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1 Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2005 Making Sense of Students' Understanding of Fractions: An Exploratory Study of Sixth Graders' Construction of Fraction Concepts Through the Use of Physical Referents and Real World Representations Veon Stewart Follow this and additional works at the FSU Digital Library. For more information, please contact

2 THE FLORIDA STATE UNIVERSITY COLLEGE OF EDUCATION MAKING SENSE OF STUDENTS UNDERSTANDING OF FRACTIONS: AN EXPLORATORY STUDY OF SIXTH GRADERS CONSTRUCTION OF FRACTION CONCEPTS THROUGH THE USE OF PHYSICAL REFERENTS AND REAL WORLD REPRESENTATIONS BY VEON MURDOCK-STEWART A Dissertation submitted to the Department of Middle and Secondary Education in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Fall Semester, 2005 Copyright 2005 Veon Murdock-Stewart All Rights Reserved

3 The members of the committee approved the dissertation of Veon Murdock- Stewart defended on October 12, Elizabeth Jakubowski Professor Directing Dissertation Florentina Bunea Outside Committee Member Leslie Aspinwall Committee Member Maria L. Fernandez Committee Member Approved: Pamela Carroll, Chairperson, Middle and Secondary Education The Office of Graduate Studies has verified and approved the above named committee members. ii

4 ACKNOWLEDGMENTS I would like to express my profound gratitude to my Heavenly Father for His constant watch care and provision throughout my entire life. All glory, praise and honor to the One whom I adore. How precious also are thy thoughts unto me, O God! How great is the sum of them (Psalm 139:17). I could not have completed this work without the help of my committee members. First, I would like to thank my major professor, Dr. Elizabeth Jakubowski for motivating me to complete this degree. Thank you for all your advice. Thank you for being there when I needed you the most. Special thanks to Dr. Leslie Aspinwall for your thoughtful instructions; Dr. Maria Fernandez for your keen eye for perfection I have really learned a lot from you; and to Dr. Florentina Bunea who made Probability come alive for me. Special thanks to the many individuals who have provided me guidance and support throughout my doctoral journey. Without you this milestone would have been more tedious. The Adamson s family has been a tower of strength and support over the years THANK YOU. Richard and Daisy Cousins, thanks for being the brother and sister that I needed and still need in Tallahassee. Words cannot express how grateful I am for ALL that you did for me. Rose, and the rest of my friends in Tallahassee thanks a million. To all my mathematics teachers from kindergarten to university, thanks for instilling in me the love for this subject. Special thanks to my mentor, Avery Thompson. My principal, Mr. Othniel Scott, thank you for believing in me. Thank you for giving me the permission to work with twenty wonderful children. I would also like to express my appreciation to those who volunteered to edit my paper: Mr. Edmund Harty, Ms. Martha Morton and Mrs. Patty Hall. May God continue to bless you. A BIG thank you to my parents who believe in education. Thank you for your support throughout my many years of schooling. To the rest of my family members and friends your prayers meant a lot. Thank you. Last, but definitely not least, the BIGGEST thanks goes to my wonderful husband, Paul Stewart. This page is not enough to list all the things that you have done to iii

5 make this moment possible. Thank you for all your care in all forms. When I felt like giving up you were there. Without your love and support I would not have had the strength to finish this project. I love you with all my heart. This project is dedicated to you. iv

6 TABLE OF CONTENTS LIST OF TABLES viii LIST OF FIGURES.. ABSTRACT ix xi CHAPTER 1 CHAPTER 2 THE NATURE AND PURPOSE OF THE STUDY Introduction 1 The Difficulty of Fractions.. 2 Statement of the Problem. 5 Purpose and Significance of the Study 8 Research Questions. 9 Summary. 11 THEORETICAL FRAMEWORK AND REVIEW OF RELATED LITERATURE Introduction 12 The Teaching and Learning of Mathematics. 13 Epistemology. 14 The Fraction Concept. 18 Models of Understanding 21 Relational and instrumental understanding. 22 Kieren s model of the growth of mathematical understanding 23 Herscovics and Bergeron s model of understanding 26 Rationale for using Herscovics and Bergeron s model. 30 Criteria for the Components of Understanding in the Herscovics and Bergeron s (1988) Model of Understanding 31 Understanding the Underlying Physical Concepts 32 Understanding the Emerging Mathematical Concept 33 Physical Referents and the Understanding of the Fraction Concept. 34 Partitioning Strategies 38 Partitive quotient construct strategies 40 Multiplicative strategies 42 Iterative sharing strategies 42 Summary.. 44 v

7 CHAPTER 3 CHAPTER 4 CHAPTER 5 METHODOLOGY Qualitative Interpretive Framework. 46 Participants 46 Research Design 48 Pretest 48 Teaching Sequence 49 Interviews.. 55 Data collection 56 Data Analysis. 57 Assumptions and Limitations 57 Research Methodology.. 58 Teacher-Researcher 59 Sampling 61 Summary 62 RESULTS: THE NATURE OF STUDENTS UNDERSTANDING OF FRACTIONS Introduction 63 Definition of the Fraction 64 Analysis. 68 Intuitive Understanding. 69 Analysis 77 Logico-Physical Understanding 78 Analysis 91 Logico-Physical Abstraction 92 Analysis 118 Logico-Mathematical Procedural Understanding 119 Analysis 126 Logico-Mathematical Abstraction/Formulization 126 Analysis 149 Summary RESULTS STUDENTS PARTITIONING STRATEGIES Introduction. 152 Summary. 162 vi

8 CHAPTER 6 SUMMARY, CONCLUSION, DISCUSSION AND IMPLICATIONS Introduction. 164 The Nature of Students Understanding of Fractions. 165 The Students Partitioning Strategies Physical and Real World Representations Significance of the Study 170 Conclusions and Discussion 171 Implications for Teaching and Future Research. 174 Concluding Remarks APPENDIX A Pretest. 177 APPENDIX B Activities for Teaching Sequence. 185 APPENDIX C Names, Categories and Group Assignments. 238 APPENDIX D Consent Forms Human Subjects Approval Letter REFERENCES BIOGRAPHICAL SKETCH. 255 vii

9 LIST OF TABLES Table 1. Fractions Sample Item from NAEP. 3 Table 2. Charles and Nason s (2000) Partitioning Strategies 41 Table 3. Outline of Teaching Episodes. 51 Table 4. Research Questions and the Methods of Data Collection 56 Table 5. Summary of Responses for Question 5 on Pretest.. 66 Table 6. Summary of Responses for Question 9 on Pretest.. 70 Table 7. Summary of Responses to Question 2 (Multiple Choice) on the Pretest 72 Table 8. Summary of Responses to Question 11 on the Pretest. 73 Table 9. Summary of Responses to Question 1 (Multiple Choice) on the Pretest.. 75 Table 10. Summary of Responses for Assessment Task for Activity 1 87 Table 11. Configurations for Task 2a Activity Table 12. Configurations for Three-Fourths Task 2a Activity Table 13. Classification of Students Difficulties 172 viii

10 LIST OF FIGURES Figure 1. Steffe and D Ambrosio s (1995) Hypothetical Learning Trajectory. 16 Figure 2. Conceptual Scheme for Instruction on Rational Numbers 19 Figure 3. Kieren s (1993) Model for the Recursive Theory of Mathematical Understanding 24 Figure 4. Potential Signing Process Appropriate for Fraction Readiness. 29 Figure 5. Herscovics and Bergeron Model of Understanding 30 Figure 6. Lesh s Translation Model.. 37 Figure 7. Diagram for Question 15 Pretest 67 Figure 8. Students Configuration of the Parallelogram 82 Figure 9. Mary s Method of Equally Sharing 12 Chips 84 Figure 10. Common Configurations for Exercise 2 Assessment Task - Activity Figure 11. Configurations for Task 2 Individual Interview.. 91 Figure 12. Partitioning the Circle in Question 2 Activity 2 Assessment Task.. 98 Figure 13. Students Drawing of the Parallelogram Item #4 100 Figure 14. Configuration for Item 4 (Second Part). 101 Figure 15. Squares Used for Activity 3 Task Figure 16. Squares Used for Task 6 Activity Figure 17. One of the Configurations Given in Assessment Task Activity Figure 18. Six-Part Configurations for Task 1b Activity ix

11 Figure 19. Common Configurations for Task 2b Activity Figure 20. Karla s Configuration of Task 3d Activity Figure 21. Configurations for Task 3d Activity Figure 22. Richy s Number Line Representation of One-Fifth. 123 Figure 23. Alton s Picture of Four Figure 24. Item 3 RNP Lesson 6 Activity Figure 25. Item 8 RNP Lesson 6 Activity Figure 26. Students Configurations for Second Assessment Task (Item 3a) Activity Figure 27. Configurations for Second Assessment Task (Item 3b) Activity Figure 28. Item 11 RNP Lesson 15 Student Page E Level One Figure 29. Example of Partitive Quotient Foundational Strategy. 154 Figure 30. Karla s Representation of the Sharing of the Pancakes. 157 Figure 31. Example of the Preserved-Pieces Strategy 157 Figure 32. Item 5 RNP Lesson 22 Student Page A - Level x

12 ABSTRACT This study was an investigative, whole class descriptive research, on the development of twenty sixth graders understanding of fractions as they interacted with physical referents, hands-on task-based activities and activities that model real life situations during eight weeks of a teaching sequence. The study was conducted in a metropolitan school situated in southeast Florida. The teaching sequence consisted of 12 task-based activities that spanned 20 sessions with each session lasting for approximately 60 minutes. Data was collected through audio- and video-recording, in addition to the numerous written tasks. The task-based activities that the students were involved with during this study were analyzed to gain an insight into their understanding of fractions in the context of subdividing, comparing and partitioning of continuous and discrete models and the connections they made with the fraction ideas generated through these activities. The study also examined how these students make sense of fractions and investigated how their performance differed when fractions were presented using different models. Herscovics and Bergeron s (1988) extended model of understanding, and the partitioning strategies identified by Charles and Nason (2000) and Lamon (1996) provided the theoretical framework through which the investigation was explored. Results from the study revealed that the participants exhibited an understanding of unit and non-unit fraction based on the components of the above-mentioned model of understanding. The students also displayed a number of different partitioning strategies. The knowledge growth that was evident in the whole class confirms earlier studies as to the significant role that partitioning plays in the basic development of the fraction concept. Although discrete models were used by the students, a majority of the students exhibited a preference for using continuous models as forms of reference for given fractions. The students appreciated working with fractions that model real world situations. Preliminary findings from this study seem to indicate that students should be introduced to fraction concepts via partitioning activities. The partitioning activities should be introduced in grades earlier than sixth grade. Further research can be xi

13 undertaken to investigate the role partitioning activities play in the development of students ability to add, subtract, multiply and divide fractions. xii

14 CHAPTER 1 THE NATURE AND PURPOSE OF THE STUDY Introduction There is a growing consensus among mathematics educators that rational number concepts, in particular the fraction concept, are among the most ubiquitous, multifarious and significant mathematical ideas that children encounter before reaching high school (Behr, Lesh, Post & Silver, 1983; Mack, 1993). The wide use of fractions in everyday life makes information about fractions necessary as early as elementary grades. Smith (2002) noted that students experience with fraction concepts begins even before formal schooling and extends well into the high school years. However, even with this early introduction, students still have trouble conceptualizing fractions, probably forming one of the most critical barriers to the mathematical maturation of children (Aksu, 1997; Behr, Harel, Post & Lesh, 1992; Bezuk & Cramer, 1989; Hope & Owens, 1987; Schminke, Maertens, & Arnold, 1978). Some researchers (e.g. Behr, Lesh, Post & Silver, 1983) went as far as attributing many of the trouble spots in algebra to an incomplete understanding of earlier fraction ideas. The students inability to perform basic operations on fractions has resulted in error patterns in the successful completion of algebraic exercises and problems. Wu (2001) underscored the idea by suggesting that no matter how much algebraic thinking is introduced in the early grades, and no matter how worthwhile this might be, the failure rate in algebra will continue unless the teaching of fractions and decimals is radically revamped. The proper study of fractions provides a ramp that leads students gently from whole number arithmetic up to algebra (p. 11). 1

15 The Difficulty of Fractions Lamon (1999) said that as one encounters fractions, the mathematics takes a qualitative leap of sophistication. Suddenly, meanings, models, and symbols that worked when adding, subtracting, multiplying and dividing whole numbers are not as useful (p. 22). This qualitative leap of sophistication seems to cause such a level of confusion not only in primary school students but has stimulated Riddle and Rodzwill s (2000) mathematical curiosity to ask, Why is it that many adults, even after years of schooling, still do not understand some mathematics topics, such as fractions (p. 202)? Ohlsson (1988) believed that the difficulty associated with fractions is semantic in nature: The complicated semantics of fractions is, in part, a consequence of the composite nature of fractions. How is the meaning of 2 combined with the meaning of 3 to generate a meaning of 2/3? (p. 53). He further purported that the difficulty encountered in fractions is partially due to the result of the bewildering array of many related but only partially overlapping ideas that surround fractions (p. 53). Results from state, national and international assessments have done very little in convincing the mathematics education arena that the teaching and learning of fractions is not a difficult and daunting task. Although the results from the 2003 National Assessment of Educational Progress (NAEP) indicate a steady increase in fourth- and eighth-graders average mathematics score, 51% of the fourth-graders tested scored below the Satisfactory level on an extended constructed-response item dealing with equivalent fractions and only 35% of eighth graders tested were able to accurately identify the correct ordering of three fractions, all in reduced form. The twelfth-graders fared even worst with only 25% of the test-takers able to write a fraction resulting from dividing a fractional part of a unit into an integral number of parts (National Center for Education Statistics [NCES], 2004, Wearne & Kouba, 2000). Although the eighthgraders were more successful at the fraction items than the fourth graders, the results show that certain fractional concepts remain problematic to these middle school students. These concepts include a) ordering a set of fractions, and b) using fractions to solve word problems the application questions. Students performance on the fraction test items suggests that they have learned computational algorithms with little understanding of the 2

16 critical building-block concepts that are needed in the application of fractions to problemsolving tasks. NAEP studies done on previous assessment years corroborated with the current findings in highlighting the difficulty elementary students encountered with elementary fraction concepts (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1981; Kouba, Zawojewski, & Strutchens, 1997). The 1996 NAEP data provide evidence that fractions continue to be difficult for students, particularly for fourth-grade students (Wearne & Kouba, 2000, p. 164). Table 1: Fractions Sample Item from NAEP (Wearne & Kouba, 2000) Item Percent Responding Grade 4 How many fourths make a whole? Answer: Correct response of 4 50 Any incorrect response 35 Omitted 16 Alongside the difficulty students faced with rational numbers, two observations of the NAEP interpretive report done by Wearne and Kouba (2000) are worth mentioning. Firstly, students at all three grade levels (4, 8 & 12) are more successful in solving routine one-step tasks than nonroutine tasks including tasks that involved more than one step. Wearne and Kouba (2000) noted that fourth-grade students encountered fewer problems representing a fraction on a region if the number of parts into which the region 3

17 was divided was equal to the denominator of the fraction than if the number of parts into which the region was divided was a multiple of the denominator (p. 166). Fundamental to the notion of the fraction a/b is that the b represents the number of equal parts the whole is divided into. An analysis of the test item related to this concept reveals that only one-half of the test-takers in grade 4 were able to correctly state that 4 fourths are needed to make a whole with one-sixth of the students not responding to the item (see Table 1 for actual percentages). This becomes rather perturbing as one considers that this item requires knowledge that should be acquired from first grade. Secondly, students are able to construct more vigorous meaning for the unit when fractions are represented as regions than the meaning attached to the unit when fractions are represented as sets, as collections of objects. This coincides with the students use of concrete and/or discrete objects to illustrate the unit fraction. Students should also be able to connect written symbols of fractions with other representations such as physical objects, pictorial representations and the spoken language. The results of the Third International Mathematics and Science Study (TIMSS) have shown that eight grade students in the United States performed below average in fractions, proportionality and algebraic concepts compared to eight grade students internationally (Jakwerth, 1996). The grim findings resulting from NAEP reports coupled with those from TIMSS have aroused the interest of mathematics educators in view of the fact that fractions occupy a great part of the middle school mathematics curriculum and are also foundational to the mathematics encountered in high school and beyond (Lesh, Post, & Behr, 1988). A number of outstanding researchers in the field of mathematics education such as Steffe and Olive (1991) and Lamon (1999) have documented various reasons primary school students encounter difficulty with fractions. These include: 1) The abstract way in which fractions are represented during classroom instruction. It is a noted fact the students in the USA score significantly lower that their Asian counterparts in the TIMSS. On observing the TIMSS videotape survey of 231 eighthgrade lessons in the United States, Japan and Germany, Geist (2000) noted that in a Japanese classroom the teacher related the topic and concept to a real-world application 4

18 while in the United States classroom, the lesson was abstract, and no links to real life or applications were made. 2) Fractions do not form a normal part of the learners environment. The most common fractions used are halves, quarters and thirds. Other researchers in the quest to provide a solution to this dilemma, have suggested that fraction is introduced too early in the school curriculum while some notable researchers in the field, such as Behr, Wachsmuth, Post and Lesh (1984) recommended that formal fraction instruction should begin in third grade instead of fourth grade. Cramer and Henry (2002) added, Students will be more successful if teachers in elementary school invest their time building meaning for fractions using concrete models and emphasizing concepts, informal ordering strategies, and estimation (p. 47). They suggested that much of the fraction symbolization done in fourth and fifth grades should be taken up in middle grades. Watanabe (2001) suggested that this topic in whatever form be eliminated from the primary school (K 3) curriculum and encouraged fraction instruction to begin when the students are developmentally ready. Statement of the Problem The importance of the understanding of fractions to mathematics learning, coupled with the poor performance on fraction items on internal and external assessments have prompted various researchers to take a keen interest in this particular area of mathematics. Most of the prominent work on fractions date as far back as the early 1980 s with one of the most significant studies done on rational number - the Rational Number Project (RNP) starting in 1979 and officially ending in August Since 1980, the RNP has graced the research literature with reports on many investigations into the teaching and learning of fractions among fourth and fifth graders (Bezuk & Cramer, 1989; Post, Wachsmuth, Lesh & Behr, 1985). Careful scrutiny of the research done on fractions to date reveals an attempt by various researchers (e.g. Boulet, 1998; Charles & Nason, 2000; Saenz-Ludlow, 1994; Tzur, 1999) to focus their study where fraction use and/or instruction begins, whether formally or informally, that is, in fifth grade or lower with sparse research on how middle school or junior high children make sense of this concept. 5

19 Various studies on fractions have focused on middle school students misconception (e.g Behr, Lesh, Post, & Silver, 1983) or on the instructional approach undertaken by the instructor to lead students to a functional understanding of the topic (e.g. Mack, 1993). One such study was done by Erlwanger (1973) with an 11 year old boy named Benny who was a student in a sixth grade class using Individually Prescribed Instruction (IPI) Mathematics. Examining Benny s concept of decimals and fractions, Erlwanger (1973) noted Benny s view about rules and answers reveals how he learns mathematics. Mathematics consists of different rules for different types of problems. Therefore, mathematics is not a rational and logical subject in which one has to reason, analyze, seek relationships, make generalizations, and verify answers (p. 54). These rote rules often result in various procedural misconceptions. Ashlock (2002) outlined a number of the misconceptions plaguing the fraction concepts and stressed the importance of students understanding the underlying concept of a fraction. These misconceptions include: 1) Students equating the number of shaded parts in a diagram to the numerator and the number of unshaded parts to the denominator. For example, when Gretchen was 3 asked to name the fraction for the part that is shaded in the diagram below she wrote. 1 2) Students associating the denominator of the fraction with the total number of parts indicated in the diagram and the numerator with the number of shaded parts disregarding the fact that the associations apply only when all the parts are equal in area. For example, 1 Carlos writes for. 5 3) When asked to express a fraction in its lowest term the students will choose a 3 1 specific factor of the numerator and one for the denominator thus will be reduced to 8 4 by dividing the numerator by 3 and the denominator by 2. According to Ashlock (2002) This procedure is a very mechanical one, requiring no concept of a fraction; however, it does produce correct answers part of the time (p. 151). 6

20 A research review reveals that most studies done on fractions at the middle school level have focused primarily on students misconception of computational procedures as the ones described above or on other representations of rational numbers such as percent, ratio, and proportional reasoning (Gay, 1997; Lamon, 1993; Singh, 2000 to name a few). Consequently, only small portion of studies focused on the conceptual issues related to the understanding of fractions (Peck & Jencks, 1981), how students make sense of the concept and the role, if any, that physical referents including real life problem-solving activities play in the students understanding of fractions. Because of the complexity of the different subconstructs (part-whole relationship, measure, operator, quotient, and ratio) that can be assigned to a fraction, more in-depth and ongoing research need to be done in the quest to enhance conceptual and functional understanding of this vital middle school topic. Boulet (1998) expounded on the importance of students understanding of mathematical ideas and what would result from the lack of it. She wrote: Understanding is certainly the goal of learning, and teachers generally believe that their pupils understand their lessons. Without understanding, the learning of mathematics is reduced to the memorization of formulae and the rules governing them. Mathematics thus learned cannot be meaningful, much less useful (p. 19). Bezuk and Bieck (1992) purported that fraction concepts, order and equivalence usually receive only shallow attention and are often taught in a meaningless manner. They believed, and I agree, that it is crucial for middle grades instruction to strengthen students understandings before progressing to operations of fractions, rather than assuming that students already understand these topics (p. 119). As stated by Aksu (1997), A common error in teaching fractions is to have students begin computation before they have adequate background to profit from such operation (p. 375). As a result of these findings, a number of recommendations have been gleaned from these studies in the effort to improve students understanding of fractions. These include (1) The use of manipulatives/physical referents and meaningful, engaging activities in promoting higher order thinking skills and conceptual understanding; (2) The need for students to develop the correct concepts and relationships among fractions necessary to fully undertake the tasks of performing and understanding operations on 7

21 fractions (Behr, Wachsmuth & Post, 1985; Bezuk & Cramer, 1989; Empson, 2003; Morris, 1995; Yang, 2002); and (3) Students should have experience working with different physical models - continuous and discrete (Behr & Post, 1992). In fact, a major goal of mathematics education programs is to develop students mathematical reasoning so that they become proficient in using the content knowledge and skills learned to solve real-life problems (National Council of Teachers of Mathematics [NCTM], 2000). Purpose and Significance of the Study The results from a study on students use of part-whole and direct comparison strategies on fractions with 4 th, 6 th and 8 th grades students indicate that at the end of middle school, students seldom recognize real-world situations in which they can apply their knowledge of fractions other than halves and quarters (Armstrong & Larson, 1995). Clearly, the mathematics education community and, in particular, practicing mathematics teachers are bombarded with the demand for finding innovative ways to teach fractions and to engage mathematics students in meaningful activities that will lead them to conceptually understand fractions and to satisfy the students thirst for fun and relevance of mathematics ideas. This exploratory, whole class research study seeks to add to the knowledge base about how students conceptualized fractions as they worked with manipulatives and participated in process-oriented activities that model real-life events. These activities that the students were involved with during this study were analyzed to gain an insight into students understanding of fractions in the context of subdividing, comparing and partitioning of continuous and discrete models and the connections they made with the fraction ideas generated through these activities. The study also seeks to examine how these students make sense of fractions and investigates how their performance differs when fractions are presented using different models. The models selected for this study include continuous quantity and discrete objects. Continuous quantity usually refers to length, area or volume. The continuous whole is made up of one single object, which can be circular, rectangular or other geometrical regions (e.g. triangles); the number line and liquid measure that are not generally used in the classroom to illustrate a fraction. Discrete objects refer to several separate objects, that is, the whole consists of more than one object (Behr & Post, 1992). 8

22 Research Questions This study is an investigative, whole class research, descriptive study on the development of sixth graders understanding of fractions as they interact with physical referents, hands-on task-based activities and activities that model real life applications during eight weeks of a teaching sequence. The following questions served as the guide in the investigation of these students understanding of this subset of rational numbers. Each question is accompanied by a brief elaboration that serves the purpose of providing explicit nuances relevant to the question. 1. What is the nature of sixth grade students understanding of fraction? Embedded in this question is the need to understand how middle school students; define fractions unitize and partition make sense of fraction symbols order fractions generate equivalent fractions. Behr, Wachsmuth, Post and Lesh (1984) concluded from the RNP conducted from 1979 to 1983 that the performance of a considerable number of the participants on fraction questions dealing with order and equivalence, demonstrated a substantial lack of understanding. Research literature also indicates that children s knowledge of fractions is mainly algorithmic and is flawed by the interference of the students whole number language (Hiebert & Wearne, 1986; Moss & Case, 1999). This study utilizes the Herscovics and Bergeron s (1988) model of understanding to analyze the sixth graders responses to the fraction tasks that are presented to them during the teaching sequence. These tasks are designed to cover the various fraction concepts that are mentioned above. 2. What strategies do sixth grade students employ to ensure partitioning or equal sharing as they engage in process-oriented activities? The action of partitioning an object or a set of objects is learned by a child in a social setting (Poither & Sawada, 1983, p. 311). Although the participants in this study worked on numerous partitioning tasks, the answer to this research 9

23 question will be based largely on the process-oriented activities that the students were involved with during the study. Process oriented activities are activities that are designed to model real-life events with a purposeful twist to glean the mathematical ideas deeply entrenched in the students solicited behavior needed to complete the tasks. One such activity was the hosting of a Fraction Breakfast where students were required to evenly share the group s food among themselves. For the purpose of this study, the partitioning activities that the students performed were observed to determine their strategies for ensuring equal sharing. Charles and Nason s (2000) and Lamon (1996) partitioning strategies will be used to analyze and interpret the partitioning activities. These categories will be discussed with more elaboration in Chapter Two. 3. How do physical and real world representations aid in the development of sixth grade students understanding of fractions? The task-based activities that the students were engaged in gave them the opportunity to work with various physical referents. The terms physical referents, physical models, concrete models, and manipulatives will be used synonymously throughout this paper. For the purpose of this research, these are defined as the concrete objects (discrete or continuous) and hands-on activities that students work with as they solve fractional problems. They include objects and activities that appeal to several senses, can be touched, handled, moved, or cut. The participants in the study worked with fraction strips, fraction circles, fraction squares, fraction triangles, fraction balance, measuring cups, chips, and regular geometric polygons among other things. These will be identified throughout the research. There is a need to understand how students performance differs when modeling fractions using continuous and discrete objects. Research has shown that the use of physical referents can, but not necessarily, facilitate an improvement in student learning (Hiebert, et al., 1997). Numerous studies (e.g. Behr, Wachsmuth, Post and Lesh, 1984; Cramer & Post, 1995; Lesh, Cramer, Doerr, Post, Zawojewski, 2003; Kolstad, Briggs and Hughes, 1993) tout the active use of physical referents as a means for students to develop sound mathematical concepts. Behr, Wachsmuth, Post and Lesh (1984) have 10

24 also noted the negative effects of students dependence on the use of manipulatives to aid in the solution of every mathematics problems they encountered. Their dependence oftentimes dwarfed the need to make generalizations. This type of dependence is not altogether inappropriate but the learner who is able to connect the mental images of relations expressed via the objects rather than on direct actions with the object is at a clear advantage. Leinhardt (1988) defined real world representations as real life classroom situations that are similar to events familiar to the students every day life. Wellchosen tasks can pique students curiosity and draw them into mathematics. The tasks may be connected to the real-world experiences of students, or they may arise in contexts that are purely mathematical (NCTM, 2000, pp ). Streefland (1982), after perusing numerous articles on fractions up to the time of his research, commented on the literature s lack of meaningful contexts both as sources and as domains for the application of fractions (p. 235). The researcher s review of current articles on the same subject reveals a similar perspective. Summary The results of assessments done locally, nationally and internationally on students fractional knowledge have displayed sufficient evidence that fraction concepts are difficult to learn. This phenomenon is occurring despite the fact that children may have informal knowledge of the concept before schooling and the fraction concept is introduced in the early grades and continues to be taught or presented throughout the school years. This has caused real concern in the mathematics education arena thus whetting the research appetite of individuals seeking to understand, explain and/or offer suggestions to alleviate this daunting problem. This chapter has outlined the necessity of undertaking a study on students understanding of the fraction concepts thus adding to the growing and varied collection of meaningful studies on this concept. The study seeks to investigate and explore the sixth graders understanding of fractions as they work with manipulatives and participate in various process-oriented activities during instruction. The process-oriented activities include situations where students participate in real-life events that model scenarios that are directly related and meaningful to them. 11

25 CHAPTER 2 THEORETICAL FRAMEWORK AND REVIEW OF RELATED LITERATURE Introduction In qualitative work, theory is used at every step of the research process. Theoretical frames influence the questions we ask, the design of the study, the implementation of the study, and the way we interpret data (Janesick, 1998, p. 5). Pirie (1998) deemed the role of theory in qualitative research as very crucial and if not always, the most dominant in the presentation of the data. This chapter serves the sole purpose of providing the theoretical lens through which this present study can be viewed and understood. In this study, the approach to understanding the dynamics of students meaning of fractions, partitioning strategies, and how they work with physical models will be directly influenced by the teacher-researcher philosophy on the teaching and learning of mathematics, the constructivist epistemological stance of Piaget (1970), von Glasersfeld (1995), Steffe and D Ambrosio (1995), and other literature relevant to the study. Among the most relevant terminologies used in this study are constructivism, conceptual and relational understanding, meaning of a fraction, unit fraction, discrete and continuous quantities, sharing and partitioning, and logico-physical and logicomathematical understanding. Good teaching sequences require both a practical and a theoretical frame of reference (Higgins, 1973, v.). The extended model of understanding developed by Herscovics and Bergeron (1988) was used as the framework to explore and interpret the depth of the students understanding of a fraction as they performed a set of task-based activities during an eight-week teaching sequence. The participants methods of partitioning were observed, analyzed and categorized according Charles and Nason (2000) and Lamon (1996) partitioning strategies. 12

