Developing Conceptual Understanding of
Running head: Conceptual Understanding of Fractional Numbers
Developing Preservice Teachers’ Procedural and Conceptual
Understanding of Fractional Number Concepts
Cheryl J. McAllister
M. N. S., Mathematics, Southeast Missouri State University, 1990
B. S. Ed., Secondary Education – Mathematics, Southeast Missouri State University, 1985
A Dissertation Submitted to the
Graduate School of the University of Missouri-St. Louis
in Partial Fulfillment of Requirements for the
Doctor of Philosophy Degree in Education
Advisory Committee:
Lloyd I. Richardson
Helene J. Sherman
Cody Ding
Tamela Randolph
December 12, 2005
Abstract
Research indicates that of the mathematical concepts taught in the K-8 curriculum, fraction concepts are one of the least understood by both students and teachers. Of secondary concern is how the attitudes towards mathematics and fractions that teachers bring into the classroom may affect the way they teach the topic. One of the avenues suggested to address this issue is to improve the training of preservice teachers on the topic teaching fractions. This study explored that suggestion by examining how two different methods of teaching the fraction concept unit in a mathematics content course for elementary teachers impacted the conceptual and procedural understanding and the attitudes of the preservice teachers in the classes. Also studied was whether adding a field experience to the content course, in which the students would teach fraction concepts to fifth and sixth graders in an after school program, would improve their conceptual and procedural understanding and attitudes towards fractions.
The study had a quasi-experimental design and was analyzed using a 2 (type of instruction) × 2 (field experience or not) × 4 (test occasion) factor analysis of variance with repeated measures on the factor test occasion (pretest, posttest, retention test 1, retention test 2) for a test of conceptual and procedural understanding of fractions and two attitude surveys related to mathematics – Aiken’s Mathematics Attitude Scale and a Semantic Differential. Data was analyzed using SPSS version 13 for within-subjects and between-subjects differences. Analysis of the conceptual and procedural understanding test scores revealed a significant difference for the factor of test occasion. Further analysis indicated that all groups improved significantly from the pretest to the other three testing occasions, but no other significant differences between groups was found. No significant differences were found at any level for the two attitude scales.
ACKNOWLEDGEMENTS
There are hundreds of people I would like to acknowledge for their contributions to this project, but space and time considerations mandate that I be concise. Failure to mention someone on this page does not mean that I have forgotten each and every encouragement and assistance I have received in achieving this goal. Dr. Lloyd Richardson recruited me to the University of Missouri – St. Louis doctoral program and has served as my advisor, mentor, teacher, and friend. Thanks for pushing me. Along the way Helene Sherman and Tamela Randolph also served as teachers, mentors, and friends and agreed to serve on my dissertation committee. Your support and encouragement were priceless. Dr. Cody Ding completes my committee and I am grateful for his expertise and what he taught me about statistical analysis. All of the faculty, staff, and my fellow students in my courses at University of Missouri – St. Louis played a vital role in helping me complete my degree and making my experience rich and wonderful.
My colleagues in the Mathematics Department at Southeast Missouri State helped me in ways I can not enumerate. Dr. Gummersheimer arranged a reduced teaching load and allowed me to teach coursework that supported my graduate work. Dr. Pradeep Singh spent several hours helping me to understand analysis of variance with repeated measure. Other members of the department served as research participants and reviewers for various projects. I appreciate everyone’s support and encouragement.
It is not possible to list all of the thousands of ways that friends and family have helped me. My most ardent supporters were my two daughters, Jacqueline and Margot, and my husband, Mark. There are not enough words in the world to tell you how grateful I am to have you in my life.
CONTENTS
Page
ABSTRACT ii
ACKNOWLEDGEMENTS iii
TABLE OF CONTENTS iv
LIST OF TABLES vii
LIST OF FIGURES viii
CHAPTER
I. PURPOSE AND RESEARCH PROBLEM 1
Purposes of Study and Hypotheses 7
Delimitations 8
Limitations 9
Definitions of Terms 9
Significance of the Study 12
II. REVIEW OF RELATED LITERATURE 13
Procedural and conceptual knowledge – why teachers need both 13
Research on U.S. elementary teachers’ understanding of rational 16
number concepts
Research on attitudes towards mathematics and fractions 19
Creating rich learning environments 24
Teaching mathematics content to US preservice teachers 30
Research on ‘teaching to learn’ content 31
Rationale for why this study is needed and how it adds to the 32
body of research
III. METHODS 34
Subjects 35
Reliability and Validity of Assessment Instruments 38
Instruments to assess conceptual and procedural knowledge 38
Instruments to assess students’ attitudes towards mathematics 40
Procedures 41
Design 46
Possible Limitations and Threats to Validity 47
IV. Results 49
Design and Hypotheses 49
Reliability of Testing Instruments 51
Assumptions Underlying the General Linear Model with Repeated 52
Measures
Analysis of Variance Results 54
Summary of Results 66
V. Discussion 68
Summary of Purpose, Hypotheses, and Design 68
Finding and Conclusions 70
Implications of Study 73
Limitations of the Study 76
Future Research Directions 78
REFERENCES 79
APPENDIXES 87
A. Version A and B of Conceptual and Procedural Understanding Test 87
B. Attitude Scales 100
C. Lesson Plans for the Traditional Lecture/Discussion Treatment 103
D. Lesson Plans for the Hands-on Treatment 116
E. Field Experience Assignment 136
F. Outside Reading Assignment 138
G. Lesson Plans for After School Mathematics Program 140
H. Raw Data 171
I. Histograms of Dependent Variables 173
J. Box Plots 178
LIST OF TABLES
3.1 Study Design
3.2 Study Timeline
4.1 Correlations Between Scores on Administrations of Version A and B of Testing Instrument
for Conceptual and Procedural Understanding
4.2 Means and Standard Deviations for Type of Instruction × Field Experience × Test Occasion
on Content Knowledge Scores
4.3 Summary Table for Type of Instruction × Field Experience on Content Knowledge Test
Scores
4.4 Means and Standard Deviations of Participants on Each Test Occasion
4.5 Analysis of Variance Table for the Main Effect of Test Occasion
4.6 Post Hoc Tests: Tukey HSD
4.7 Means and Standard Deviations for Type of Instruction × Field Experience × Test Occasion
for Aiken’s Mathematics Attitude Scale
4.8 Summary Table for Type of Instruction × Field Experience on Aiken’s Mathematics
Attitude Scale
4.9 Means and Standard Deviations for Type of Instruction × Field Experience × Test Occasion
for Semantic Differential Scale
4.10 Summary Table for Type of Instruction × Field Experience on Semantic Differential Scale
LIST OF FIGURES
1.1 Lesh’s Model of Multiple Modes of Mathematical Representation
2.1 Lesh’s Model of Multiple Modes of Mathematical Representation
3.1 Study Paradigm
4.1 Comparison of mean content test scores for all combinations of treatments
4.2 Comparison of mean Aiken Scale scores for all combinations of treatments
4.3 Comparison of mean Semantic Differential scores for all combinations of treatments
Chapter I
In 1992 the University of Chicago Conference report issued on mathematics education in the United States stated, “…by nearly every measure, mathematics education today is a failure” (p.3). In 2000 the National Commission on Mathematics and Science Teaching for the 21st Century (NCMST) issued a report that included the conclusion that “Our children are falling behind; they are simply not ‘world-class learners’ when it comes to mathematics and science” (p. 4). From these reports and others, it is obvious that the level of U.S. students’ mathematics achievement is below what most U.S. policy makers deem satisfactory for the leading technological nation in the world. Results from such sources as the National Assessment of Educational Progress (NAEP), individual state assessments, and the Third International Mathematics and Science Study (TIMSS) indicate that while some progress has been made in mathematics achievement for some groups in some areas, overall, students in the United States are not learning and understanding mathematics at sufficiently high enough levels to reassure policy makers that our educational system is producing future citizens with the skills and knowledge needed to compete in a global economy (CBMS, 2001; NCMST, 2000).
Research by Liping Ma (1999) and others indicates that U.S. elementary teachers do not have the understanding of basic arithmetic needed to help children develop deep, conceptual understanding of mathematics. In spite of a variety of efforts to provide professional development for teachers, to hold schools accountable for student achievement, and to fund a variety of programs aimed at raising test scores, there is little evidence to believe much has changed since the 1992 report.
Improving mathematics education in the Unites States is a complex problem. A few of the factors that make a quick and clear-cut solution impossible include: the realities of federal No Child Left Behind policies; state assessments for students which don’t align well with what they are taught; non-uniform state teacher certification requirements; independent school district curricula; issues of providing an equal educational opportunity to students with a variety of special needs; and difficulty hiring and keeping qualified teachers willing to meet the challenges of today’s classrooms. One area that has been targeted by researchers and policy makers as a means of improving the quality of mathematics education in the U.S. is the improvement of teaching through enhancing teacher content knowledge (NCMST, 2000). “Simply put we have failed to give elementary school teachers the kinds of mathematical experiences that will equip them to teach mathematics (University of Chicago Conference, 1992, p. 3).” Ma’s study (1999), entitled Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States, demonstrated the vast differences in procedural and conceptual knowledge of mathematics of Chinese and U.S. elementary teachers and suggested that the most logical place to begin to address the quality of elementary teachers’ knowledge of mathematics is in their teacher preparation programs.
The issue of the mathematics preparation of elementary school teachers is almost as complex as the problems of mathematics education in this country. Mathematics preparatory standards and requirements for receiving a teaching certificate vary from state to state. How many and what types of college mathematics course hours should be required for elementary teachers are questions that are answered differently in different states, as well as at different teachers’ colleges and universities within those states. Within the mathematics education community there is an ongoing debate about the types and criteria of courses elementary teachers need to be prepared for today’s mathematics classroom (CBMS, 2001). Some believe that taking more advanced mathematics courses will help teachers develop the levels of understanding they should have to teach for understanding. Others believe that elementary teachers do not require higher level mathematics, but should take courses that reexamine the mathematics they will teach at the elementary level, with the goal being more depth and breadth of knowledge about the arithmetic, geometry, and data analysis topics common to elementary mathematics curricula (CBMS, 2001; Ma, 1999; Schifter, 2001; Sowder, Bezuk, & Sowder, 1993; Spungin, 1996; Tirosh, 2000).
One elementary mathematics topic that is consistently identified as a topic for which elementary teachers in the Unites States are ill prepared is that of rational number concepts. High school students’, beginning college students’, and elementary teachers’ understanding of rational numbers and operations on rational numbers have been shown to be weak in a variety of studies (Brown, Cooney, & Jones, 1990; Carpenter, Corbitt, Kepner, Lindquist, & Reyes, 1981; Ma, 1999; McDiamid, Ball, & Anderson, 1989; Post et al., 1988). Tirosh (2000) stated, “Division of fractions is often considered the most mechanical and least understood topic in elementary school” (p. 6). Ma’s study (1999) illuminates many of the issues addressed above, but a few points bear special consideration here. Ma concluded that the U.S. elementary teachers, as a group, had weaker procedural knowledge of fraction division than the Chinese teachers and almost no conceptual knowledge of fraction division. She also provided evidence that suggests the U.S. elementary teachers were weaker in procedural and conceptual knowledge than a group of Chinese preservice teachers and a group of Chinese ninth grade students.
Research seems to indicate the majority of preservice elementary teachers have negative attitudes toward mathematics in general and rational number concepts specifically (Brown, Cooney, & Jones, 1990; CBSM, 2001). “Students’ attitudes towards math are often increasingly negative as they progress through elementary into secondary grades. These students form the pool of potential elementary school teachers and bring their negative attitudes into teacher preparation programs (Putney & Cass, 1998, p. 627).” Preservice teachers report limited elementary and middle school experiences with manipulatives in learning mathematical concepts, especially fractional number concepts. They have a weak grasp of the procedural algorithms for operations on fractions, limited conceptual understanding of fractional number concepts, and low standardized test scores to demonstrate this deficit (Cramer & Lesh, 1988). Because of the difficulty, lack of understanding, and poor teaching many students experience with fractions, it is not surprising that research has shown that many preservice teachers hold attitudes toward this topic that are negative. Research suggests that preservice teachers usually complete their teacher preparation programs without addressing these issues and carry their negative attitudes and weak skills into their own classrooms (Karp, 1991). These negative attitudes are then recycled to the next generation of elementary school students as these preservice teachers graduate and take jobs in U.S. classrooms (Putney & Cass, 1998).
It is important that preservice teachers identify their negative attitudes and actively work to change those attitudes (Stuart & Thurlow, 2000). Teacher preparation programs can assist these students by providing mathematics content and techniques courses that supply the rich environments for mathematical exploration, problem solving, and success that have obviously been lacking in their prior experiences (Post et al., 1988). If mathematics education in the United States is ever to raise the achievement scores of students to levels that are considered satisfactory, steps need to be taken to break the cycle of poor teaching producing poor teachers (CBMS, 2001).
Mathematics students of all ages need rich learning environments that promote improved attitudes and learning activities that support flexible and reliable procedural and conceptual knowledge of mathematics. But what constitutes a rich environment for learning mathematics? Piaget posited that children need active interaction with their environment and physical objects to build conceptual understanding of mathematics (Piaget, 1941, 1972). Dienes theorized that children must experience and create multiple embodiments of mathematical concepts using a variety of physical objects and manipulatives to help them generalize concepts (Dienes, 1970). Bruner identified three stages through which children progress in learning mathematics: the enactive, the iconic, and the abstract or symbolic (Bruner, 1968). He advised teachers to use manipulatives to help students move from the enactive (concrete) stage to the iconic (semi-concrete) stage. Lesh revised Bruner’s theory from three stages to a model with six systems of representation that are interactive rather than linear, each contributing to a student’s developing mathematical understanding (Behr, Lesh, Post, & Silver, 1983). Lesh’s model (illustrated in Figure 1.1) includes the following modes or systems of representation: spoken symbols, manipulatives, pictures and diagrams, written symbols, and real world situations.
