best work, a number of problems, or parts of problems, were left blank by each group of students. Approximately twenty to twentyfive percent of all students left at least one problem blank. The instruction (experimental) group left an average of 2.125 questions blank on the pretest and 1.67 problems blank on the posttest. The without instruction (control) group left an average of 2 problems blank on the pretest and 1.625 problems blank on the posttest. More blanks occurred for problems 11, 12, and 13, although all problems from 3 to 13 had at least one blank per group. These blanks were scored as a zero and had an affect on the calculation of mean scores and standard deviations. Interpretation of the blanks was difficult for the researcher to assess without followup. No answers to the problems may have indicated 85 confusion about the question, no knowledge of how to answer the question, or no desire to answer the question. Earlier examination of the pretests could have allowed the researcher time to question students about the problems they did not complete. Unfortunately, all tests were examined at the end of the semester to help with the rater reliability regarding scoring. Due to already small sample sizes, it was not possible for the researcher to discard tests with blanks. Analyses indicate that for these 55 students, the treatment of base numeration system instruction did not significantly impact understanding as measured by the APVU. There was, however, growth in each class’ place value understanding. The change in the means from pretest to posttest for the total, in every cell, and in every rubric indicate increases in understanding for both groups of students. These results do suggest that instruction in this content area had a positive influence on the development of understanding in place value and numeration systems. Student responses from Problem 1 show this increase in instruction. Problem 1 on the APVU asked students to replicate and explain Caroline’s addition error pattern. One student on the pretest wrote, “She is adding from left to right instead of right to left.” On the posttest the same student answered, “Caroline doesn’t understand to add from right to left. She is adding as if she is reading a book. She also carries the wrong numbers to the next digits place” The posttest response is still not at a pedagogical level, but the student has begun to include place value language in her response. Authors of the APVU, in their studies ten years ago, had similar findings. Rusch’s research with place value understanding, constructivist, and direct instruction concluded that “the participants’ failure to use place value concepts in even moderately 86 sophisticated ways suggests that, by the end of the course dealing with place value concepts, they had not yet achieved the recommended ‘explicit’ level of understanding” (Rusch, 1997, p. 104). Gains made by the constructivist class of students were not significantly greater than the gains made by the direct instruction students. Hannigan’s study with place value understanding and writing prompts concluded that students did show some growth in their place value understanding. However, understanding was still not at the desired level for teachers, and the treatment of freewriting made no difference in the students’ scores (Hannigan, 1998). The current study’s findings support the results of these two research projects. Recent research in elementary preservice teachers’ place value understanding by FarroLynd found a “lack of foundational knowledge of the structure of place value” (FarroLynd, 2003, p. 104). Her study focused on misconceptions of place value concepts by preservice elementary teachers. Although the current study’s focus was on numeration system instruction and used a different assessment instrument, this researcher found a similar lack of understanding. Skipper’s study in 1972 using manipulatives with numeration systems concluded instruction methods provided no significant difference in place value knowledge. The current research studied understanding, not just knowledge, and found no significant different in place value understanding. Haukebo’s research in 1967 was similar to this project in that the research focused on place value under 



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