26 The Teaching and Learning of Mathematics Mathematics is a language that is used to describe the physical and non-physical aspects of the world we are living in. This quotation from Everybody Counts summed it up nicely: Mathematics is a living subject which seeks to understand patterns that permeate both the world around us and the mind within us. Although the language of mathematics is based on rules that must be learned, it is important for motivation that students move beyond rules to be able to express things in the language of mathematics. It involves renewed effort to focus on seeking solutions, not just memorizing procedures; exploring patterns, not just memorizing formulas; and formulating conjectures, not just doing exercises (National Research Council, 1989, p. 84). Fueled by this definition, I have sought in my teaching experiences to explore and find new ways to energize my mathematics classroom in the quest of understanding how my students make sense of the mathematical ideas presented in the class. Fundamental to my credo is the fact that all students can learn. This thinking conforms to current ideologies of students learning. The National Council of Teachers of Mathematics (NCTM) in its 2000 publication of Principles and Standards for School Mathematics, outlined as one of the six principles for school mathematics: Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge. Mathematics can and must be learnt by all students (pp. 11, 13). The National Research Council in its 1989 Nation Report stated: Only in the United States do people believe that learning mathematics depends on special ability. In other countries, students, parents, and teachers all expect that most students can master mathematics if only they work hard enough. The records of accomplishment in these countries--and in some intervention programs in the United States--show that most students can learn much more mathematics than is commonly assumed in this country (p. 10). It is therefore the teacher s responsibility to provide the opportunity for all students to learn this subject. This includes mathematical concepts that are consistently difficult for students to understand such as fractions, ratio and proportion. One aspect of 13

27 the teacher s role is to provoke students reasoning about mathematics. Teachers do this through the tasks they provide and the questions they ask. Teachers are responsible for the quality of the mathematical tasks in which students engage. Teachers should choose and develop tasks that are likely to promote the development of students understandings of concepts and procedures in a way that also fosters their ability to solve problems and to reason and communicate mathematically. There are documented evidences that direct instruction may not provide an adequate base for students development and for students use of higher cognitive skills (Confrey, 1990). Burns (2000) believed that for students to learn mathematics it becomes imperative for them to create and re-create mathematical relationships in their own minds. Therefore, when providing appropriate instruction, teachers cannot be seduced by the symbolism of mathematics. Children need direct and concrete interaction with mathematical ideas. Continuous interaction between a child s mind and concrete experiences with mathematics in the real world is necessary (Burns, 2000, p.24). As a practicing teacher, I believe that it is through interactions with children and deliberately seeking ways to understand mathematics from their lenses that as teachers we begin to make crucial distinctions between how children view mathematical situations and the best way to teach them. Epistemology Epistemology is the branch of philosophy that deals with the underpinnings of how we know what we know, and in particular the logical (and sometimes the psychological) bases for ascribing validity or truth to what we know (Goldin, 1990, p. 32). On this premise, the epistemological basis for this study is operated from a constructivist view of knowledge. Constructivism focuses on how knowledge is acquired. It emphasizes knowledge construction rather than knowledge transmission. Constructivists believe that all humans have the ability to construct knowledge in their own minds through a process of discovery and problem solving. Piaget, a proponent of constructivism, believed that children construct knowledge through the process of internalizing the physical operations on objects. As they move sets of objects about by putting them together, arranging them and/or separating them, they internalize properties of mathematical operations rather than the objects themselves 14

28 (Noddings, 1990). In the process of constructing meaning, the learner must actively strive to make sense of new experiences. This epistemological view is based on an assumption that knowledge is not passively received but is actively constructed by the learner. This ideology is supported by numerous studies on constructivism (Confrey, 1990; Maher & Davis, 1990; Romberg & Carpenter, 1986). Ideas and thoughts cannot be communicated in the sense that meaning is packaged into words and sent to another who unpacks the meaning from the sentences. That is, as much as we would like to, we cannot put ideas in students heads, they will and must construct their own meanings. Our attempts at communication do not result in conveying meaning but rather our expressions evoke meaning in another, different meanings for each person (Wheatley, 1991, p. 10). Simon (1995) purported that constructivism does not stipulate a particular model to be used in the teaching and learning of mathematics, in other words, it does not tell us how to teach mathematics. He believed that the sole purpose of constructivism is to provide a useful framework for thinking about mathematics learning in the classroom. Steffe and D ambrosio (1995) rebutted with the conjecture that there is a kind of teaching that can be legitimately be called constructivist teaching (p. 146). From their point of view, a teacher regards the students mathematical language and action to constitute a living mathematics and interacts with students in a learning space whose design is in part based on that language and action (p. 152). Simon s (1995) view of constructivism is implicitly embedded in their stance on constructivism. This view becomes particularly interesting to the teacher-researcher as the fraction teaching sequence conducted in the present study was designed to conform to Steffe and D Ambrosio s (1995) working model of constructivist teaching. In conducting teaching sequences modeled from a constructive standpoint, the teacher s description of the students schemes of action and operation would be at the forefront. This ideology forces the mathematics teacher to make a concerted effort in reforming the classroom environment to provide optimal learning. The underlying assumption pertaining to this study is that mathematics learning can be facilitated by having the students participating in preplanned, specific task-based activities. Figure 1 shows the working model 15

29 representing Steffe and D Ambrosio s (1995) hypothetical learning trajectory (HLT) that formed the threshold for the learning environment that the sixth graders worked in. The phrase zone of potential construction is used to refer to a teacher s working hypotheses of what the student can learn, given her model of the student s mathematics. The zone of potential construction is determined by the teacher as she interpreted the schemes and operations available to the student and anticipates the student s actions when solving different tasks in the context of interactive mathematical communication (p.154). Though not a quantitative study, the working hypothesis for this study is that the sixth graders participating in this study will display an understanding of fraction concepts as outlined by Herscovics and Bergeron s (1988) model of understanding as they work through a set of task-based activities. Of major significance was the way in which the information was presented, the scenarios were situated and how learners were supported in the process of constructing knowledge. Von Glasersfeld (1995) was convinced that within a constructivist learning environment students will be more motivated to learn something. Mathematics of students Student actions and modifications of actions Figure 1. Steffe and D Abrosio s (1995) Hypothetical Learning Trajectory 16

30 This learning environment contains: The learner, the setting or space where the learner acts, using tools and devices, collecting and interpreting information and interacting with each other (Wilson, 1995). According to Piaget (1970), knowledge arises from progressive social interactions that actively take place between the subject and the outside world. This social interaction is important and must be negotiated (Cobb, Yackel & Wood, 1992; Lo & Wheatley, 1992; Yackel, Cobb, Wood, Wheatley & Merkel, 1990). Social interaction plays a significant role in the development of mathematical knowledge in the individual, and of the corpus of mathematics available to the human race as a whole (Hunting & Davis, 1991). These researchers purport that a major if not determining role of constructing the fraction one-half and other fractions, is played by the social activity of sharing. Dialogues, which were encouraged through the act of sharing in the task-based activities done in the study, provided the catalyst for knowledge acquisition. Oral exchanges facilitated mathematical understanding that occurred through social interaction, through questioning and explaining, challenging and offering timely support and feedback. The classroom was eventually transformed into a community of learners, a knowledgebuilding community where students had firsthand knowledge of their peers thinking processes. Besides the learning communities, concepts such as scaffolding, cognition and cooperative learning lend support to the constructivist s learning environment (Brown, 1994; Rogoff, 1990). Burns (2000) and Mack (1990) have documented the need for teachers to organize and plan classroom instruction that will build on children s previous experiences with fractions and help these children clarify the ideas they have encountered while grappling with the fraction concept. The teacher in a constructivist environment will draw on the culture (ethnomathematics) and resources available to the students. In the realm of the study that was undertaken, it was crucial for the teacher to recognize that children s introduction to fractions occurred primarily outside of the mathematics classroom and that active knowledge construction was taking place as the students interacted with each during the activities. 17

31 The Fraction Concept Fractions can be assigned various meanings according to the context in which the concept is used. According to Ohlsson (1988) in order to understand the meaning of fraction one has to pay attention to the mathematical theory in which fractions are embedded, to which fractions are applied, and to the referential mapping between the theory and those situations (p. 54). Fractions form a part of a subset of rational numbers which form a subset of a larger set of real numbers. Freudenthal (1983) viewed fractions as the phenomenological source of rational numbers. Kieren (1976) divided the concept of fraction into five major subconstructs: part-whole relationship, measure, operator, 3 quotient, and ratio. Each subconstruct will be briefly described and the fraction, where 4 3 is called the numerator and 4 the denominator, is used to give a clearer idea of the description. Part-whole relationship: This depends directly on the ability to partition either a continuous quantity or a set of discrete objects into equal sized subparts or sets (Behr & Post, 1992). In this study the part-whole relationship is considered to be 3 the ratio of the part to the whole. The fraction representing three equal slices 4 out of a cake cut into four equal pieces (continuous model) or three eggs from a carton with 4 eggs (discrete model) can be shown diagrammatically as Measure: This refers to the position of a number on the number line or a ruler and for the purpose of this research will also be applied to the measure on measuring cups or cylinders. 18

32 Operator: In this context, the fraction is considered as a transformation (Behr, Lesh, Post, & Silver, 1983). The fraction 4 3 as an operator may be perceived as 3 finding three-quarters of some quantity. For example, of 12 will result in nine. 4 Quotient: The quotient subconstruct of a fraction focuses on the operation. The 3 fraction can be interpreted as 3 divided by 4 or the result of sharing 3 cookies 4 among four people. Ratio: This subconstruct expresses a relationship between two quantities, for example the relationship between the number of nickels and the total number of coins in a purse, that is, there are three nickels out of the four coins in the purse. A fraction is only a ratio if it is related to the ratio between the part and the whole and not between part and the part such as the relationship between the number of boys and the number of girls in the sixth grade classroom. This means that not all ratios are fractions. Ratio Operator Quotient Measure Equivalence Multiplication Problem Solving Addition Figure 2. Conceptual Scheme for Instruction on Rational Numbers 19

33 Behr, Lesh, Post and Silver (1983) deemed it plausible that the part-whole subconstruct, based both on continuous and discrete quantities, represents a fundamental construct for rational number development. It is, in addition, a point of departure for instruction involving other subconstructs (p. 100). Figure 2 depicts the preliminary conceptualization of the interrelationships among the various subconstructs. This diagram suggests that partitioning and the part-whole subconstruct of rational numbers are basic to learning other subconstructs of rational number (p. 100). The solid arrows represent established relationships while the dashed arrows represent the hypothesized relationships among the subconstructs and operations on the subconstructs. Studies (Behr & Post, 1992; Charles & Nason, 2000; Lamon, 1996; Poither & Sawada, 1983) of children s partitioning and fraction knowledge suggested that children use different mental processes according to whether the physical expressions of fractions are discrete or continuous quantities. Behr, Wachsmuth and Post (1988) gave the cognitive distinctions between continuous and discrete models: There are significant similarities and differences between a continuous model and a discrete model for showing rational number concepts. To represent rational number concepts, each requires: (a) the identification of a unit and (b) partitioning of the unit into parts of equal size (i.e. equal measure); for continuous model, each part is again a single continuous piece and contiguous to other part(s); for the discrete model, the parts may be a single discrete object, or several discrete objects, and, in general, are not contiguous with other parts. Thus, there seem to be distinctly different cognitive demands for representing fractions with discrete and continuous models (p. 65). Students partition discrete and continuous quantities in real-life situations, for example when they have to share a chocolate bar (continuous model) or a set of cookies (discrete model). Behr and Post (1992) claimed that geometric regions, sets of discrete objects and the number line are the most frequently used models in representing fractions in middle school. Based on the subtle differences with the fraction subconstructs as mentioned above, deliberate effort must be made by the mathematics teacher in the presentation of this vital middle school topic. This study focused on the unit fraction, other proper 20

34 fractions, and the order and equivalence of fractions since these concept areas play a critical role in the understanding of other topics relating to fractions. For the purpose of this study, unit fractions are considered to be fractions of the type n 1 where n represents any natural number (1, 2, 3, 4 and so on). 2 1 { }is an example of a unit fraction. 3 Non-unit fractions represent fractions whose numerator is greater than 1. For example 4 { } is considered a non-unit fraction. Proper fractions refer to those 3 fractions that are less than the whole (the numerator is smaller than the denominator). 4 is also a proper fraction. Improper fractions are fractions that are greater than the whole. Improper fractions can be expressed as mixed numbers. Thus 4 9 { }can be 1 expressed as 2. 4 Models of Understanding The major goal for every practicing mathematics teacher is that mathematics be learnt with understanding. Teaching for understanding is not a new goal of instruction: School reform efforts since the turn of the 20 th century have focused on ways to create learning environments so that students learn with understanding (Carpenter & Lehrer, 1999, p. 19). But, what is understanding and how is it developed? Carpenter and Lehrer (1999) suggested five forms of mental activity that contributed to the development of mathematical understanding. These are a) constructing relationships, b) extending and applying mathematical knowledge, c) reflecting about experiences, d) articulating what one knows, and e) making mathematical knowledge one s own. There exists, however, a number of seemingly differing and overlapping models of understanding presented in the mathematics education arena in the pursuit of answering the first part of the question posed above. Among these are Skemp s (1987) relational and instrumental understanding, Kieren s (1993) levels in the growth of understanding and Herscovics and Bergeron s (1988) two-tiered model of understanding. This section includes a detailed description of the models of understanding mentioned above and a rationale for using the 21

35 Herscovics and Bergeron s (1988) model for analyzing the students understanding of fractions as they worked with physical referents. The tasks designed for the study aimed at generating the students understanding and thinking on unit and non-unit fractions, and the criteria needed for each component of the two-tier model of understanding. Skemp s Relational and Instrumental Understanding Skemp (1987) grouped the varied views of mathematics understanding under two broad and overlapping headings, namely relational and instrumental understanding. Relational understanding includes both understanding of concepts and procedures. The core of conceptual knowledge is the understanding of relationships or connections among mathematical ideas and concepts (Hiebert & Lefevre, 1986). This type of understanding can be perceived as a connected web of knowledge. A unit of conceptual knowledge cannot exist as an isolated piece of knowledge but should have some relationship or connection to other pieces of information. Procedural knowledge in its purest form focuses on symbolism, skills, rules and step-by-step algorithms used in accomplishing a mathematical task (Aksu, 1997; Ashlock, 2002). Conceptual and procedural knowledge are considered crucial aspects of mathematical understanding. Students should learn concepts concurrently with learning procedures so they can see the connections and relationships (Carpenter, 1986). Instrumental understanding or learning technique without reason focuses on calculation, computation and problem solving without sense making. Students are unable to make connections to a conceptual rationale. The goal of the student, who believes that mathematics is just a set of formulas to be followed, is to understand a mathematics problem instrumentally. Skemp (1987) cited three advantages for this type of understanding. 1. Instrumental mathematics is usually easier to understand. 2. The rewards are more immediate and more apparent. 3. One can often get the right answer more quickly. In contrast, relational understanding provides opportunities for the students to adapt methods learnt to accomplishing new tasks which students who exhibit only instrumental understanding rarely can do. Since the students who possess relational understanding are constructing knowledge from their own experiences, social interaction and peer 22

36 negotiation, it becomes easier for them to remember methods and strategies employed in solving the problem. Skemp (1987) also believed that more mathematical content might be involved when teaching for relational understanding. For this type of mathematical understanding, the learner is able to build up conceptual structures (or schema) from which the possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point (p. 163). To achieve relational understanding, it becomes pertinent for the mathematics teacher to help the students to develop a conceptual and procedural knowledge of mathematics and the connections and differences between them. Both Skemp (1987), and Hiebert and Lefevre (1986) advocated for linking both types of understanding which will aid in the storage, retrieval and effective use of mathematical procedures. Teachers are urged to shift their classroom practices away from the focus on computational accuracy and toward a focus on deeper understandings of mathematical ideas, relations, and concepts (NCTM, 2000). It therefore becomes imperative for teachers who want to support conceptual mathematics understanding in their students to have knowledge about mathematics teaching and children s mathematical thinking and the pedagogical skills needed to put the knowledge to practice. Kieren s Model of the Growth of Mathematical Understanding Pirie (1988) and Kieren (1988) viewed understanding as a dynamic, nonlinear process and not as a single acquisition, or linear combination as suggested by Skemp s (1987) model. This view on understanding led them to collaborate on a model of understanding which characterizes understanding as an organized and recursive process which responds to the constructivist description of the understanding process (Kieren, 1993; Pirie and Kieren, 1989). This model describes mathematical understanding as recursive in that it involves the use of the same sequence of processes, but at a new level, with new elements of action a dynamic process that involves folding back on itself for growth, extension, and re-creation (Kieren, 1993, pp. 72, 73). In this model, mathematical understanding is represented using eight levels" in concentric circles (see Figure 3). The movement between the levels is not necessarily from the smaller circle to the largest one but learners increase their knowledge base by 23

37 returning recursively to previous levels of understanding. A brief description of each level is given to provide a theoretical comparison and contrast to other models of understanding that are mentioned. Inventing Structuring Observing Formalizing Doing Property Noticing Image Having Image Making Figure 3. Kieren s (1993) Model for the Recursive Theory of Mathematical Understanding Primitive Doing. This is the core of action and involves the initial knowledge that the learner possess. This forms the basis or starting point for the development of a 24

38 particular mathematical understanding. In reference to fractions, students ability to partition an object (e.g. sharing a pizza) forms the primitive acts of doing that will eventually form the core for understanding this daunting mathematical topic. Image Making. At this level, images begin to form through specific actions where the learner is able to apply the informal knowledge to new situations. With respect to fractions, this may mean working on various sharing problems and making a record of such actions, using fractional language in a way that is very closely tied to results (Kieren, 1993, p. 73). Image Having. From the activities included in the image making level, the learners form mental objects and pictures an idea they have, rather than simply an action they can take in response to a call to action (p.73). Fractions become a mental object for the learner. Property Noticing. This is a recursion of the previous levels. The learner is able to develop patterns, properties, conjectures and connection from and between the mental objects formed before. For example, the learner may figure that a string of equivalent fractions can be generated from a given fraction. Some of these fractions may be fractions, which the learner cannot mentally see. Formalizing. At this level, the learner engages in self-conscious thinking. Knowing moves beyond specific actions and mental images mathematical definitions can be formulated. For instance, the learner is able to see things in terms of for all a fractional numbers, and know that can be any fractional number where b 0 b (Kieren, 1993, p. 73). This is considered to be the level where mathematics is no longer a metaphor for events in a physical or image world. It is abstract, although not necessarily expressed in generalized mathematical terms or symbols (Pirie & Kieren, 1994, p. 40). Observing. The learner, at this level, is able to observe his or her conscious thought processes and have the ability to consistently structure and organize them. For example, the learner will observe that there is no least positive fractional number (p. 73). Structuring. The learner can now structure his or her own observations made at the previous level into a set of systematic assumptions. Formal proofs can be formulated 25

39 at this level. With respect to fractions, the learner will not view the addition of fractions as an action of combining quantities, or a property of equivalence, or as a relationship between the numerator and denominator but as a logical consequence of the field properties and the nature of formal equivalence. Inventing. At this level, the learner possess the ability to change the entire way of thinking about a mathematical topic without destroying and eliminating the structure and understanding of previous ways of knowing. New questions can be generated which might grow into a totally new concept. Kieren (1993) described the levels in this model as knowing organized by understanding (p.74). Learners are able to fold back to the inner level of knowing thus resulting in an interweaving of intuitive (informal) and formal understanding. This model suggests four bases for mathematical knowing, namely, action, image, form, and structure. These bases are not the same as concrete, pictorial and symbolic modes. Under this model, an understanding of rational numbers is characterized as a recursive, dynamic whole and of knowing rational numbers at many levels simultaneously (Kieren, 1993). Herscovics and Bergeron s Model of Understanding Herscovics and Bergeron (1981) suggested a model which identified four levels of understanding, namely, intuitive understanding, initial conceptualization, abstraction and formalization. This model, however, provided little help in analyzing the formation of mathematical concepts since in some cases it was not clear how to characterize and distinguish between the different levels. This led Herscovics and Bergeron (1988) to refine and clarify this model of understanding and forced them to extend their model to include the physical aspects of the concept under consideration the number scheme. This extended model of understanding (Herscovics & Bergeron, 1988) divides the construction of mathematical concepts that stem from actions performed on objects in the real world into a two-tiered model. The first tier identifies three different levels of understanding of the preliminary physical concepts. The second tier identifies three distinct constituent parts of the comprehension of mathematical concepts. An understanding that results from observing and thinking about situations involving physical objects (logico-physical understanding) is crucially different from an 26

40 understanding resulting from observing and thinking about situations involving mathematical objects (logico-mathematical understanding); the first kind is natural and real, the latter is artificial and abstract (Boulet, 1998, pp. 20, 21). The three components or levels of the first tier are intuitive understanding, logicophysical procedural understanding, and logico-physical abstraction. Intuitive understanding. This refers to a global perception of the notion at hand; it results from a type of thinking based essentially on visual perception; it provides rough non-numerical approximations. Logico-physical procedural understanding. This is the acquisition of logicophysical procedures which the learners can relate to their intuitive knowledge and use appropriately. Logico-physical abstraction. This type of understanding refers to the construction of logico-physical invariants or the reversibility and composition of logico-physical transformations, or as generalization. Since the teaching and learning of mathematics is usually characterized by symbolic and numerical manipulation and not the handling of physical objects, the second tier of Herscovics and Bergeron s model (1988) deals with the understanding of the mathematical concept itself (Boulet, 1998). It is worthy to note that the levels of understanding in the first tier are essential to the constituent parts of understanding in the second tier. Boulet (1993) put it this way: without the understanding of such concepts as those describe in the first tier of the model, understanding at this level can be both superficial and fleeting. In fact, it is this very connection with the real, physical world which gives meaning to elementary mathematical notions (p. 33). The second tier consists of three constituent parts: logico-mathematical procedural understanding, logico-mathematical abstraction and formalization. At this point in the model Hersovics and Bergeron (1988) replaced the word level with constituent part with the purpose of preventing an overly hierarchical interpretation (Herscovics & Bergeron, 1988, p.20). Logico-mathematical procedural understanding. At this constituent part of understanding, the learner possesses explicit logico-mathematical procedures that the learner can relate to the underlying preliminary physical concepts and use appropriately. 27

41 Logico-mathematical abstraction. This refers to the construction of logicomathematical invariants together with the relevant logico-physical invariants, or the reversibility and composition of logico-mathematical transformations and operations, or generalization. Formalization. This component of understanding involves a gradual development of various mathematical notations. This constituent part is defined as the axiomatization and formal mathematical proof (Boulet, 1998, p. 23). At the middle school level, it can be viewed as the discovery of axioms and the ability to justify these axioms logically. Formalization also includes the enclosure of a mathematical notion into a formal definition and the use of mathematical symbolization for notions for which prior logicomathematical procedural understanding or abstraction already exist to some degree. The research literature (e.g. Saxe, Taylor, McIntosh & Gearhart, 2005) does indicate, however, that the students knowledge of the standard notation of fractions and the partwhole relationships can develop to a certain extent independently. Chaffe-Stengel and Noddings (1982) seeking to clarify the symbolic process involved in the treatment of fractions made mentioned of the connection that Charles Pierce (Eisele, 1962) made with the visual, written and spoken representation of a mathematical concept that form an integral part of the formalization constituent part of the Herscovics and Bergeron s (1998) model of understanding. Pierce characterized the signing process as the triadic interrelation between the concrete objects involved in the problem, the sign which stands for those objects, and the signification, or meaning, of the sign. The signification serves as a mediator, making sense of the object-sign relation. The function of the sign is to stand in place of the object, to allude to the object but remain more easily available than the object (Chaffe-Stengel & Noddings, 1982, p. 43). Figure 4 illustrates Chaffe-Stengel and Noddings (1982) process of signing which corroborates with the connections that exist between the two tiers of Herscovics and Bergeron s (1988) model of understanding. 28

42 Verbal mediator Sign Part of a thing 1 1 1,( ),,etc Concrete representation Figure 4. Potential Signing Process Appropriate for Fraction Readiness (Chaffe- Stengel & Noddings, 1982) Although the understanding of a mathematical concept relies heavily on the understanding of the preliminary physical concepts, this does not suggest that the understanding of a mathematical concept needs to await the prior three levels of understanding of the preliminary physical concepts (Herscovics & Bergeron, 1988, p. 20). This means that the construction of a mathematical concept does not necessarily progress in a linear fashion but are interconnected as Figure 5 shows. This model has several pedagogical implications as suggested by Herscovics and Bergeron (1988). 1. It explicitly links up the children s mathematics to their physical world. 2. Thus, the physical world should be used as a starting point in the construction of the children s mathematical concept. 3. Teachers can develop tasks related to every aspect of understanding of a given concept. The broad range of activities that the students are engaged in will complement the rich cognitive environment provided in a constructivist learning environment. 29

43 UNDERSTANDING THE UNDERLYING PHYSICAL CONCEPTS Intuitive Logico-physical Logico-physical understanding procedural abstraction understanding Logico-mathematical Logico-mathematical Formalization procedural abstraction understanding UNDERSTANDING THE EMERGING MATHEMATICAL CONCEPT Figure 5. Herscovics and Bergeron s (1988) Model of Understanding Rationale for using Herscovics and Bergeron s (1988) Model of Understanding Close examination of Kieren s (1993) recursive theory of mathematical understanding reveals the possible steps a child encounters in the learning process of a mathematics concept. It does not explicitly provide an instrument for analyzing the development of a mathematical concept. This model is mainly geared towards problem solving and does not adequately provide the tools needed to describe the comprehension involved in concept formation. In particular reference to the fraction concept, Kieren s (1993) model does not specify the content of the fractional understanding needed at each level. Skemp s (1987) categorization of understanding is too broad and general to use for the analyzing of the students concept of the unit and non-unit fractions. 30

44 Although Herscovics and Bergeron s (1988) do not claim that their model of understanding is suitable to describe the understanding of all mathematical concepts, there have been successful attempts to use this model to analyze the addition of small numbers, measurements, surface areas, algebraic concepts and coordinate geometry. Boulet (1993) used this model and applied it to the unit fraction as she worked with a set of fourth-graders. In this study, this model was used to interpret the sixth-graders construction of the unit and non-unit fraction from a physical and mathematical standpoint. From my review of the different discussions on models of understanding Herscovics and Bergeron s (1988) model of understanding that was partially influenced by Skemp s (1987) instrumental and relational understanding, provides a more refined, detailed and systematic model for interpreting the students actions on the tasks provided to analyze their mathematical behavior. Though this model does not provide an explicit didactical approach to the handling of the fraction it does provide for the leeway for the building of one. Herscovics and Bergeron s (1988) model is not concerned with the pedagogical aspects of the fraction but it can be used to supply concrete cues for the formation of an instructional program in that activities can be designed specifically for each of the components (Boulet, 1993, p. 45). This model of understanding does not provide any inkling of how the fraction concept is learned, however, it is able to clarify the components needed to interpret the understanding of the unit and non-unit fraction as the students work with the physical referents. It can also be used as a template or frame of reference for the development of tasks and activities that are designed to expunge some of the cognitive obstacles students encountered while working with the fraction. Criteria for the Components of Understanding in the Herscovics and Bergeron s (1988) Model of Understanding Having established the rationale for the use of the Herscovics and Bergeron s (1988) model of understanding, an outline of the criteria necessary for each level of understanding is included below. These criteria are adapted from Boulet (1993) study of the unit fractions with two sets of fourth graders. 31

45 Understanding the Underlying Physical Concepts The physical concepts underlying the emerging mathematical concepts are the focal points of the first tier of the Herscovics and Bergeron (1988) model. Based on the assumption that partitioning and the part-whole subconstruct are fundamental to learning the other subconstructs of rational numbers, a teaching sequence was designed to develop an understanding of fractions based on partitioning and the part-whole relationship. These concepts were expanded to the order and equivalence of fractions, and the quotient, measure and operator subconstructs. Similar sequencing of the concept was done by the Rational Number Project (RNP). Intuitive understanding of the unit fraction. The sole criterion necessary for intuitive knowledge is the awareness of the role of partitioning in part-whole relationships. There are four conditions that are necessary for this awareness. Students should recognize: 1. the equality of the parts in a partitioned whole despite the position of the parts in the whole. 2. the inequality of the parts in similar wholes regardless of the equal number of parts (shaded or unshaded). 3. the inequality of the parts of similar wholes that are partitioned differently. 4. the similarity in the part-whole relationships in spite of the inequality of the size of the parts. Logico-physical procedural understanding. The most fundamental criterion for this level of understanding is the ability to partition a whole into a given number of equal shares. The methods to ensure equality of the parts can vary with the students freedom to use whatever materials (e.g. pipe cleaners, rulers, cutouts) present at hand to aid in their estimation. Logico-physical abstraction. According to Boulet (1993): It is within the context of this component in the model of understanding that a first glimpse of the breadth and depth of a child s understanding of the unit fraction is given (p. 39). At this level the fraction concept is distinguished from the simple action of dividing a whole into equal parts but rests upon the understanding of the relationship between the part and whole. There are five criteria related to this component. 32