[pic]
Figure 1.1 Lesh’s Model of Multiple Modes of Mathematical Representation
Lesh’s model, implemented with a constructivist viewpoint (Herscovies, 1996; Nickson, 1992; Thompson, 1992), embodies the rich learning environment mentioned previously.
There is an old adage that ‘you don’t know a subject thoroughly until you teach it’. Research (Evans & Flower, 2001; Lowery, 2002) supports the notion that teaching subject content to another person actually improves the conceptual understanding of the teacher. Ma’s (1999) conjectured that “the key period during which Chinese teachers develop a teachers’ subject matter knowledge of school mathematics is when they teach it – given that they have the motivation to improve their teaching and the opportunity to do so” (p. 147). Ball and McDiarmid (1990) wrote, “As they struggle to teach their subject in ways that make it meaningful to the students, the beginning teachers … draw on their growing knowledge of students, of the context, of the curriculum, and of pedagogy. In short, they evolve a new understanding of the content…” (p. 445). Ball (1993) wrote, “It was in working with third graders … that the most crucial issues of content and learning emerged for me. Looking at rational numbers from the perspective of a nine year old whose familiar mathematical domain is being stretched and transformed, I saw aspects of rational number thinking that I had not noticed before” (p. 173). Deborah Schifter (2001) relates experiences with tutoring a student she called ‘Wanda’ and working with a teacher she called ‘Mary Ryan’ and how those episodes changed her way of thinking about elementary mathematics. “When I consider my work with Wanda or Mary Ryan, I recognize how many mathematical insights came, and still do come, to me in analogous ways and how they contribute to my own growth as a teacher” (p. 115). Research needs to be done to further understand how teaching subject matter content to others can increase or improve teachers’ subject matter knowledge and understanding.
Purposes of Study and Hypotheses
Since a case has been made that teacher education programs need to be improved if teachers’ attitudes, content understanding, and procedural skills are to be improved, a study that would suggest a model for an improved mathematics content/methods course would contribute to the body of literature currently available. Hence, the main purpose of this study is to explore how the conceptual knowledge of preservice teachers presented a unit of study on fractions (based on a learner-focused, constructivist view (Thompson, 1992) and utilizing Lesh’s model of a rich learning environment) in a mathematics content course, followed by a field experience in which the preservice teachers would interact with the mathematical content while working with fifth and sixth grade students compares to the conceptual understanding of fractions retained by preservice teachers who are exposed to a more traditional, teacher directed presentation of fractions and who do not participate in the field experience. A secondary purpose of the study is to determine if the attitudes of preservice teachers exposed to a learner focused unit on fractions, followed by a related field experience are more positive than preservice teachers exposed to a more traditional delivery of the content material and who are not required to participate in a field experience. More formally stated the research hypotheses are as follows:
• Preservice teachers who have been through a unit of study on fractional number concepts utilizing a learner focused approach based upon Lesh’s model of six interactive systems of representation will develop deeper procedural and conceptual understanding of fractions compared to preservice teachers who are presented the material in a more traditional, teacher-directed environment.
• Preservice teachers who have been through a unit of study on fractional number concepts utilizing a learner focused approach based upon Lesh’s model of six interactive systems of representation will show an improved attitude towards mathematics in general and towards rational number concepts specifically when compared to preservice teachers who are presented the material in a more traditional, teacher-directed environment.
• Preservice teachers who explore fraction concepts with age appropriate students (fifth and sixth graders) will develop a deeper, conceptual understanding of fraction concepts than preservice teachers who have not worked with the students.
• Preservice teachers who explore fraction concepts with age appropriate students (fifth and sixth graders) will develop a more positive attitude towards the topic of fraction concepts than preservice teachers who have not worked with the students.
Delimitations
This study will be conducted at a Midwestern regional university with a total approximate enrollment of 9500 students and a well established, NCATE accredited teacher education program. The study will utilize as participants preservice elementary education majors enrolled in the second of a series of three mathematics content courses. This mathematics course covers the content area of fractional numbers. The two treatment units developed for the preservice teachers will consist of a unit based on a traditional lecture/discussion format with some work with manipulatives compared to a unit consisting of constructivist based activities utilizing Lesh’s model of multiple representations, with most of the format being student driven and the teacher acting as facilitator. A random sample of the preservice teachers from both groups will participate in an enrichment program for fifth and sixth grade students at a local middle school in an after school ‘academic club’ situation. The students not chosen to participate in the field experience read a series of articles from the current research on teaching fractions to children.
Limitations
This study may not be generalizable to the overall population. The preservice teachers in the study will be selected from those opting to attend a regional university in the Midwest. Since colleges and universities have different entrance requirements and every state has different criteria guiding mathematics education, the mathematical preparation of the education majors selected for the study may not be representative of the overall population of elementary education majors. Some participants will be students who are able and willing to commit a significant amount of their own time to be in the field experience portion of the study and that type of student may not be representative of all education majors. Students in the study may have scheduling conflicts that prohibit them from being selected for the field experience.
Definitions of Terms
For the purposes of this study the following operational definitions will be used:
• Preservice teachers or teacher candidates will be considered to be education majors who are required to complete this particular university’s mathematics content courses for elementary mathematics teachers. This includes elementary education majors, special education majors, middle school education majors, and some early childhood education majors.
• A field experience is a situation in which uncertified preservice teachers are allowed to work with and practice teaching to school children. An appropriately certified teacher usually supervises a field experience.
• Rational numbers will refer to the subset of the real number system characterized by numbers that may be represented as a quotient of two integers. Fractional numbers will refer to the set of non-negative rational numbers.
• Fractional number concepts will include: the definition of fractions; equivalent fractions; ordering fractions; adding, subtracting, multiplying, and dividing fractions; and changing improper fractions to a mixed numeral representation and back. Also included will be the ability to solve word problem situations involving fractions, formulating word problems using fractions, and developing multiple physical and pictorial representations of fractions.
• Manipulatives will be defined as physical objects designed and used to model abstract mathematics concepts. This includes (but is not limited to) fraction circles and squares, pattern blocks, tangrams, counters, and Cuisenaire Rods.
• Computational tools are devices such as calculators and computers that assist in computation.
• Measurement tools include (but are not limited to) rulers, protractors, measuring cups, scales, clocks, thermometers, and other items used to measure length, area, volume, mass, weight, time, or temperature.
• Instructional aids are items such as hundreds charts, number lines, graph paper, flash cards, and worksheets.
• Procedural understanding of fractions is knowledge of and ability to use appropriate algorithms to perform accurate calculations of the operations of addition, subtraction, multiplication, and division of fractions, simplify answers, and move easily between improper form and mixed numeral form of fractions. The definition of procedural understanding will be more fully developed in Chapter II.
• Conceptual understanding of fractions is knowledge of and ability to correctly apply procedural knowledge to real world problems solving situations involving fractions, the ability to create real world problems involving fractions, and the ability to apply the concepts associated with fractions in multiple ways with various representations. The definition of conceptual understanding will be more fully developed in Chapter II.
• Constructivism is “the assumption that any learning of higher order concepts involves some kind of integration by the learner into his or her existing cognition” (Herscovies, 1996, p. 351). When actualized in a teaching environment this involves determining the student’s current understanding of the material, then providing activities which allow the student to ‘construct’ an understanding of the concepts being taught. When the activities involve working with small and large groups, discussing the concepts until a mutual understanding is achieved, this concept is referred to as ‘social constructivism’. In this definition the ‘truth’ is the concept the group constructs and accepts. The definition of constructivism for purposes of this study will be more fully developed in Chapter II.
• Traditional, teacher-directed environment refers to the model of instruction in which the teacher, in a lecture/demonstration or lecture/discussion mode, presents information and models appropriate problem solving behaviors for the students. In this model the ‘truth’ of a concept is what the teacher determines it to be.
• Real world applications are mathematical problems based in the normal, daily life experiences of human beings.
• Spoken symbols are vocabulary, both formal and informal, used to describe, discuss, and explain mathematical ideas. Spoken symbols may be conveyed in written form.
• Written symbols are special mathematical symbols used to describe, discuss, and explain mathematical ideas.
• Rich learning environment is a learning setting that includes opportunities for students to construct knowledge through multiple representations of concepts and through social interactions and exchanges with fellow students and the teacher.
Significance of the Study
This study is intended to add to the current literature on the teaching and learning of rational number concepts. Since rational number concepts appear to be a challenge for U.S. mathematics education, any significant finding could improve the current approach taken to the teaching and learning of rational number concepts and to preparing teachers to teach these concepts. The study has implications for the body of research on teacher education and could serve as a means of identifying ways to improve mathematics content and methods courses for preservice and in-service elementary teachers. Improved teacher preparation programs could help break the cycle of ineffective teaching producing the next generation of ineffective teachers.
Chapter II
Review of Related Literature
Procedural and conceptual knowledge – why teachers need both
In a review of research on novice and expert teachers, Leinhardt and Putnam (1986) concluded that expert teachers have “specialized knowledge about the specific topics they teach” which they integrate with their knowledge of how students learn to produce “cohesive, well-structured lessons” (p. 28). But what constitutes this ‘specialized knowledge’ in the current mathematics teaching and learning environment?
There has been debate during the past five decades over whether students’ and teachers’ time and energy in mathematics classes should be spent developing computational skills or understanding the underlying principles and concepts of mathematics. Movements such as ‘The New Math’, and ‘Back to Basics’ grappled with the issue of which type of understanding is essential for producing citizens that will be able to function mathematically in a highly technological society. The current reform movement, spearheaded by the National Council of Teachers of Mathematics (NCTM, 1989, 1991, 2000), has encouraged a balanced view in which an understanding of the fundamental principles and concepts of mathematics supports and reinforces the algorithmic strategies and procedures used to calculate answers to mathematical problems. Teaching in the current reform environment requires elementary teachers to possess both well-developed conceptual understanding and mastery of procedural skills of the mathematics they teach to adequately prepare students for future learning experiences in mathematics (Ma, 1999; McDiarmid et al., 1989).
In 2001 the National Research Council published Adding it up: Helping Children Learn Mathematics. The purpose of the report was to summarize the available research literature on what mathematics children need to learn, how they learn it, and how it should be taught (NRC, 2001, p. xiv). In this report a five-strand model of mathematical proficiency for children was promoted as the goal of school mathematics. The five strands are interwoven and all are necessary for producing the type of mathematical understanding the editors of the report believe should be the goal of school mathematics. The five strands of mathematical proficiency are: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition (p. 5). While all the strands are equally important, the first two strands are of particular concern for this study. Conceptual understanding in this model is defined in the report as a “comprehension of mathematical concepts, operations, and relations” (p. 5). One indicator a student has developed this type of understanding is the ability to “represent mathematical situations in different ways and knowing how different representations can be useful for different purposes” (p. 118). This would include being able to model a mathematical situation with concrete manipulatives, drawing a visual representation of the problem, creating a word problem from which the mathematical relationship would develop, or correctly representing the situation symbolically (p. 118). Procedural fluency is “skill in carrying out procedures flexibly, accurately, efficiently, and appropriately” (p. 5). Students who are procedurally fluent have mastered basic number facts, the algorithms for performing operations on multi-digit whole numbers and rational numbers, estimation skills, and mental math skills (p. 118). It is possible to have procedural fluency without conceptual understanding, but to have the flexibility to apply mathematics to non-routine problems, both types of understanding are important (p. 122).
As an extension of the five-strand model of mathematical proficiency, a similar five-strand model of proficient teaching of mathematics was offered in the NRC report as a goal for teacher preparation and development. The five strands for proficient teaching are: conceptual understanding, fluency, strategic competence, adaptive reasoning, and productive disposition (p. 380). In the context of this model, conceptual understanding refers not only to mathematical content knowledge, but also to knowledge of how children learn and teaching practices that facilitate that learning. Fluency for teachers is the development of a “repertoire of instructional routines” that teachers can utilize to adapt to differing student learning needs (p. 382). It would seem obvious that unless teachers have mastered the five strands of mathematical proficiency for students, they will have a difficult time developing the five components deemed necessary to become proficient teachers.
Post and Cramer (1989) discussed the procedural and conceptual knowledge needed by teachers of mathematics in a chapter of the book Knowledge Base for the Beginning Teacher. They described conceptual knowledge of mathematics as consisting of understanding two levels of relationships: understanding of an idea within a specific context and understanding of ideas in an abstract sense (Post & Cramer, 1989, p. 222). For example 3 + 2 = 5 can be extracted from the context of a word problem in which the question posed is to determine the total number of objects when three objects are joined with two objects or it can be extracted from determining the cardinality of the union of two disjoint sets, one of which has cardinality 3 and the other 2. Conceptual knowledge can be considered as “a connected web of knowledge, a network in which the linking relationships are as important as the discrete pieces of information” (p. 222). Procedural knowledge is composed of knowledge of the symbolic language of mathematics and the “rules, algorithms, or procedures used to solve mathematical tasks” (p. 222). Procedural knowledge can be present without conceptual understanding of the underlying principles and concepts (p. 222). Division of fractions is a common example of a concept that many students and teachers can deal with procedurally, but don’t understand conceptually (Tirosh, 2000). In the past, school curricula emphasized procedural knowledge and as long as teachers were proficient with the various rules, algorithms, and procedures in those curricula, their knowledge base was sufficient. In the current reform environment, procedural knowledge alone is not sufficient for providing students with the conceptual understanding that is one of the strands of mathematical proficiency described earlier. Unfortunately many potential and current teachers not only lack in procedural skills, but also have limited conceptual understanding of the mathematics they will teach.