46 1) The child s ability to accurately understand the equivalence of the part-whole relationship in spite of a variation in the physical attributes of the whole, that is, the wholes can be of different shapes and/or sizes. 2) The child s ability to reconstitute the whole from its parts. 3) The child s ability to recognize the equivalence of the part-whole relationship regardless of the physical transformations of the whole. For example, will the student recognize the part-whole relationship in these identical whole to be the same? 4) The child s ability to recognize the relationship between the size of the parts and number of equal shares. 5) The child s ability to repartition already partitioned whole. Understanding the Emerging Mathematical Concept The quantification of the fraction is the main consideration in the second tier instead of the part-whole relationship. Logico-mathematical procedure. The procedures that the child uses correctly in association with the underlying physical concepts gained in the first tier become the focus of analysis at this level of Herscovics and Bergeron (1988) model of understanding. The criterion set for this component include the child s ability to quantify part-whole relationships depicted in all sorts of situations and to produce illustrations of orally given unit fractions (Boulet, 1993, p. 41). Logico-mathematical abstraction. The criteria for this level is similar to those of the logico-physical abstraction level in that the main objective is to determine the scope of the child s ability to comprehend and understand the invariance, reversibility and generalization aspects of unit and non-unit fractions. The main difference between both levels is the fact that at the logic-mathematical abstraction level, the child is able to quantify the part-whole relationships connected to the physical attribute of the fraction. Mainly, the student is now able to verbally associate a numerical value to the physical referent illustrating the fraction. For example, in examining the child s apprehension of the invariance (i.e. the part-whole relationship remains unchanged regardless of the physical operations and/or transformations) of the fraction as demonstrated using discrete 33

47 objects, the child is shown a set of 10 chips 4 white and 6 black. The chips are firstly arranged similar to this representation. Using the chips to represent cookies, the student will be asked what fraction of the set of cookies is eaten if only the white ones are eaten. The chips will then be rearranged to look like this. At this point the child can be asked to decide if the same amount of the set of cookies would be eaten if only the white ones are eaten (logico-physical abstraction invariance) and what fraction of the set of cookies was eaten if only the white ones are eaten (logicomathematical abstraction invariance). Boulet (1993) noted that children may be able to recognize a logico-mathematical invariant while not recognizing the underlying logicophysical one. Consequently, at this constituent part both the logico-physical and logicomathematical abstractions need to be verified. Formalization. At this final constituent part of the model of understanding, the formal aspect the fraction is considered. Embedded in this component of Herscovics and Bergeron s (1988) model is the need to symbolize the fraction that has been internalized and spoken. Boulet (1993) defines formalization (at the elementary level) to mean the discovery of axioms which will lead to the discovery of logical mathematical 2 justifications. For example, is two-thirds as well as what it represents. Included in the 3 set of criteria for formalization is the students ability to symbolically order a set of fractions, and symbolically reconstitute wholes as well as parts of given fractions. Physical Referents and the Understanding of the Fraction Concept The issue of the use of manipulative materials in the mathematics classroom is hotly debated. Some research articles reported on the ineffectiveness of physical models in helping the students understand the underlying mathematical concept involved with a fraction, and the difficulty children have in moving from using physical models to using 34

48 symbolic notation (Bezuk & Bieck, 1992; Thompson & Lambdin, 1994). Majority of the research conducted on physical referents in the mathematics classroom, however, (e.g. Hall, 1998; Hasemann, 1981; Hinzman, 1997; Post, Cramer, Behr, Lesh & Harel, 1993) have shown that the active use of physical referents may aid in the conceptual development of the fraction concept. These concrete analogs are used by teachers with the conviction that they will facilitate the construction, understanding and retrieval of mathematical concepts, reduce learning effort, serve as memory aids, provides a means of verifying the truth, mediate transfer between tasks and situation, and indirectly facilitate transition to higher levels of abstraction (Boulton-Lewis, 1998, p. 220). Physical referents play a pivotal role in the acquisition and retention of vital mathematical concepts. Wearne (1990) from a study on decimal fractions with a set of students, argued that the participants were able to use their reasoning abilities about decimal fractions at the same level or better as they appeared to be using the meanings derived and developed from the firsthand experiences the students had with the physical referents they used as aids to justify processes and solve problems. NCTM (2000) in its Representation Standard for Grades 6-8 stated that representation is central to the study of mathematics. Students can develop and deepen their understanding of mathematical concepts and relationships as they create, compare, and use various representations. Representations such as physical objects, drawings, charts, graphs, and symbols also help students to communicate their thinking (p. 280). From the students perspective the representation could be their reconstruction or drawing of a contextualized problem situation or their interpretation or use of a representation that was structured or designed by the teacher-researcher. From the very onset of fraction instruction students are introduced to this concept using circular and/or rectangular objects such as a pizza, a cake or sheet of rectangular paper used in paper folding. Thus, when asked to represent a fraction pictorially or diagrammatically, students most frequently resort to drawing and partitioning these geometric regions. The task-based activities that the students worked with during this study gave them the opportunity to manipulate physical referents of different shapes, sizes and quantity besides the ones mentioned above. Their behavior were observed as they labored with continuous wholes that are not confined only to circular or rectangular 35

49 regions but triangular regions and solid/liquid quantities. Discrete models consisted of objects of varying shapes - circular, triangular and rectangular in appearance. Behr, Wachsmuth and Post (1988) purported that discrete-embodiment tasks are more difficult for children than continuous-embodiment tasks (p.73). One the major work done on rational numbers, of which fraction is a subset, is the Rational Number Project (RNP) pioneered by Merlyn Behr (now deceased), Richard Lesh and Thomas Post and funded by the National Science Foundation (NSF). From this extensive work a RNP curriculum was developed and used which reflected the following beliefs: (a) Children learn best through active involvement with multiple concrete models, (b) physical aids are just one component in the acquisition of concepts verbal, pictorial, symbolic and realistic representations also are important, (c) children should have opportunities to talk together and with their teacher about mathematical ideas, and (d) curriculum must focus on the development of conceptual knowledge prior to formal work with symbols and algorithms The NCTM Principles and Standards (2000) recommends that physical models be used in the classroom to aid in the development of students understanding of fractions. As a matter of fact, representations are ubiquitous in the middle-grades mathematics curriculum (p. 280) that is proposed by NCTM. Most often than not, young children are presented with physical referents in mathematics learning that are expected to facilitate the learning process. These include everyday materials, structure or semi-structured concrete embodiments of mathematical concepts such as Cuisenaire Rods bundles of tens. However, as the students get into middle grades teachers seem to eliminate these concrete representations for one reason or the other. NCTM 2000 and other researchers (e.g. Boulton-Lewis, 1998; Cramer, 2001; Kamii, Lewis, Kirkland, 2001) encourage the use of manipulatives at all levels of education suffice these physical referents aid the students to make relationships through constructive abstraction in problem solving. Building on the theories of Piaget, Bruner and Dienes, the Lesh s translation model (first cited in Lesh, Landau, & Hamilton, 1983) suggested that a deep understanding of mathematical ideas can be developed by involving students in activities that embed the mathematical ideas to be learned in five different modes of representation 36

50 with an emphasis on translations within and between modes (Cramer, 2003, chap. 24). These five modes - manipulatives, pictures, real-life contexts, verbal symbols, and written symbols - stress that understanding is reflected in the ability to represent mathematical ideas in multiple ways, plus the ability to make connections among the different embodiments (Lesh, Cramer, Doerr, Post & Zawojewski, 2003). Translations within and between various modes of representation make ideas meaningful for students. Lesh s translational model (Lesh, Post & Behr, 1987) shows the connections among different external representations including real world situations and manipulative aids. This instructional model (see Figure 6) necessitates the active involvement of students as well as the extensive use of manipulatives and other learning aids such as pictures, diagrams, student verbal interactions, and real life situations simulations modeled in the classroom. Each mode is connected to every other mode forming a translation such as picture-to-symbolic. Each connecting link is bidirectional. This signifies the interconnection with the different modes evidenced during problem solving. Pictures Spoken Symbols Written Symbols Manipulative Aids Real World Figure 6. Lesh s Translation Model (Lesh, Post & Behr, 1987) 37

51 Among the pedagogical beliefs that emerged from the literature, the use of multiple physical models such as fraction strips, fraction circles, Cuisenaire rods, and chips over an extended period of time is important in optimizing children s learning of fractions. The art of paper folding is also useful in developing and extending students understanding of the unit fraction (Cramer and Henry, 2002). In Piaget s stages of cognitive development he proposed that children s learning is enhanced and optimized when they are able to manipulate concrete objects in the development of the concept then further extending his knowledge to include abstract ideas (Higgins, 1973). Partitioning Strategies Partitioning is defined as the act of dividing a quantity into a given number of parts which are themselves qualitatively equal (Kieren & Nelson, 1981, p. 39). This process produces equal parts that are not overlapping (Lamon, 1999). An understanding of partitioning is fundamental in the development of students understanding of fractions (Poither and Sawada, 1990). It forms the basis for equivalence that in turn forms the foundation for operations on fractions. Partitioning also plays an active role in the development of the fraction concept as it relates to the part-whole, measure and quotient subconstructs. Fair sharing has been part of most students everyday experiences, thus, partitioning activities should build on what the students already know and provide the opportunity for them to extend their knowledge to include more complex tasks (Mack, 1995). Findings from several studies on fraction understanding show that students who fail to extend their knowledge usually exhibit a lack of understanding about partitioning (Behr, Lesh, Post & Silver, 1983; Lesh, Landau & Hamilton, 1983). Behr and Post (1992) recommended that children be given the opportunity to partition various types of objects (p. 211). This act of partitioning should go beyond the visual separation of a regular polygonal region into congruent parts. The focus should be on the appropriate interpretations of the children s actions as they operate on the partitioning activities (Steffe, 2002). The students partitioning experiences should include partitioning of objects in even and odd number of parts. The process of unitizing and the recognition of the unit play a crucial role in partitioning. Children s understanding of fraction, for the most part, is delayed due to the 38

52 failure to recognize the importance of the unit. The unit is different in every context. For example in equally sharing 12 quiches among 4 people (Activity 12 see Appendix B), 3 1 the unit is the twelve quiches. Each person would get 3 quiches, that is or of the 12 4 quiches. However, in dividing a circular birthday cake among 4 friends, the cake is the unit. The process of operating on the unit is called unitizing (Lamon, 1996). In the case of the 12 quiches, a child might consider the unit as one unit consisting of 12 quiches continuous while another child may think of the unit as 12 separate and distinct quiches. This study will not delve into the process of unitizing but will make mention of it as is necessary in explaining the students partitioning strategies. Poither and Sawada (1983) and Lamon (1996) suggested that children use an assortment of intuitive strategies when given partitioning activities. A number of these partitioning strategies are identified in the literature such as Armstrong and Larson (1995) part-whole and direct comparison strategies and Poither and Sawada s (1990) algorithmic halving and evenness. However, as these strategies proved to be too general in their scope, the more explicit partitioning strategies that Charles and Nason (2000) and Lamon (1996) had generated from their study on young children s partitioning strategies were used in this study as a tool to analyze and interpret the participants sharing behavior as they participated in the task-based activities. The strategies provide an explicit framework or template based on the observation of partitioning strategies in children and is therefore suitable for use in this research. Charles and Nason (2000) identified twelve partitioning strategies, six that emerged from the literature and the remaining six emerged from the strategies used by the students in their study that were not reported in the literature. They categorized the twelve strategies into three areas: Partitive Quotient Construct Strategies Multiplicative Strategies Iterative Sharing Strategies. They identified the steps that are involved in the utilization of each strategy. A deliberate attempt is made to include the steps for each strategy similar to ones identified in Charles 39

53 and Nason (2000) with the aim of providing the readers of this study with the explicit tool used as the frame of reference for the interpretation of the students partitioning behavior. Table 2 shows how the partitioning strategies identified by Charles and Nason (2000) are categorized and whether they are gleaned from the research literature on fractions or not. The authors referred to these as a taxonomy for classifying young children s partitioning strategies in terms of their ability to facilitate the abstraction of the partitive quotient fraction construct from the concrete activity of partitioning objects and/or sets of objects (p. 192). Partitive Quotient Construct Strategies Strategies in this category utilize the relationship between number of people sharing and the fractional name to generate the denomination for each share (Charles & Nason, 2000, p. 199). Partitive quotient foundational strategy. This strategy involves the application of the following steps: Step 1: Step 2: Step 3: Step 4: Recognition of the number of people (n) Generation of fraction name from the number of people Recognition of relationship between fraction name and number of equal pieces in each whole object (n equal pieces) Partitioning of each whole object into equal pieces (n equal pieces) Step 5: Sharing the pieces ( 1 n of each object to each person) Step 6: Quantifying each share (addition of 1 n pieces) Proceduralised partitive quotient strategy. This is a reduced version of the strategy mentioned above. The utilization of this strategy to formulate a solution to the task results in three fewer steps than the foundational strategy. They are: Step 1: Step 2: Recognition of number of people (n) Recognition of number of analog objects (m) Step 3: Quantification of each person s share ( m n ) Partitive and quantify by part-whole notion strategy. This strategy is one of the six strategies that was not reported in the literature but emerged from the researchers observation of some of the students partitioning behavior. It is similar to the foundational strategy without Steps 2 and 3. 40

54 Step 1: Step 2: Step 5: Step 6: Recognition of the number of people (n) Partitioning of each whole object into equal pieces (n equal pieces) Sharing one piece from each object to each person Quantification of each person s share by application of part-whole system mapping to determine fraction name. Table 2: Charles and Nason s (2000) Partitioning Strategies Partitive Quotient Construct Partitive quotient foundational Proceduralised partitive quotient Partition and quantify by part-whole notion Regrouping Horizontal partitioning Multiplicative People by objects Iterative Sharing Halving object then halving again and again Half the objects between half the people Whole to each person then half remaining objects between half the people Half to each person then a quarter to each person Repeated sizing strategy Repeated halving/repeated strategy Regrouping strategy. The steps for this strategy are: Step 1: Recognition of the number of people (n) Step 2: Recognition that the number of people gives the fraction name (nths) Step 3: A realization that the fraction name gives the number of pieces in the whole (n) Step 4: A realization that the total number of nths to be shared can be generated by multiplying the objects (m) by the number of nths in each whole. 41

55 Step 5: Step 1: Step 2: Step 3: Step 4: Quantification of each share through a recognition that the number of nths for each person can be calculated by dividing the total number of nths by the number of people (m) Horizontal partitioning strategy. The steps for this strategy are: Recognition of the number of people (n) Generation of fraction name from number of people (nths) Recognition of relationship between fraction name and number of pieces in each whole object (n) Horizontal partitioning of each whole circular object into pieces (n pieces) Step 5: Quantification of each share ( 1 n piece) Step 6: Multiplicative Strategies Recognition that shares are equal. For this type of strategy, a multiplication algorithm is used to find the number of parts in each whole. Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: People by objects strategy. This strategy involves the following steps: Recognition of the number of people (n) Recognition of the number of objects (m) The number of pieces in each whole is generated by multiplying (n) by (m) Partitioning each whole into (nm) pieces Sharing the pieces between people Quantification of each share Iterative Sharing Strategies are: Step 1: Halving object then halving again and again strategy. The steps for this strategy Iterative halving of each whole object into eight equal pieces Step 2: Recognition that each piece is 1 8 Step 3: Step 4: Sharing out the pieces Quantification of each share 42

56 Half the object between half the people strategy. This strategy is slightly different from the halving strategy identified in the literature. For the purpose of their study, Charles and Nason (2000) noted that this strategy was employed for the sharing of two objects between four people. These are the steps for this strategy. Step 1: Recognition of the number of people (4) Step 2: Recognition that the number of objects (2) Step 3: Realization that halving the objects will generate enough pieces to share between all the people. Step 4: Partition of the objects into halves Step 5: Sharing the halves between all the people Step 6: Quantification of each share Whole to each person then half remaining objects between half people strategy. This strategy is an extension of the above strategy and involves the following steps: Step 1: Recognition of the number of people (n) Step 2: Recognition of the number of objects (m) Step 3: Realization that the number of objects (m) is greater than the number of people (n) Step 4: Sharing of one whole object to each person [Step 3 and 4 are repeated until m < n] Step 5: Application of half objects between half the people strategy Step 6: Quantification of each share Half to each person then a quarter to each person strategy. This strategy is also an extension of the half the objects between half the people strategy. The steps utilized in the strategy are: Step 1: Application of the half objects between half the people strategy Step 2: Realization that there are not enough remaining objects to reapply the half the objects between half the people strategy Step 3: Application of the partitive quotient foundational strategy Step 4: Quantification of each share Charles and Nason (2000) study was conducted in Eastern Australia. They reported that pizzas were hardly ever cut into equal slices in that country. This facilitated the 43

57 observation of the next two partitioning strategies. The strategies were not used in this study but are included for the complete citation of Charles and Nason (2000) twelve partitioning strategies. Repeated sizing strategy. This steps involved in this strategy are: Step 1: Partitioning of each whole object into an even number of unequal pieces Step 2: Sharing of pieces using attribute of area rather than attribute of number in an attempt to achieve equal shares. Repeated halving/repeated sizing strategy. The steps for this strategy are as follows. Step 1: Application of the halving the object again and again strategy Step 2: Application of the repeated sizing strategy The next three partitioning strategies were used in Lamon (1996) study of 346 children from grades four through eight. These strategies were used alongside Charles and Nason (2000) strategies to categorize the sixth graders partitioning acts. Preserved-pieces strategy. In sharing situations where a person should receive more than one whole of the total quantity, the students will leave the required wholes intact and then mark and partition the whole that requires cutting. Mark-all strategy. As the name of this strategy suggests, the students will mark all the pieces but at the time of sharing only the pieces that require cutting will be cut. Distribution strategy. All the pieces of the whole are marked and cut. Each piece is then distributed. Summary This study was set within a constructivist environment where notions about the teacher and students role in the learning environment, how they construct fraction knowledge as they participate in the activities and the role of the physical referents and real world representations in this construction of knowledge come together. Kieren (1976) divided the concept of a fraction into five subconstructs: the part-whole, quotient, measure, operator and ratio subconstruct. Herscovics and Bergeron s (1988) model of understanding and the partitioning strategies identified by Charles and Nason (2000) and Lamon (1996) were explicitly discussed as they provided the theoretical frame of reference in investigating the students method of determining fraction size, 44

58 partitioning and repartitioning the whole, and generating equivalent fractions. These partitioning strategies are definitive and are well suited for the purpose of the present study. 45

59 CHAPTER 3 METHODOLOGY Qualitative Interpretive Framework Current trends in research in mathematics education reflect a paradigm shift from an emphasis on scientific or quantitative studies to the use of qualitative, interpretative methodologies (Teppo, 1998). Qualitative research is primarily concerned with human understanding, interpretations and intersubjectivity and is a field of inquiry in its own right (Denzin & Lincoln, 2003, p. 3). This type of research necessitates the mingling of the researcher with the everyday life of the setting for the study where the researcher would enter the participants world. Through the ongoing interactions with the participants, the researcher sought to obtain a clear understanding of the participants perspectives on a particular phenomenon. The research questions posed in this study are based largely on students understanding of fractions as they engage in certain processoriented activities, thus necessitating a qualitative study. The methods used in this qualitative study are better able to provide a deeper understanding of social phenomena than could be obtained from purely quantitative data (Silverman, 2001). One such social phenomenon is the act of fair sharing. Besides highlighting the technical aspects of this study, this chapter also includes the limitations and assumptions associated in conducting the type of research design mentioned above. Participants The participants involved in this study come from a private middle school situated in a metropolitan area of southeast Florida where the teacher-researcher currently works. The school serves a predominantly Hispanic community with a number of students from West Indian origin. The participants for this study were members of the sixth grade class during the school year the study was conducted. The study took place between the 46

60 months of February and April, The researcher focused on sixth grade students for various reasons. Many students at this level exhibit insufficient understanding of fractions despite prior instruction in lower grades. This fact is based on the teacherresearcher s years of experience in teaching sixth grade, comments from other sixthgrade mathematics teachers, and reports from literature (e.g. Lamon, 1993; Mack, 1995) dealing with this vital middle school topic. Another reason for choosing this group was the minimal attention and instruction that are given to fractions in grades higher than sixth, thus making it a priority for teachers in sixth grade to seek an in-depth understanding of how students at this level make sense of the fraction concept with the aim of providing meaningful engaging activities that will promote relational understanding of the concept. Two colleagues who have experiences in teaching middle school mathematics, served in the role of teacher-observer. These teacher-observers scored the pretest and made the different group assignment. One of them regularly attended and observed the activity sessions. Informal conversations were held with the teacher-observers that help the teacher-researcher in planning review sessions and interpreting the findings of the study. The sixth grade class involved in the study consisted of 10 males and 10 females ranging from ages 10 to 12. The results of the pretest that was administered prior to the teaching sequence were used to categorize the students on a scale from 1 to 3 with 1 representing very little understanding of the fraction concept and 3 representing a basic understanding of fractions. For the purpose of the cooperative activities, the students were divided into five heterogeneous groups with at least four students in each group. The members of the group were selected randomly by the teacher-observers based on the categories defined by the pretest results. At least one member from each category was in each group. Appendix C shows the students by category the different group assignments. The teacher-researcher intentionally chose to involve the whole class in this undertaking since studies (e.g. Boulet, 1998; Mack, 1990) have shown that research done in a natural classroom setting yields more accurate data which are of more educational value. Written permission was granted to the teacher-researcher to pursue this study from the administration of the school. Permission forms and letters were sent to the sixth- 47

61 grade parents explaining the purpose and nature of the study and were returned prior to the beginning of the study. The teacher-researcher met with individual parents who had questions concerning their child s participation in the research. Their main concern was what they needed to do to aid in the study. The tasks and process-oriented activities that were used in this study did not yield a grade for the students. They were used for the sole purpose of seeking to understand how students make sense of fractions. A pseudonym was given to each student to aid in the non-recognition of the participants involved in the study. Research Design The research design involved three phases that lasted for a period of eight weeks. The phases comprised the pretest, teaching sequence packed with task-based activities, and selected interviews. Classroom based research has become more accepted in the mathematics education arena due to criticisms of the limited applicability of the findings of studies conducted in out-of-school laboratory controlled environments (Confrey & Lachance, 2000). Teachers fear that these findings have very little use in the classroom and lack the vigor and practicality of the everyday classroom, that is, they are theoreticalbased and not classroom-based. As Steffe, Thompson and von Glasersfeld (2000) wrote a large chasm existed between the practice of research and the practice of teaching (p. 271). Conducting a whole-class research study within the confines of the researcher s class allows the researcher to experience, firsthand, students mathematical learning and reasoning about fractions. In this section, each phase will be described and the rationale for including these phases in the study will be established. No instruction on fractions was given in sixth grade prior to the study. Pretest At the beginning of the study a pretest consisting of seven multiple choice and 18 free-response items was administered to the 20 participants. A copy of the pretest is included as Appendix A. One of the main purposes of this test was to obtain a detailed diagnostic skill profile for each learner as it related to the students current understanding of fractions. Some of the items from this test were also used to verify the intuitive understanding of fractions as detailed in the Herscovics and Bergeron s (1988) model of understanding and to obtain the students definition of fractions prior to the 48

62 administration of the teaching sequence. The first seven items (multiple-choice) of the test are NAEP and TIMSS sample test questions. The Free-Response items are gleaned from research related to fractions (Boulet, 1993; Mack, 1990) with the last seven questions coming from the written test databank of the Rational Number Project [RNP] (Cramer, Behr, Post & Lesh, 1997). To ensure the feasibility and fairness of the test items for the sixth grade level, the pretest was administered to two sets of sixth-graders from two schools two private institutions with the same affiliating body as the school in which the study was conducted and with similar demographics, with the exception that one of the schools had a predominantly West Indian population. The results of the tests and suggestions from the volunteer teachers aided the teacher-researcher in the final draft of the pretest items. Teaching Sequence The teaching sequence consisted of 12 task-based activities (see Table 2). The students performed two types of tasks during the course of the study. Some tasks were individually done while others were completed in cooperative groups. The activities with the relevant tasks were designed and arranged particularly for this study. The fraction tasks were adapted from Boulet s (1993) study on the unit fraction with the researcher modeling them to suit the sixth grade audience. An assessment task was given at the end of most of the activities and served the purpose of verifying or determining whether the students comprehended the concepts that were highlighted in the tasks for that particular activity. Some of the assessment and activity tasks were from RNP, while others, including the interview protocols, emerged from other fraction-related literature. The tasks and interview protocols are found in Appendix B. The final activity of the teaching sequence was centered on a Fraction Breakfast. The idea for conducting an actual breakfast emerged from a similar task Cake Problem used in Poither and Sawada s (1983) study. Unlike the Cake Problem, the Fraction Breakfast activity used real food that the students were allowed to eat at the end of the partitioning tasks. The questions given on the activity sheet were fashioned from similar questions used by Behr and Post (1992), Lamon (1993), Mack (1995) and Moss and Case (1999) in their quest to obtain students meaning of fractions and its operations. Each breakfast table was set with the utensils for eating and sharing 49

63 including gloves to be used while partitioning the food. Items for the breakfast were distributed to each group. Each group had the same food items in the same quantity. They were required to share the food equally among the group members and then answer the questions on the activity sheet that accompanied the breakfast. One member of each group was chosen to initiate the discussion, but each person was required to write on the individual task sheet. The Fraction Breakfast activity had been field tested and had proven to be successful in facilitating children s partitioning capabilities and techniques. This task also provided key insights into the partitioning process. Table 3 gives an outline of the actual timeline for the study. The order of the sessions was determined from the field tests that the researcher conducted with the previous set of sixth graders at the same school as the present study. It should be noted that there was a review session of the previous class topic at the beginning of every session. This lasted for approximately 10 minutes and was in the form of an oral exercise on the previous tasks. The middle school where this study was conducted operated on a block schedule with mathematics occurring for sixth-grade three times for a Block B week and two times for a Block A week. The eight week period resulted in 20 sessions. Each session lasted for 60 minutes except the Fraction Breakfast which lasted for one and a half hours. In addition to the task-based activities that formed the major component of the teaching sequence employed in this whole class exploratory study, other aspects of the sequence need to be considered. These include the teaching agent (the researcher), the group of participants (sixth grade) that were dealt with earlier in the chapter, the learning space (classroom setting), witnesses of the teaching sequence (peer mathematics teachers), and the various methods of data collection that were used to answer the research questions posed in Chapter 1. Initial conjectures about the teaching sequence were based on the research literature, however, modifications were made daily based on the ongoing analysis of the students activity (Ellerbruch & Payne, 1978; McClain, 2002). The activities done during the teaching sequence were adapted and designed by the researcher in an attempt to provide opportunities for the child to learn by judiciously selecting tasks, offering suggestions, and posing questions (Cobb, Wood & Yackel, 50

64 1990). The teaching sessions were flexible and were susceptible to the students grasp of the concept in the subsequent sessions. Table 3: Outline of the Teaching Sequence Types of Understanding/Session Logico-Physical a. Intuitive Pretest Tasks b. Logico-physical procedural ACTIVITY 1 (Partitioning the whole) 1, 2 & 3 c. Logico-physical Abstraction ACTIVITY 2 (Reconstitution of the whole from its parts) 4 i. Partitioning the continuous whole in odd and even parts: circle into 3 parts; triangle into 6 parts; rectangle into 8;parts; parallelogram into 4 parts; L into 5 parts ii. Partitioning discrete whole (12 chips per child) 4, 6 3 and 5 equal shares iii. Partitioning a 4 line (measure) using a ruler 2, 5, 8 equal parts iv. Liquid partitioning (measuring 1 cup) 1 per child 2, 3 and 5 equal shares v. Verification Task i. Making a whole out of given parts (12 circular parts) ii. Given 1-fourth of circular whole to predict the missing parts to complete the whole (One part of the whole) iii. Given 5-eighths of a triangle to predict the missing parts to complete the whole (Many parts of the whole) iv. Given 1-fifth of a circle to predict the missing parts to complete the whole (One part of the whole) v. Given 8-twelfths of a square to predict the missing parts to complete the whole (One part of the whole) vi. Give each child 1-eighth (2 oz) of a cup and then predict the missing parts to complete the whole (1 cup) vii. Verification Task 51

65 Table 3: Continued ACTIVITY 3 (Equivalence of the part-whole relationship in spite of variation in physical attributes transformation of the whole) 5 & 6 ACTIVITY 4 (Repartitioning already partitioned whole) 7 Logico-Mathematical a. Logico-Mathematical Procedural ACTIVITY 5 (Quantification of part-whole relationships in unit and non-unit fractions) 8 & 9 i. Given four different sizes of rectangles with a third shaded in each ii. Given two different sizes of circle with a half shaded in each iii. Given 1-fourth shaded in different shapes and sizes: circle, triangle, square, rectangle, L, star iv. Given four identical squares with differently partitioned halves v. Finding equivalent part-whole relationship in discrete whole (circles) with differing amount in each set (10 chips with 4 shaded and five with 2 shaded) and different shapes (diamonds 10 with 4 shaded) vi. Transforming a quarter of a square (continuous) vii. Transforming shaded parts in discrete whole 8 chips with 2 shaded - transform the set viii. Verification Task i. Partition a circle divided into half into 4,8,16,32 equal parts and a rectangle already partitioned in 3 into 6, 12 equal parts ii. Partition a rectangle and a circle already partitioned in 3 into 4 equal parts iii. Repartition a circle divided into 8 equal shares for 4 persons, a circle divided into 5 to repartition into 3, a rectangle divided into 7 parts for 3 persons, a rectangle divided into 8 to share with two and the letter L divided into 7 to share with 4. iv. Verification Task i. Naming unit fractions in continuous wholes of various shapes: circles, squares, rectangles, parallelograms; star and others and unit fractions in discrete wholes ii. Orally quantify part-whole relationships other than unit fractions. iii. Naming unit fractions in continuous and discrete wholes from part-whole relationships such as 2- fourths 3-sixths iv. Give concrete illustrations of given unit fraction vi. Verification Task 52

66 Table 3: Continued b. Logico-Mathematical Abstraction & Formalization ACTIVITY 6 (Writing conventional symbols for unit & non-unit fractions- Formalization) 10 [Spring Break] ACTIVITY 7 (Mathematical reconstitution of wholes from unit fractions the whole is made up of its parts) 11 & 12 ACTIVITY 8 (Relative size of unit and nonfractions) 13 & 14 i. Use paper folding to model and name unit and unit fractions ii. Writing symbols for unit and non-unit fractions in continuous wholes iii. Writing symbols for unit and non-unit fractions in discrete wholes iv. Verification Task i. Orally determine how many of the unit fractions are needed to make a continuous whole use different shapes and different sizes ii. Orally determine how many of the unit fractions are needed to a make a discrete whole iii. Determine how many of the symbolic unit fractions are needed to make a whole iv. Reconstitution of a part from symbolically given unit fractions v. Verification Task i. Orally determine by comparison the size of unit and non-unit fractions using models and stories ii. Determine size of unit and non-unit fractions symbolically iii. Verification Task ACTIVITY 9 (Quantifying equivalent fractions) 15 & 16 i. Finding equivalent fractions of one-half using models and symbols ii. Symbolically represent equivalent fractions RNP Lesson 10 Student Page A, and Lesson 15 Student Page B. iii. Orally and symbolically find equivalent fractions of unfamiliar unit and non-unit fractions iv. Verification Task 53

67 Table 3: Continued ACTIVITY 10 (Ordering proper fractions) 17 & 18 i. Ordering fractions by comparing them to half [Rational Number Project level 2 lesson 8] ii. Ordering fractions by comparing them to 0, ½, 1 iii. Verification Task ACTIVITY 11 (Problem Solving Tasks- Examining partitioning strategies) 19 i. The Cake Problem ii. Problem solving tasks from the Rational Number Project ACTIVITY 12 Fraction Breakfast 20 Fraction Breakfast where the students will use the knowledge gleaned throughout the study to share food items equally amongst themselves. A range of instructional approaches were used during the study. Small group discussion, whole class discussion, teacher interviewing and the use of hands-on activities formed an active part of the daily classroom experience. The learning space set up for this study did not only include the actual physical setting of the classroom. It also encompassed the setting where students were active in the fraction environments. In this environment they engaged in tasks, talked and wrote about what they know as they attempted to reason about their actions and their thinking. The learning space was open in nature thus resulting in a number of responses that provided variety in the understanding of the students actions. To foster a healthy creation of fraction learning space, it was important for the teacher-researcher to understand the different subconstructs of fractions as mentioned in Chapter 2. With this understanding, the teacher was able to design fraction tasks and activities that allow the students to build and effectively use fractions in a variety of settings. Consequently, the students were able to reflect on the various uses to which fractional numbers can be applied (Kieren, 1995). In conjunction with the constructivist epistemological perspective for this study, the space for learning fractions should deliberately foster the students construction of 54

68 their own meaning for fractional numbers and operations on them (Kieren, 1995, p. 45). It should provide ways for the students, if necessary, to change from a more to less sophisticated activity, to extend their knowledge from working with the physical models to symbolic notation and for students at different levels of understanding to function together (Kieren, 1995). The teacher-researcher, therefore, made deliberate effort to model the learning environment to be compatible to the suggestions given by Kieren (1995). In this learning space the participants were allowed to express their fraction number ideas using physical referents which included models (continuous and discrete), diagrams, pictures and charts. The role of the researcher (teaching agent) in this study was twofold. As the researcher, the students mathematical behavior was observed documented, analyzed and interpreted as they participated in the different activities. The students critical thoughts on the fraction tasks presented to them were probed and recorded. As the teacher, fraction concepts were presented using the theoretical foundations set by Steffe and D Ambrosio (1995) hypothetical learning trajectory and Lesh s model of translation (Lesh, Post & Behr, 1987). Students interactions among themselves and with the teacher were facilitated and encouraged. The focus of this study was not on the method of instruction but on the students as they performed the tasks given them. With that in mind, no formal lecturing was given. The teacher, instead, moved around, visiting groups and individuals conducting spontaneous on-the-spot interviews and asking questions that encouraged the students to think critically. Interviews The initial plan was to conduct two sets of interviews during the study. The first set of interviews was aimed at the tasks that were done individually. The second set of interviews would be geared towards the group activities including Activity 12. As the study unfolded, it became possible for the researcher to visit the groups and question the students while they were on task, thus gleaning valuable information on their thought processes. Two sets of individual interviews were done with students who experienced difficulty while working on particular tasks. These interviews lasted for approximately 35 minutes and were conducted during the student s afternoon elective sessions. The interview sessions were held in their classroom with the researcher serving as the 55

69 interviewer. Information gathered from the interviews provided further inferences about the students cognitive processes while sharing and constructing fractions. De Groot (2002) believed that, Students talk about their experiences of learning in unstructured, in-depth individual interviews (p. 42). Due to time restraints no individual interviews were conducted at the end of the study. Data Collection Various methods were used for collecting data for the duration of this study. Besides the observation of the students mathematical behavior, each session and interview were video- and audio-taped as the students verbal and nonverbal behavior constituted a major part of the database. Students written work from the classroom activities and tasks were collected, reviewed and analyzed continuously by the researcher after each session. Audio-tapes and transcripts of the teacher-researcher s reflection were recorded and transcribed after every session and played a pivotal role in the planning of the next activity. The conversations between the teacher-observer who attended the sessions and the teacher-researcher were also documented and used as an aid in validating the interpretations of the findings of the study. Table 4 shows the method(s) of data collection used to gather information that was pertinent in answering each of the research questions. The audio column represents the taped individual interviews, and whole class and group discussions done during the study. The test represents the pretest that was administered at the beginning of the study. Table 4: Research Questions and the Methods of Data Collection Methods Research Questions Observations/Journals Audio Video Test Tasks Sheets Researcher Teacher-Observer

70 Data Analysis Data analysis involves organizing what you have seen, heard, and read so that you can make sense of what you have learned (Glesne, 1999, p. 130). During this process you discover themes and concepts embedded throughout the interviews and discussions (Rubin & Rubin, 1995). As mentioned earlier, data for the study consisted of the students verbal and written responses to the task-based activities and the participants responses to the semi-structured interviews and any other relatable responses from the researcher s reflection and teacher-observers. The analyses of the information gathered during the teaching sequence were done in three phases. (1) The students (individual or as a group) responses whether verbal or written were continuously analyzed during and immediately after each session. The teacherobserver s observation played an important role at the end of sessions that she attended. (2) There were daily analyses of the entire class responses coupled with the teacherresearcher s observation. (3) At the conclusion of the study the analyses that were done in (1) and (2) were combined with the final analysis to create a rich, descriptive narrative of authentic insight of the sixth graders understanding of fractions. Pertinent solution behavior patterns were noted and responses to interview questions were transcribed. Careful notes were taken of the mathematical language the students employed during the discussions of the taskbased activities. The following steps were used in the analysis of the data: (a) coding of each group s discussion, interviews and task sheet; (b) grouping excerpts per categories revealed from the coding; and (c) organizing the categories in a coherent manner, following an interpretive, narrative approach. The major aim of the data analysis was to seek an in-depth understanding of the participants understanding of the fraction concept as they worked with the task-based activities individually and within groups. Assumptions and Limitations One of the characteristics of qualitative research is that the researcher is the primary instrument for data collection and analysis. This can result in certain limitations and biases that will impact on the study. Instead of trying to eliminate these shortcomings and biases, this study seeks to identify them and monitor them as to how they may be shaping the collection and interpretation of the data (Merriam, 2002). The 57

71 limitations will be discussed under the following sections: research methodology, the active involvement of the researcher as the main teacher, and the sample. The assumptions are embedded in the discussion for each section. Research Methodology Due to the nature of the research questions, a qualitative approach was used as the lens through which the data acquired from this study were viewed and interpreted. A summative description of this research methodology is given at the beginning of this chapter. In this section, the major issues concerning this type of research methodology are highlighted. This includes the steps that were taken to check for accuracy and credibility of the results of the study. One of the major setbacks for using this design is the need to validate the findings of the study. Creswell (2003) explained: Validating does not carry the same connotations as it does in quantitative research, nor is it a companion of reliability or generalizability (p. 195). Instead, he saw validity as one of the strengths of qualitative research that should be used to determine whether the results harvested from the data are accurate from the viewpoint of the researcher, the participant or the readers to which the study is targeted. Hammersley (1990) defined validity as truth interpreted as the extent to which an account accurately represents the social phenomena to which it refers (p. 57). This issue of validity in qualitative research is a hotly debated topic (Lincoln & Guba, 2000; Silverman, 2001). Suffice it to say, Creswell (2003) offered a number of synonyms for validity. These include trustworthiness, authenticity and credibility. Addressing the issue of generalizability in qualitative studies, Stake (2003) wrote, Damage occurs when the commitment to generalize or to theorize runs so strong that the researcher s attention is drawn away from features important for understanding the case itself (p. 141). A major strategy to ensure for generalizability in this qualitative research is the provision of narrative with rich, thick description. Enough information including excerpts from students interaction with the researcher and each other are included in the research so that the readers are able to determine how closely their situations match, and therefore determine whether the interpretations of the study can be transferred to their own situation. As the researcher, I aspired to use the data collected in this study to produce qualitatively valid and reliable information in an ethical manner that provided 58

72 the groundwork for the development of further study in the understanding of students fractional scheme. One the most significant limitations of this study is that it involves inferences about the learners observed behavior. Although the twelve activities done during the teaching sequence were audio- and video-taped, with such a large number of students working at the same time, there existed moments when clear cut audible discussions were not possible and reasonable inferences had to be made from the visual data. Since the inferences drawn were based on the teacher-researcher s background and experience with these students, the study is limited in objectivity. Being a mathematics teacher for over eighteen years has possibly shaped the way the teacher-researcher viewed and interpreted the data thus devaluing the validity of the study. Every effort was made, however, to be objective and a number of strategies were employed to ensure the validity of the study. Besides using rich, thick description to convey the findings, there was a triangulation of the different sources of data with the aim of building a coherent and justifiable interpretation of the results. The different methods of data collection were compared to see whether they corroborate. The interpretation of the findings of this study was also subjected to the reviews of the teacher-observers who helped to determine whether the findings are plausible based on the data (Merriam, 2002). Related literature also played a significant role in supporting the interpretation of the data. Teacher-Researcher In defending researchers acting as teachers, Cobb and Steffe (1983) believed that the activity of exploring children s construction of mathematical knowledge must involve teaching (p. 83). Among the reasons given to substantiate the position of teacher-researcher is the idea that teachers are able to form close relationships with the students thereby helping them to reconstruct the contexts within which they learn mathematics. Other advantages include a) the teacher has access to the detail of the class that others would not; b) she may understand the students talk and mannerisms in ways that an outside researcher would not, and c) she knows the history of shared examples, problems, and discussions that she can use to probe the children s ideas, be better able to frame the tasks and interpret the students problem solving activities. This study did not attempt to examine the students construction of fractions due to the instructional 59

73 processes. Instead, the teaching sequence provided the setting or learning space for fractions where the participants could construct their knowledge of fraction as they interacted with their peers and teacher while performing the assigned task-based activities. This first-person perspective, often refered to as backyard research, has its downside. The data collected can be biased, incomplete, or compromised (Creswell, 2003). The teacher-researcher commitment to helping her students can impede the capacity to see and hear the students problems and difficulties. Aptly stated by Ball (2000, pp. 389, 390) the teacher may not be able to notice the subtle ways in which her manner affects particular children and, hence, may not be able to probe their responses. To minimize these discrepancies, Ball (2000) included three crucial questions that a person who wishes to undertake the role as both teacher and researcher should ponder. The researcher will provide the answers to these questions, thus providing the rationale for conducting this study and interpret the data collected during the study. Question 1(a). Does the researcher have an image of a kind of teaching, an approach to curriculum, or a type of classroom that is not out there to be studied? Given the complexity of the teaching process and the unique learning environment of every classroom, this qualitative study does not claim to be projecting or highlighting any teaching approach or pedagogical skill on the part of the teacher-researcher. Instead, the researcher was simply interested in what goes on in the mind of her students as they engaged in meaningful activities that are relevant to their everyday lives and as they made sense of fractions while working with physical models. Teachers are always encouraged to understand what students know and need to learn and then challenging and supporting them to learn it well (NCTM, 2000, p. 11). Question 1(b). Given that there is a need for a first-person research, is the researcher well-equipped to be the designer and developer for this study? The researcher has taken several courses that dealt with conducting qualitative studies. She has also conducted small-scale studies that have been reviewed by professors who have experiences in qualitative research. In addition, the teacher-researcher has been engaged in an extensive review of related literature to cull tasks, activities and designs that will prove beneficial and significant for this study. 60

74 Question 2. Is the phenomenon to be studied only uniquely accessible from the inside? Seeking to understand how students make sense of fractions is not uniquely accessible from the inside. However, the researcher believes that a natural setting environment will add a fresh view on this cannot-be-exhausted topic. Question 3(a). Is the phenomenon one in which other scholars have an interest? Numerous studies have focused on the fraction concept. This subset of rational numbers, along with its counterparts such as ratio, has been hotly discussed and has also been the main focus of the 2002 NCTM yearbook. Question 3(b). Will conducting the study within the teacher s natural environment offer perspectives crucial to a larger discourse? Besides adding to the reservoir of fraction studies done over the decades, this study can possibly trigger a discussion of the advantages of doing similar research within the teacher-researcher s natural environment. Sampling The participants in this study do not represent a randomly selected group as the teacher-researcher is working with her own group of students. This research is not testing a well-defined theory, but seeks to add to the knowledge base about how students conceptualize fractions as they work with manipulatives and participate in processoriented activities that represent real life events, thus the nature of the sample is deemed appropriate. One of the underlying assumptions was that the students would perform to the best of their ability as no grades were assigned to the tasks and activities used in the study and that the tasks the students did elicited information that aided in answering the research questions. With the exception of one student (Ben) who was sometimes lowkeyed, the participants were enthusiastic about the activities and possessed the same energy level throughout the eight weeks. The students could express themselves verbally through written and non-written responses using the English language. A simple survey of the students involved in this study showed that they were all born in the United States of America and spoke English fluently. Only a few misspellings prevailed throughout the written tasks that were clearly understood by the researcher. The written tasks always corroborated with the video- and audio-taped recordings. 61

75 Summary The chapter began with a brief description of the type of study that will be pursued by the researcher. All research design that includes the qualitative design, calls for a clear outline of the sample, methods of data collection and how the data will be analyzed. This whole-class research project included a teaching sequence, a pretest, and task-based activities. All these formed the network through which the students understandings of fractions were scrutinized and interpreted. Despite the limitations mentioned above regarding this study, there is sufficient reason for the use of this type of study and setting to investigate students understanding of fractions. The teacher acting as the researcher had first-hand knowledge of the students ability to do the work thus providing the opportunity for her to provide tasks that were suitable for the different levels of understanding in the classroom. 62

76 CHAPTER 4 RESULTS - THE NATURE OF STUDENTS UNDERSTANDING OF FRACTIONS Introduction The discussion of the results of this study is organized around the research questions. Chapter 4 is solely dedicated to the answering of research question number one and the remaining two research questions will be dealt with in Chapter 5. Being a whole-class study, thousands of statements generated during the teaching sequence, and tens of written tasks were examined to distill patterns that were relevant to the nature of the study. The participants written and oral behaviors were observed for commonalities in the way they approached, worked with, understood and talked about fractions. The first ten task-based activities done during the teaching sequence formed the core of the coding for the levels and constituent parts of the Herscovics and Bergeron s (1988) model of understanding and were designed to deduce the nature of the students understanding of fractions thus serving the purpose of answering the first research question. The remaining activities were designed to investigate the participants partitioning strategies and the way they solved problems with real world applications. At the end of each activity, an individual and/or group assessment task was given to determine the level of each student s understanding or misunderstanding of the concepts introduced during the activity. An outline of the activities done during the teaching sequence is shown in Table 3 in the previous chapter. An analysis of the activity and its accompanied assessment task was done instantaneously with the aim to inform the subsequent session. As Merriam (1998) aptly put it: Data collection and analysis is a simultaneous activity in qualitative research. Analysis begins with the first 63

77 interview, the first observation, the first document read. Emerging insights, hunches, and tentative hypotheses direct the next phase of data collection (p. 151). This study was done with a set of six-graders who attended a private metropolitan school in southeast Florida. The class was comprised of 10 males and 10 females. The sequence of activities was done during the last two quarters of the school year. At the beginning of the study each student was given a transparent bag containing a ruler, flexible wires, eraser, two sharpened pencils and a set of circular red and yellow chips The students were at liberty to use any of manipulatives in their bag and those assigned during the specific activities to help them with the assigned tasks. Research Question #1: What is the nature of sixth grade students understanding of fractions? The various task-based activities were designed and adopted from the research literature to glean an in-depth view of the nature of the participants understanding of fractions. As mentioned in Chapter 1, Herscovics and Bergeron s (1988) model of understanding was used to analyze the sixth graders responses to the fraction tasks that were presented to them during the teaching sequence. These tasks were designed to solicit the participants definition of a fraction, their process of unitizing and partitioning, their sense-making of fraction symbols, and the order and equivalence of fractions. Each level and constituent part of Herscovics and Bergeron s (1988) model will be highlighted in the effort to answer the first research question. Items from the pretest (Appendix A) and the Fraction Breakfast task sheet (Appendix B) were used to verify the participants definition of a fraction and the first level of Herscovics and Bergeron s (1988) model of understanding intuitive understanding. An interpretative analysis of the data will conclude each section. Definition of the Fraction Prior to scrutinizing some of the pretest items for verification of the participants intuitive knowledge of fractions, an analysis of the students definition of a fraction was done with the purpose of revealing their perception of what constituted this mathematical concept. These students had previously completed two years of formal fraction instruction and thus it was the researcher s intent to gain insight into the students sense 64

78 making of the fraction before they began formal work with the concept in sixth grade. This sense making is deeply entrenched in the meaning they assigned to the fraction. Four questions (1, 5, 6 & 15 Free Response Section) from the pretest were used as the lens through which the students definition of a fraction was verified. These four questions were chosen on the merit that they could solicit from the students any of the major subconstructs of a fraction as outlined by Kieren (1976). Question 1 directly asked the participants to give their verbal definition of a fraction and to make different representations of a fraction of their choice. Question 5 gave a fraction and then similar to Question 1, asked the students to generate two different representations of this fraction. Question 6 hinged on the fact that the sum of the equal parts make up the whole which is an essential point in the understanding of the fraction, and Question 15 investigated the students knowledge of the role equal parts play in the naming of a fraction. A detailed summary of the responses to each question will be done, followed by an analysis in reference to the students meaning of a fraction. The first question required the students to tell a friend, who did not know, what a fraction was and to illustrate a fraction in more than one ways. 45% (or 9 out of 20) of the participants made no attempt to answer any part of the question. Here are some of the responses. The number beside the name represents the category that the student was placed in by the teacher-observers after the completion of the pretest. The results of the pre-test that was administered prior to the teaching sequence were used to categorize the students on a scale from 1 to 3 with 1 representing very little understanding of the fraction concept and 3 representing a basic understanding of the fraction. Each student was assigned a pseudonym. Carol was the only student placed in Category 1, seven students were placed in Category 3 while the remaining twelve students were placed in Category 2. Appendix C shows the students by category. Ashley (2): Clauia (1): I would tell them sorry cause I don t know it either. Let s say we bought 4 slices of pizzas, if I took 2 we would have a fraction of two pieces left. Polly (3): Mary (3): A fraction shows what part of a whole has been taken or shaded. A way to divide things. 65

79 Marla (3): A fraction is when something is divided into 1 or more parts and then a certain amount is taken away from it to show what is left. Dave (3): Fraction is when you divide things into even pieces. Bob (3): A fraction is a piece of something. Two students Richy (2) and Paul (3) were able to produce a diagram and a symbolic notation of a fraction but did not give a word definition for a fraction. All the participants who attempted to illustrate the fraction using diagrams used a continuous 1 2 model (square or rectangle). Jay (3) used the fractions and as two ways of 2 4 representing a fraction. Question 5 What does the fraction five-eighths look like? Using a diagram, show this fraction in two different ways. produced some rather noteworthy responses. A summary of the responses is shown in Table 5. Table 5: Summary of Responses for Question 5 on Pretest RESPONSE Number of Students (Total = 20) 5 2 Fraction Symbol only 8 Diagram and Fraction Symbol 8 Circle and Rectangle 5 Circle and Triangle 1 Rectangle and Discrete Chips 1 2 Sets of Rectangles 2 No Response 1 Ashley showed three representations of five-eighths. She used the unshaded parts to indicate the numerator. SYMBOL CONTINUOUS MODEL DISCRETE MODEL

80 Question 6 asked the students how many fourths are in a whole? They were required to produce a diagram to illustrate this. 35% of the participants did not respond to the question. The remaining participants correctly answered 4 and used either a circle or a rectangle divided into four parts as the illustration. In Question 15, the students were asked to determine the fraction that was represented by c in Figure 7 below. 40% of the students did not respond to this question. 1 Two students (Bob and Chrissy [3]) gave the correct answer. Five students (Carol [2], 6 Alton [2], Jay, Dave and Kisha [2]) named the fraction as 5 1 with three students (Josh [3], Paul and Richy) writing. Mary wrote 2 1 while Marla wrote Figure 7. Diagram for Question 15 Pretest As the students participated in 12 task-based fraction activities their formal understanding of unit and non-unit fractions should have strengthened. No post-test or follow-up activity was given but Item 7 on the Fraction Breakfast task-sheet asked a similar question to Question 1 on the pretest free response section. It should be noted that none of the previous activities solicited or gave a formal definition of a fraction. Here are some examples of the students emerging definitions by the end of the study: Ashley: A piece of a whole. Josh: Dividing things into equal parts. Dahlia: A piece or pieces that come from dividing into partitions. 67

81 Alton: Chrissy: Carol: My definition of a fraction is the equal shaded parts of a whole. A fraction is a part of an equal partitioned square (doesn t have to be a square) the part that is shaded. A fraction is a group of numbers that helps you to divide and multiply equally. Claudia: A fraction is something that we use everyday. All the students had a definition with the majority of them hinting towards the part-whole subconstruct with one student (Carol) connecting the concept of equivalence in her definition. Analysis Fundamental to the meaning of a fraction is the fact that the parts that constitute the whole must be equal in area. This is also an important component in intuitive understanding of fractions. From the responses for Question 15, it became apparent that the majority of the students (with at least two years of formal fraction instruction) did not consider that the parts represented by a, b, c, d, and e were not all equal. Bob, Richy, Paul, Chrissy, and Josh recognized the need for equality of the parts but unlike Bob and Chrissy who ignored the previous divisions and partitioned into six equal parts, Richy, Paul and Josh chose to ignore half of the whole and named their fractions using the three equal parts. Marla s problem was simply that of over counting while Mary had no idea how she came up with that answer she just guessed. Dave alluded to the equality of parts in his written definition of a fraction when he referred to even pieces. Careful scrutiny of the results for the four questions showed that the participants in this study had very little or no understanding of the meaning of a fraction except for the symbolic representation of a fraction. Mary, Marla and Dave s definitions hinted towards the quotient subconstruct. Interestingly, with the exception of Polly, none of the participants used the part out of a whole definition that is prevalently used in textbooks and often used by mathematics teachers (Kieren, 1993). Deliberate efforts were made not to influence the students towards any definition of a fraction with the hope that their meaning of the fraction would continue to emerge as they participated in the meaningful, task-based activities. As previously mentioned, by the end of the study all the students 68

82 had a personal definition of a fraction, which, incidentally made reference to the partwhole subconstruct. In making reference to the numbers constructing the fractions, the students mostly refered to the top number instead of numerator and bottom number instead of denominator. When illustrating a fraction, the participants mostly used the common continuous geometric regions (circles, squares or rectangles) with little use of other shapes or discrete models. One participant (Ashley) used a discrete model to illustrate the fraction five-eighths. Intuitive Understanding Items from the pretest were used to verify the students intuitive understanding of the fraction. As mentioned in Chapter 2, the sole criterion for this type of understanding is the awareness of the role of partitioning in part-whole relationships. There are four conditions necessary for this awareness. Each condition will be examined with the relevant question(s) followed by a table summary of the responses to the question(s) and a summative narrative of the results. Condition 1: Students should recognize the equality of the parts in a partitioned whole despite the position of the parts in the whole. Question 9: Here are some cookies: A. Suppose you eat only the dark ones. How much of the cookies would you eat? Why do think so? If the cookies are rearranged as follows: 69

83 B. How much of the cookies would you eat now if you eat the dark ones? Why do you think so? (adapted from Boulet, 1998). Table 6: Summary of Responses for Question 9 on Pretest RESPONSE Number of Students (Total = 20) 2 17 (Part A) (Part B) 8 Other (Part A) 3 Other (Part B) 3 The students had very little difficulty in recognizing the equality of the parts in the discrete whole despite the rearrangement of the cookies. Jonah (2) correctly put 8 2 for both parts of the question giving the same reason because there are 8 dots and only 2 are shaded. Alton (2): (Part A) 2 out of 8. Because you ate 2 and there are 8 cookies. (Part B) 2 out of 8. Because it s the same thing but you ate the pizzas in different ways. Chrissy: (Part A) 8 2 because 2 cookies were eaten out of 8. (Part B) 8 2 because you are still eating 2 cookies out of 8. None of the students wrote the equivalent unit fraction 4 1 to represent the amount of cookies eaten. Carol answered the first part of the question correctly but wrote the 70

84 answer to Part B as 8 1 because there is one cookie eaten out of four and when you add all the cookies together they add up to 8 cookies. Richy wrote 8 1 for Part A counting the two shaded dots as one since they are lined up under each other but then failed to count how many groups of twos were present to arrive at the denominator of 4. He wrote 2 for Part B. When questioned about his answers, he remarked that he remembered 8 learning something similar in a lower grade but had forgotten how to do it. Brian (2) and Ashley wrote 8 6 for both parts of the question misinterpreting the question thus giving the answer for the fraction of non-dark cookies that were not eaten. Brian initially wrote adjacent to the diagram but scratched it out and then wrote below Parts A and B of the 8 question. Ashley s misinterpretation was consistent with her response to Question 5 on the pretest. No interventions were planned for the students who did not arrive at the 2 correct solution of since the researcher was simply examining the presence of the 8 students intuitive knowledge of fractions based on the test items. The responses to this item indicated that the sixth graders exhibited the ability to recognize the equality of the parts in a partitioned discrete whole despite the position of the parts in the whole. Tasks involving continuous wholes were used during the teaching sequence where all the students also displayed much ease at recognizing the equal parts of the whole even though the parts were transformed (Activity 3 Appendix B). Condition 2: Students should recognize the inequality of the parts in similar wholes regardless of the equal number of parts shaded or unshaded. Two items were used to verify this condition. 71

85 Question 1: Which shows 4 3 of the picture shaded? (NAEP Sample Item) A. B. C. D. Table 7: Summary of Responses to Question 2 (Multiple Choice) on the Pretest RESPONSE Number of Students (Total = 20) C 17 C and D 3 This item was chosen to examine the second condition despite the fact that choice A is not a similar whole to the other three choices and choice B is not divided into four parts. None of the students attempted to chose this option (see Table 7). Polly originally chose C and D as her response with words main answer pointing to C. She eventually eliminated D leaving C as her only choice. The three students (Carol, Dahlia and Jonah) who chose C and D were placed in category 2 indicating that they had difficulty with some of the basic concepts embedded in the understanding of fractions. If you recall, Carol also displayed some form of confusion in her response to Question 9. 72

86 Question 2: The circles below represent two pies of the same size one for you and one for your friend. you eat this much your friend eats this much Did you eat as much pie as your friend? Why do you think so? (Boulet, 1993) Table 8: Summary of Responses to Question 11 on the Pretest RESPONSE Number of Students (Total = 20) Yes 12 No 6 No Response 2 The responses to this question suggested that at least 70% of the participants experienced difficulty in recognizing the inequality of the parts in similar wholes regardless of the equal number of parts. Here are some of the reasons for answering Yes to the question. Kisha (2): As you can see that my friend s part is cut in different shapes and you can also see that mine is cut differently but we still ate the same amount. Chrissy: There are three pieces in each pie and both of us ate one piece. 73