Research on U.S. elementary teachers’ understanding of rational number concepts
Warfield (2001) suggests that the level of a teacher’s mathematical content knowledge is important as she tries to discern the mathematical thinking of her students and make instructional decisions appropriate for her students’ understanding (p. 153). Unfortunately there is ample evidence that many preservice, as well as in-service, teachers have not mastered the concepts associated with teaching the topic of fractional numbers in a way that would lead to their students developing a solid conceptual understanding of fractions (Carpenter et al., 1981; Brown et al., 1990; Ma, 1999; Spungin, 1996; Tirosh, 2000). The literature also provides evidence that this has been a problem for many years.
Ginther, Pigge, and Gibney reported in 1987 the results of a comparison study of mathematical understanding of preservice and in-service teachers based on testing 1075 participants in 1967-69, 978 participants in 1975-77, and 750 participants in 1983-85. The test covered the basic principles of mathematics that elementary teachers need to know to teach the topics in the elementary mathematics curriculum, including fractional numbers (Ginther et al., 1987). While the primary purpose of this study was to determine the relationship between the number of high school and college level mathematics courses taken by the participants and their level of mathematical understanding, a secondary finding was that the teachers in the 1983-85 group had significantly lower levels of mathematical understanding than their counterparts in the 1967-69 group or the 1975-77 group with the same number of mathematics courses. They concluded the preservice and in-service teachers in the 1983-85 group “did not possess as much mathematical understanding as did similar teachers” in the other groups (Ginther et al., 1987, p.594), even though greater proportions of the 80’s group had taken more high school and college level mathematics courses (p. 597).
The Rational Number Project (RNP) is a project funded by the National Science Foundation since 1979 to investigate the teaching and learning of rational number concepts. A group of mathematics education researchers associated with the RNP turned their attention to the level of understanding of rational numbers held by middle school teachers. With the scope and sequence in which most school districts’ mathematics curricula are organized, middle school teachers are the group primarily responsible for teaching fractional number concepts to children. The RNP group developed an assessment tool for teachers, based on previous work that had been done to understand the ways children learn and understand fractions. The assessment tool consisted of a short answer test designed to determine the teachers’ conceptual understanding of basic fraction content, a six item assessment in which the teachers were to write as much as they could about how they would solve six rational number problems and how they would explain their solutions to children, and a structured interview that probed at responses the teachers gave to answers on the short answer assessment and the six item problem set. This assessment was given to 167 teachers in Minnesota and 51 teachers in Illinois. Post, Harel, Behr, and Lesh (1988) presented some of the findings of the RNP on this subject. The authors shared several disheartening results:
• Teachers displayed some of the same error patterns that children exhibit.
• On even the most basic conceptual items, 10 to 25% of the teachers gave a wrong answer.
• On some items considered fundamental by the researchers, only about half of the teachers answered correctly.
• In general 20 to 30% of the teachers scored less than 50% on the overall instrument (Post et al., 1988).
The researchers identified two major concerns: many teachers do not know enough mathematics related to rational numbers and those that do show proficiency have difficulty explaining their solutions to problems dealing with rational numbers in a manner that is conceptually and pedagogically appropriate for children (Post et al., 1988). Similar results from a related study were found when a group of 48 junior and senior level elementary education majors were assessed after all but one had completed their required mathematics courses (Cramer & Lesh, 1988). The overall mean correct score for this group was 63%. The study concluded that 20% of these future teachers did not possess a level of conceptual understanding necessary to adequately teach a topic (fractions) that they would be certified to teach (Cramer & Lesh, 1988).
Liping Ma (1999) compared the knowledge level of Chinese and American elementary school teachers utilizing an in-depth interviewing technique, which asked the teachers to demonstrate their ability to solve four types of problems found in an elementary mathematics curriculum, as well as asking for various ways they would explain this material to children. One of the four problems dealt with division of fractions, specifically 1 ¾ ( ½ . All 72 of the Chinese teachers in the study could correctly perform this operation, 90% of them could provide a correct representation (story problem) for the concept, and most tended to provide ‘proofs’ for their calculation methods. Of the 23 American teachers (who were described as being experienced, above average teachers), only 9 (43%) could perform the calculation correctly and only 1 (4%) could provide a correct representation. Ma concluded that the Chinese teachers had a more “profound understanding of fundamental mathematics” than their American counterparts (Ma, 1999, p. 120). From the available research it is clear that finding ways to improve preservice and in-service teachers’ understanding of this complex and important mathematical topic is important.
Research on attitudes towards mathematics and fractions
According to Aiken, “[a]ttitudes are learned predispositions to respond positively or negatively to certain objects, situations, institutions, concepts, or other persons” (1979, p. 207). It is not surprising, based on the difficulty American mathematics students and teachers have with mathematics in general, and fractions specifically, that many preservice and in-service elementary teachers have developed negative beliefs and attitudes about this subject. They may have learned these attitudes from previous teachers (Post et al., 1988). There is evidence to support the idea that teachers with negative attitudes and beliefs towards mathematics and those with positive attitudes and beliefs actually employ different approaches and techniques in presenting mathematics lessons to their students. Furthermore these different approaches may lead to different attitudes towards mathematics in their students.
Karp (1991) conducted a case study of the teaching methods employed by two fourth and two sixth grade teachers, one at each grade level deemed to have a negative attitude and one with a positive attitude towards mathematics. Karp concluded that teachers with negative attitudes display teaching practices that place the teacher at the center of instruction, as the source of knowledge, and promote “learned helplessness” from their students (p. 267). Teachers with positive attitudes employ instructional methods that support the development of conceptual understanding and self-reliance in their students. Although this case study focused on the behaviors of the teachers, the researcher conjectured that the different teaching styles of these individuals had attitudinal consequences for their students.
The research on preservice and in-service teachers’ attitudes is not as overwhelmingly clear as the research on their level of content knowledge. Becker (1986) administered a revised version of the Fennema-Sherman Mathematics Attitude Scales (Fennema & Sherman, 1976) to 81 elementary education majors enrolled in a required mathematics course and 71 students enrolled in a general astronomy course to determine if elementary education majors differed from other students in attitudes toward mathematics. She determined that her sample of elementary education majors held attitudes towards mathematics that were not particularly positive or negative and were comparable to the students in the general astronomy course. Becker acknowledged that the sample of education majors in her study was taken from a university that was ‘selective’ and her results might not be generalizable. (Becker, 1986, p. 51)
Sherman (1989) studied how three different instructional methods for a unit on fractions related to content knowledge, knowledge of teaching methodology, and attitudes towards mathematics of preservice elementary teachers. One group was taught a unit on fractions using various manipulatives to support the concepts. A second group was taught the unit through the use of various diagrams to explain the concepts. The third group had a unit that was taught utilizing mathematical symbols only, without diagrams or manipulatives. The study utilized a pretest/posttest design. An analysis of variance indicated no significant differences between the treatment groups on the Dutton attitude instrument (Dutton, 1962), but significant differences were found for portions of a Semantic Differential Test (Osgood, Suci, & Tannenbaum, 1964). Other than the different instructional treatments, nothing else was introduced into the study to address change in the students’ attitudes.
Stevens and Wenner (1996) studied the relationships between teacher content knowledge of mathematics and science, beliefs regarding mathematics and science instruction, and number of mathematics and science courses taken in high school and college. No significant relationship between the number of high school or college mathematics and sciences courses taken and beliefs about teaching mathematics and science were found. However, Stevens and Wenner reported a contrast between the students’ demonstrated content knowledge (evidenced by the students’ scores on an assessment instrument designed to measure their level of understanding of basic mathematics and science concepts) and the students’ beliefs about their ability to teach this content. While these students demonstrated a lack of mastery of the content they would be responsible to teach, the students believed they had (or could work to acquire) adequate content knowledge to teach mathematics and science at the elementary grade level. Yet they were not confident about “their ability to teach at a conceptual level or to conduct process-oriented, hands-on experiences in classroom settings” (Stevens & Wenner, 1996, p. 5). The authors were concerned that the students held unrealistic beliefs about their readiness to teach elementary science and mathematics.
A study in England addressed this issue of students’ confidence levels compared to their assessed understanding of content. Sanders and Morris (2000) tested 53 students in a graduate teacher preparation program on their mastery of five elementary level mathematics content areas and administered an instrument utilizing a Likert scale to measure the students’ level of confidence in solving certain types of mathematics questions. Students who did not score at least 67% correct on any section of the mathematics test were required to retake that section, after time was given for the students to address the deficiency through either independent review or scheduled review sessions. Sanders and Morris reported the students who scored poorly had three main reactions to the results of their mathematics tests. Thirty-two percent of the students simply didn’t accept the results of the math test and did little to prepare for the retest. Another 28% acknowledged they had a deficiency, but kept putting off preparing for the retest. The remaining 40% acknowledged there was a problem and diligently worked to prepare for the retest. The students’ levels of confidence were not lowered by poor test results, except for students who tested poorly in the areas of addition, subtraction, and division. The researchers noted that students who were going to teach a certain topic for their fieldwork that semester were more motivated to address the knowledge gaps they had in their understanding of that topic (Sanders & Morris, 2000). They suggested future research needs to explore how teaching a topic impacts both content knowledge and confidence levels of preservice teachers (Sanders & Morris, 2000).
Foss and Kleinsasser (1996) studied 22 preservice elementary teachers in a mathematics methods course to determine if the philosophy of and attitudes towards mathematics education promoted by the instructor of the course were adopted by the students, as evidenced by their own teaching practices during a series of short, mandatory student teaching experiences. What emerged from this study was that the students tended to cling to the earlier patterns and practices of teaching that were part of their own experiences as students in elementary and secondary mathematics classes – they taught as they were taught. The authors concluded “that changes are not likely to occur without creating a teacher education ethos where attention to preservice teachers’ beliefs is in the forefront” (Foss & Kleinsasser, 1996, p. 441).
Putney and Cass (1998) undertook a study to see if a hands-on, manipulatives based methods course that stressed positive attitudes towards mathematics, could effect change in the attitudes of preservice teachers. These researchers used Aiken’s Mathematics Attitude Scale (Aiken, 1972) to measure the students’ positive and negative attitudes towards mathematics. Putney and Cass found that the students in the methods course did improve their attitudes towards mathematics (1998), but there were some deficiencies in the study. The study did not utilize a control group, so it is impossible to determine whether the manipulatives approach or the encouragement of the instructor to address negative attitudes effected the positive change in attitudes exhibited by the students. A study that compared different instructional methods or different instructors would be needed to more clearly determine what change agent is essential in improving preservice teachers’ negative attitudes.
Another approach to effecting change in preservice teachers’ attitudes was explored in a study by Stuart and Thurlow (2000). They designed a science and mathematics methods course to specifically challenge preservice teachers’ “beliefs regarding the nature of mathematics, themselves as learners, and the teaching-learning process” (Stuart & Thurlow, 2000, p. 114). This was accomplished through having the students write mathematical autobiographies, journal responses to prompts about their beliefs about mathematics, in-class writings, exam questions, and interviews. The researchers concluded that the revised course did help the preservice teachers recognize that their beliefs about mathematics were an important factor in how they approached teaching the subject. Recognition of these beliefs is the first step in an attempt to change these beliefs, but actually changing beliefs is a much more difficult process. The authors indicated that they had plans for a longitudinal study to follow these students into their student teaching and professional teaching experiences to determine whether and how these individuals would act upon the recognition of their own beliefs in their teaching practices (Stuart & Thurlow, 2000).
It is evident that preservice and in-service teachers often have weak skills and negative attitudes, but how can this be changed? They are being asked by the reform movement to teach content and use methods for which they are under-prepared (Brown et al., 1990; Post et al., 1988). Even teachers who profess commitment to teaching mathematics for understanding have a limited notion of what that actually entails. Two factors contribute to this dilemma: their own mathematical experiences have never exposed them to the type of teaching necessary for developing conceptual understanding and their own knowledge base is so limited that they have trouble creating the numerous representations essential to help students understand the connections between concepts (Ball, 1993). Any attempts to address this problem, must address these two factors. Teacher preparation programs can address both issues by providing mathematics content and techniques courses that supply the rich environments for exploration, problems solving, and success that have been lacking in preservice teachers’ prior educational experiences (Post et al., 1988). The next section will discuss what theory and research have to say about what constitutes a ‘rich learning environment’.
Creating rich learning environments
Piaget’s work with children and his theories about how children learn and develop intellectually served as a foundation and catalyst for much of the research and theory currently guiding today’s thinking about mathematics education (Ernest, 1996). Piaget’s developmental stages (sensori-motor intelligence, intuitive thought, concrete operations, and formal operations) provided guidelines for the creation of curricula appropriate for children’s cognitive abilities (Muller-Willis, 1970). His theory of equilibration, in which he posited that learners faced with new information or experiences either assimilate this information into their current cognitive structures or accommodate this information by reorganizing and expanding their current cognitive structures, became the basis for a view towards learning called constructivism (Ernest, 1996; Herscovics, 1996; Janvier, 1996). Constructivism has become a commonly held teaching philosophy among mathematics educators (Booker, 1996).