87 Dahlia (2): Dave: We both ate the same amount of pie because both of the pies are half of 3 pieces of pie which means we both ate the same amount of pie. Because even though the shapes and sizes are different we both ate one piece. Ashley: Because there are 3 slices, I ate one and she ate 1. Polly: We both ate 3 1. Josh (3), Brian and Claudia gave similar responses to Polly. Two of the students who answered No offered no suggestion as to why they arrived at that conclusion with the other four students making some form of implicit reference to the shape and size of the slices. Paul: No, because the one I ate is longer and fatter than the one my friend ate. Jay: Bob: The thirds were not equal. They are unequal pieces. Mary: No, more. It s like my friend ate maybe 2 pizzas and I look like I 1 ate 3 pizzas. 2 Although Mary s response seemed to indicate a lack of understanding of the unit fraction in reference to the whole, it was obvious that she recognized that the slices are different with one slice bigger than the other. A review of the combined results of these two items reveals that at least 50% of the students did not recognize the role that equal parts played in the construction of the fraction. Polly, for example, assigned one-third to the shaded parts in both circles disregarding the fact that the parts were not equal and was initially confused with choice D as a possible option for the fraction three-quarters. Dahlia, Ashley, Carol, Brian, Jonah and Claudia were not sure of the necessity of the equality of the parts. Condition 3: Students should recognize the inequality of the parts of similar wholes that are partitioned differently. 74

88 Question 1: Which rectangle is not divided into four equal parts? (NAEP Sample Test Item) Table 9: Summary of Responses to Question 1 (Multiple Choice) on the Pretest RESPONSE Number of Students (Total = 20) D 17 Other Choices 3 Most of the students recognized the inequality of the parts of similar wholes that were partitioned differently. It is noteworthy to mention that the question never used the word fraction or had any fraction symbol. It is therefore possible that the students were able to make their choices without any recognition of the role equality of parts plays in the fraction concept. Based on the evidence shown above, however, the participants were able to recognize the inequality of the parts of similar wholes that were partitioned differently. Two students chose option B (Carol and Dahlia) while Kisha chose option A. Carol and Dahlia chose C and D as responses to Question 2 (Multiple Choice) on the pretest. As mentioned before, this could be an indication that they had little understanding of the role that the equality of parts played in the generation of a fraction. 75

89 Condition 4: Students should recognize the similarity in the part-whole relationships in spite of the inequality of the size of the parts. Question: Think carefully about the following question. Write a complete answer. Be sure to show all your work. Jose ate 2 1 of a pizza. Ella ate 2 1 of another pizza. Jose said that he ate more pizza than Ella, but Ella said they both ate the same amount. Use words and pictures to show that Jose could be right. Karla, Ben (2), Ashley, Alton, Claudia and Brian did not respond to this question. Below are some of the varied responses. Polly: Jose could be right because 2 1 is equivalent to saying half. The smaller amount of slices of pizza there are, the bigger the slices. Jose Ella Carol: Jose could have gotten a huge 2 1 of the pizza, while Ella could have gotten a medium 2 1 of the pizza. Ella s Pizza Jose s Pizza Kisha: I think that they both ate the same because the pizza can be the same size

90 Bob: Jose could be right. One pizza could be smaller. Paul: It depends on the size of the pizza. Jose s pizza Ella s pizza Dave: They both bought an 8 slice pizza. If Jose eats half and Ella eats half, they both eat 4 slices of pizza. Jay: Josh: Dahlia: Ella could have eaten a smaller pizza. Jose s pizza might have been bigger than Ella s pizza. To me, I think Jose is wrong. Ella is right. They both ate the same amount. At least 50% of the students who responded to this question made reference to the number of slices of pizzas eaten instead of referencing the size of slices. Although the exercise asked them to defend Jose s statement twelve of the respondents supported Ella s position. Chrissy wrote: Jose is wrong! Analysis The combined results shown for the four conditions indicated that at least 85% of the students (e.g. Bob, Jay, Dave and Paul) possessed an intuitive understanding of fractions. All of the students satisfied at least two or more of the conditions which indicated some form of intuitive knowledge. This intuitive knowledge could possibly be a result of previous lessons done on fractions in the lower grades prior to Grade 6. No instructions/lessons on fractions were done within the sixth grade curriculum prior to the study. 77

91 Logico-Physical Procedural Understanding Logico-physical procedural understanding is exhibited when the student can relate and use intuitive knowledge suitably. As partitioning plays a critical role in fraction understanding (Behr, Lesh, Post, & Silver, 1983), the first four activities of the teaching sequence were centered on this fundamental building block. The main criterion at this level of understanding is the learner s ability to partition a whole into a given number of equal shares. Activity 1 (see Appendix B) formed the central activity for this level of understanding and was done on an individual basis. A whole class discussion followed at the end of the task. This activity occupied three sessions where each session was geared mainly towards the students practicing their partitioning skills. The activity consisted of four tasks with each task having at least 3 subtasks. Some of the tasks were adapted from Boulet s (1993) work on the unit fraction. The researcher developed the tasks involving the number line and the cup of water. At the beginning of the activity each student was handed his/her personal bag, a folder containing the task sheets, drawn figures, a cutout L, and a cup of water with five empty plastic containers. Tasks 1, 3 and 4 allowed the students to work with continuous models (geometric regions, lines, and liquid measure) while Task 2 gave the students the opportunity to work with a discrete model. The tasks were designed to engage the students in partitioning in odd and even shares. It is noted in the literature that students seemed to experience more difficulty partitioning into odd shares (depending on the shape of the region) than even shares (Poither & Sawada, 1983). This was very evident as the sixth graders attempt to partition the circle into three equal parts (Task 1 Activity 1). At the onset, Ashley, Claudia and Alton partitioned the circle using vertical lines ( ). Claudia and Alton eventually erased their configuration and divided their circle similar to most of the students ( ) but Ashley was adamant that her way of partitioning the circle produced equal shares. I (Interviewer): Are they equal parts? Ashley: I think so. I: Why do you think so? Ashley: I measure them. I: What did you measure? 78

92 Ashley: Two inches for each part. [She measured along the diameter and divided the diameter into three equal parts.] I: What about the curved sides? Mary, who was sitting at the same table, butted in. Claudia also chimed in. Mary: Claudia: The two sides are fatter than the one in the middle so I don t think they are equal. (She pointed to the curved arcs of the circle.) Also because the middle part is not even equal. The middle is not like the two sides. I: What if your pizza represents a real pizza [pointing to Ashley s circle]. I cut the pizza just the way you have it. I give you the middle slice; I give Karla one of the outer slices and give Ben the other outer slice, who would get the most? Ashley: Ben and Karla. I: Would each of you get the same amount? Ashley: No I: So, are the 3 parts equal [pointing to Ashley s circle]? Ashley: No. In subsequent activities, when given the task to partition a circle in 3 parts, Ashley proceeded to the conventional configuration ( ). 85% of the students divided their circle in the conventional manner but more than three-quarters of them had no idea how to check for equality of the parts. When it was stressed that the three parts needed to be equal the students began to use items from their bag such as the ruler, flexible wires and even the chips to verify equality. I: [Speaking to Jay] Does your circle have 3 equal parts? Jay: I got three parts but I not sure they are 3 equal parts. I: Why aren t you sure? Jay: I don t have a protractor. I: What would you use the protractor to do? Jay: I would have to measure each angle to see if they are all equal. 79

93 The sixth grade class had previously finished a unit on geometry where the students worked with angles and protractors. They also worked with different types of polygons. I: Is there anyone else not sure of the equal parts. Alton: I m not sure. It doesn t look equal. I: What can you do to let your parts be equal? Alton: I could use a protractor. I: Did you think of protractor because of what Jay said. Alton: Marla intervened: Marla: I was thinking about the circle thing. Not a protractor, a compass! I measured the lines from one point to the next. On the teacher-researcher s invitation she proceeded to illustrate her strategy on the circle drawn on the transparency. She used the ruler to draw three radii and then draw the chords connecting the outer endpoints of the radii ( to confirm equality of parts. Chrissy: ). Chrissy used the same method I got my ruler and I measure it and put it on each line to see if it is the same and one of them was different. I: What did you do? Chrissy: I fix it. The remaining students proceeded to check their circle using Marla s strategy and made the necessary corrections. The students were then asked to divide the given equilateral triangle into six equal parts. At first Polly, Brian, Bob and Josh had difficulty in perceiving how this could be done. At the first attempt, Polly s triangle looked similar to this. She then realized that she had different shapes: two small triangles, two large triangles and two trapezoids. She sat at her desk in frustration, then started to measure each side of the triangle and realized that they had the same measurement. Marla, sitting next to her, had already begun to join the midpoints of each side of the triangle to the opposite vertex of the triangle to produce six equal parts: 80

94 Polly then copied Marla s strategy. I: How do you know that the six parts are equal? Polly: They look alike, uhm, they have the same shape. At least I know that they have one side in common [referring to half line on each side of the original triangle]. To partition her triangle in 6 parts, Carol put a point in the center (her perception) of the triangle. She then drew lines from the point to touch the sides of the triangle. She estimated where the lines should end without any idea of finding and using the midpoint of the sides of the triangle. She contended that her parts were equal because they all looked alike. She had no suggestion as to how to check for equality of parts. Josh joined the midpoints of the sides of the triangle but discovered that it only produced four equal parts:. Bob, on the other hand, sought to do his partitioning a little different from the other resulting in a diagram looking similar to this:. He recognized that his division produced seven parts instead of the required six and then proceeded to partition a new triangle similar to Marla s. Brian, Mary and Jonah did not complete their partitioning considering it an impossible task. Partitioning the rectangle into eight parts and the parallelogram into four parts proved to be an easy task for majority of the students. Various configurations for each shape were observed. The given rectangle measured 4 by 1 thus making it an easy task for the students who measured the length of the rectangle. They marked points ½ apart on both sides of the triangle and then continued to draw vertical lines to construct 8 small rectangles ( ). They were pretty sure that these rectangles were equal considering the measurements they used. Dave partitioned his rectangle into 4 equal squares and then drew the diagonals to create eight equivalent triangles ( ). Dave: I know the squares are equal so this line [pointing to the diagonal] will make two equal triangles. Ashley decided she was going to think outside the box in partitioning her triangle. She divided the rectangle lengthwise and widthwise to produce four equal rectangles. She then drew the diagonals for each rectangle to produce a rectangle looking like this lines at each turn.. She argued that the parts were equal because she drew only drew the half 81

95 Ben and Karla partitioned their rectangles in a similar fashion. To verify the equality of the parts, they each looked at the small rectangles and decided that each triangle divided the rectangle into two equal parts. They were very confident with their decision. The class had previously worked with rectangles and triangles in a geometry unit. Alton divided his rectangle lengthwise into four rectangles and then used one line in the center of the rectangle widthwise to create his eight rectangles. He did not use his ruler for measurement and thus his rectangles were not equal. He, nonetheless, insisted that if he spent a little more time with it he would be able to produce eight equal rectangles with this configuration:. Marla s rectangle produced a replica of this configuration. Partitioning the parallelogram (4 x 4 ) into four shares generated various configurations. 70% of the participants partitioned their parallelogram similar to Figure 8(a). Jay and Polly were the only students to divide their parallelogram resembling Figure 8(b). Neither of them used their ruler to mark off 1 points on top and bottom sides of the parallelogram and then joined them to form parallel lines thus forming four equal parallelograms. Instead, they used the estimate-erase-readjust (Boulet, 1993) method of partitioning the shape. Kisha drew one diagonal of the parallelogram but failed to recognize that drawing the other diagonal would produce four equal triangles. She did not complete her partition but continued to the next task. Karla, although admitting that her division did not result in equal parts, left her diagram looking like Figure 8(c). (a) (b) (c) Figure 8. Students Configuration of the Parallelogram 82

96 The last shape to be partitioned in Task 1 of Activity 1 is the letter L. They were required to partition the L into five equal parts. 17 out of 20 students did the partition in a similar fashion by closing off the bottom of the L. They found that the dimensions of the square was 2 x 2 and then proceeded to mark off four 2 x 2 squares to complete five equal squares ( ). Jonah closed the bottom of his L but continued to divide the long side without using any measurement derived from the closed square. His diagram, therefore, showed five unequal parts. Kisha and Claudia never completed their partition. The second task of Activity 1 required the participants to partition twelve chips into four, six, three and five equal shares respectively. The first three shares never posed any difficulty for the students who displayed a variety of ways to do their equal groupings. After using the chips to form the required partition they then made a model on the task sheet. Jay, Josh, Polly, Bob, Paul and Chrissy employed the algorithmic approach of division to decide how many chips would be in each group. Polly: 12 divided by 4 gives three, therefore I will have 3 chips in a group. Others such as Kisha, Ben, Mary, Richy, Brian, Carol and Dahlia simply placed four chips on the table to represent the four groups and then continued to place a chip in each group until there were no more chips. This strategy is called dealing (Davis & Hunting, 1990; Davis & Pitkethly, 1990). Similar methods were used for partitioning in six and three shares. The last subtask proved rather tricky for the students. Proceeding as they did in the previous subtasks, those who were using division to aid in forming their groups found out that 5 is not a factor of 12. Polly said: You cannot do the problem. 5 times nothing equals 12. All students made some attempt to form the five groups. I: How many chips do you have there? Jay: 10 I: Where do the other two chips go? Jay: I m trying to find out where to put them. It has to be five equal shares. 83

97 Marla and Paul resolutely stated that only 10 chips could be partitioned. Chrissy gave this configuration choosing to assign four chips to one group. Mary, Kisha, Richy, Bob, Ben and Paul decided to make the groups of five with two chips in each group with remaining two chips set aside as remainders (see Figure 9 for Mary s distribution). Ashley considered this task to be impossible and 25% of the students agreed with her making no suggestion as to how this task could be accomplished. S. 1 S. 4 S. 2. S. 5. S. 3 Rem. 2/ Figure 9. Mary s Method of Equally Sharing 12 Chips The subtask was then placed in the context of dividing 12 cookies with five persons. Jay then offered this suggestion for doing the partition: Jay: Take each cookie and divide each cookie into 5 pieces and then give each person one-fifth. Carol agreed with him. Claudia decided she would break each of the two remaining cookies in halves but quickly retracted when she recognized that each of the five persons would not get a half. Dave had a different idea. Dave: You only need to divide the leftover 2 cookies in five equal parts each and give one piece to each person. Each person would get, uhm, 2 wholes and 2 small pieces. He made this sketch on the task sheet:. represented one of the pieces from the leftover cookies. Task 3 required the sixth graders to partition the number line into two, five and eight equal parts. The given line measured 4. Dividing the line in halves posed little 84

98 problem although the students employed different tactics to accomplish the task. After measuring the number line, which was labeled 0 and 1 respectively at the endpoints, most students simply used their ruler to guide them to the half of 4. Others used the estimateerase-readjust method to find the center of the line. Partitioning the line in eight parts never posed much problem as the subtask before it partition into five parts. Still using the 4 measure, Marla, Karla, Jonah, Alton, Jay, Mary, Carol and Brian recognized that they could not easily partition into five parts therefore they resorted to estimating the parts to be less than an inch. Claudia, Ben, Dave, Josh, Bob, Mary and Paul decided to use the centimeter (cm) side of the ruler showing 10cm in close proximity to 4. Using the division property, they assigned approximately 2cm to each part to produce the five equal parts on the number line. Polly and Dahlia completed this task outside of the regular session due to their absence at the last part of the session. The last task in this activity proved to be a rather interesting one. Some students were a little baffled as to the meaning of a cup of water. Does it mean that the measure of the water was one cup, or the container was referred to as a cup? They were told to use the former meaning. At each of the three subtasks, they assumed that there was no water loss. They were asked to equally share the cup of water into two, three and five shares. I: Can anyone share how they partition their water into three equal parts? Karla: Dave: Josh: I count the stripes around the cup. I kinda did what Amanda did, except this time I just pour the water into the cup, look at each one until only two stripes are left. I put water to four stripes, another one to four stripes and another to four stripes. I: Does it work exactly? Josh: Yes 85

99 Chrissy, Kisha and others decided to use some innovative ways to ensure their equal partition. This included using the chips, the erasers, the rulers and the graduated measurement on the cup. I: Does anyone want to share their method of making sure the parts are equal? Kisha: My water is up to 250 ml. I divide it by 5 and get 25 and then give everyone 25 ml. Ashley drew my attention to what she was doing by exclaiming that she had water left over. I: What should you do with your extra water? Ashley: I should drink it [laughing]. I will use all of it. [She poured and observed until all her water was used up contending that she partitioned her water into five equal parts.] The students found this task an interesting one as they worked with non-conventional ways to verify the equal sharing of the cup of water. At the end of the three sessions for Activity 1, students were given the opportunity to share their work with the whole class and a review timeslot was scheduled at the beginning of the next session. As mentioned earlier, each student participated in individual assessment tasks at the end of some of the activities. These tasks were designed to verify the students learning gains that were captured during their participation with the meaningful, engaging task-based activities coupled with their interaction with the physical referents at their disposal. Keep in mind that the main objective for this level in the model of understanding is the discernment of equal and unequal parts. For each of the tasks in Activity 1 some students displayed signs of miscomprehension and uncertainties as to the way in which the partitions should be done. The assessment task required the students to recognize and partition discrete and continuous wholes. There were three exercises in the task sheet (see Appendix B Assessment Task Activity 1). On the first exercise, the participants were presented with four circles divided into three parts. The students were asked to put a check mark beside the ones that were partitioned. Choices b and c were pre-designed 86

100 to be partitioned but with different orientation while a and d were not partitioned. Table 10 gives a summary of the students choices. Table 10. Summary of Responses for Assessment Task for Activity 1 RESPONSES Number of Students N = 20 b only 1 c only 2 b and c 17 Some of the reasons for choosing b and/or c were: Dahlia: Claudia: Richy: Mary: Kisha: Marla: Polly: Josh: I know that d and a are not equal because I observe both of them and I measure b and c in my head. c seems to be equal and a looks really messy. I know that circle c and b are partitioned because I measured from point to point using centimeters and if the space is the same size. Circle b looks like its partitioned because the angles are the same size and circle c looks partitioned because the edges are all the same size. Well, I saw that letter b was equal because I measure it with my ruler. Also letter c. Both letter b and c has 4 cm from point to point. You can solve this by drawing an equilateral triangle in to the middle to measure each side. For each circle I would measure the three lines with the ruler. If all three lines were the same length, they were partitioned. I used the ruler for circle b and c. I measured from one point to the other. Jay: c is just an upside-down b. 87

101 None of the students chose a indicating that they were not confused with the conventional orientation of a circle dividing in three equal parts. Ashley, who had earlier in the activity divided her circle similar to d, chose b and c without offering any explanation for her choices. On the second exercise the students were asked to partition a rectangular-shaped cornbread among six people. The three common configurations were: (a) (b) (c) Figure 10. Common Configurations for Exercise 2 Assessment Task for Activity 1 Bob and Jonah were the only students who differed from one of these configurations. For Bob s second configuration he partitioned his rectangle by first dividing the figure into three squares and then drew one of the diagonals for each square resulting in a configuration looking similar to this. His first partition looked similar to Figure 10(b). Jonah chose to draw the two diagonals of the rectangle and then a line cutting vertically through the center of the rectangle to form 6 triangles. He had no idea of how to check if the triangles were equal. He was satisfied with the fact that he could divide his rectangle into six parts. The final exercise asked the students to equally divide eight cookies among three persons. Here are some of the students responses: Dave: I gave each person two cookies and later broke the two extra cookies into three parts. Karla: I give two cookies to all of the three people then two cookies are left. I break one cookie in half and give one piece to the first and the other half to the 2 nd kid. I break the last cookie into four pieces and give one to the 3 rd kid and one to the 1 st kid another piece to the 2 nd kid and the last piece to the 3 rd kid. 88

102 Alton: Jay: Claudia: Ben: Kisha: Bob: I get two cookies for each person and then divide the two by three slices. I counted the circles and there were two left and I divided them into thirds. So each person gets two and two-thirds cookies. I give everyone one-half of a cookie. I put two cookies each then split up the other two into three and give each person two pieces. Well each person had two cookies so I divided the other two cookies into three halves so everyone could get two cookies and two thirds. I divide the cookies into thirds so each one could get equal amount of cookies. All the students, except Jonah, Karla, Claudia and Kisha figured that they would have to split the remaining two cookies in three equal parts. They distributed the thirds in two different ways, either by giving each person one-third of each cookie or putting the thirds together to form six thirds and then gave each person two thirds. Jonah did not make any attempt to share the cookies with three persons instead decided to share the cookies with four persons thus distributing two cookies to each person. Karla focused mainly on sharing all the cookies making sure that each person had the same number of pieces of cookies instead of ensuring that each person got the same amount of cookie. So for her, each person got four pieces of cookie disregarding the inequality of the pieces. While referring to halves in her explanation of how she performed the task, Kisha partitioned the circle into three equal parts using the conventional method ( ). This was evident in the diagram she drew to accompany her explanation. Based on the less than satisfactory results on the individual assessments tasks, individual interview sessions were planned with Claudia, Ashley, Carol, Kisha, and Dahlia. These sessions were held during their elective session and lasted for approximately 30 minutes. They were designed to provide re-verification tasks that allowed the students to talk aloud their strategy for completing them. The interview protocol and tasks are shown in Appendix B. The first task invited the interviewees to 89

103 divide 11 cookies among three friends. Using whatever physical referents they needed, they were encouraged to talk aloud as they solved the problem. Dahlia: Uhm Yea. I am going to give three cookies to each person then I have two remaining cookies, and from those two cookies I break them up in three equal parts and two parts go to each person. So it s three cookies and two parts of a cookie to each person. At first Dahlia divided the two remaining cookies using vertical lines but when questioned about the equality of the parts, she figured that the parts were not equal and then proceeded to partition the circle the conventional way. She was able to verify the equality of the parts using the strategy that Marla shared in a previous session. Kisha: I found out that each person will get three cookies but two are left and so I broke them in three parts and gave each person one part from each cookie. Kisha proceeded to do the partition in a similar fashion as Dahlia but without any intervention she employed the regular strategy of dividing the circle into three equal parts. Carol, Ashley and Claudia distributed the cookies in the same manner giving three whole cookies to each person and then dividing the last two cookies into three equal parts and then gave two parts to each child. Claudia used the chips to aid in her distribution. She even attempted to partition the last two remaining chips. Carol made a table with the three names as a header then put a stroke ( ) to represent a whole cookie and a dot ( ) to represent the third of the circle. Carol used the word half to mean the third as she did in a previous task. I: So each of the part of this cookie is a half. Carol: Yes. I: So this cookie has how many halves? Carol: Three. Hmm! But something can t have more than 2 halves. I: How do you define a half? Carol: Two equal pieces. I think I am using the wrong word. 90

104 To verify that Carol knew what she was doing, she was asked to share seven cookies among three friends. She was able to do so this time using the word one-third to represent one equal part. The second task required the interviewees to partition a square cake among eight friends. They were expected to show the partition in two different ways. None of the students experienced any difficulty in partitioning and verifying the equal parts of the square. They suggested measuring the sides of the parts, cutting out and placing the part on top of each other (superimposing) and visual observation as means through which they could verify that the square is partitioned into eight parts. A summary of their configurations is shown in Figure 11. Figure 11. Configurations for Task 2 Individual Interview Analysis As you may recall, the most fundamental criterion for the logico-procedural understanding is the ability to partition a whole into a given number of equal shares. The twenty students in this study, during the various Activity 1 tasks, assessments tasks and interviews demonstrated that they possessed this level of understanding. At the end of the three sessions dedicated to the partitioning activities, the students were eager to share their strategies for partitioning the continuous and discrete wholes. The word partition became a standard word in their mathematics vocabulary. By the end of Activity 1, the participants exhibited much ease with working with continuous and discrete models. Dividing the circle into three equal parts and sharing discrete objects among individuals where the number of objects was not a multiple of the number of equal parts were the two most trying tasks for the students. As the participants of the study 91

105 interacted with each other they were able to agree on acceptable methods of ensuring equal parts when dividing the whole. Those who had difficulty in authenticating the equality of the three parts eventually abandoned their methods and adopted the method that Chrissy and Marla used measuring the chords formed at the end of the partitioned lines. They remarked that the chords should form an equilateral triangle. The students employed various methods in verifying equality of parts in continuous models. These methods ranged from measuring with whatever physical referents (e.g. rulers, flexible wires, fraction circles, fraction squares, fraction triangles, graduated marks on the objects, their fingers) made sense to them at the time. They also relied on visual observation (especially when working with the parallelogram) and little facts that they gleaned from the unit previously done in geometry. The participants experienced little to no difficulty when partitioning discrete objects where the number of objects was a multiple of the number of partitions. The two most common methods that these students used were the dealing strategy and the use of their multiplication/division facts. The main manipulative used to aid them in their strategy was the circular chips that were provided in their bags. Some students also used the circle cutouts that were given during the tasks. Partitioning a number of discrete objects where the situation was different than what is cited above posed considerable struggle for the participants. Some of the students, initially, regarded the task as impossible but were later able, with the help of their classmates, to accomplish this feat. There were open class reviews of partitioning ideas the students developed in the previous sessions. These review sessions were student-based and student-centered where the students shared what they learned. The teacher played the part of a facilitator at these review sessions by asking leading questions and opening the floor for discussion using excerpts from the previous tasks. This practice became a standard feature throughout the teaching sequence. Logico-Physical Abstraction This level of understanding is the last component in the first tier of Herscovics and Bergeron s (1988) model of understanding. At this level the fraction concept is distinguished from the simple action of dividing a whole into equal parts (logico-physical procedural) but rests upon the understanding of the relationship between the part and the 92

106 whole. The five criteria related to this component (see Chapter 2, page 42) will be evaluated through three activities (see Appendix B). Each activity consisted of several subtasks which covered the participants perception of the relationship between the part and the whole in spite of the transformation, reversibility and repartitioning of the part or whole. Activity 2 majored on the reconstitution of the whole from its part, Activity 3 focused on the equivalence of the part-whole relationship regardless of variation in physical attributes and/or transformations of the part or the whole, while Activity 4 centered on repartitioning an already partitioned whole. Activity 3 spanned two sessions. The three activities were group-based. An assessment task accompanied each activity. At the beginning of each activity the students were handed their personalized manipulative bag which included all the materials they needed for the tasks at hand. They were given twelve one-twelfth pieces of a circle (green or black) and one-quarter of circle part (yellow). Each group was also given a bag containing five one-eighth parts of a triangle, one one-fifth part of a circle, eight one-twelfth part of a square. There were also provided with two ounces of water in a small plastic container and a measuring cup. They sat in their group at the beginning of each group session. The first two tasks in Activity 2 were done individually. For Task 1 each student was expected to use the twelve equal parts of a circle to reconstruct a whole circle. The sixth graders had no difficulty in accomplishing this task. Task 2 required the participants to determine how many of the given part (one-fourth) of the circle would be needed to reconstitute the whole circle. Again, this task posed no problem for all the students who recognized that four parts were needed in total. To verify this, some of the students (e.g. Alton, Paul, Bob, Cindy, Karla, Ashley, and Carol) placed the given part on a blank paper and traced it. They repeated this action three more times laying the given part adjacent to the drawn part until the circle was complete. Others such as Ben, Jay, Dave, Mary, Chrissy, Marla, and Jonah immediately recognized that the piece represented one-fourth or one-quarter (Jay s terminology) and stated that it would take four of the pieces to complete the circle. A number of students linked Task 1 with Task 2 using the circle formed in Task 1 to recognize that three additional pieces are needed to complete the circle. 93

107 The remaining six tasks required the students to work in their groups to discuss the solution to each problem. The members of each group were previously assigned by the teacher-observers. Each task was accompanied with the required props necessary to arrive at the solution. Without any prompting from the researcher, the students assigned roles to each member. They took turns in reading the questions and in monitoring the tape recorder placed at the center of the table. Because of the relative ease at which the groups completed the tasks for this activity only the results from two of the groups will be displayed. These two groups represented the overall work done by the members of the different groups. The excerpt below is pulled from the conversation among Bob, Kisha and Jay. Dahlia was absent that day. Task 3: You are given five parts of a triangle. How many more pieces of the same size are needed to complete the whole triangle? Jay: We need about four more. Kisha: We could measure the sides to help us. We could make it equal sides with our ruler. The others ignored what she was saying. Jay: We need about two more, no, three more. Kisha: That s true, we need three more triangles. Bob agreed with them. Task 4: Bob: Kisha: Jay: Bob: Jay: You are given one part (green) of circle. How many pieces of the same size are needed to complete the whole circle? I think we need about (pause) 3 or 4 more circle parts to complete the circle. I don t agree that we need 4 more. I think we only need 3 more because we already have one. We need only 3 more parts to complete the circle. Actually, the answer can t be four or three. It is five. Why? 94

108 Bob: Jay: Bob: Kisha: Bob: Kisha: Jay: Bob: If you take this part of the circle and trace it (he traces the given part of the circle on the paper) you will see that you need about five more. So each piece is a sixth? Yes. Four more pieces? Yes, four more pieces. I guessed it right. So each piece is a fifth. I made the mistake cause I count the one we already had. They all agreed that four more pieces are needed. Task 5: You are given eight parts of a whole square. How many more pieces of the same size are needed to complete the whole square? Kisha put the pieces together. The group decided that they needed four more little rectangles to complete the square. Task 6: You are given one part of a cup of water. Predict how many more similar parts will give you a whole cup of water. They poured the water in the measuring cup. Kisha: We have one-fourth of a cup. Bob and Jay agreed with Kisha that three more fourths of the part of the cup was needed based on their observation. Alton s group had quite an interesting twist to Task 6. Here is the excerpt of their discussion from Task 3 to Task 6. Task 3: Paul: We need 3 more pieces. Chrissy: That s impossible. Ashley: We need six more pieces. Chrissy: We need seven more pieces to do this triangle. They reassembled the parts until they all agree that three more pieces are needed. 95

109 Task 4: Chrissy: I think we are missing five pieces from this one. There are six pieces in all. They all agreed and moved on. Task 6: Paul: Chrissy: How much is the whole cup? 8 oz. or 250 ml and the part is75 ml. Ashley: What is 75 times 3? Chrissy: 75 times 4. They struggled with the multiplication realizing that they were not getting the 250 they desired. Chrissy reread the question. Alton: I m lost. Paul: Wait, why don t we do 250 take away 75? Ashley: That s what I did and get 175. The group decided the answer was 175 ml. Dave had quite a task in convincing his group members that Task 6 required only three more parts to complete the whole cup. Dave: I think 4 more parts of the cup because on the metric side of the cup, 1 cup is 250 ml and we have 50 ml already so we need 4 more 50 ml to get the whole cup. Carol: I also get 4 more parts by observing it. Karla: It could be 5 more parts or even 4 ½. Claudia: I agree. Dave: It doesn t need so much. It only needs 4 more parts. (He said the last part with much emphasis). Claudia and Karla were satisfied with Carol s observation method but needed more tangible proof that four more parts were needed. They decided to listen as Michael repeated his method of determining that four more parts were needed to obtain a cup of water. The group then agreed. From the discussions and conclusions reached by the students concerning Activity 2 it became evident that the students experienced little difficulty. They employed 96

110 various strategies to ensure the completion of the whole whether they are given many parts of the whole or one part of the whole. Some of the strategies included observation, drawing pictures, working with the manipulatives and in the case of the liquid measure, graduated marks on the container. The students were adequately able to reconstitute the whole from its parts. To further investigate this ability, however, an assessment task was given before the start of Activity 3. The assessment task for Activity 2 was a paper-pencil task-sheet that required the participants to determine how many of the shaded parts were need to complete the whole object. The group members were at liberty to use any physical referents to aid in the derivation of a solution. Only continuous models were used. The first exercise proved fairly simple to all the students. The dotted frame used to complete the whole provided an easy scaffold thereby enabling all the participants to recognize that one more part of the shaded region was needed. After agreeing that the first exercise was simple, Chrissy s group grappled with the circle. Here is an excerpt from her group s discussion on the entire task-sheet. Chrissy: That was easy. [Speaking of the first task.] For number 2, the circle, I started with the three lines, put the ruler on each line and then make a line half way through the circle and I did that for each three lines. Therefore 4 more pieces are needed. Ashley: Paul: Alton: That s what all of us did. What I did, I draw the other three lines and made them longer so the other half [meaning the other half of the circle] could have the same amount of lines. That gives me four pieces. What I did was I split them in half to make each a triangle and them make them equal cause I saw that each of them was a triangle. [He used the word triangle to refer to the sector of the circle.] Alton, Ashley, Chrissy and Paul drew lines in the diagram to partition the circle into eight pieces (see Figure 12). 97

111 Figure 12. Partitioning the Circle in Question 2 Activity 2 Assessment Task This would mean that they needed seven more of the shaded part to complete the circle. The group focused, however, on the half circle that did not have any lines thus obtaining four as the needed number of parts. The four remaining groups concluded that seven more of the shaded part was needed. They verified their decision by completing the circle as shown in Figure 12. Dahlia stated: 1 The circle needs 7 more parts because the shaded part is only. 8 The circles in Task 3 posed some challenge to some of the students as they recognized that they could partition the shaded part into any number of desired pieces and then use one piece of the shaded part to determine how many of that piece was needed to complete the circle. Josh: This circle needs sixteen pieces. Richy: Mary: Richy: Josh: Richy: Josh: Mary: That s not true. Or a thousand million little pieces. This little circle only needs one more piece. Richy, it doesn t need one more piece. It needs sixteen more pieces. Okay, it doesn t matter. We could still divide it into sixteen pieces. You put that and I will put the one piece of fourth. As a matter of fact, it can take any amount. But the pieces have to be equal pieces. 98

112 Richy: It just means that the shaded parts would have to be divided into smaller equal parts. Josh: It can be done but it is not going to be easy to draw the parts. You could put even 16 billion equal pieces in your one part, 16 billion there, 16 billion there and 16 billion there [making reference to the three parts of shaded fourths.] Josh and Alton s groups chose to partition the whole differently. Each member in both groups divided his or her circle into different equal parts. Josh decided to divide the first circle into twelve parts instead of the sixteen parts he argued about. He first divided his circle into four equal parts, and then partitioned each fourth into three equal parts thus resulting in a circle that was partitioned into twelfths. From this Josh concluded that If you put three equal parts in the blank spot you will get 12 equal pieces all around. Threetwelve more pieces are needed. Alton did a similar partition. Paul partitioned the circle into sixteen equal pieces thus determining that four more pieces of sixteenths were needed. Paul: I divided each four into four little pieces which comes out to be 16 pieces. Chrissy and Mary divided their circle into eight equal parts and deduced that two-eighths were missing. Karla, Kisha and Brian s groups (see Appendix C for group members) partitioned their circle in four thus stating that one more fourth (¼ or quarter) was missing. Dave: This one is made up of four parts [referring to the first circle in item 3.] We re guessing that they cut it up into four parts. So we are guessing that we need one fourth more. The other group members chimed in with their confirmation of Dave s guess. The groups mentioned above expressed similar reasoning. Ashley also opted for one missing part of fourth. Ashley: For the first circle, I close it and then I put it for four equal pieces so one piece of four is missing. 99

113 Jonah misunderstood what his group members were discussing concerning the number of equal parts that could be drawn in the missing part and chose to divide his circle into five parts. He drew three lines in the shaded region resulting in four parts plus the blank part completing the five parts. Armed with some insight from the first circle of item 3, most of the participants moved ahead to divide their circles into parts with no apparent method of ensuring the parts were really equal. There was little group discussion on this exercise. 45% of the class partitioned the circle into eight equal pieces by completing the fourths and then drew center lines through the quarters resulting in three more fourths needed to complete the entire circle. Josh, Polly and Brian made six parts and figured that two more sixths were needed while Kisha and Dahlia simply drew a line to divide the missing part into two equal parts stating that two more fourths were missing. Paul did the same as Kisha and Dahlia but concluded that the shaded part consisted of three parts. Paul: I divided the circle into five and so two pieces are needed. Bob is the only student who divided his circle into three parts while Richy divided his circle into 10 pieces. Bob: Richy: One more third is missing because I noticed that 2/3 were shaded. Four more pieces of ten are missing. The responses to the tasks revealed that the students were aware that the parts must be equal but were still unable to apply suitable methods of checking that the parts were actually equal. Most of the students relied on the method of observation making little use of the manipulatives that they worked with throughout the previous activities. Two parallelograms were used as the whole in item 4. All the students stated that two more fourths (quarters) were needed to complete the first parallelogram. They drew the two lines needed to form the parallelogram (see Figure 13a) as a form of verification with Jay adding that he used his common sense to know the answer. Dave: I think in the parallelogram we need two more pieces cause all we have to do is to complete it to make slanted lines. Carol: And it s obvious too. 100

114 For the second parallelogram, all the students drew the necessary lines to complete the figure with 60% of the entire class dividing the parallelogram into six parts as shown in Figure 13b. (a) (b) Figure 13. Students Drawing of the Parallelogram Item #4 Dahlia and Kisha who were working together concluded that only one more third was needed giving no explanation of how they arrived at their solution. Richy drew lines to make eights parts in his parallelogram thus needing One more piece of eight. Mary and Jonah simply closed their figure without making any suggestion to how many parts were missing. Josh and Paul made two different configurations after completing the parallelogram (see Figure 14). Paul s Figure 14. Josh s Configuration for Item 4 (Second Part) Paul: For the box, I just close it in. I divided it into twelve equal parts. 2 pieces of 12 are missing. 101

115 Josh: I divide the parallelogram into 24 equal pieces so I need four more pieces of twenty-four to complete the parallelogram. 80% (four groups) of the participants completed the existing lines in the first circle of item 5 to form six parts thereby determining that four more sixths were needed. Brian wrote that 4 6 more pieces are needed indicating his confusion with the positioning of the shaded parts with the total number of equal parts in the whole. No intervention was planned at this point as formal fraction language and symbolization would not be considered until further activities in the logico-mathematical tier of the model. Dave and his group members divided the circle into five parts and unanimously agreed that three more fifths were needed. Most of them did not offer a method of ensuring equal parts. Bob argued with Jay that his parts looked larger than the ones that were shaded and then concluded we need a protractor to be exactly sure. Kisha used the words equal and half synonymously. All the students except Mary responded that two more thirds were needed for the second circle in item 5. Mary, possibly trying to be different from the others, said that 12 more pieces of twelfths were needed. She drew lines to divide the blank portion of the circles into twelve parts. The last item on the assessment sheet for Activity 2 never posed much challenge to the students who were able to use their ruler to measure one of the shaded parts of the rectangle and then used that measurement to mark off the remaining parts. They all concluded that seven more pieces of tenths were needed in the first rectangle and three more pieces of fourths were needed for the second rectangle. Chrissy: Ashley: For the rectangle, I measured how big the shaded piece is and then I made the same size on the rectangle and for the last rectangle I did the same thing. For the first one of the rectangles, I measure how wide and then I put my ruler to the length and I put it equal, then lay it straight down. For the second one, I measure it again, I made a line with 10 equal parts. Based on the responses to the tasks in Activity 2 and the assessment sheet, the participants displayed the ability to reconstitute the whole from its part. This includes 102

116 making the whole given many of the parts and making the whole given only one part. Although majority of the students were not very clear in their oral discourse about their method of ensuring that the parts were equal, on observing their behavior on the video clip they could be seen using the chips, ruler, and even the flexible wires to aid in their decision. Others just simply used the method of observation or common sense especially on the circular diagrams, which proved more of a challenge to them than the parallelograms. There was evidence that the students made effort to show that the parts were equal. There was very little teacher intervention as future activities would clear up some of the discrepancies. At this level the students were expected to employ the whole number language in their responses as the use of the fraction language would elevate the task into the logicomathematical tier of the model (Boulet, 1993 p. 121). However, as sixth graders who had been introduced to fractions at least two years earlier, it was anticipated that the fraction language would be used widely in some of their responses. This does not outlaw the activity as logico-physical as emphasis was placed only on the numerical aspect of their solutions and not on the language itself. Careful scrutiny of most of the responses revealed that the students mostly used the whole-number language to express their idea both in their written and oral work. These include three pieces of ten and four pieces of 24. Activity 3 was the second activity at the logico-physical abstraction level. The main objective of the seven tasks comprising this activity was to foster awareness and understanding in the students of the equivalence of the part-whole relationship in spite of variation in physical attributes and transformation of the whole. The first three tasks were aimed at determining the students understanding of the preservation of the partwhole relationship regardless of the physical attributes of the whole while the remaining four tasks focused on gleaning the students comprehension of the preservation of the part-whole relationship despite the transformation of parts in the whole. In five of the seven tasks the students worked with continuous models (circular and rectangular regions). At this point, the students were clear about the term part-whole relationship since it was discussed during one of the review sessions. 103

117 For the first two tasks, the students were presented with two stories accompanied with the necessary materials (see Appendix B) to aid in their solution. As there were four persons in a group, in solving task 1, all the students within the groups assigned to him/her one of the names in the task. In Claudia s group consisting of two boys and two girls, they changed one of the male names to a female name. All the members of each group easily recognized that one part out of three parts was eaten. Polly and Ashley s groups, after acknowledging that all four persons had the same amount of his/her chocolate bar, made note that each person had a different size. Paul: Each student didn t eat the same size because they have different sizes but they are all going to eat one-third of their chocolate. So they didn t really eat the same because there are different sizes but they each ate one-third. It still doesn t mean that they ate the same size. Chrissy: Ashley: Alton: What Paul said is true but if you re saying that they are eating the same amount as in fractions then they did eat the same amount. They eat the same amount in fractions but they have different sizes so it isn t the same amount as for each. If you put it into regular pieces it wouldn t be the same amount but when you put it into fractions they each have the same amount of pizza to eat. The students continued to use fraction terminologies though the teacher did not use the terms and the word fraction was not mentioned in the tasks. Bob: All of us ate the same amount of chocolate one third. Polly: Everybody ate 1/3 of their chocolate bar, but each ate different size. Claudia: We still ate the same 1/3. Even though they are different sizes its still the same fraction. The second task was similar to the first task except that the continuous wholes were circles. Similar responses were given where students made note of the part-whole relationship of the pizza that was eaten (one-half) but mentioned that Mitch s portion was bigger than Kris. Here is an excerpt from Carol s group. Claudia: Mitch had a lot more to eat. 104

118 Dave: Carol: Claudia: But if you put it fraction-wise it s the same fraction. Mitch had to lose weight. He had to do exercising. Our conclusion was that even though they are different sizes it is still the same fractions. Task 3 required the students to look at a set of figures (circle, square, rectangle, equilateral triangle, L and a star) and to determine if the part-whole relationship was the same for all the figures. Polly was confused with the term part-whole relationship but Marla using two of the figures on her desk, explained the term to her. The part-whole relationship was the same for the circle, star, rectangle, square and triangle with the L- shape being the odd one out. The students quickly recognized the difference and readily determined that the part-whole relationship was not the same in all the figures. Carol, along with most of her classmates, made note that the circle, triangle, rectangle, square and the star had the same part-whole relationship. Task 4 focused on the part-whole relationships in wholes where the parts were transformed. The students were presented with four squares (see Figure 15) where halves were shaded using varying configurations. They all figured that the part-whole relationship was the same after some discussion and negotiation. Polly: Look at these two squares [referring to the first two squares in Figure 15], do you think the part-whole relationship is the same? Marla: I think so because they look equal, the only thing that is different is that they are on different side of the square but they are still half of the square. (a) (b) (c) (d) Figure 15. Squares Used for Activity 3 Task 4 105

119 The group agreed that the part-whole relationship was the same. They examined the remaining squares: Marla: The other squares are still the same, this one is divided into four but it is still missing two parts (same as shaded part for her) which is the same as a half and this one is divided into two but it is still the half of the square. This discussion was similar to the ones shared in the other groups. Ashley read the question in her group and laid the squares on the table so all the group members could see them. Alton: No, the part-whole relationship is not the same. Paul: Chrissy: Ashley: Alton: No, they are not the same because they all have different shaded parts. Yes, it is cause each one is half of a whole. The question asks if the part-whole relationship is the same. It doesn t matter if this is two and this is one. It is still half of the whole. It is the same. If we put it together it is still a half. Yea, right, it doesn t matter how it looks it s still going to be a half (nodding in agreement). Josh and his group members agreed that the part-whole relationship was the same in Task 4 because each shaded part represented half of the square no matter the size or how the shaded parts were placed. They were able to arrive at a consensus with the transformation parts by mentally flipping the parts to match and compare with the other parts. Based on the responses mentioned above, the students exhibited some knowledge of the preservation of the part-whole relationships in wholes that vary in shape, size and transformation of the parts within the whole. Although they consistently mentioned the size of the part, for example, Mitch gets a bigger slice, they were able to differentiate between continuous wholes that possessed the same part-whole relationship and those that did not. 106

120 Tasks 5 and 7 were designed to examine the participant s understanding of the equivalence of the part-whole relationship in the case of discrete wholes despite physical attributes and transformation of the parts. There were two subtasks in Task 5. Task 5a had the students working with sets of circles and diamonds. Each set of 10 had four of them shaded. The students agreed within their groups that the part-whole relationship was the same four out of 10. A third set of circles was added with two of five of the circles shaded. Initially some of the students refuted the others who claimed that the partwhole remained the same. With much discussion and negotiation the groups said Yes to the equivalence of the part-whole relationship. Here are some of the strategies used to verify this equivalence. Richy: Kisha s group: Bob: Jay: Dahlia: Yes, they are all equal because you can divide the black circles and diamonds into twos. They both can be simplified to two-fifths. (Covering one set) Take out this part. That s two-fifths. They are not equal. Jay and Bob: They are equal. Jay: Dahlia: Jay: Dahlia: Marla s group: Polly: Marla: Brian: Polly: Brain: Okay, just divide by two. What? You divide both numbers by two and you get two-fifths. Okay. So two-fifths is a half of four-tenths. Is it the same part-whole relationship? I don t think so. I don t think so either. Why? Because they are not they are not all equal amount. Marla looked again at the set of 10 circles with four shaded and the set of five circles with two shaded. Marla: Oh yes I do. I think they are the same because they are all missing one fifth even though there are ten here, if you make one row you can see that s like one fifth. There s two rows of five and it looks 107

121 the same as this one which has one row of fives you see. This one is like four tenths, two fifths is half of this. This is the same thing. If you move these here and these here [pointing to the set of 10 circles] it will look the same as this ones. Polly: Marla: Polly: Marla: Polly: I disagree with you. These two are divided into ten pieces and they are all missing four while this is divided into 5 pieces and only missing two. I know it s half of this. The 10 diamonds and 10 circles with the four parts shaded are the same but the ones that have the five circles and only two are shaded is not the same as this one. This is five circles and missing two and this is ten. Yea but when I was in fifth grade they taught me that if you have two fourths it is the same as one half? [She makes the illustration 2 1 on her paper =.] 4 2 Yea because it s half. When you have 4-tenths, it is the same as two-fifths because you can divide the four in half and get two and divide the ten in half 4 2 and get five. [She wrote on her paper =.] 10 5 That s right. Polly s acceptance of Marla s explanation could be based on the fact that Marla used the word half in simplifying four-tenths. Previous activities revealed that the word half for some students did not necessarily mean two equal parts but could be used for different connotation such as even parts and in Polly s case may mean the only fraction that another fraction can be reduced to or reduced by. The last two tasks focused on the preservation of the part-whole relationships in spite of physical transformation of the parts. In Task 6 the students worked with continuous models. The part-whole relationship was the same in all the squares though some of the parts were transformed (see Appendix B). Again the students employed different strategies to determine this fact. Each person easily identified Figure 16(a) as having one part shaded out of four making note that the parts were equal. At this point Polly expressed that she now understood the term part-whole relationship. Referring to Figure 16(b): 108

122 Polly: They are the same. If you put this part [pointing to the half of the quarter that is shaded] with the other part it creates one square. Oh I know that the part-whole relationship is now one-fourth yeah! Alton s group was talking about Figure 16(d): Paul: Alton: Chrissy: Ashley: Alton: This one is not the same. Yes, it is. No, it s not. It is Chrissy. Yes it is, cause if you take out these two lines and put this piece right here and like this then it will be just like this one. Chrissy: Ashley: They are not the same because you can t just move things around like that. But we move the others [referring to the square shown in Figure 16(b).] Chrissy: Okay, I guess they have the same part-whole relationship. Using similar strategies the concluded that all five squares possessed the same part-whole relationship. (a) (b) (c) (d) Figure 16. Squares Used for Task 6 Activity 3 Task 7 (discrete model) posed no problem to any of the students despite the transformation of the parts as evident in some of the students dialogue given below. In this task the students determined if the same share of cookies was eaten in the two instances where the shaded cookies changed relative positions in the whole. A similar 109

123 task was included in the pretest and was used to test the students intuitive knowledge of the equality of the parts in a partitioned discrete whole despite the position of the parts in the whole. Karla: It s equal cause in each one there is 6 cookies and I ate 2 cookies. Dave: There are 2 cookies eaten and in each set you have 4 left. Josh: It s the same. There s two out of six in each one. Jonah: Mary: Yes, because there is 6 2 in each set of cookies. Yes, because there s two out of six in each set just not in the same positions. Polly: The part-whole relationship is the same because they are all missing two. That was an easy task one of the easiest ones. The group assessment task for Activity 3 required the students to use all the figures they worked with including some new ones and sort them according to their partwhole relationships. They found this task rather exciting as they negotiated where each representation of the part-whole should go. An onsite interview with each group revealed that they were able to identify diagrams showing the same part-whole relationships and those that did not. They were also able to discard those figures that show unequal parts though these figures possessed the same number of divided parts as other diagrams. Much discussion took place as they collaborated on the part-whole relationships depicted on the diagram shown in Figure 17. Figure 17. One of the Configurations Given in Assessment Task Activity 3 Brian: That s one-half. I: Why one-half? 110

124 Brian: Carol: It is one part shaded out of two. It s one-fourth Carol used the transparency copy of the diagram and shared with the class how she got one-fourth as her solution. She connected the center line to make four equal parts. Brian ignored the unshaded half and looked at the whole as the two equal parts. Activity 4 - the final activity of the logico-physical abstraction level of understanding - was used to investigate the students understanding of the relationship between the part and the whole at the logico-physical abstraction level. The main objective of this activity was to examine the students ability to repartition wholes that were already partitioned. This activity formed the prerequisite physical basis for understanding equivalent fractions. Partitioning continuous wholes were only considered in this activity as Boulet (1993) noted that in repartitioning discrete wholes that were already partitioned the original partitioning would have been lost. There were three main tasks with each task comprising of at least two subtasks. Task 1(a) required the students to repartition a circle that was already partitioned in half (see Appendix B). It aimed to evaluate how the participants would partition the circle into an even number of parts. The students found this task rather easy as they drew the perpendicular bisector ( cutting the circle in the middle ) of the diameter of the circle. The remaining three subtasks then asked the students to share the assumed circular pie among eight, sixteen and thirty-two people. They were to show the shares for eight and sixteen while they were expected to extrapolate the shares to thirty-two people. The students utilized the halving algorithm to obtain the required number of equal shares. To share the pie for 32 people Carol remarked that she would draw one line through all the sixteen lines to get thirty-two. 3 Richy started to use the division algorithm ( 8 32 ) to aid him in acquiring the number of thirty-two equal shares. He recognized that his solution was incorrect, erased it, and then confirmed that he could cut each eighth in four parts to get the desired shares. There was clear evidence from the whole class that they knew that cutting the parts in half would double the number of parts. Karla: Brian: I split all of the parts into halves. I would divide each piece in half. 111

125 To verify that they understood what they doing, the teacher extended the exercise to sharing the pie already partitioned into thirty-two shares among sixty-four people. They all used the doubling algorithm to get the required sixty-four shares. The rectangle for Task 1(b) was pre-partitioned in three parts and the participants had to further partition this rectangle into six and twelve parts respectively. This task proved to be of little challenge to them with each student partitioning the rectangle using one of the configurations shown in Figure 18. Bob was the only student who did the partition shown in Figure 18(c). The dark lines represent the original partitioned lines. They applied the halving algorithm recognizing that if you double the 3 you will get 6 (Josh). The students used their rulers to measure the sides of the rectangle and then proceeded to do the required partitions. They applied the halving algorithm in partitioning into twelve parts by drawing the vertical center lines for the three original given parts in Figure 18(a) and for each created part in Figure 18(b). To complete his twelve equal parts, Bob simply drew the diagonal for each of the three given parts. (a) (b) (c) Figure 18. Six-Part Configurations for Task 1b Activity 4 For the first part of the second task, the students were asked to share equally a rectangular chocolate bar already partitioned into three parts for four persons. At first this task caused a lot of perturbation as the students negotiated among themselves how should this be done. Jonah: This is impossible. Josh: Ben: I don t think it is impossible, but it is going to be hard to do. Can t we just add another chocolate bar to it? 112

126 Dave: Kisha: That s what I was going to do it [referring to Ben s method.] I find a way. I give everybody one piece of the larger part and then break up one piece into four equal parts and give everybody one piece of that. Table 11. Configurations for Task 2a Activity 4 CONFIGURATONS NAMES OF PARTICIPANTS Richy Jonah Kisha Josh, Brian A A A B B B C C C D D D Ashley, Polly, Jay, Alton, Dahlia, Mary, Bob, Marla, Claudia, Chrissy, Paul, Ben Dave Carol 113

127 Kisha s verbal explanation of her solution started a chain reaction of solutions. Richy chose to divide the rectangle into 24 equal pieces. Each person gets four equal parts but I have eight left and I give each person two more pieces. Ashley partitioned each of the three equal parts into four equal shares. She then assigned the shares as shown in Table 11. A, B, C, and D represented the four people she shared the chocolate bars with. Karla partitioned the rectangle similar to Ashley but chose to distribute the parts in a different manner. She gave two pieces to each person than gave one piece each of the last four parts to the four persons. Dave divided each third into eight. It was 24 so 24 divided by four is equal to six pieces for each person. Table 11 gives a summary of the different configurations accompanied with the names of the participants who drew that particular configuration. Of the other configurations not previously described, one share of the larger piece along with one of the smaller piece was distributed to each person. Task 2(b) also requested the students to divide a circular whole already partitioned in three to be divided into four equal shares. With the insight gained from the previous subtask, this assignment posed little difficulties to the sixth-graders. 50% of the participants divided the circle into twelve pieces by using the halving algorithm and distributed three parts to each person [see Figure 19(a)] while 35% of them partitioned the thirds in half and then divided the last third into four pieces distributing one large piece of the third and one of the small piece of the third to each person [see Figure 19(b)]. They made much effort in ensuring that the parts are equal. Each person got a half of the third and a small piece of the third. Richy partitioned his circle into twentyfour shares halving each part continuously until he got 24 parts and then distributed six shares to each person. Polly chose a unique way to share her circular pie. She halved each third and then halved one of the sixths in two of the original third. Each person would get half of a third and a little piece as shown in Figure 19(c). Dave struggled with the exercise as he divided it into eight slices and each person gets two slices. He did not produce a diagram to match his statement. 114

128 (a) (b) (c) Figure 19. Common Configurations for Task 2b Activity 4 The four subtasks of Task 3 sought to evaluate the sixth-graders ability to repartition a continuous whole already partitioned in larger parts to smaller parts. For the first task, the children would equally share a circular birthday cake already divided into eight equal slices among four persons. This task proved to be a very simple one as they easily recognized that each person would get two slices. Most of them applied the division algorithm (eight divided by four equals two) while others used the counting principle to arrive at their solution. Task 3b presented the birthday cake now cut in five equal slices to be shared equally with 3 persons. Four different configurations emerged from the students work. 1. Partitioned each of the five parts into three parts producing fifteen parts. Each person would get five slices. 2. Partitioned two of the original parts into three parts each. Each person would get one of the original parts with two of the smaller pieces. 3. Partitioned two of the original parts into six parts each. Each person would get one of the original parts with four of the smaller pieces. 4. Partitioned four of the original parts into two pieces. The last original part would be partitioned into three pieces. Each person would get three of the larger slices and one piece of the smaller slice. A rectangular whole previously divided in seven equal parts was used for Task 3c. The students were to share this rectangular pizza among three persons. This task is similar to the previous one with the exception that the continuous whole was now rectangular. The participants were now comfortable in doing the partitions with some of them looking for innovative ways to do their equal sharing. One of the most prominent 115

129 configuration was the one in which each person got two of the original slices plus one of the slices from the last original slice which was divided into three equal slices. Ashley chose to divide four of the original slices into twelve equal pieces and then gave each person one of the large slices plus four pieces from the twelve slices: Jay divided the seven parts into 21 parts and gave each person seven pieces : Alton divided the first piece into three pieces and make a long line across : Paul had a slightly different configuration than the ones above: In Task 3(d) the students were asked to shade one person s portion of a rectangular cake already sliced in eight pieces if the cake should be shared with only two persons. C C D C D D D C Figure 20. Karla s Configuration of Task 3d Activity 4 116

130 Three different methods of assigning the shares were observed from the students work. Karla noted: eight divided by two is four so each person would get four. She then proceeded to assign the shares as shown in Figure 20.The rest of the students had configurations similar to or. The final task for Activity 4 gave the participants the opportunity to partition an L-shaped candy that was already striped into seven equal parts. The candy should be shared equally among four people. This task was given to aid the students in their practice of reconstituting larger shares from smaller shares. With the skills they acquired in the previous tasks, the sixth-graders got creative in their partitioning. The most popular strategy of sharing, however, had the students assigning one of the original pieces of the candy to each of the four individuals and then dividing the remaining three pieces into four equal parts each with every one getting three of those parts. The partitioning of the seventh was done one of three ways, (a) using intersecting perpendicular bisectors, (b) four vertical lines, or (c) drawing the two diagonals. The distribution of three small parts varied with the students. Some distributed the three smaller pieces from one of the original part with the remaining quarter counting with two pieces from the next partitioned part to complete the three parts for the next person as shown in Figure 21. Others assigned one of the quarters from each partitioned seventh to each person. Below are some of the configurations that were employed in completing the task. (a) (b) (c) (d) (e) (f) (g) Figure 21. Configurations for Task 3d Activity 4 117