Constructivism has more than one interpretation (Ernest, 1996; Herscovics, 1996), but common to each is the idea that individual students learn by building on previous knowledge by taking new experiences and either fitting them into current conceptual structures or reconstructing current understanding to allow the new information to become part of their conceptual structures (Ernest, 1996; Herscovics, 1996). Among the types of constructivism are radical constructivism (which envisions the mind as an “evolving, adapting organism”) and social constructivism (which views the mind as “persons in conversation” (Ernest, 1996; Herscovics, 1996). Herscovics (1996) promotes a view of constructivism he calls “rational constructivism” which allows for a variety of metaphors to explain how learning can occur in different ways, including the metaphors evoked by radical and social constructivism. Rational constructivism allows for the learning of different types of knowledge specific to mathematics, i.e. functional, cultural, pedagogical, professional, and epistemological knowledge, to be approached from different models or metaphors for how that knowledge is received and learned (Herscovics, 1996). For example a young child can be given manipulatives and through exploration ‘construct’ for himself the concepts of odd and even numbers (an example of radical constructivism), while at the post secondary level a student should be able to read a formal definition of even and odd numbers and accept that definition based on his previous understanding, without the need to discuss the definition with a community (which would be dictated by social constructivism) (Herscovics, 1996, pp. 353-354). This view of constructivism allows teaching and learning activities to take a variety of forms and approaches appropriate to the developmental level of the learner and the nature of the concept or material to be learned.
Janvier (1996) stresses that constructivism is not about teaching, but about learning. Yet teachers holding a constructivist philosophy need a framework to help them organize lessons and materials in such a way that students are able to reconstruct knowledge in an efficient manner, without the need to reinvent it (Herscovics, 1996). Janvier (1996) defines teaching “as a set of actions intended to guide the knowledge acquisition of someone else … through adequate interventions on [the learners’] experiential reality” (p. 453). For mathematics education this is achieved, in part, by recreating “rich situations” which are “complex situations involving the students in problem solving cycles or inducing reflections based on trial and error” (p. 453). Teachers need to determine the goal level of mathematical understanding they wish their students to have, determine the level of existing mathematical understanding their students possess, and then create situations that take the learners from their present level of understanding to some level closer to the goal level (p. 454). This can sometimes be accomplished through social interactions where groups of students, or the whole class, explore and discuss an issue, problem, or concept and come to a consensus of meaning (p.455). To aid in arriving at a consensus of meaning for the group, a variety of representations are necessary (p. 455) which may include formal language, diagrams, concrete manipulatives, real life situations, and symbolic representations, but caution must be taken not to provide so many different representations at once that students lose sight of the concept being taught (p. 456). Enough time must be spent with various types of representations to allow students to make the appropriate connections to prior knowledge (p.456).
Janvier (1996) provided several goals specifically addressing the training of mathematics teachers. These include:
• Providing preservice teachers with the abilities required to intervene in the pupils’ learning process so that they can listen to and observe pupils to determine current levels of understanding, carefully use tools to introduce conflicts that might provoke accommodation, bring about small changes leading to assimilation, and fruitfully make use of open-ended questions and large tasks unfolding over many lessons.
• Changing preservice teachers’ conceptions or beliefs about mathematical knowledge, and learning and teaching in general.
• Helping preservice teachers to note the central role of representations in learning and doing mathematics.
• Making preservice teachers acknowledge the importance of several affective factors (such as motivations) often resulting from implicit goals.
• Focusing preservice teachers’ attentions on the importance of social contexts in the process of learning, discussions, and negotiations.
• Making preservice teachers sensitive to the influence in the whole process of the learner’s conception of knowledge and mathematics. (p. 457)
An area where many preservice teachers are deficient is the ability to represent fractional concepts with multiple representations. Dienes (1970) conjectured that learners abstract concepts by drawing out from different situations and representations that which is common to all. This is most likely to happen in learning mathematics if multiple embodiments of mathematical ideas are presented (Dienes, 1970, p. 52). Bruner (1960) stated, “The task of teaching a [any] subject to a child at any particular age is one of representing the structure of that subject in terms of the child’s way of viewing things” (p.33). This not only involves specific pedagogical approaches to the subject, but recognizing that the student must have some sense of why the subject is important. “The first object of any act of learning, over and beyond the pleasure it may give, is that is should serve us in the future. Learning should not only take us somewhere, it should allow us later to go further more easily” (Bruner, 1960, p.17). It would seem appropriate to develop approaches that help students conceptualize mathematical concepts in multiple ways, but also in ways that are meaningful in the students’ lives.
To assist the child in understanding new concepts, Bruner (1968) suggested a three-stage model through which children progress in learning mathematics: the enactive, the iconic, and the abstract or symbolic. In the enactive stage, learners interact with concrete objects to learn concepts. This is abstracted to the level in which the student is able to represent the concrete objects with drawings or diagrams. Finally, the learner completes the abstraction process by being able to represent the concepts strictly by symbolic means without the need for a concrete reference (Bruner, 1968). Lesh (Behr, Lesh, Post, & Silver, 1983) expanded Bruner’s model and conceptualized it as being interactive, rather than linear. Lesh’s model (illustrated in Figure 2.1) includes the following modes or systems of representation: spoken symbols, manipulatives, pictures and diagrams, written symbols, and real world situations (Behr et al., 1983). Using this model a learner is said to understand a mathematical concept when she can express that concept in each of the various modes of representation (Lesh, Landau, & Hamilton, 1983). For example, if a student is verbally asked to represent the fraction ½, she may do so in written word form (one half) or in fraction notation (1/2), she may model it with concrete materials or draw a diagram. She should also be able to give a real life example of ½. This model addresses how students may translate a problem situation into different modes of representation in order to solve it (Behr et al., 1983). Asked to add ½ + ¼, the student may model the problem with fraction circle pieces, draw a pictorial representation of the sum, or use only the symbolic notation and a learned algorithm to solve the problem.
[pic]
Figure 2.1 Lesh’s Model of Multiple Modes of Mathematical Representation
Thus a rich learning environment is one in which concepts are presented with multiple representations, in problems and situations that are meaningful for the learner. Students need the opportunity and the time to work with concrete objects, pictorial representations, and symbolic notation. They need to read, write, and talk about the concepts they are learning and understand why these concepts are linked and why they are important in the learner’s future. With these goals in mind, Lesh’s model of multiple representations has been chosen for this study as an experimental approach to teaching fractional number concepts to preservice teachers.
Teaching mathematics content to US preservice teachers
Sowder, Bezuk, and Sowder (1993) discuss applications from research that need to be present in mathematics content courses for prospective elementary teachers, if these students are to be expected to teach mathematics (and specifically rational number concepts) for understanding. Teacher candidates need to reflect on their own understanding of rational number concepts, identify misconceptions and deficiencies, and work to build new, deeper understandings (Sowder et al., 1993, p. 243). These preservice teachers need evidence of their deficiencies and they need to be convinced that a complete understanding of fractional numbers is essential for their futures as teachers (p. 253). This requires that a safe atmosphere be developed in the classroom, so students will not feel threatened when revealing lapses in their understanding. If preservice teachers’ previous learning experiences with fractions have been largely formal and symbolic, then they need to be presented with experiences designed to assist them in linking their symbolic understanding to concrete and pictorial representations of fractions (p. 244). This would ideally include opportunities to discuss their understandings with others in whole class or small group encounters. Teacher candidates need experiences with all of the major interpretations (subconstructs) of fractions, as well as with a variety of models and representations (p. 246). More class time needs to be spent in mathematics content courses for elementary majors developing number sense for rational numbers, ordering rational numbers, and developing estimation skills with operations for fractional quantities (p. 248). Prospective teachers need experiences with choosing the correct operations to solve problems dealing with fractional numbers, specifically with multiplication and division of fractions (p. 249). The goal of this type of environment for learning about rational number concepts would be teachers who understand the relationships between, and operations on, fractions (p. 255). They should be able to choose models and representations appropriate to the concepts they intend to teach and be able to discuss these concepts confidently (p. 255).
Research on ‘teaching to learn’ content
In addition to offering content courses designed to provide preservice teachers with a rich learning environment, there is evidence to support that facilitating the learning of another may increase the content knowledge of the facilitator. Ma (1999) observed “the key period during which Chinese teachers develop a teachers’ subject matter knowledge of school mathematics is when they teach it …” (p. 147). Ball and McDiarmid (1990) wrote, “As they struggle to teach their subject in ways that make it meaningful to the students, the beginning teachers … draw on their growing knowledge of students, of the context, of the curriculum, and of pedagogy. In short, they evolve a new understanding of the content…” (p. 445).
Evans, Flower, and Holton (2001) designed a qualitative study to investigate the impact peer tutoring had on the learning of mathematics of 24 freshman preservice teachers enrolled in two teacher preparation programs in England. While the researchers claimed that the learning of the students involved in the study was enhanced by the tutoring experience, no assessment was made of their actual content knowledge either before or after the tutoring experience. There was no control group comparison in the study. Analysis of students’ writings and comments suggests the tutoring experience may have improved attitudes towards teaching and learning mathematics, but even this conclusion is tenuous from the data gathered in the study.
Lowery (2002) investigated how situating a mathematics and science methods course in an elementary school, with immediate field experiences directly related to material covered in the course, would impact preservice teachers’ knowledge of mathematics and science content and pedagogy. This study was qualitative and was focused on mapping how the students constructed knowledge from the content course, working with in-service teachers, working with children, and working with each other. The researcher reported gains in positive attitude and confidence to teach mathematics and science content (Lowery, 2002), but did not provide any empirical evidence that the level of content understanding of the students in the study improved or that this situated experience produced greater gains in mathematics and science understanding than traditional, university-based methods courses. An experimental study in which education students who tutor are compared to students who do not tutor would provide empirical evidence that this approach to designing content and/or methods courses is a valuable avenue to explore for improving teacher education programs.
Rationale for why this study is needed and how it adds to the body of research
There is much that is known about how students and teachers learn and feel about mathematics. Research supports the view that having procedural knowledge does not necessarily indicate the individual has a conceptual understanding of the mathematics that supports the procedures, but a good conceptual understanding will support procedural understanding (Ma, 1999; NRC, 2001; Post & Cramer, 1989; Tirosh, 2000). Today’s reform mathematics curriculums require teachers that have both procedural and conceptual understanding in order to teach using methods and activities that research indicates are the most effective in helping children develop the level of mathematical proficiency required by a highly technological society (NRC, 2001; NCTM, 1989, 1991, 2000; Post & Cramer, 1989). There is abundant evidence that preservice and in-service teachers need better preparation to teach mathematics (Carpenter et al., 1981; Brown et al., 1990; Cramer & Lesh, 1988; Ginther et al., 1987; Ma, 1999; Post et al., 1988; Spungin, 1996; Tirosh, 2000). There is research that indicates preservice and in-service teachers may need guidance in recognizing and correcting negative attitudes and faulty beliefs about the subjects of mathematics and how to teach it (Foss & Kleinsasser, 1996; Karp, 1991; Post et al., 1988; Putney & Cass, 1998; Sanders & Morris, 2000; Sherman, 1989; Stevens & Wenner, 1996; Stuart & Thurlow, 2000). Theory and research support the view that knowledge is constructed by the individual based upon previous understanding gained from concrete to abstract experiences (Bruner, 1960; Dienes, 1970; Ernest, 1996; Herscovics, 1996; Janvier, 1996).
Research studies on following preservice teachers’ classroom training on specific mathematical content and concepts by field experiences in which students work with children on the content just studied, are few and tend to be qualitative (Evans et al., 2001; Lowery, 2002) and leave many unanswered questions that require further research. Unfortunately, there are few empirical or comparative studies that provide those responsible for preparing elementary teachers with the research evidence to support or discard content specific field experiences as an approach to improving teacher education programs, to support deeper understanding of content, and to help preservice teachers develop positive, realistic attitudes and beliefs about mathematics and mathematics education. The purpose of this study is to investigate such an approach to teacher preparation and add to the body of research related to teacher education.
Chapter III
Methods
The primary purpose of this study is to explore how two different instructional methods, used in a mathematics content course during a fraction content unit, might improve the procedural and conceptual understanding of fractions of preservice teachers enrolled in the course. In addition the study explores how some participants, assigned to a voluntary field experience associated with the recently studied fraction content, compare on procedural and conceptual knowledge of fractions to participants in the study who are assigned a series of readings from the research literature on teaching fractions to children. A secondary purpose of the study is to determine if the attitudes of preservice teachers in the study are affected by the different instructional methods or the participation in the field experience. More formally stated the research hypotheses are as follows:
• Preservice teachers completing a unit of study on fractional number concepts utilizing a learner-centered approach based upon Lesh’s model of six interactive systems of representation will develop deeper procedural and conceptual understanding of fractions when compared to preservice teachers who are presented the material in a more traditional, teacher-directed environment.
• Preservice teachers completing a unit of study on fractional number concepts utilizing a learner-centered approach based upon Lesh’s model of six interactive systems of representation will show an improved attitude towards mathematics in general and towards fractional number concepts specifically when compared to preservice teachers who are presented the material in a more traditional, teacher-directed environment.
• Preservice teachers who explore fraction concepts with age appropriate students (fifth and sixth grade students) in a field experience will develop deeper procedural and conceptual understanding of fraction concepts than preservice teachers who have completed a series of readings from the recent research literature on teaching fractions to children.
• Preservice teachers who explore fraction concepts with age appropriate students (fifth and sixth grade students) in a field experience will develop a more positive attitude towards the topic of fraction concepts than preservice teachers who have completed a series of readings from the recent research literature on teaching fractions to children.