131 The individual assessment task for Activity 4 modeled the previous tasks requiring the students to partition already partitioned continuous wholes. The first exercise asked the students to repartition a parallelogram, already divided in halves, into twelve pieces. The source of challenge for the students was the need to make sure the lines were parallel and the same distance apart to produce the required equal parts. Some of the sixth-graders drew five lines to cut the given horizontal to arrive at their solution. Others did similar to Jonah who put two lines going down so that made 6 pieces then on the top I put a line across then I did the same thing on the bottom one. He was referring to the top of the given line. The second exercise sought to assess their ability to repartition a smaller share from a larger using a circular continuous whole already partitioned in halves. Dave chose to divide his circle into twelve equal pieces then gave each person four, while Alton made it equal pieces by making an X and the two sides will have three pieces and each person will get two. Chrissy divided the cookie in four equal pieces by drawing the perpendicular diameter and dividing one of the quarters into three equal pieces. She gave each person one of the quarter and a small piece. It was evident from the peer observer and teacher s observation including the students drawings that care was taken to ensure that the parts were equal. Bob made note of his method: I measure the circle to find the exact center then drew lines through it. Erased marks could be seen all over their diagrams. For the final exercise the students shaded one child s share of a rectangular chocolate bar that was previously partitioned into six pieces which should now be shared for four people. The sixth graders completed this task with relative ease with the majority splitting two of the original slices into halves each and then assigned one of the larger slices plus one of the smaller slices to each child. The children apparently recalled what they did while performing similar tasks. Analysis The three activities that comprised the logico-physical abstraction level of Herscovics and Bergeron s (1988) model of understanding covered the breadth of the five criteria related to this component. The responses given on the tasks coupled with the results of the assessment tasks indicated that the sixth-graders were capable of 118

132 reconstituting the whole from its parts, were aware of the equivalence of the part-whole relationship in spite of a variation in the physical attributes of the whole, recognized the equivalence of the part-whole relationship regardless of the physical transformations of the whole, recognized the relationship between the size of the parts and the number of equal shares, and were capable of repartitioning already partitioned whole. The students readily figured whether the part-whole relationships were the same or not but consistently noted that the sizes were not the same in relevant tasks. This awareness fulfilled one of the criteria for this component of understanding. Physical transformations of the part or differently shaped continuous wholes with the same partwhole relationship did very little in deterring their newfound concept as they negotiated and agreed on their solutions. The students frequently and freely use fraction language on most of the tasks. At this physical tier of the model of understanding the fraction language was not required and was not encouraged. The students, possibly because of the previous instructions they had in fractions, had no difficulty in expressing their solutions using the fraction language. Besides the usual stopping at the group tables and/or individual desks to make sure the students were on task or clarifying any misunderstanding of the tasks, teacher interventions at this point were seldom. Partitioning the circle and the parallelogram gave more challenge to the students than the rectangular whole. They were aware, however, of the importance of the equality of the parts and made much effort to ensure that this was done using whatever they had in their manipulative bag or just their imagination. Jay remarked that if he had a protractor he would be able to make sure the parts in the circle are equal. When repartitioning an already partitioned whole, the sixth graders displayed much excitement in finding creative ways of doing the sharing as shown in some of the configurations above. Logico-Mathematical Procedural Understanding This is the first constituent part in the second tier of Herscovics and Bergeron s (1988) model of understanding. The quantification of the fraction is the main consideration in the second tier instead of the part-whole relationship. One highlight of this model is that it extracts the mathematical from the physical in the analysis of a particular mathematical concept (Boulet, 1993, p. 165). As a reminder, at the logicomathematical procedural understanding the learner possesses explicit logico- 119

133 mathematical procedures that the learner can relate to the underlying preliminary physical concepts and use appropriately. This takes into the account the ability to orally quantify concretely presented part-whole relationships and to also generate part-whole relationships from concrete representations of the fraction. Activity 5 was designed to assess the sixth graders ability to orally quantify partwhole relationships depicted in all sorts of situations and to produce illustrations of orally given unit fractions. This day s session started off with the usual review session where the students shared what they learned from the previous activities, followed by an introduction to Activity 5. Based on previous misspellings and wordings used by the participants in an effort to express part-whole relationships using the fraction language, it became necessary for the teacher at this point in the study, to lead the students in a discussion of the naming of unit and non-unit fractions, for example, one-twelfth and two-fifths. This was followed by the tasks for Activity 5. For the first task, the students were shown a set of diagrams on transparencies depicting unit fractions. These included new diagrams as well as some that they had worked with on previous tasks. Both discrete and continuous representations were shown. They should determine what the part-whole relationships were, orally quantified them and then wrote the words on their task sheet. The following is an excerpt of the discussion that ensued after the first diagram [a parallelogram partitioned into sixths with one-sixth shaded] was shown: Mary: One sixth. Paul: One-third [not agreeing with Mary]. I: Give us a reason why you think it is one third, Paul. Paul: The top part has 3 with 1 shaded and the bottom part also has three parts. Dave and Mary partially agreed with Paul. Dave: I think it could be one-third. It is wrong but it does look like onethird. Mary: [She repeated Jeffrey s explanation.] All of us could be right - one sixth and one third because there are six pieces that can be divided into threes. 120

134 The teacher asked the class to identify the number of equal parts in the whole and the number of shaded parts. Jay and Richy then quantified the part-whole relationship. To show that they now understood how to orally quantify part-whole relationships, Paul and Dave chose to give the answer one-seventh for the second diagram: and then noted that they could not tell the part whole relationship for the fifth diagram because the parts were not equal:. The students never had any difficulty in naming the unit fractions in the first task and the non-unit fractions presented in the next task. This second task was very similar to the first one in that the students had to orally quantify non-unit fractions. The class easily recognized that the diagram showing a circle divided into 12 equal pieces with two of the pieces shaded represented the non-unit fraction, two-twelfths. Marla said it could be one-sixth. If you recall from the excerpts included from Activity 3, Marla had previous knowledge of reducing fractions. The students were expected to say what the unit fraction was for each of the diagram shown to them in the next task. The first diagram shown was a set of eight diamonds with two of them shaded. Claudia said it could not be expressed as a unit fraction. Others disagreed. Chrissy shared her reason for believing that two-eighths can be expressed as the unit fraction one-fourth. Chrissy: It s eight and if you cut that in half it will be four and then the two if you cut that in half it will be one [the two shaded regions]. So it will be one-fourth. Richy: I divided by two. To verify if the students got the concept, a diagram with ten discrete circles with two of the circles shaded, was shown. The class agreed that the diagram represented the unit fraction one-fifth. Jonah made up his own scenario to validate his understanding. Jonah: So, if you have six circles and two of them are shaded, that would be one-third, right? The class members gave their approval. Jonah: Oh, I get it. Because the first two diagrams involved the familiar halving algorithm the next diagram depicted twelve arrows with three of them shaded. More than half of the 121

135 students shouted one-fourth. Ashley s reasoned that there are three rows of four that s twelve and each row has one shaded. So it is one-fourth. The next diagram showed fifteen heart-shaped discrete objects arranged horizontally with three of the hearts shaded. Initially some of the students called the non-unit fraction name three-fifteenths and then renamed the fraction as a unit fraction. Some students exhibited signs of confusion, as they were not able to group in sets of twos or divide by two. This was an indication that they never fully understood the solution given by some on the previous diagrams. Polly: Why is it one fifth? Josh explained how he got one-fifth. Josh: Three times five equals fifteen. Then I give one little heart to each little five. Polly: Claudia: Oh I get that. [She proceeded to explain to her tablemates who were puzzled at Josh s solution.] I rearranged the hearts to make it three rows of five. O I get it. I: Explain what you got. Claudia: Dave shared his method with the class. Dave: You group them in fives and give each group one. I divided it. I divided fifteen by three and it comes out to be five. The remaining discrete model diagrams were no challenge for the students so diagrams depicting continuous wholes were introduced. The first continuous whole was a circle partitioned into sixteen parts with four shaded. Jay: Chrissy: Jay: One-eighth How did you get one-eighth? I mean one-fourth. I: How did you get one-fourth? Jay: If you divide 4-sixteenths by 2 you get 2-eighths and then you simplify again and it comes out as one-fourth Brian: [Interrupting Jay] There are sixteen equal parts and if you divide it by four you get four and the four shaded ones, if you divide it by four it equals one. 122

136 The last diagram showed a rectangular whole partitioned into seven parts with two shaded parts. The sixth graders readily expressed that they could not say the unit fraction for the shaded part as they could not find any common number that two and seven are divisible by. The final task of Activity 5 sought to examine how the students would produce concrete illustrations of orally given fractions. They were advised to use any of the manipulative in their bag or draw pictures to represent the fractions. The three fractions were one-half, one-sixth and one-fifth. 5 out of 20 students used the discrete model to illustrate the three fractions. Each of these students formed the fractions using the chips and then reproduced the model using paper and pencil. Bob portrayed his fractions as an equivalent fraction of the particular unit fraction, that is, for one-half he drew six circles including three shaded ones. Two of the students (Claudia and Ashley) represented at least one of the unit fractions as a discrete model. The remaining thirteen students utilized the continuous model (circle, rectangle, L, number line) to represent the fractions. The students used their rulers to ensure equality of the parts. Richy and Jay were the only students who used the number line to illustrate at least one of the fractions. Figure 22 shows one of Richy s representations. Figure 22. Richy s Number Line Representation of One-fifth As some confusion was evident at the start of Activity 5, the assessment task was designed to determine whether or not the intended objectives of the activity were accomplished. Were the sixth-graders capable of orally quantifying unit and non-unit fractions? Can they reproduce the fraction given an oral representation of the fraction? The paper and pencil assessment task was made up of four exercises. The students were given the opportunity to work with their groups as they orally quantified the part-whole relationships shown. The first exercise was the simple task of determining whether or not 123

137 the diagram shown represented one-half. The students readily recognized that the parts were not equal and unanimously said so. They were then required to show one-half whichever way they chose to. This was done with relative ease. Some students (such as Karla, Dahlia, Jay, and Richy) used two red and yellow chips with a chip showing one color and the other chip showing the next color. Josh, Dave, Paul, and Mary used more than two chips to successfully represent one-half. Carol, Ashley and Marla used other manipulatives such as the fraction circle, fraction squares or fraction triangles to illustrate the fraction. The remaining students drew continuous models splitting them in halves to depict the required fraction. Exercise 2 had three subparts where the students had to identify parts of the whole and verbalized the solutions. The students immediately recognized the part-whole relationships in the first two. The third subpart proved a bit tricky for the students. The scenario presented a patio in the form of a half circle. The students were expected to partition the patio in two and then determine whether the patio was one-half or a whole. A number of students drew a semi-circle to complete the circle, thus determining that Mary was right in saying that the patio was really one-half and not a whole. After Ben read the question, the group members unanimously said one-half for the first part of question. Brian reread the last part of the task. Here is an excerpt of the conversation within the group and across groups as they negotiated whether Mary was right or wrong. Polly: Ben: Brian: Polly: You re supposed to cut this half into half again. She is right. Me, too. I think she is right also because a whole [completing the circle] would be the whole circle. Marla: Polly: Marla: Polly: Marla: I don t think she is right because it is a whole. That s the shape of his patio. You maybe right. It s not half of the patio, that s the size, that s the shape. Okay, I think Marla is right. It may be shaped like this but that doesn t mean it is a half. 124

138 Polly: Dave: Chrissy: Paul: Chrissy: So I would tell her that she is wrong cause that s the shape, even though it looks like half a circle. It s half of a circle. But the picture shows half of a whole. That could be the whole. How can half of a circle be a whole? I: Did the question say the patio is a circle? Marla: Mary: No, they said the patio looks like that. It looks like half of a circle. It did not say it is a whole circle. It can t be a whole because obviously they don t show a whole there. Polly & Marla: Yes they do. Brian: It is really weird to see a patio looking like a full circle anyway. Marla repeated her strategy to the class in an effort to convince them that the whole was not a circle but was the shape that was shown on the paper. To confirm that she understood what Marla was explaining Dahlia used a model square, cut off a part of it, resulting in a rectangle, which she said, was a whole in itself. She mentioned that any object whatever the shape or size could be considered a whole. Consequently, majority of the class agreed with Marla. Chrissy, Alton and Ashley reluctantly went along with the majority thus confirming the assumption that most students got stereotyped into believing the whole must either be a circle or a rectangle. The third exercise required the students to make concrete representations of onehalf, two-thirds and one-fifth. The students had no difficulty in completing this exercise as they used both continuous and discrete models to illustrate the fractions. Mary used circles to represent discrete wholes as well as continuous wholes. The frequency of the usage of discrete representations increased in comparison to the related tasks in Activity 5. Careful consideration of the equality of parts was given while showing the part-whole relationships using continuous models. The final task asked the students to identify and 125

139 say the unit fraction related to a set of discrete objects ten cookies with two shaded. The ease at which the students did this exercise bore evidence to the fact that they were comfortable in identifying unit and non-fractions in discrete wholes. Chrissy expressed the answer shared by all the students: one-fifth or two tenths. Analysis The main purpose of Activity 5 and the accompanying assessment task was to reveal and provide opportunities where the students can orally quantify part-whole relationships and also make concrete representations of part-whole relationships from orally given fractions. It is evident from the responses shared above that the sixthgraders demonstrated logico-mathematical procedural understanding of unit and non-unit fractions. The major hurdles encountered in this activity were a) renaming a non-unit fraction as a unit fraction where possible and b) the assumption by some students that a semicircle could not be a whole. The students were able to discuss, negotiate and agree on solutions. They seemed to work comfortably with discrete and continuous models. Based on the responses to the different tasks, the students were capable of extracting the mathematical from the physical in the quantification of the unit and non-unit fraction. The responses corroborate with the possession of logico-mathematical procedural understanding where the learner possesses explicit logico-mathematical procedures that he/she can relate to the underlying preliminary physical concepts and use appropriately. Logico-Mathematical Abstraction/Formalization The logico-mathematical abstraction constituent part of the second tier of Herscovics and Bergeron s (1988) model of understanding is analogous to the logicophysical abstraction level in the first tier. The criteria for this component is similar to those of the logico-physical abstraction level in that the main objective is to determine the scope of the child s ability to comprehend and understand the invariance, reversibility and generalization aspects of unit and non-unit fractions. The main difference between both components was the fact that at the logic-mathematical abstraction level, the child was able to quantify the part-whole relationships connected to the physical attribute of the fraction. The use of the fraction language was expected and encouraged as students displayed the ability to obtain unit fractions from already partitioned wholes, 126

140 mathematically reconstitute wholes from unit fractions, determine the size of unit and non-unit fractions, quantify equivalent fractions and order fractions that are less than one. Formalization is the final constituent part of the Herscovics and Bergeron (1988) model of understanding. Besides the formal meaning of formalization (discovery of axioms and using these axioms to justify mathematical arguments), this component embodies the use of mathematical symbolization for concepts for which the previous levels and/or constituent parts hane already established. As noted in Chapter 2, this model of understanding does not necessarily function in a linear fashion yet as Boulet (1993) noted the connections among the components are by no means arbitrary. Some components are necessary for other components (p. 232). For the purpose of this study, the first four levels and constituent part were used as the prerequisites for formalization. Logico-mathematical abstraction and formalization will be considered simultaneously in this section of the paper. This decision was based on the pervading presence of the fraction symbol in the responses of the participants ever since the start of the study and the relative ease at which they were able to orally quantify unit and non-unit fractions thus confirming logico-mathematical procedural understanding. With this in mind, Activity 6 was designed to aid the students in quantifying oral and written part-whole relationships in symbolic form. Paper folding was introduced as a way to create fractions from continuous wholes. The students were to orally quantify the fractions modeled and generate convention symbols for these fractions. Activities 7-10 included tasks relevant to both logico-mathematical abstraction and formalization constituent parts. For example, in tasks that related to the size of fractions, the students were expected to orally compare and order given fractions and then order unit and nonunit fractions given in symbolic form. Numerous tasks from RNP were used in the effort to assess the students for the last two components of understanding in Herscovics and Bergeron s (1988) model. References to the relevant RNP tasks are noted in Appendix B. Prior to the students working on Activity 6 the students were asked to give their 7 understanding of the fraction. Activity 6 formally introduced the students to fraction 8 127

141 symbols so it was pertinent for the teacher to know what understanding the students had of these symbols. Ashley: Seven is how much is shaded and the eight is the number of equal parts in the whole. The other sixth-graders agreed with her. After working with two more fractions the group then went through the exercise of creating fractions via paper folding. This was done with the entire class as a means of providing one more strategy that they could use to decipher fraction problem-tasks. Besides, the art of paper folding provides an innovative way of generating equivalent fractions. The students used triangular, rectangular and circular shaped papers to form fractions such as,,, and. The rectangular-shaped papers were among the favorite for the students who were free to 2 choose which of the shapes they wanted to work with. gave them the most challenge 5 as they could not successfully apply the halving algorithm that they used in creating the half and the fourths. The teacher modeled how to create sevenths and then allowed the students to create the desired fifths. There were required to write the fraction name and the fraction symbol for the fraction part that was shaded. Karla noted that she had to make her half before she could make the one-fourth. Kisha said that it is easier to make the fractions on the paper because it s easier to make equal parts. The task sheet used in Activity 6 was copied from RNP s level one activities designed for middle school students. Lesson 5 Student Pages A E were assigned with the main purpose of investigating the students understanding of three different representations of a fraction, namely, fraction name, fraction symbol and a pictorial representation. For the first task on Page A the sixth-graders had to express given fraction symbols in words. The next two tasks reversed the instruction with each of the tasks expressing the fraction name in two different formats. For example, item 2a had 3 of 5 equal-size parts are shaded, while 3a used the format 9-tenths. All the students were able to complete the tasks with no apparent difficulty. On items 4 and 5, the students were expected to name the fraction that was depicted in the narrative and then draw a picture to illustrate the fraction. Instead of writing a fraction to tell how much is 128

142 shaded in all, Alton simply wrote the number of parts that were shaded, for example, 3 are shaded in response to item 4 which asked them to tell what fraction was represented 1 if three parts are shaded. His pictorial representation showed three separate 4 rectangular wholes with one-fourth shaded in each whole. He did similarly on item 5. 1 Figure 23 shows Alton s picture of four parts. He used his ruler to make equal parts. 5 He read two-fifths instead of one-fifth. Figure 23. Alton s Picture of Four 5 1 Parts Dahlia ignored the words three and four in items 4 and 5 respectively thus drawing pictures depicting 4 1 and 5 1. She repeatedly used the phrase the top number in her response to the fraction representing the total shaded parts in each item. Alton and Dahlia blamed misreading and misunderstanding the questions for the responses they initially gave for items 4 and Ben used mixed number fraction symbols ( 3,4 ) to name the fractions that 4 5 represented the shaded parts. He drew a blank circular whole partitioned into four for item 4 and a blank rectangular whole partitioned into fifths for item 5. He was unsure what the whole numbers (three and five) meant in front of the fraction and how they related to the number of shaded parts. When questioned, he said he was confused. He could not remember what he did in earlier grades and cannot recall the difference between and. No intervention was done as mixed numbers and improper fractions 4 4 were outside the scope of the study. Instead, the teacher-researcher relied on the 129

143 remaining task-based activities in the study to aid in the clearance of this misconception. 3 Ben defined as three parts shaded out of four equal parts. The responses to the 4 remaining tasks indicated that the students including Ben were successful in representing fractions in three different ways. On Lesson 5 Student Page C the students matched pictorial representation of a fraction with its symbol or word name. Page D asked them to shade each continuous circle to show the designated fractional amount while on Page E they were required to write the word name and symbols for the shaded part of shown in the rectangular whole. Four simple assessment tasks were used to assess the students use of the symbols associated with the fraction notation. These tasks were done with relative ease indicating that the sixth-graders were capable of linking pictorial representations of a fraction with its fraction name and fraction symbol. Task 2 depicted eight discrete circles as the whole with four shaded ones. Marla gave both the general fraction name (4-4 eighths) accompanied with its symbol ( ) and the unit fraction name (1-half) and 8 1 symbol ( ) associated with the diagram. The responses to these tasks including the ones 2 done during Activity 6 indicated that the students possessed a level of competence in working with fraction symbols coupled with the meaning attached to these symbols. The remaining activities will continue to assess the students formal understanding as they worked to generate equivalent fractions and order symbolic unit and non-unit fractions. Activity 7 was done over a period of two sessions. The main objectives of this activity were to have the participants determine the number of unit fractions needed to reconstitute the whole (continuous or discrete) and to construct given unit and non-unit fractions from preset discrete wholes. In Task 1 the sixth-graders were given four unit fractions to determine how much of each would compose the whole. If you recall, similar tasks were done at the logico-physical abstraction level. At the logicomathematical abstraction constituent part, however, the students were expected to use the underlying physical concept garnered at the parallel level to quantify (orally and written) the situations under consideration. Some of the sixth-graders (e.g. Jay, Marla, Bob, Dave, Josh and Polly) were already exhibiting logico-mathematical understanding and 130

144 these tasks posed no problem to them. Others such as Ashley, Carol, Claudia, Brian and Mary showed signs of growth in the understanding of basic fraction concepts. For 6 instance, Brian who earlier had written to indicate four more pieces of sixth was able to 4 complete Activity 6 and its accompanied assessment task and Activity 7 with little difficulty. All the students were able to determine that four fourths are needed to make the whole (Task1a). They used the quarter pieces of the fraction circle to make the whole. They easily recognized that seven pieces of one-seventh (Task 1b) and six pieces of onesixth (Task 1d) were needed to make the whole. When asked how many were needed 5 1 to complete the whole (Task 1c), Chrissy and Dahlia answered confidently that seven more pieces are needed. A pictorial representation of 5 1 ( ) was given. This apparently posed a possible distracter for a number of the students whose first initial step to solve the problem was to construct the complete circle. Chrissy and Dahlia drew similar pieces of the given one-fifth to make a complete circle thus resulting in their answer. Others erased their drawing when the number of pieces they drew conflicted with their desired answer five. The misunderstanding that the whole must be a circle if the given part was a portion of a circle was evident in Task 2c of the Assessment Task for Activity 5. Dave reminded the class that the whole could be of any form. Jay, Josh, Jonah and Karla were the only students who did not attempt to complete the circle or draw any form of diagram to answer the question. Karla seemed to have discovered a pattern to find out how many of the unit fractions would make the whole: one-fifth would need five, one-sixth would need six and one-twelfth would need twelve. An obvious highlight of this task was the students awareness that the parts had to be equal. Task 2a - 2c ended the first session for Activity 7. For the first two parts of the task, the students were to use their chips to model the given wholes and then used the red side of the chips to indicate the given fraction. The teacher with Claudia s help, modeled three-fifths on transparency at the beginning of the session. The students modeled the fraction using their chips and moved on to Task 2a, which asked them to show threefourths using twelve chips. Alton, Brian and Ashley initially formed four rows and 131

145 shaded three chips in each row to represent three-fourths. Paul explained to Alton that he was only showing one-fourth. He argued that he was showing three-fourths because he had four rows and shaded three in each row. Brian and Ashley gave similar reasons for their configuration. Brian: I have four rows and three shaded ones. These students were able to identify the unit fraction (one-fourth) from their configurations but argued that there were three one-fourths thus resulting in three-fourths. Bob showed Ashley his model; she looked at hers and then exclaimed: Oh, the three means I should shade three groups out of the four groups and that would be threefourths. I get it. She changed her configuration (see Table 12). Carol was unable to use her chips to display the solution she had in mind. She decided to draw two rows of six and then shaded the first three circles in each row. She then split the next circle in each row into halves as shown below. Karla tried to correct her but she was adamant that she had three-fourths. When interviewed she pointed to the three shaded ones to represent the three in the fraction and the three unshaded ones to represent the fourth. The halves were counted as two to complete the fourths. The second row was a duplication of the first row. Carol displayed a common error noted by Ashlock (2002) where students equated the number of shaded parts to the numerator and the number of unshaded parts to the denominator. This is usually the case in continuous wholes but Carol used discrete whole to display a similar misconception. She was able to do a three-fourths configuration after working with her in a one-on-one session. In this session the teacher used a modified version of one of Ashlock s (2002) strategies to aid Carol in clearing up the misconception. In this strategy start with the whole (Ashlock, 2002, pg 153), Carol was asked to identify the whole. She quickly grouped all 12 chips as the whole. After explaining her understanding of the meaning of three-fourths four chips with three shaded, she was prompted to make the three-fourths. Reminding herself that the fourths mean four 132

146 equal parts she took four chips and created three-fourths; she did the same with the next set of four chips and the next set until there were no more chips to work with (see Table 12). Kisha and Karla did a similar configuration. Table 12. Configurations for Three-Fourths Task 2a Activity 7 CONFIGURATIONS NAMES Dahlia Carol, Karla, Kisha Jonah, Jay, Richy, Polly, Mary, Alton Marla, Bob, Claudia, Chrissy, Paul, Dave, Josh, Brian Ashley The remaining set of students focused on the four equal parts by creating four groups first and then shaded three parts in each of the group to successfully representing three-fourths using twelve chips. A summary of the different configurations accompanied with the names of students are shown in Table 12. Ben never completed Task 2. He was not in the mood to work that day. 133

147 The students experienced no difficulty in completing Task 2b and 2c. However, for Task 2b which required the students to model two-sixths with six chips some of the students including Alton, Ashley and Dahlia were not sure how to group the chips should all six chips be laid side by side or should they be grouped in two rows? They were not able to resolve those issues but decided on shading only two of the chips no matter the configuration. Alton: I have three rows with two chips in each row or six chips with two rows of chip shaded. Some students resorted to drawing discrete or continuous models to describe 3 the steps they would take to show (Task 2c) while others used words I ll get 7 chips 7 and shade in 3 chips, (Alton), I would draw seven equal parts and shade in three (Paul). They were all successful in completing this task. Before the students completed the final part of Activity 7 in the next session they had to do an assessment task based on first two tasks done in the previous session. The students were able to complete the paper and pencil work within minutes of getting the work. Majority of them were able to arrive at a solution without the aid of the chips. The second part of Activity 7 contained problems that were used to verify the students understanding of making concrete representation of discrete wholes given parts of wholes. The students worked in groups and were encouraged to verbally discuss the problems before writing their solution on paper. As all ten questions (RNP Lesson 6 - Student Pages A and B Level 2) were based on the same principle only the responses to two of the questions (Items 3 and 8) will be considered. These questions were chosen based on the level of difficulty the students experienced while working on them. One of the criteria necessary to complete the unit is the understanding wrapped up in the meaning of the numerator and denominator of a fraction. Based on the previous assessment task, the students seemed to have a basic understanding of the meaning of both the top number stands for the number of shaded parts while the bottom number represents the total number of equal parts. Their first step in solving the different tasks 134

148 was to put the circles in groups of twos. This worked fine for the first two tasks. Item 3 (see Figure 24) caused some perturbation. Brian, Richy, Alton and Carol exhibited signs of confusion while working on this item revealing the need for understanding the underlying principles involved in the meaning of both the denominator and numerator of a fraction other than the meaning mentioned above. Figure 24. Item 3 RNP Lesson 6 Activity 7 Alton: That s impossible. There s one left over [after grouping in twos]. I: How many groups should you have in all? Alton: 3, no 5. I: How many groups do you have there? Alton: 4 with one left over. I: What does the three mean? Alton: How many circles are shaded. I: If you shade 3 circles here, will you have three-fifths? Alton: No. I: You said you should have 5 groups in all because of the five at the bottom. Since shading 3 circles did not give you your answer, do you have any other idea for the three/ Alton: Uhm, no. I: Can you get three equal groups here? Alton: Yes. [He regrouped the circles to get three groups of threes.] 135

149 I: How many groups do you need in all? Alton: 5, so I will make two more groups of three. [He completed the whole.] I: Very good. I ll come back to you later. Similar conversations were held with Brian, Richy and Carol. Figure 25. Item 8 RNP Lesson 6 Activity 7 Alton was still a little unsure of what do with item 8 (see Figure 25). He interpreted the numerator two to mean groups of twos instead of two equal groups of fours. Seven students (Richy, Carol, Jonah, Ben, Claudia, Brian and Kisha) had difficulty with item 8. Two features about the problem caused the difficulty the students faced: 1. There was an even number of x s so the students could easily group in twos. 2. The two in the numerator enforced their misconception. Dave shared his method of knowing how many discrete objects to group together: I count how many x s there are and then divide by the top number. Jay and Paul agreed with him. The students regrouped using Dave s strategy. Due to the recurring misconception a second assessment task for Activity 7 was administered. The students (except Kisha) readily determined that nine one-ninths and 13 one-thirteenths were needed to complete the wholes. The teacher worked with Kisha in a follow-up interview that took place after the verification task. Item 3 was designed to 136

150 evaluate the students understanding of making the whole (continuous and discrete) given parts of the whole. For the first subpart of Item 3 the students were required to fold a square sheet of paper into eight equal parts and shade three-fourths while for the second part the students folded a rectangular sheet of paper into nine equal parts and then 1 shade of the parts. A deliberate attempt was made to use the fraction word name and 3 the fraction symbol in Items 3a and 3b respectively. This was to verify whether the students had a working understanding of both forms of fraction representation as these were necessary conditions for the last two constituent parts of the Hercovics and Bergeron (1988) model of understanding. All the students were able to accurately partition the square paper using one of two sets of configurations as shown in Figure 26(a) and 26(b). There were some students who folded their paper similar to Figure 26(b) but chose to shade the parts differently [see Figure 26(c)]. The shaded display shown in Figure 26(c) verified that the students were able to identify and use the equivalence of the part-whole relationship despite the physical transformation of the parts in the whole. (a) (b) (c) Figure 26. Students Configurations for Second Assessment Task (Item 3a) Activity 7 Item 3b produced similar results as the previous one with the exception of Brian who shaded one of the parts (one-ninth) to represent one-third. Initially, Brian exhibited lack of interest in the tasks but recognizing that his classmates were deeply involved in 137