• Preservice teachers completing a unit of study on fractional number concepts utilizing a learner-centered approach based upon Lesh’s model of six interactive systems of representation and explore fraction concepts with age appropriate students (fifth and sixth grade students) in a field experience will develop deeper procedural and conceptual understanding of fractions when compared to other groups in the study.
• Preservice teachers completing a unit of study on fractional number concepts utilizing a learner-centered approach based upon Lesh’s model of six interactive systems of representation and explore fraction concepts with age appropriate students (fifth and sixth grade students) in a field experience will show an improved attitude towards mathematics in general and towards fractional number concepts specifically when compared to other groups in the study.
Subjects
This study was conducted during the spring semester of 2005 at a regional Midwestern university with a well-established teacher preparation program. Students that participated in the study were at least 18 years of age, admitted to the university where the study was conducted, and had declared majors in elementary education, special education, middle school education, or early childhood education. These students were enrolled in two sections of the course Mathematics II, a mathematics content course elementary education majors are required to take at this university. The Mathematics II course includes among the syllabus topics the study of fractional numbers as one unit of the course. Enrollment in Mathematics II is dependent upon completion of Mathematics I with a grade of C or better.
The participants for the study were solicited through three means: inviting students enrolled in Mathematics I (the course that is a pre-requisite for Mathematics II) in the fall semester of 2004 to be part of the study by enrolling in selected sections of Mathematics II during the spring 2005 semester, notifying the academic advisors for the elementary majors about the study and asking them to encourage students to participate and enroll in the sections selected for the study, and directly appealing to students enrolled in Mathematics II for spring 2005 during the first week of class. All members of the two sections were apprised of the study the first day of class and told that some students would be selected to participate in a voluntary field experience. The students were informed of their rights not to participate in the study and to drop out of the study at any time. The students were asked to give written consent to participate in the study. Students were told that if they did not wish to be part of the study’s participants, they would be allowed to change to a section of the course not in the study before the deadline for schedule changes or they could opt to remain in the original section, but request that none of their test scores or other data be included in the study data. Both instructional treatments used for the study are valid approaches to presenting content material to preservice teachers, so regardless of which section a student was enrolled in or whether they chose to be a participant in the study, all students received a valid educational experience. All activities the participants of the study were asked to perform are consistent with regular coursework for education majors. Choosing to be part of the study had no effect on the final grades students received in the course. All subjects were volunteers and no compensation was paid to participants, although students who participated in the field experience did receive a letter from the principal of the school where the field experience was held recognizing them for their participation. The students who did not wish to participate in the study had exactly the same experience as the students in the study who were not chosen to participate in the field experience. The study was carefully designed so students who chose not to participate would not be penalized in any way by the instructor of the course or the university.
Two sections of Mathematics II were utilized in the study. The researcher taught both sections of the Mathematics II course. The two sections both met on Tuesdays and Thursdays for 75 minutes each day. The sections were randomly assigned to two different treatment conditions. One section, Group A, received a traditional, teacher directed unit on fraction concepts and consisted of 15 participants. Two additional students enrolled in the section chose not to participate in the study, and one student in the section dropped out of the course due to personal reasons. The other section, Group B, received a laboratory based unit on fraction concepts, utilizing a learner-centered constructivist approach with activities based on Lesh’s model of multiple representations and consisted of 26 participants. Two additional students enrolled in this section chose not to participate in the study. All students enrolled in this section completed the course. Of the 41 participants in the study, 4 were male and 37 were female, 1 student was African-American, 1 student was a Caucasian-international student, and the remainder were Caucasian-American. There were no statistically significant differences between the students in the two sections on the factors of number of hours completed before taking Mathematics II, ACT mathematics sub-scores, or final grades in Mathematics I.
Reliability and Validity of Assessment Instruments
Instruments to assess conceptual and procedural knowledge.
There are two types of understanding of fractions that preservice teachers need to develop: procedural and conceptual. For this study it was necessary to find an instrument that would measure students’ mastery level of both types. As defined earlier, procedural understanding of fractions is knowledge of and ability to use appropriate algorithms to perform accurate calculations of the operations of addition, subtraction, multiplication, and division of fractions, simplify answers, and move easily between improper form and mixed numeral form of fractions. Conceptual understanding of fractions is knowledge of and ability to correctly apply procedural knowledge to real world problem solving situations involving fractions, the ability to create real world story problems involving fractions, and the ability to apply the concepts associated with fractions in multiple ways with various representations. No commercial tests were found that measure students’ fraction conceptual understanding as defined for this study. The instruments chosen to determine the students’ level of understanding on procedural and conceptual knowledge of fractions were two equivalent or parallel versions of an adaptation of instruments developed by the Rational Number Project (Cramer, Behr, Post, & Lesh, 1997a; Cramer, Behr, Post, & Lesh, 1997b). Copies of the instruments developed for this study can be found in Appendix A.
Pilot Version A of this instrument was given to 47 students in Mathematics II early in the fall 2004 semester to determine the reliability of the instrument. The results of the student responses to the instrument were analyzed using SPSS to determine a Cronbach’s Alpha coefficient of 0.92. The original Version A of the instrument required students to solve fraction word problems, but did not require students to create appropriate word problems for fraction equations. Based on the research of Ma (1999), the ability of students to write reasonable word problems for a fraction equation such as [pic] would be a valid indicator of the depth of their conceptual understanding of operations on fractions. Pilot Version B of the testing instrument was created by slightly varying the questions on Pilot Version A, then adding four questions requiring students to create appropriate word problems for each of the four operations. Pilot Version B with this modification was given to the same group of students as Pilot Version A in October of the 2004 semester. Pilot Version B had a Cronbach’s Alpha coefficient of 0.83. Pilot Version A was revised to have the word problem items added. A couple of very slight modifications to the directions and formatting of both pilot versions led to the Versions A and B of the testing instrument for conceptual and procedural knowledge found in Appendix A. Reliability of these instruments with the study participants will be discussed in Chapter 4.
The instruments for testing the students’ conceptual and procedural knowledge have both face validity and content validity. Since these instruments were adapted from assessment tools developed by researchers from the Rational Number Project (Cramer, Behr, Post, & Lesh, 1997a; Cramer, Behr, Post, & Lesh, 1997b) and the research conducted by Ma (1999), similar items have been used before to measure procedural and conceptual knowledge of fractions. The items on the testing instruments also represent the types of problems that these students will someday teach. The instruments have content validity because the items chosen for the instruments test all of the procedural fraction skills, i.e. simplifying, ordering, adding, subtracting, multiplying, and dividing of fractions, as well as testing the ability of the student to understand and represent fractions in various of the representational modes indicated by Lesh (Behr, Lesh, Post, & Silver, 1983). With the addition of the items asking students to write word problems for the four different operations with fractions, the instruments test all aspects of the definitions of conceptual and procedural knowledge as defined for this study.
Instruments to assess students’ attitudes towards mathematics.
The Mathematics Attitude Scale (Aiken, 1972), found in Appendix B, and a Semantic Differential Scale (Osgood, Suci, & Tannenbaum, 1964) dealing specifically with attitudes towards fractional numbers (Appendix B) were selected to assess students’ attitudes towards mathematics. The Aiken Mathematics Attitude Scale is a twenty item, single dimensional Likert scale that was chosen because it has been used in previous studies of students’ attitudes towards mathematics. Aiken stated in 1972 that his scale was reliable and valid with high school and college age students, but gave no specifics (Aiken, 1972, p. 230).
One problem with the Aiken Scale type of instrument for measuring attitude is that participants may try to anticipate which answers the researcher is seeking and answer accordingly. Thus a second attitude instrument, the Semantic Differential Scale, was also used in this study to protect against subjects trying to please the researcher, who was also the instructor for this study. The semantic differential technique consists of presenting subjects with bipolar adjective pairs such as good/bad or red/green separated by a number of spaces (Osgood, Suci, & Tannenbaum, 1964). The subjects are asked to indicate by placing an X on one of the spaces between the adjectives how they feel about a particular topic, which for this study was fractions. Since the adjective pairs selected for the instrument are often seemingly unrelated to the topic, the subjects are less likely to try to anticipate the response the researcher is seeking (Leder, 1985; Osgood, Suci, & Tannenbaum, 1964). Osgood, Suci, and Tannenbaum claimed that test-retest reliability data for various forms of semantic differential scales ranged from .87 to .93 (1964, p. 192). They also determined that subjects’ scores on a semantic differential scale and a Thurstone scale were highly correlated on a test-retest study (.74 ≤ r ≤ .82, p < .01) and both instruments were reliable (Osgood et al., 1964, p. 193).
Procedures
Just before the spring 2005 semester began, the two sections of Mathematics II targeted for the study were randomly assigned to instructional treatments. When the Mathematics II classes chosen for this study met for the first time, the instructor/researcher briefly described the course and the requirements, benefits, and obstacles for participating in the study. During the second class meeting of the spring 2005 semester, consent forms were distributed, students’ rights were explained – including the right to change sections during the first week of class or drop out of the study, but remain in the class at any point. Signed forms were collected from all students with either ‘yes I agree to participate’ or ‘no I do not wish to participate’ options checked to indicate a students’ willingness to be part of the study. Collecting forms from all students helped to insure that other students couldn’t readily tell who was, and who was not, part of the study. Students who agreed to participate in the study were asked for permission to access ACT mathematics sub-scores, records of mathematics courses taken in high school and college, and high school GPA for sample description purposes.
Both classes (participants and non-participants) were tested four times on procedural and conceptual understanding of fractions using two versions of an adaptation of instruments developed by the Rational Number Project (Cramer, Behr, Post, & Lesh, 1997). A copy of the instruments is found in Appendix A. The students in all groups were also given the Mathematics Attitude Scale (Aiken, 1972 found in Appendix B) and a Semantic Differential Scale dealing specifically with attitudes towards fractional numbers (Appendix B) at four different times immediately after completing the content knowledge instrument. Version A of the instrument was given as a pretest during the first week the Mathematics II classes met. The students were given version B after the fraction unit was completed (the first six weeks of the semester) as a posttest. Version A was given again at the class meeting following the last field experience as a retention measure, and version B at the end of the semester as part of the final examination, again to measure retention. A summary of the study design is presented in Table 3.1.
Table 3.1
Study Design
|Group designations |Pretest |Delivery of instruction |Posttest |Field experience |Retention test 1|Retention test 2 |
| |1/20 |1/25 – 2/25 |3/1 |2/9,2/16,2/23, |4/14 |5/24,5/26 |
| | | | |3/2,3/9,3/23, | | |
| | | | |3/30,4/6,4/13 | | |
| | | | | | | |
|Group A |O |5 weeks of traditional |O |6 subjects |O |O |
|(15 subjects) | |instruction | |Yes | | |
| | | | |9 subjects | | |
| | | | |No | | |
| | | | | | | |
|Group B |O |5 weeks of laboratory |O |8 subjects Yes |O |O |
|(26 subjects) | |experience | | | | |
| | | | |18 subjects | | |
| | | | |No | | |
O = Collection of data
Group A experienced the fractional number content in the Mathematics II course in a traditional, teacher-directed environment. Most of the content material on the definitions and conceptualizations of fractions, equivalent fractions, ordering of fractions, algorithms for fraction operations, demonstration of concrete and pictorial models, and construction of word problems for fractions was presented in a lecture/discussion/demonstration format. The students were given two class periods of laboratory exploration working in small groups with fraction manipulatives. An outline of this unit is found in Appendix C. This unit on fractions was presented the first 5-6 weeks of the semester for approximately 15-16 hours of instruction.
Group B was taught the fractional number content in the Mathematics II course in a unit that utilized activities developed with a social constructivist view and reflected Lesh’s model of mathematical learning. The students were placed in learning teams of three or four students and guided through a sequence of lessons that allowed them to explore the concepts of fractional numbers through various problem situations. The instructor/researcher acted as a facilitator, clarifying and guiding the discussion, to allow students to construct an understanding of fractions. The students were required to share verbally and in writing their methods for solving the various problem situations with which they were presented. Most of the activities were student-directed, but some teacher led lecture and discussion was used to help summarize and clarify the concepts. The students had multiple experiences with manipulatives, pictorial representations, real world application situations, and alternative computational algorithms. An outline of the unit Group B experienced appears in Appendix D. This unit on fractions was presented the first 5-6 weeks of the semester for approximately 15-16 hours of instruction.
Approximately three weeks into the semester, prior to February 9, 2005, students in Groups A and B were randomly assigned to either Group 1 to complete a field experience with fifth and sixth grade students or to Group 2 to complete a series of reading assignments dealing with teaching fraction concepts to children. This selection was not completely random, as many students in the study had scheduling conflicts. Many students had a class during the time set up at the middle school for the after school mathematics program arranged to provide the field experience for the study. It became necessary to ask the students in the study whether their schedules would permit them to be in the group from which the field experience participants would be chosen. Once that pool of students was identified, a random selection of those students was assigned to the field experience (a copy of the field experience assignment is located in Appendix E). Those who identified scheduling conflicts automatically became part of the study group assigned to do the readings assignment (a list of the readings for this assignment may be found in Appendix F.)
Group 1 spent approximately forty-five minutes a week for nine weeks at a local middle school working with fifth and/or sixth-grade students on the topic of fractions. The fifth and sixth grade children involved in the field experience were not assessed or used in this study in any way other than as part of a normal after school academic enrichment program. This is not unlike children being taught by student teachers or uncertified volunteers who present drug prevention lessons. The principal of the school chosen for this part of the study was informed of the details of the study and gave permission to allow the university students to work with the middle school children. In addition, a certified middle school teacher attended each of the after school sessions and provided advice on school procedure, behavior management issues, and safety of the children. The school and the children benefited by receiving a free after school enrichment program targeted at a topic that is traditionally recognized as difficult for students. The after school program was scheduled for Wednesday afternoons, starting the second week in February and continuing through the weeks leading up to the middle school’s Terra Nova standardized assessment in April 2005. The preservice teachers and the middle school students were put in learning teams and the preservice teachers assisted the middle school students as facilitators and co-learners in activities, games, and problems related to fraction concepts. Group 1 students were required to submit short one-page reflective papers after each session with the middle school students and a final reflection on the overall experience (see Appendix E). Since the students in the study may or may not have completed previous education courses related to development of lesson plans and activities, the researcher developed and provided the field experience participants with all lesson plans, materials, and activities. A copy of the curriculum materials used in the after school mathematics program may be found in Appendix G.
Group 2 students did not participate in the after school enrichment program, but were required to read at least 8 articles on teaching fractions to children from the current research literature. They were required to submit short summaries of the articles they read. Both Group 1 and Group 2 received course credit for these writing activities as part of their participation grade in the Mathematics II course. Students who did not participate in the study were required to read and summarize the same articles on teaching fractions required of Group 2 as part of the regular requirements of the course.
At the end of the fraction units for both groups, the students were given Version B of the testing instrument as a posttest for both conceptual and procedural knowledge of fractions. The score on this instrument was used as part of the unit exam for the university mathematics course. Immediately after completing the content knowledge test, students were also asked to complete the attitude surveys. At the conclusion of the 9 weeks of the after school program, all preservice teachers in both sections of Mathematics II took Version A as a retention assessment and the attitude survey. Students were given points for completing this assessment as part of the participation activities required for the course. In other words, the students were given course credit for taking the tests, but not for how well they performed. At the end of the semester, as part of the final exam for the Mathematics II course, all students in the two Mathematics II sections took Version B of the testing instrument as a final retention test. Again the attitude survey was administered immediately after testing. The attitude survey was not counted as part of the course grade, but the content and procedural knowledge test score became part of the final exam grade. A timeline for the study is found in Table 3.2.
Table 3.2
Study Timeline
|January 17-21 |direct appeal to Mathematics II students to participate in study |
|January 20 |pretests (Version A) given to both groups A and B (Hands-on vs. Lecture/discussion) |
|January 24 –February 25 |fraction units presented to both groups A and B |
|February 9 – April 13 |Group 1 completed field experience, Group 2 completed reading assignment |
|March 1 |posttests (Version B) given to both groups |
|April 14 |retention test 1 (Version A) given to both groups |
|May 2 – May 6 |retention test 2 (Version B) given to both groups |
Note: Dates in bold type are testing dates.
All student information collected for this study was coded so that no individual scores or information may be linked to a particular student. This coded data and the data from the assessment instruments has been stored in SPSS files and analyzed using SPSS software. Information linking individual students to demographic and test score information will be destroyed once the data files are constructed and successful analysis is completed.
Design
The study utilizes a quasi-experimental design. The overall study design is represented in Figure 3.1. The data from the study was analyzed using a 2 × 2 × 4 factor analysis of variance with repeated measures. Analysis was used to test the null hypothesis of no difference between treatment groups. The factors are Type of Instruction (Group A - traditional or Group B – hands-on), Field Experience (Group 1 - yes or Group 2 - no), and Test Occasion (Pretest, Posttest, Retention test 1, Retention test 2). Test Occasion was the repeated measure. Type of Instruction and Field Experience will be the independent variables and Test Occasion will be the dependent variable. The data was analyzed using SPSS for between group and within group differences.
[pic]
Figure 3.1 Study Paradigm
Possible Limitations and Threats to Validity
This was an ambitious project and the likelihood of everything going according to the original plan was practically zero. The proposal estimated 60 participants in the study. Unfortunately there were only 41 participants, due to lower than expected enrollment in the sections and only being able to use two sections of Mathematics II for the study. The students in Mathematics II are representative of students from the population of elementary education majors, since all students in this program must take the course, but there is no specified time that they must do this. This means that some students will have had previous field experiences and some may have had none. The participants in the study are volunteers, and this may indicate they are not a true representation of the population of elementary education majors. Although there was no way to control for the amount of mathematics preparation participants of the study had before the study was conducted, other than the fact that members of the Mathematics II class have completed Mathematics I (or its equivalent) with at least a grade of C or better, an examination of the demographic information about the students indicated they were representative of the population of elementary education majors in general. One student was dropped out of the study, because this student dropped out of school. No other students dropped out of the study or were dropped by the researcher.
The decision was made for the researcher for the project to teach both sections of the Mathematics II course. This was both a control and a limitation. The instructor/researcher’s personality and teaching style may have impacted how both teaching methods were delivered and served to make the two instructional methods less distinct. Another design decision that might have impacted the outcome was the administration of the attitude surveys. The surveys were always given after the students had completed a forty-five minute test and the results of those surveys may have reflected test anxiety as well as attitudes about mathematics.
Threats to internal validity that may be of concern for this study would be pretest sensitization and mortality. Since students normally take a unit exam and a comprehensive exam in Mathematics II, only the pretest, retention test 1, and the attitude surveys will be different than a normal experience in the course, which reduces the threat of pretest sensitization. Threats to external validity that may be of concern would be treatment diffusion (since many of the students in the two sections would have other classes with participants in the other section) and multiple-treatment interference. Asking participants not to discuss their treatment experiences with others until the completion of the study will help to minimize treatment diffusion. The study design randomized the participants as much as practically possible to help minimize the threats to validity.
Chapter IV
Results
Design and Hypotheses
A quasi-experimental design was used for this study, due to the use of participants in a regular university classroom setting. Subjects were randomly assigned to treatments where possible and groups were randomly assigned to treatments when individual random assignment was impossible. The research hypotheses tested were as follows:
1. Preservice teachers completing a unit of study on fractional number concepts utilizing a learner-centered approach based upon Lesh’s model of six interactive systems of representation will develop deeper procedural and conceptual understanding of fractions, as demonstrated by a significantly higher mean score on a test measuring conceptual and procedural knowledge of fractions, when compared to preservice teachers who are presented the material in a more traditional, teacher-directed environment.
2. Preservice teachers who explore fraction concepts with age appropriate students (fifth and sixth grade students) in a field experience will develop deeper procedural and conceptual understanding of fraction concepts, as demonstrated by a significantly higher mean score on a test measuring conceptual and procedural knowledge of fractions, than preservice teachers who have completed a series of readings from the recent research literature on teaching fractions to children.
3. Preservice teachers completing a unit of study on fractional number concepts utilizing a learner-centered approach based upon Lesh’s model of six interactive systems of representation will show an improved attitude towards mathematics in general and towards fractional number concepts specifically, as demonstrated by a significantly higher mean score on a mathematics attitude scale, when compared to preservice teachers who are presented the material in a more traditional, teacher-directed environment.
4. Preservice teachers who explore fraction concepts with age appropriate students (fifth and sixth grade students) in a field experience will develop a more positive attitude towards the topic of fraction concepts, as demonstrated by a significantly higher mean score on a mathematics attitude scale, than preservice teachers who have completed a series of readings from the recent research literature on teaching fractions to children.
5. Preservice teachers completing a unit of study on fractional number concepts utilizing a learner-centered approach based upon Lesh’s model of six interactive systems of representation and explore fraction concepts with age appropriate students (fifth and sixth grade students) in a field experience will develop deeper procedural and conceptual understanding of fractions, as demonstrated by a significantly higher mean score on a test measuring conceptual and procedural knowledge of fractions, when compared to other groups in the study.
6. Preservice teachers completing a unit of study on fractional number concepts utilizing a learner-centered approach based upon Lesh’s model of six interactive systems of representation and explore fraction concepts with age appropriate students (fifth and sixth grade students) in a field experience will show an improved attitude towards mathematics in general and towards fractional number concepts specifically, as demonstrated by a significantly higher mean score on a mathematics attitude scale, when compared to other groups in the study.
The data from the study was analyzed using a 2 × 2× 4 factor analysis of variance with repeated measures. Analysis was used to test the null hypothesis of no differences between treatment groups. The factors are Type of Instruction (Group A - traditional or Group B – hands-on), Field Experience (group 1 - yes or group 2 - no), and Test Occasion (pretest, posttest, retention test 1, retention test 2). The score on the test was the repeated measure. Type of Instruction and Field Experience are the independent variables. The test scores for each student on the four different testing occasions are the dependent variables. The data was analyzed using SPSS version 13 for between group and within group differences. The factors Type of Instruction and Field Experience are the between-subjects factors. Test Occasion is the within-subjects factor.
Reliability of Testing Instruments
The two versions of the instrument used to test participants’ conceptual and procedural understanding of fractional number concepts were piloted during the fall 2004 semester, but slight variations were made to those pilot versions. When these instruments were used with the study participants in the spring of 2005 Version A, given as the pretest on January 20 and as Retention test 1 on April 14, had Cronbach’s Alpha coefficients of 0.88 and 0.88 respectively on the first and third administrations of the testing instruments. Version B, given as a posttest March 1 and as Retention test 2 May 24, had Cronbach’s Alpha coefficients of 0.87 and 0.86 respectively on the second and fourth administrations of the testing instruments. These figures indicate that the testing instruments for conceptual and procedural knowledge used for this study meet acceptable levels for internal reliability (Aiken, 1979; Gay & Airasian, 2000).
Since the participants would be tested a total of four times during the semester long study, two versions of the content knowledge instrument were used alternately to reduce pretest sensitization. The pilot versions of the two testing instruments were different enough in design that it was determined that calculating a correlation coefficient for the two pilot versions was inappropriate. Correlation coefficients comparing the two versions used in the study over the four testing periods are represented in Table 4.1. The two versions of the testing instrument have significant correlations (( = .01) for all combinations of version and testing sequence, thus it was determined that the two instruments have sufficient alternate forms reliability.
Table 4.1
Correlations Between Scores on Administrations of Version A and B of Testing Instrument for Conceptual and Procedural Understanding
______________________________________________________________________________
Version A1 Version B1 Version A2 Version B2
Version A1 -- .787** .853** .762**
Version B1 -- .804** .785**
Version A2 -- .833**
Note. Participants (n = 41). ** indicates a correlation that is significant at the 0.01 level (2-tailed).
Assumptions Underlying the General Linear Model with Repeated Measures
When a repeated measures analysis is approached from a univariate perspective, the dependent variables are considered responses to the levels of within-subjects factors. From this perspective the following assumptions are made:
• Participants must be chosen randomly and independently from the population of interest
• The dependent variables have a multivariate normal distribution
• The covariance matrix of the dependent variables is constant across cells formed by the between-subjects effects
• The covariance matrix of the dependent variables is spherical in form (Cohen & Lea, 2004).
The independence assumption was met by randomly assigning different treatments to the two sections of Mathematics II that were used for the study. While students who enrolled in these sections were aware that the two sections were part of a study, they had no way of knowing which treatment either section would be receiving. They enrolled in the sections based on the normal conditions of individual scheduling needs. The two sections can be assumed to represent the total population of students eligible to take Mathematics II.
The distribution of scores for the dependent variables of the content instruments as well as the attitude surveys are found in Appendix I. While these distributions are not perfectly normal, they are not extremely non-normal. Since analysis of variance is robust for normality, the data for this study is suitable for analysis without transformation (Norusis, 2002; Tabachnick & Fidell, 2001).
Since the sample sizes are unequal, the Box’s M test was performed on all data sets to check the assumptions for the homogeneity of the covariance matrices across cells. Box’s M is extremely sensitive and has significance level of ( = 0.001 (Tabachnick & Fidell, 2001). The Box’s M results for each set of dependent variables was as follows: content test scores, F(30, 1470) = .995, p = .47 > .001; Aiken Mathematics Attitude Scale scores, F( 30, 1470) = 1.786, p = .006 > .001; Semantic Differential scores, F(30, 1470) = .677, p = .907 > .001. The Box’s M was not significant for any of the data sets, so the homogeneity assumption is met.
The sphericity assumption can be checked by performing a Mauchly’s W (Cohen & Lea, 2004). This test was applied to all three sets of dependent variables with the following results: content test scores, Mauchly’s W = .864, p = .391; Aiken Mathematics Attitude Scale scores, Mauchly’s W = .844, p = .301; Semantic Differential scores, Mauchly’s W = .373, p = .000. The content test scores and the Aiken Mathematics Attitude Scale scores are not significant for Mauchly’s W, but the Semantic Differential scores are significant at the ( = .01. When this occurs an adjustment to the degrees of freedom, called epsilon, is used to analyze the data. SPSS offers three types of adjustments: Greenhouse-Geisser epsilon, Huynh-Feldt epsilon, and the lower-bound epsilon. The Greenhouse-Geisser epsilon is conservative for small sample sizes and was used during the evaluation of the analysis of variance with repeated measures for the Semantic Differential scores to insure that the sphericity assumption was met.
Analysis of Variance Results
To test the hypothesis of no differences between groups on the participants conceptual and procedural knowledge of the fraction concepts, the content test scores were entered into a 2 (Type of Instruction) × 2 (Field Experience) × 4 (Test Occasion) between-subjects and within-subjects analysis of variance with repeated measures on the third factor (Test Occasion) to determine whether any combination of factors produced a significant difference. The group means and standard deviations of the fraction content test scores for each cell of the variables Type of Instruction × Field Experience × Test Occasion model are reported in Table 4.2. Figure 4.1 shows a graph of the mean test scores for all combinations of Type of Instruction × Field Experience × Test Occasion. Table 4.3 summarizes the results of the between subjects and within groups analysis of variance. The analysis revealed a significant difference of the within-subjects factor of Test Occasion. There were no other significant between-subjects differences for any of the four testing occasions for any of the treatment combinations.
Table 4.2
Means and Standard Deviations for Type of Instruction × Field Experience × Test Occasion on Content Knowledge Scores
| | | |Test Occasion |
|Treatment Condition| | |Pretest |Posttest |Retention Test 1 |Retention Test 2 |
| | | | | | | |
|Hands-on |Field Experience |Mean |43.38 |55.29 |55.38 |54.88 |
|Instruction |n = 8 |sd |9.410 |5.824 |7.070 |6.875 |
| | | | | | | |
| |No Field Experience|Mean |42.22 |51.94 |53.94 |51.11 |
| |n = 18 |sd |9.384 |8.908 |8.640 |8.181 |
| | | | | | | |
|Traditional |Field Experience |Mean |41.17 |51.17 |53.00 |54.50 |
|Instruction |n = 6 |sd |10.458 |5.456 |5.865 |4.370 |
| | | | | | | |
| |No Field Experience|Mean |39.89 |52.33 |52.63 |52.78 |
| |n = 9 |sd |9.171 |7.467 |9.300 |7.429 |
[pic]
Figure 4.1. Comparison of mean content test scores for all combinations of treatments
Table 4.3
Summary Table for Type of Instruction × Field Experience on Content Knowledge Test Scores
|Source |df |SS |MS |F |Significant |
| | | | | | |
|Between | | | | | |
|Type of Instruction |1 |62.13 |62.13 |.273 |.605 |
|Field Experience |1 |77.17 |77.17 |.339 |.564 |
|Type of Inst × Field Exp |1 |30.53 |30.53 |.134 |.716 |
|Error within |37 |8434.13 |227.95 | | |
| | | | | | |
|Within | | | | | |
|Test Occasion |3 |3531.62 |1177.21 |83.113 |.000** |
|Type of Inst × Test Occasion |3 |46.62 |15.54 |1.097 |.354 |
|Field Exp × Test Occasion |3 |18.80 |6.266 |.442 |.723 |
|Type of Inst × Field Exp × Test Occasion |3 |25.36 |8.46 |.597 |.618 |
|Error within |111 |1572.19 |14.16 | | |
| | | | | | |
| | | | | | |
|Total |163 | | | | |
Note. Participants (n = 41). ** indicates a correlation that is significant at the 0.01 level (2-tailed).
A significant difference was found in the analysis of the content test scores for the main effect of Test Occasion. Further analysis revealed that all groups did significantly better (p < .01) on the posttest, retention test 1 and retention test 2 than on the pretest. There were no significant differences of the means of any other combinations of the Test Occasions. Table 4.4 provides the means and standard deviations for content test scores at each test occasion. Table 4.5 summaries the analysis of variance findings. Table 4.6 gives the results of the post hoc Tukey HSD.
Table 4.4
Means and Standard Deviations of Participants on Each Test Occasion
| |Test Occasion |
| |Pretest |Posttest |Retention Test 1 |Retention Test 2 |
| | | | | |
|Mean |41.78 |52.56 |53.80 |52.90 |
|sd |9.213 |7.513 |7.935 |7.562 |
Table 4.5
Analysis of Variance Table for the Main Effect of Test Occasion
|Source |df |SS |MS |F |Significant |
| | | | | | |
|Between Groups |3 |3966.56 |1322.19 |20.226 |.000** |
|Within Groups |160 |10459.17 |65.37 | | |
| | | | | | |
|Total |163 |14425.73 | | | |
Note. ** indicates mean difference is significant at the .01 level.
Table 4.6
Post Hoc Tests: Tukey HSD
|(I)Test Occasion |(J) Test Occasion |Mean Difference |Std. Error |Significance |
|Pretest |Posttest |-10.780** |1.786 |.000 |
| |Retention 1 |-12.024** | |.000 |
| |Retention 2 |-11.122** | |.000 |
|Posttest |Retention 1 |-1.244 |1.786 |.898 |
| |Retention 2 |-0.341 | |.998 |
|Retention 1 |Retention 2 |0.902 |1.786 |.958 |
Note. ** indicates the mean difference is significant at the .01 level.
To test the hypothesis of no differences between groups on attitude toward mathematics, the scores on the Aiken’s Mathematics Attitude Scale were entered into a 2 (Type of Instruction) × 2 (Field Experience) × 4 (Test Occasion) between-subjects and within-subjects analysis of variance with repeated measures on the third factor (Test Occasion) to determine whether any combination of factors produced a significant difference. The group means and standard deviations of the Aiken Scale scores for each cell of the variables Type of Instruction× Field Experience × Test Occasion model are reported in Table 4.7. Figure 4.2 shows a graph of the mean Aiken Scale scores for all combinations of Type of Instruction× Field Experience × Test Occasion. Table 4.8 summarizes the results of the between subjects and within groups analysis of variance. The analysis revealed no significant differences of the within-subjects factor of Test Occasion. There were no significant between-subjects differences for any of the four testing occasions for any of the treatment combinations.
Table 4.7
Means and Standard Deviations for Type of Instruction × Field Experience × Test Occasion for Aiken’s Mathematics Attitude Scale
| | | |Test Occasion |
|Treatment Condition| | |Pretest |Posttest |Retention Test 1 |Retention Test 2 |
| | | | | | | |
|Hands-on |Field Experience |Mean |63.88 |59.71 |63.63 |63.13 |
|Instruction |n = 8 |sd |23.339 |21.191 |21.554 |20.153 |
| | | | | | | |
| |No Field Experience|Mean |63.89 |63.33 |63.67 |65.39 |
| |n = 18 |sd |19.439 |20.434 |17.723 |21.582 |
| | | | | | | |
|Traditional |Field Experience |Mean |64.33 |66.00 |66.00 |67.50 |
|Instruction |n = 6 |sd |20.176 |24.339 |24.207 |25.532 |
| | | | | | | |
| |No Field Experience|Mean |53.33 |56.12 |56.83 |56.22 |
| |n = 9 |sd |19.761 |18.820 |18.645 |19.318 |
[pic]
Figure 4.2. Comparison of mean Aiken Scale scores for all combinations of treatments
______________________________________________________________________________
Table 4.8
Summary Table for Type of Instruction × Field Experience on Aiken’s Mathematics Attitude Scale
|Source |df |SS |MS |F |Significant |
| | | | | | |
|Between | | | | | |
|Type of Instruction |1 |224.23 |224.23 |0.139 |0.712 |
|Field Experience |1 |683.02 |683.02 |0.423 |0.520 |
|Type of Inst × Field Exp |1 |1218.51 |1218.51 |0.754 |0.391 |
|Error within |37 |59771.46 |1615.45 | | |
| | | | | | |
|Within | | | | | |
|Test Occasion |3 |80.43 |26.81 |1.002 |0.395 |
|Type of Inst × Test Occasion |3 |93.55 |31.19 |1.165 |0.326 |
|Field Exp × Test Occasion |3 |24.80 |8.27 |.309 |0.819 |
|Type of Inst ×Field Exp × Test Occasion |3 |28.93 |9.64 |.360 |0.782 |
|Error within |111 |2970.47 |26.76 | | |
| | | | | | |
| | | | | | |
|Total |163 | | | | |
To test the hypothesis of no differences between groups on attitude towards the content of fractions, the scores on the Semantic Differential Scale were entered into a 2 (Type of Instruction) × 2 (Field Experience) × 4 (Test Occasion) between-subjects and within-subjects analysis of variance with repeated measures on the third factor (Test Occasion) to determine whether any combination of factors produced a significant difference. The group means and standard deviations of the Semantic Differential scores for each cell of the variables Type of Instruction × Field Experience × Test Occasion model are reported in Table 4.9. Figure 4.3 shows a graph of the mean Semantic Differential scores for all combinations of Type of Instruction × Field Experience × Test Occasion. Table 4.10 summarizes the results of the between subjects and within groups analysis of variance. Since Mauchly’s W for this data set was significant, the Greenhouse-Geisser epsilon was used to adjust the degrees of freedom to avoid a Type I error. The analysis revealed no significant differences of the within-subjects factor of Test Occasion. There were no significant between-subjects differences for any of the four testing occasions for any of the treatment combinations.
Table 4.9
Means and Standard Deviations for Type of Instruction × Field Experience × Test Occasion for Semantic Differential Scale
| | | |Test Occasion |
|Treatment Condition| | |Pretest |Posttest |Retention Test 1 |Retention Test 2 |
| | | | | | | |
|Hands-on |Field Experience |Mean |63.88 |64.25 |59.19 |59.00 |
|Instruction |n = 8 |sd |16.805 |13.541 |14.531 |13.352 |
| | | | | | | |
| |No Field Experience|Mean |61.17 |62.49 |58.78 |56.44 |
| |n = 18 |sd |16.681 |14.508 |11.247 |15.764 |
| | | | | | | |
|Traditional |Field Experience |Mean |61.17 |57.17 |58.20 |54.17 |
|Instruction |n = 6 |sd |11.669 |7.91 |8.709 |15.237 |
| | | | | | | |
| |No Field Experience|Mean |64.11 |62.78 |64.67 |58.89 |
| |n = 9 |sd |15.894 |22.643 |22.433 |13.280 |
[pic]
Figure 4.3. Comparison of mean Semantic Differential scores for all combinations of treatments
Table 4.10
Summary Table for Type of Instruction × Field Experience on Semantic Differential Scale
|Source |df |SS |MS |F |Significant |
| | | | | | |
|Between | | | | | |
|Type of Instruction |1 |8.98 |8.98 |0.021 |0.885 |
|Field Experience |1 |82.82 |82.82 |0.197 |0.660 |
|Type of Inst × Field Exp |1 |402.56 |402.56 |0.956 |0.335 |
|Error within |37 |15580.49 |421.09 | | |
| | | | | | |
|Within | | | | | |
|Test Occasion |1.902 |598.01 |314.35 |1.153 |0.319 |
|Type of Inst × Test Occasion |1.902 |156.62 |82.33 |0.302 |0.729 |
|Field Exp × Test Occasion |1.902 |40.23 |21.15 |0.078 |0.918 |
|Type of Inst × Field Exp × Test Occasion |1.902 |4.09 |2.15 |0.008 |0.990 |
|Error within |70.387 |19182.53 |272.53 | | |
| | | | | | |
| | | | | | |
|Total |117.995 | | | | |
Summary of Results
The primary hypothesis of this study was that presenting preservice teachers a unit on the concepts of fractions utilizing a hands-on laboratory approach, together with a highly-related field experience over the same content, would improve their mastery of the content as evidenced by producing significantly higher mean scores on a conceptual and procedural content knowledge test when compared to students presented the material in a more traditional setting with assigned readings over teaching fraction content. The results of this study do not support that hypothesis. No significant differences were found for content test scores for any of the treatment combinations used in the study. There was a significant difference for the within-subjects factor, Test Occasion, for which the means for the posttest, retention test 1 and retention test 2 were all significantly different from the pretest, but not from each other.
Secondarily, it was hypothesized that presenting preservice teachers a unit on the concepts of fractions utilizing a hands-on laboratory approach, together with a highly-related field experience over the same content, would improve their attitudes toward mathematics in general and towards fractions specifically by producing significantly higher mean scores on two different attitude surveys (Aiken’s Mathematics Attitude Scale and a Semantic Differential Scale) when compared to students presented the material in a more traditional setting with assigned readings over teaching fraction content. The results of this study do not support that hypothesis. No significant differences were found for any of the treatment combinations used in the study. Scores on both attitude surveys remained relatively steady over the course of the study.
Chapter V
Discussion
Summary of Purpose, Hypotheses, and Design
This study was motivated by the widely recognized deficiencies of the mathematical understanding of American students and elementary teachers, when compared to elementary level students and teachers in other countries (Brown, Cooney, & Jones, 1990; Carpenter, Corbitt, Kepner, Lindquist, & Reyes, 1981; Cramer & Lesh, 1988; Ma, 1999; McDiamid, Ball, & Anderson, 1989; Post et al., 1988). As a teacher of elementary education majors, the researcher frequently encounters the lack of understanding of basic mathematical concepts many current college students bring to mathematics content courses. Most of these students are products of American public education and their poor quality performance lends support to the argument that mathematics educators in this country must do a better job of teaching mathematics to students of all levels.
This study was supported by the current body of research into how best to teach and learn mathematics. Several major themes run through the support of this study. Learning theory supports the development of conceptual understanding of mathematics through the use of a constructivist approach (Behr, Lesh, Post, & Silver, 1983; Herscovies, 1996; Nickson, 1992; Piaget, 1941, 1972; Thompson, 1992). This approach allows students to construct new knowledge based on current understanding through interaction with concrete, verbal, pictorial, and abstract means within a community of learners. As students progress from learners to teachers, they reshape their understanding of mathematical content in ways that allow them to communicate these concepts effectively (Ball and McDiarmid, 1990; Evans & Flower, 2001; Lowery, 2002; Ma, 1999). This process provides the teacher with a deeper understanding of the mathematical concepts. A teacher’s attitude towards mathematics also plays a role in how she communicates that content knowledge to her students (Brown, Cooney, & Jones, 1990; CBSM, 2001; Putney & Cass, 1998; Stuart & Thurlow, 2000). Attempts to improve current and prospective teachers’ attitudes toward mathematics can only help improve the teaching of that subject to American students.
Based on these premises, this study sought to determine if a change in the approach taken to teaching a unit on fractions in a mathematics content course for elementary education majors would improve the students’ conceptual and procedural understanding of the content, as well as improve their attitudes towards the content. It was hypothesized that adopting a constructivist, hands-on approach to a unit on fractions, supplemented with a related field experience wherein the preservice teachers would help fifth and sixth grade students understand fractions, would produce measurable gains in the conceptual and procedural understanding of fractions the elementary education majors developed and retained when compared to students taught the unit in a more traditional, teacher-centered lecture and discussion format. In addition, this combination of classroom instruction plus the field experience was expected to provide the education majors with an improved attitude towards mathematics generally and fractions specifically.
The study was designed to determine the effect of a change in teaching approach in a regular, university classroom environment. Thus a quasi-experimental design was necessary, but every opportunity to randomize the participants, without disrupting the normal educational experience they were receiving, was employed. Two sections of a mathematics content course taught by the same instructor/researcher were used to compare a traditional lecture/discussion format course with a hands-on, constructivist format course. In addition, students from both sections were selected to participate in a related field experience in the form of an after school mathematics program focusing on fractions designed for middle school students. The students who were not chosen for this field experience were provided with a series of readings related to the teaching of fraction content. Measures of the participants content understanding and attitudes were taken four times during the semester the course was taught: as a pretest before the unit was taught, as a posttest after the unit was completed, about three-fourths of the way through the course as a retention measure, and as part of a final exam as a second retention measure. Data from the various testing sessions was coded into an SPSS file and analyzed using a three factor analysis of variance with repeated measures for within-subjects and between-subjects differences.
Finding and Conclusions
The results from this study do not support the hypothesis that a constructivist approach to teaching the mathematics content course, coupled with a related field experience, will help develop the conceptual and procedural understanding of fractions of education majors better than a more traditionally taught course. Both methods of teaching the course produced a statistically significant increase in content test scores for the students from the pretest to the posttest, and this increase was maintained by all groups over the period of the course as evidenced by two retention tests given periodically throughout the semester. Students that participated in the field experience, while gaining a better understanding of the difficulties and practicalities of teaching fraction content to children, did not significantly improve their understanding of fraction concepts when compared to the participants who read about teaching fraction concepts.
While research supports the idea that preservice (as well as in-service) teachers often lack the deep conceptual understanding of fractions needed to teach that concept to children in a way that produces conceptual understanding of the concept (Carpenter et al., 1981; Brown et al., 1990; Ma, 1999; Spungin, 1996; Tirosh, 2000), there is no agreement or specific level of conceptual understanding that is considered the minimum needed to teach fractions adequately. In the current study, the instrument used to measure the level of understanding of the participants (see Appendix A) was adopted from an instrument provided with a curriculum from the Rational Number Project that was intended to be administered to fourth, fifth or sixth graders (Cramer, Behr, Post, & Lesh, 1997a; Cramer, Behr, Post, & Lesh, 1997b). This instrument was chosen for this study, because it specifically tested the concepts the participants might one day be expected to teach. There were four items added to the instrument involving the writing of appropriate word problems that were not part of the original, elementary level test. The participants in this study (overall) had a mean score of 41.78 points out of 63 (66.3%) on the pretest, with a minimum score of 18 and a maximum score of 53. If mastery of the fraction content material would be considered a score of 90% on the testing instrument (a reasonable level considering the original test was designed for elementary age students), then no participants entered the course with a mastery level of understanding of fractions. Even the best students in the class only scored around 80% on the pretest, implying that these college students had never mastered the understanding of fractions that might be expected at the fourth, fifth, or sixth grade level. At the bottom range of scores, four students scored less than 30 out of 63 on the pretest, indicating they understood less than 43% of basic elementary school understanding of fractions. These findings are consistent with other research into the level of understanding of preservice and in-service teachers (Ma, 1999; Post et al., 1988)
Subsequent testing of the participants in the study revealed an improved understanding of the fraction content for all of the students. After the unit on fractions was completed, the mean score (overall) on the posttest instrument was 52.57 out of 63 (83.4%). While this was an improvement, only 15 of the 41 participants (37%) reached the mastery level of 90% on the posttest. Furthermore, 8 of the 41 participants (20%) still scored lower than 75% on the posttest. The scores for the students on the two retention tests remained relatively stable. The overall mean for the first retention test over content was 53.80 out of 63 (85.4%). The overall mean score on the second retention test was 52.71 out of 63 (83.7%). The students were motivated to do their best on all of the testing occasions, because the scores did have an impact on their grade for the course. It seems the concepts the students already knew or learned, they retained throughout the course, but only 37% of the students achieved and maintained a 90% mastery level of the material.
In addition, the change in teaching method did not appear to improve the teacher candidates’ attitudes towards mathematics and fractions. The results of the data analyses indicate that the attitudes of the students remained stable over the duration of the course. This supports previous research on changing preservice teachers’ attitudes (Foss & Kleinsasser, 1996; Kagan, 1992; Stuart & Thurlow, 2000). The importance of teachers’ attitudes was addressed in both sections of the course via discussions on a website for the course. Students were encouraged to share and discuss their attitudes on this website through instructor provided prompts, but no measurable, significant improvement of students’ attitudes resulted from either section of the course. Two testing instruments were utilized in this study to measure students’ attitudes. The range of the scores possible on the Aiken Mathematics Attitude Scale is a low score (indicating a very negative attitude toward mathematics) of 20 to a high score (indicating a very positive attitude towards mathematics) of 100. The mean Aiken Scale scores for all students were: (pretest) 61.63, (posttest) 61.43, (retention 1) 62.50, (retention 2) 63.24. A score of 60 would indicate an undecided or neutral attitude toward mathematics, which is similar to Becker’s conclusion about the attitudes towards mathematics of education majors (Becker, 1986). So while the mean Aiken Scale scores did slightly improve over the time the students were in the course, the improvement wasn’t significant.
Because the participants of this study were students working to obtain credit in the course that was the focus of the study, there was a reservation about how honest the students would be about their attitudes toward mathematics when completing the Aiken Scale. Therefore, as a further and less transparent measure of their attitudes specifically towards fractions, a Semantic Differential was also utilized. The results from this instrument confirmed the validity of the results from the Aiken Mathematics Attitude Scale. The Semantic Differential Scale can have a low score of 10 (indicating a negative attitude toward fractions) and a high score of 110 (indicating a positive attitude toward fractions). As with the Aiken Scale, a score of 60 indicated a neutral view of fractions. The mean Semantic Differential scores for all students were: (pretest) 62.34, (posttest) 62.12, (retention 1) 60.07, (retention 2) 57.15. These results indicated that the attitudes the students held towards fractions may have shifted slightly, but not enough of a shift to be considered statistically significant. It would appear that nothing done during the mathematics content course altered the attitudes students had about mathematics coming into the course.
Implications of Study
This study sought to find a means of improving the delivery method of mathematics content knowledge to preservice elementary educators. While this study was looking specifically at a measure of how well the participants mastered the objectives of the fraction unit, there are other instructional objectives in this course. One important objective is to model for the prospective teachers methods and processes they will need to be able to employ in their own teaching practice. While the traditional teacher-centered, lecture format was as effective as the hands-on constructivist approach of presenting the content knowledge to the students, it is less clear how well either approach prepared the students to teach children fraction concepts using manipulatives and problem situations, which are major pedagogical methods emphasized in current reform elementary level mathematics curriculums. It was noted by the researcher that the preservice teachers in the field experience from the hands-on section of the course seemed to have less difficulty introducing and using the manipulatives provided to work with the fifth and sixth graders than the teacher candidates from the traditional section. (Although, it should be noted that students from both sections had various problems and issues determining the best way to help the fifth and sixth graders understand the objectives that were targeted for the after school program.) It was not part of the scope of this study to make systematic observations of the participants as they worked in the field experience, but it would certainly be worth consideration for future study to make observations of teacher candidates’ practices in the after school program after being trained in the two different types of content courses.
Another issue that was noted by the researcher was that students in both sections already had very definite, preconceived ideas about how to work with fractions. The participants in both groups had previously learned this content from an algorithmic perspective during their own elementary education experiences. For those participants in the hands-on group, most were not familiar with a method of teaching and learning mathematics that took them back through a process where they were expected to use manipulatives, draw pictures, model word problems concretely, write word problems, and talk about the concepts in small and large groups with the instructor as a facilitator and coach, not as the dispenser of correct answers. The researcher noticed frustration and confusion on the parts of some of these students as they tried to reason through the material from a concrete perspective. They repeatedly ask for the instructor to ‘give us the right answer’ or ‘show us how to do it’. During the course, these students had extensive first-hand exposure to various types of fraction manipulatives.
The traditional course covered the same concepts as the hands-on course, but the instructor modeled various processes and ideas for the students using manipulatives on the overhead or drawing pictures on the board. In this section the teacher acted in the traditional ‘dispenser of knowledge’ role. While the students were encouraged to answer questions and ask questions at any time through the lectures, they had limited direct interaction with manipulatives. Yet they seemed to master the fraction concepts just as well as the hands-on group. The students from this group chosen to do the field experience were initially reluctant to use the provided manipulatives when working with the middle school children. But after some encouragement when faced with activities that could only be done with manipulatives, they became more comfortable with using the fraction manipulatives to explain the concepts to the children.
From the investigator’s point of view, preparing lecture and discussion type lesson plans was much more efficient in terms of being sure to cover all of the required concepts. When teaching the hands-on section, the students often failed to arrive at the conclusions and ideas the instructor was trying to guide them through. The instructor ended each class period with a whole class summary session where, first the students were asked to contribute what they had discovered, then the instructor filled in any concepts or ideas that the students failed to construct on their own. If the only objective of the content course is to insure the students have mastered a certain body of mathematical knowledge and if both methods work equally well, then it would seem that staying with the traditional method of presenting the material would be more efficient. From another perspective, the students in the hands-on section did become much more comfortable with using the manipulative materials to explain and understand concepts. The students that had this experience combined with the field experience got an opportunity to immediately practice using what they had done with the manipulative materials in teaching the fraction concepts to fifth and sixth graders. Many of these students expressed feeling much more comfortable and confident about teaching fractions to children.
Limitations of Study
When this study was planned, the researcher wanted to involve 4 sections of the mathematics content course in the study. Due to scheduling issues at the university where the study was done, only two sections of the course offered the semester the study was conducted were appropriate. All other sections of the course were scheduled for delivery via ITV, which would not have been appropriate for this study. One of the sections that was chosen had a lower than expected enrollment (18 instead of 30). So this study did not involve the number of participants the researcher originally anticipated. There also arose a scheduling issue with the time set for the after school field experience. Many of the teacher candidates had a class during that time and were unable to be selected for the field experience. These two issues limit the generalizability of the results.
The researcher planned the field experience with the principal of the middle school. He wanted the after school program to culminate during the middle school students’ mandated state testing, so the field experience was shorter than the researcher had originally planned (9 weeks instead of 10-12 weeks) and each session was only 45 minutes long. The teacher candidates indicated they did not always have adequate time to complete all of the activities planned for each session. This may have limited the impact the field experience had on the content understanding and attitudes of those participants involved.
The two versions of the testing instrument used in this study were not as highly correlated as the researcher expected. The instruments should have undergone more pilot testing prior to the study. While the testing instrument used in this study reflected the content the participants need to master in order to teach fractions to children, the instrument itself may not have been sensitive enough to measure subtle differences in understanding. Also, repeated exposure to the same instrument may have sensitized the participants to what specific types of questions they would be asked. Ma (1999) used an interview technique which revealed how each of the participants actually thought through the mathematical processes and explored the various ways the participants understood the mathematical concepts they taught. While that method of measuring understanding detects subtle differences in individual’s thinking, it is time consuming and not practical for large numbers of items. An instrument combining both types of assessment techniques might be more sensitive to subtle differences in the level of conceptual and procedural understanding.
The instrument used to measure the attitudes of the teacher candidates may have overlooked an important aspect of their development as teachers. While the Aiken Mathematics Attitude Scale and the Semantic Differential Scale addressed how the students felt about doing mathematics themselves, there were no questions that addressed their attitudes about teaching mathematics or how confident they felt about teaching fractions. As the study progressed, that issue became an obvious factor that needs further study.
Future Research Directions
More research needs to be done to develop methods that will allow more prospective teachers to master the mathematics content they will one day teach. Research also needs to determine what minimal level of mastery of mathematics content on the part of teachers is needed for them to adequately teach this content to children. Additional research on helping prospective teachers identify negative attitudes and beliefs and addressing such is also needed.
Several areas of this study have been targeted for future research efforts. While no significant differences in the groups were found during the semester time frame of the study, there may be long term differences in the retention and mastery levels of the students in the study. A follow-up study is planned for the fall 2005 semester to determine how well the students in the study retained the concepts they were taught over the summer months. As part of that follow-up study, a different attitude instrument may be utilized to determine how the students feel about teaching mathematics and fractions.
The two versions of the testing instrument used in this study to measure the concept knowledge of teacher candidates needs to be improved and research needs to be done on what minimal level of content mastery is needed for teachers to be able to teach the material to children. Research should also continue on methods of improving preservice and in-service teachers’ attitudes towards mathematics.
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Written symbols
Pictures and diagrams
Spoken symbols
Manipulatives
Real world situations
Real world situations
Written symbols
Pictures and diagrams
Spoken symbols
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