151 their work he decided to spend more time on his tasks. He was able to accurately show the required one-third. Two different configurations (shown in Figure 27) were observed as the children folded the rectangle. Figure 27(b) and 27(c) show the different patterns of shading using the same configuration. Figure 27(b) depicts three equal groups of threes with one of the groups shaded while Figure 27(c) shows one-third shaded in each group of threes. (a) (b) (c) Figure 27. Configurations for Second Assessment Task (Item 3b) Activity 7 The remaining three tasks of Item 3 required the students to reconstitute discrete wholes 1 given parts of the wholes. Item 3c required the students to draw the whole if is of 4 whole. Interestedly, Chrissy and Kisha drew a rectangle and circle respectively to represent the whole ignoring the smiling circles but possibly revealing that they knew that four fourths were needed to complete the whole. The remaining students easily did this item possibly because the part of the whole was already in a pair. Although item 3d was a duplicate of Item 8 in Activity 7 similar misconceptions prevailed necessitating a follow-up interview with Dahlia, Karla, Kisha and Carol. All the students were able to give an accurate diagram of item 3e. Dahlia, Karla, Kisha and Carol were interviewed an hour after they completed the second assessment task for Activity 7. Three tasks were designed as follow-up tasks. 1 The first task required them to draw the whole if represented a of whole. 4 I: What is the meaning of the four in the fraction, anyone can answer. Carol: It means how many groups. 138

152 I: What if I make four groups like this [I made four groups with each group having different number of chips], could that be the four groups in the fraction. Dahlia: No. I: Why not? Dahlia: The groups are not equal. The others agreed with her. I: You are all saying the four groups must be equal. They nodded in agreement. I: What does the 1 mean? Karla: How many groups are supposed to be shaded. I: The faces are not shaded so tell me how many of them should be in the group. Kisha: Four. I: Why? Kisha: Cause it says one group. I: So the one means that all of this is one group [reiterating what Kisha said.] Go ahead and complete the task and then we will continue to talk about it. They worked on the task individually. Carol and Karla started off by grouping in twos thus creating two groups of twos. I: How many groups have you formed? Carol and Karla: Two. I: How many groups should you have? Carol: Karla: Uhm, one. [Ignoring the question] We don t need three more groups only two more. Dahlia: The happy faces in the problem are counted as one group. Carol and Karla corrected their work to reflect the four equal groups of four. They moved onto the second task where they were expected to create the whole if 10 x s 139

153 2 represented of the whole. They were advised to share their response orally before they 7 began to draw the whole. This time it was Dahlia who grouped in twos and added two additional groups of twos to complete the whole. I: How many groups did you form? Dahlia: Five. I: So what does the 2 in the numerator mean? Dahlia: Two equal groups. I: Did you form two equal groups? Dahlia: No. I get it. I will group by fives to form two equal groups. She redrew her groups of chips and then successfully completed the last task. The other three students also completed the last two tasks with relative ease. They were then given the second verification task of Activity 7 to redo Items 3c, 3d and 3e, which they did successfully. Kisha worked on Items 1 and 2 which she had previously done incorrectly. Using commercial plastic circular wholes partitioned into fixed number of parts such as tenths, fourths and sevenths, Kisha was asked to count the number of tenths that were in the whole. She did the same for the next two wholes. She discovered the pattern and then made the changes to her activity sheet. Activity 8 was designed to determine whether the sixth-graders could order unit fractions according to their size. The first task of this activity required the students to paper fold and shade one-half and one-quarter on separate sheets of rectangular wholes (same size). Using the fractions they created they were able to determine that one-half is bigger than one-fourth. Jay and Josh said they knew that because they were taught that in fourth grade. Bob: is bigger than because is half of a

154 The students were then asked to determine if one-third was larger than one-fifth. Dave: One-third is larger than one-fifth because dividing a circle into 3 parts gives a bigger piece than dividing it into 5 parts. Brian: I don t get it. One-third is smaller than one-fifth. I: Why do you think so? Brian: Because 5 is a higher number than 3. Jonah: [Butting in] But you said half is bigger than one-fourth and the four is bigger than two. Brian: I know that one-half is bigger than one-fourth because one-fourth is a half of a half [repeating what Bob said earlier.] I: Look at the one- half and the one-fourth you made. Which one is bigger? Brian: One-half is bigger. It has a bigger piece shaded. I: Use the pieces on the Fraction Stack Bar and try to find out which fraction is bigger, one-third or one-fifth. Brian looked at the two pieces on the Fraction Stack Bar and then decided that one-third is larger than one-fifth because a third has a bigger piece than one-fifth. The entire group of students was able to determine that one-seventh is larger than one-eleventh. As Josh put it: I would prefer to share my pizza with seven people than 11 because I would get a bigger piece. The students were able to order the fraction according to the instructions with very little problem. Most of them applied Josh s method in determining the relative size of the fraction in comparison to other fractions. In Task 2 the students were to help Tyra decide which table she should sit if she wanted to get the most pizza to eat the table with two friends or the table with three friends. The entire group decided for one reason or the other that Tyra should sit at the table with the two friends. Here are some excerpts of the conversations that accompanied the task: Marla: With the group of two because since there are only two people that s less people to share with. 141

155 Richy: 2 because 2 is bigger than 3. Karla: At table 2 with the two friends cause she would get a bigger piece. Jay: The table with 2 friends because 3 1 is greater than 4 1. Polly: The group with 2 people because her slice is larger. They drew pictures of two circles divided into thirds and fourths to verify their decision. Marla was the only one who showed a method of verifying that the thirds are equal. The last two tasks (RNP Lesson 6 Student Pages B and C Level One) of this activity had the students identifying larger unit and non-unit fractions from given pairs of fractions. The students were able to use Fraction Stack Bar, Fraction circles, squares, and triangles in their comparison. They reasoned aloud in their groups. The excerpts below represent some of the rationale the students used in determining their solutions to Student Page B. Ben: His group members agreed. 2/3 is larger than 1/3 because there are two pieces of 1/3. So 2/3 has to be larger than 1/3. Dave: Jay: Ben: 4/5 is bigger is bigger than 3/5, the parts are equal shapes, there is just one more in 4/5 therefore 4/5 is bigger. I say 6/7 is bigger than 2/7. Once again the parts have the same shape and size but 6 is more than 2. 11/12 is bigger. You have more twelfths in this fraction. Paul: 2/9 is bigger than 2/7. Jay: Dave: I don t think so: 2/7 is bigger. It s like sharing a pizza. You get more to a slice when you have to share for 7 people instead of 9. You re going to have 2 bigger sevenths than 2 ninths. I m guessing 2/7 cause for 2/9 you are breaking up into more pieces which are smaller. They proceeded to identify the others using the methods they discussed above with the aid of fraction circles. The last two problems posed no challenge to them. The other students used one or more of the strategies shared in Ben s group to decide which of the 142

156 fraction was larger in each question. Claudia initially said that two-ninths is a larger fraction than two-sevenths because the ninths give more pieces, but Bob reminded her that the sevenths are larger than the ninths. Chrissy explained to Ashley why 10 9 is 9 larger than : 100 If you divide something in one hundred pieces you get very small pieces than if you divide into ten pieces. So if you shade 9 of the ten pieces and 9 of the hundred pieces you will get more with the ten pieces. There were four items on Task 4 - Student Page C. Because of the relative ease at which the sixth-graders completed this page, only the responses to items 4 and 5 will be considered. The students all agreed that Matthew ate more candy than Cassandra because two-thirds is larger than two-fifths (item 4). In item 5 the students had to determine whether it is possible that Ellen spent more money than Andrew if she spent one-third of her allowance on a movie and Andrew spent one-half of his allowance on candy. Dave s group assumed an allowance amount. Dave: Jay: Dave: Paul: Josh: I think Andrew spends more. Let s say the allowance is $6.00, she gets $2 and he gets $3.00. It s impossible. What if the allowance is $ Then Ellen gets $ And then he gets $23.50 (excited). He spent $23.50 on candies. He got a lot of cavities. She spent $15 on movie. She s smart. I would prefer to go to the movie than spend my money on candies. It s impossible. Andrew will spend more. Andrew spent more money. Bob, Jonah and Claudia reasoned that Ellen may have more money because she may have a bigger allowance. Carol believed that the possibility that Ellen spent more than Andrew depended on the amount of money she had in comparison to Andrew. With the 143

157 exception of Dave s group, the students believed that Ellen could have spent more money. RNP s Lesson 7 Student Pages A and B were used as the individual assessment task for Activity 8. The students (except Brian) experienced little difficulty in completing Pages A and B which required the students to determine the larger of two fractions. Brian continued to choose the larger number in the denominators to be the reference point thus he chose one-third to be greater than one-half and one-twelfth to be larger than one-half. The teacher used paper strips, Fraction Stack bar and fraction circles to help in clearing up his misconception. At this point Brian clearly displayed a dependency on the physical aspects of the fraction to aid him in making crucial decision relating to the tasks at hand. Both the tasks on Activity 8 and the relevant assessment task revealed that the students possessed the ability to order symbolic and non-symbolic fractions. The students used various manipulatives and mental image to order the given fraction (orally and/or written). Their success at completing the tasks showed that they possessed one of the key criteria for logico-mathematical abstraction understanding. Activity 10 contained more challenging tasks on ordering symbolic fractions with the aim of assessing students formal understanding in a more specific way. Tasks 1, 2 and 3 of Activity 9 had the students discovering equivalent fractions via paper folding. Equivalent fractions of one-half (Task 1) were created by folding the paper in two equal parts at each turn. The students wrote the equivalent fractions that they generated and were advised to write three more equivalent fractions without folding the paper. The students easily recognized the pattern: = = = = = = Mary: We just need to multiply the top number and the bottom number by two. The next two tasks allowed the students to generate two equivalent families of onefourth. The sixth-graders never encountered any difficulty when determining the first three equivalent fractions from their folded paper. They easily recognized that they can obtain the required fractions by folding the quarter in half and third respectively. It took quite a time before Claudia recognized that she could multiply both numbers in the 144

158 fraction by two to get the next one (Task 2). The students responses revealed that they were capable of forming equivalent fractions without the use of manipulatives. As verification, the students were asked to generate five equivalent fractions for two-thirds. All the students generated equivalents fractions derived from multiplying two-thirds by two. When asked if six-ninths was an equivalent fraction of two-thirds, most of the student hesitated to given an oral answer. Polly: No, because you cannot divide nine by two. Mary: Dave: Chrissy: I don t think so. If you multiply the numbers by two you will not get six-ninths. It is. It s like you take the paper with two-thirds and fold into thirds. If you multiply the numbers by three you will get six-ninths so I guess they are the same. The teacher drew nine circles with three in each row on the chalkboard. Two circles in each row were shaded. I: What fraction is represented here? Alton: Six-ninths. Most of the students agreed with him. I: Very good. Is there another fraction that could be used to describe this same diagram? Marla: Yes. Two-thirds I: Does anyone agree with her? Dave, Jay, Paul, and Chrissy agreed with her. Marla reminded them of the previous tasks (Activity 5) they did. Dave chimed in: take one row at a time and you will see one twothirds in each row. Karla, Carol and Kisha identified with Dave s explanation and nodded in agreement. Karla: So I can multiply by any number to get equivalent fractions? I: Let s see. If you multiply one-fourth by four what will you get? Brian and Josh: Four-sixteenths. I: Use your chips to form four-sixteenths. Make four equal groups. How many will be shaded in each group? 145

159 Class: One. The students formed four-sixteenths with the chips as they were instructed to do. I: Based on what you know and what you can see from your model are one-fourth and four-sixteenths the same. Class: Yes. I: So, Karla, do you think if I multiply both the numerator and denominator by five, I ll get an equivalent fraction? Karla: Polly: Jay: Yes. We can divide by the same number too, right? Cause if you divide four-sixteenths by four you will get one-fourth. I am going to multiply by one million. The students started suggesting numbers that they could multiply one-fourth by to get an equivalent fraction. RNP Level One Lesson 10 Student Page A and Lesson 15 Student Page B completed the set of tasks for Activity 9. The first task sheet was used to form a link between the students logico-physical abstraction understanding and their logicomathematical abstraction and formal understanding. The students had already encountered partitioning already partitioned wholes. After symbolically and orally identify the original fraction shown in the diagram the students were expected to create equivalent fractions by repartitioning the given diagram to reproduce the required equivalent fraction. The students found this task interesting and completed it with ease. Carol used the whole number to represent the fraction shaded. When questioned, she said that she knew it should be a fraction but she could not be bothered to write so much. Lesson 15 Student Page B questions were modeled similarly to the second part of Activity 7. Given the number of chips to use, the students were expected to show the given fraction and then give an equivalent fraction. These tasks proved challenging to some of the students. With no teacher intervention, the students were able to negotiate among themselves and then arrived at agreed solutions. For example, on item 2 which asked them to show two-thirds using 15 chips and then give another name for two-thirds, the students wondered if they should make groups of threes and shade two in each group 146

160 or make three groups of five and shade two of the groups. After deciding on the configuration it became a simple task to name the equivalent fraction. Most of the students chose to make three groups of five and then shade two of the groups. Initially, Alton displayed similar confusion as experienced in Activity 7 but as he worked with his classmates he was able to arrive at the correct solution. RNP s Lesson 15 Student Pages D and E Level One were used as the assessment task for Activity 9. These lessons were chosen as they reflected the goal of Activity 9, that is, to assess the students ability to determine equivalent fractions of fractions given through varying media pictures, orally and symbolically. An example of one of the items on the task sheet is shown in Figure 28. The students regrouped and quantified fractions as they completed the number sentences and write equivalent fractions from the discrete wholes shown in each item (11items in all). Figure 28. Item 11 RNP Lesson 15 Student Page E Level One The final activity used to complete the levels and constituent parts of Herscovics and Bergeron s (1988) model required the students to use the underlying physical properties of unit and non-unit fractions to order fractions mentally by comparing them to one-half and the whole. This activity was designed to specifically assess the students formal understanding. RNP s Lesson 18 Student Page A Level One and Lesson 8 Student Pages A and B Level 2 were used to achieve this goal. The first three items on Lesson 18 presented the problems using real-life situations. In item 1 the students had to 5 6 determine if Margo who ate of a large pizza had more to eat than Jose who ate16 of 8 147

161 another large pizza. The excerpt below represents the typical responses given by the sixth-graders. Josh: Margo because with 8 pieces the pieces are bigger and because she ate more than one-half. Dahlia: Jose ate less than one-half while Margo ate more than one-half. Carol: Margo ate more pieces because 16 6 would be smaller than 8 5. None of the students made note of the fact that 16 6 is equivalent to 8 3 which is smaller 5 than. This was probably due to the title of the lesson: Comparing to 1-half. Some of 8 the students (e.g. Brian, Dave and Jay) drew diagrams to illustrate their decisions. They were able to accurately complete the other items of Lesson 18 using similar strategies as employed on the first item. RNP Lesson 8 Student Pages A and B Level Two required the students to determine larger fractions by picturing the fractions in their minds. Page A asked them to 1 decide whether a given fraction is greater than or less than. The students found this 2 task relatively simple. Most of them found the half of the denominator of the fraction that is compared to one-half while others picture the slices of a pizza or a rectangular bar. Page B made no reference to the one-half but the students had to order six sets of familiar and unfamiliar fractions. At the end of ordering the fractions, they were to determine which fractions were close to 0, ½ or 1. Here are some of the strategies they used. All the students used at least one of these strategies. Polly: I picture them being pies, and I tried to find the bigger slice. Richy: I just compared all the fractions to ½ or 2/4. Jonah: Chrissy: I thought of them as rectangle. I picture them as a fraction that was divided into whatever the fraction was. Alton: I tried to find which fraction is closer to 0, ½ or

162 Analysis Karla: I looked at the top numbers to see which one was bigger than I looked the bottom numbers and compared to see how big it was and how much would be shaded. By the end of the last four activities cited above, the participants of this study demonstrated logico-mathematical abstraction and formal understanding as they performed tasks specifically designed to examine their manipulation of unit and non-unit fractions smaller than the whole. They were able to write conventional symbols for unit and non-unit fractions, mathematically reconstitute wholes from unit fractions by recognizing and using the fact that the whole is made up of its parts, use the relative size of unit and non-unit fractions to order fractions and generate equivalent fractions. The most challenging task that the students encountered as they were assessed for these two types of understanding was the reconstitution of discrete wholes given parts of the whole. The major deterrent was the students insistency to inadvertently group in twos despite the number of parts mentioned in the numerator. After a succession of assessment tasks and follow-up interviews the students who experienced the difficulty were able to perform similar tasks accurately. They were able to represent fractions in three different formats: fraction name, pictorial representation and fraction symbol. They were capable of orally quantifying the part-whole relationships depicted in any of the formats as they generated equivalent fractions and unit fractions from non-unit fractions. SUMMARY This chapter was dedicated to the answering of research question number one, which sought to investigate and examine the nature of the participants understanding of the fraction concept. Specific aspects of this vital middle school topic, such as the students definition and understanding of the term fraction, their method of unitizing and partitioning, and the strategies they employed when ordering fractions and generating equivalent fractions, were spotlighted through the medium of ten task-based activities some of which were adapted from Boulet (1993) work on the epistemological analysis of the unit fraction with a set of fourth-graders. Herscovics and Bergeron s (1988) model of understanding and the pretest provided the crucial lens through which the teacherresearcher scrutinized and gained a structured insight to these students sense making of fractions. 149

163 At the beginning of the study, most of the students showed signs of very little understanding of this concept as became evident from the results of the pre-test. Their definitions were superficial. As they participated in the activities, there appeared more structured definitions of fractions in one form or the other. Brian, for example, used mixed numbers to name fractions smaller than the whole at the early part of the study but showed signs of fraction maturity as he worked with the task-based activities. This maturity was not only seen in Brian but also in Carol, Claudia, Dahlia, Alton, Karla, Kisha, Ben and Richy. The other students were able to perform most of the tasks with relative ease with just some misconceptions here and there. These misconceptions were addressed through either one-to-one follow-up interviews, group sessions, and/or review sessions at the beginning of the previous activity. Based on the items on the pretest, the participants possessed level of intuitive knowledge, as they were able to satisfy the criterion for this level of understanding the recognition of equal parts. An analysis of ten task-based activities was carried out in accordance with the Herscovics and Bergeron (1988) model of understanding to explicate what encompassed the understanding of the fraction concept among the participants of the study. These students were able to partition discrete and continuous wholes of different shapes and sizes with very little difficulty thus fulfilling the only criterion for the logico-physical procedural understanding. The main difficulty encountered by the students occurred when they had to share a discrete set of objects if the number of objects is not a multiple of the number of equal shares. This difficulty was alleviated as the students worked through the various tasks that involved partitioning. In a number of instances the students existing network of fraction knowledge conflicted with the new knowledge gained through their construction of the fraction. Three activities (2, 3 and 4) were used to assess for logico-physical abstraction understanding the last task in the logico-physical tier of Herscovics and Bergeron s (1988) model. The task-based activities up to this point dealt exclusively with the nonquantified part-whole relationship. Some students, however, exhibited aspects of the logico-mathematical tier due to fraction instructions prior to grade six. Based on the responses (verbally and written) to the tasks for these activities the students had satisfactorily fulfilled the five criteria related to this component of understanding. The 150

164 students had no difficulty reconstituting the whole from its part given one of the parts or many of the parts. One misconception became dominant when part of the whole resembled a sector of a circle. Some of the students claim that the whole must be a circle. Group discussion was used to circumvent this misconception. The students were also able to recognize the part-whole relationships in discrete and continuous wholes despite the transformations and physical attributes of the different parts. There was much perturbation as the students repartitioned already partitioned wholes. This formed the concrete foundation for finding a common denominator a necessary concept in the addition and subtraction of fraction. They eventually developed several partitioning strategies to divide the varying figures. The next six activities allowed the students to perform tasks geared at assessing their logico-mathematical understanding. It was obvious from the results of Activity 5 that the majority of the students were able to quantify part-whole relationships in unit and non-unit fraction. The major hurdle encountered at the logico-physical mathematical constituent part was the renaming of unit fraction from non-unit fractions. The participants logico-mathematical and formal understandings were assessed simultaneously due to the pervading presence of the fraction symbol since the beginning of the study. There were clear evidences of these types of understandings as the students (orally and written) quantified equivalent fractions and ordered unit and non-unit fractions. There were signs of formal understanding as the students were able to generate equivalent fractions and order fractions without the aid of physical referents. 151

165 CHAPTER 5 RESULTS STUDENTS PARTITIONING STRATEGIES Introduction The remaining two research questions will be addressed in this chapter. Based on the students responses on the ten activities that were discussed in the previous chapters, the sixth graders seemed to exhibit the components of understanding as outlined by Herscovics and Bergeron (1988). As the students engaged in these task-based activities, plus two additional activities not discussed before, they displayed varying partitioning strategies that will be scrutinized and discussed in an effort to answer the second research question. These task-based activities also gave them the opportunity to work with various physical referents. These include continuous and discrete objects. The last two activities depicted problem-solving sessions that allowed the students to apply their fraction knowledge in solving problems that model real life situations. The participants behaviors and discussions as they manipulated the objects were carefully observed and recorded and were paramount to the answering of the final question. Research Question #2: What strategies do sixth grade students employ to ensure partitioning or fair sharing as they engage in the processoriented activities? By the end of Activity 3, a majority of the students were aware of the significant role equality of the parts played in determining the part-whole relationship. As the students divided the given wholes, they employed varying strategies to guarantee that the parts were congruent. The act of partitioning or fair sharing is essential in the development of students understanding of fractions and has been part of most students everyday experiences. Lamon (1996) purported that students partitioning strategies were most times influenced by social practices and the commodity being shared. 152

166 It forms the basis for equivalence that in turn forms the foundation for operating on fractions. Partitioning also plays an active role in the development of the fraction concept as it relates to the part-whole, measure and quotient subconstructs. As mentioned in Chapter 1, the students partitioning habit will be explored using the partitioning strategies outlined by Charles and Nason (2000) and Lamon (1996) as the frame of reference. Charles and Nason s (2000) strategies involved the quantification of the part-whole relationship which was not required for the first tier of Herscovics and Bergeron s model (1988) where much of the partitioning tasks were done. Consequently, a purposeful partitioning activity dubbed Fraction Breakfast (see Appendix B) was planned. During this final activity of the study Activity 12 the students participated in an actual breakfast where they had to share food items representing continuous and discrete objects among their group members. There were five groups with four members to a group. Despite the lack of the necessary step of quantifying the part-whole relationship, partitioning tasks from Activities 1 and 4 will be referenced with the aim of confirming the students partitioning habit. For the purpose of this study each strategy will be listed followed by the findings generated from the students observed behaviors. The first nine strategies are the ones mentioned by Charles and Nason (2000) while the last three are gleaned from Lamon s (1996) study on young children s process of unitizing. Three of the strategies reported by Charles and Nason (2000) were not observed during the study and were, therefore, omitted from this chapter. A complete listing of the strategies is included in Chapter Partitive quotient foundational strategy: This strategy involves recognizing the number of people the object(s) must be shared for, generating the fraction name from the number of people, partitioning of the object into that number of equal pieces, and sharing and quantifying each partition. This strategy was used by the members of three groups (Bob s, Mary s and Polly s) in accomplishing Task 3 of Activity 12. This task required them to share three rectangular tostados [toasted bread with cheese] among the four group members. Each tostado was considered to be a whole. Ben (Polly s group): We divided each tostado into fourths to get twelve pieces of fourths. Each person would get one-fourth from each tostado and three-fourths in all. (See Figure 29). 153

167 Figure 29. Example of Partitive Quotient Foundational Strategy 2. Proceduralised partitive quotient strategy: This strategy is a condensed version of the one above. The students exhibiting this strategy would take fewer steps in quantifying the fraction. A person demonstrating knowledge of this strategy would be able to quantify the fraction without actually partitioning the whole. Task 2 in Activity 12 required the students to share a 64 fluid ounces container of orange juice equally among the four group members. Each student had a nine fluid ounces plastic drinking container. Due to the liquid nature of the whole, the students did not attempt to make any drawing to help them decide what fraction of the orange juice each person would get at the first serving assuming that each person would be served the capacity of the container. All five groups readily demonstrated knowledge of this strategy by quantifying each 9 person s share as before partitioning and sharing the orange juice Partitive and quantify by part-whole notion strategy: The steps for this strategy involves the partitioning of each whole object to match the number of parts needed, sharing one part from each object to each person before quantifying the part-whole relationship. This strategy is different from the first strategy in that for the partitive quotient foundational strategy the learner would start the quantification process from the onset of the partitioning procedure whereas for the strategy mentioned here the quantification of each share could not be achieved until after the sharing had been completed (Charles & Nason, 2000, p. 201). This strategy was evident in the completion of Task 3 in Activity 12. Two groups (Dave s and Chrissy s) assigned each person their share of the tostado before concluding that each person got

168 Carol[ Dave s group]: We cut all the wholes into 4 pieces each to get 12 pieces in all and gave each person 3. [She paused while the tostado was shared.] So each person gets Regrouping strategy: After recognizing the number of parts and that the number of parts gives the fraction name, the students using this strategy will realize that the number of pieces to be shared can be obtained by multiplying the number of discrete objects by the number of parts in each whole. This is done prior to the quantification of each share. This strategy was utilized by four of the five groups in determining the fraction of the set of quiches each person should get if 12 quiches should be divided equally among four persons (Task 1 Activity 12). The following two excerpts represent the responses received from the students who used this strategy. Mary: I multiply by 3. And since there are four people and there are 12 quiches so you also divide Richy: The way I got 3 parts is that I divided 12 by 3, also because 4 people multiply by 3 quiches equal Horizontal partitioning strategy: On recognizing the number of parts needed to partition the whole, the students employing this strategy will horizontally divide the circular whole into the desired number of parts; quantify each part before recognizing that the parts are unequal, for example using vertical lines to divide a circle into four equal parts. Considering earlier misconceptions displayed by Ashley, Kisha and Dahlia concerning partitioning a circle in three shares, it is worthy mentioning that none of the students used this strategy when sharing the pancakes in Task 4 of Activity People by objects strategy: In sharing circular objects among a number of people, the students using this strategy will recognize the number of people and the number of objects. The number of pieces that each whole will be divided into is generated from the product of the number of people and the number of objects. This is followed by the quantification of each share. Although this partitioning strategy was not evident in any of the tasks that involved circular food items (quiches and pancakes), Mary s group employed this strategy in determining each group member s share of tostados for Task 3 Activity 12. They figured that there were three tostados and four group members so 155

169 each whole needed to be partitioned into twelfths. Each person would get one twelfth of 3 each tostado. They quantified each person s total share as Halving object then halving again and again strategy: This strategy involves an iterative halving of the object until the desired number of parts is obtained. This strategy is referred to as algorithmic halving by Poither and Sawada (1983). Although this strategy was not evident in the Fraction Breakfast, some students repeatedly used this strategy in partitioning tasks for Activities 1 and 4. For example in Task 2b Activity 4 where the students had to repartition a circular pie previously partitioned in thirds to share for four people, some of the sixth-graders halved the thirds, then halved each of the halves of the thirds resulting in four equal parts to the third [12 parts to the whole.] During the paper-folding exercises done during the study, it was observed that the initial procedure to obtain a fraction for a majority of the students was to fold the paper in halves. For instance, in one session the students were asked to fold one-half followed by a quarter. They were all successful in completing this task. They were then asked to fold the same paper, already folded in fourths, to obtain twelve equal parts. Most of the students proceeded to fold the already folded paper in half. This was done three times. They recognized that 3 4 = 12 but could not understand the reason for obtaining thirtytwo equal parts on the paper instead of the desired twelve parts. They were encouraged to think aloud about what they did. Polly spoke up: Every time we fold, we are doubling the parts. They discovered Polly s pattern after much practice. Jay suggested and then demonstrated how to make a tri-fold. 8. Whole to each person then half remaining objects between half people strategy: This strategy and the one immediately below are extensions of the halving strategy. Karla and Claudia deviated from their group members who displayed the mark-all strategy mentioned below. During the group discussion on the strategy that could be used to share the pancakes in Task 4 Activity 12, Karla reasoned that each person can get a whole. Take two of the pancakes that are left over and cut them in half, give each person one piece and then cut the last pancake in four and give each person one of those. She made a pictorial representation of her strategy (see Figure 30). Claudia nodded in agreement. 156

170 Claudia: Each person gets one whole, one half and one fourth. Figure 30. Karla s Representation of the Sharing of the Pancakes 9. Half to each person then a quarter to each person strategy: This strategy is selfexplanatory. Dahlia s group utilized this strategy in assigning the amount of pancakes each member should receive in Task 4 of Activity 12. They halved each pancake, further divided the last pancake into fourths. Each person in the group received three halves and a fourth as shown here:. 10. Preserved-pieces strategy: Whenever the sharing involves more than one of the discrete objects of the total number of objects to be shared, the student exhibiting knowledge of this strategy would partition only the object(s) that needed dividing leaving the other objects intact. This strategy was widely used in completing Task 4 of Activity 12 where the students had to equally share seven pancakes among the four of them. Marla s group left four of the pancakes intact and then partitioning each of the remaining three pancakes into fourths with each person getting one-fourth of each. By the end of the process, each group member would receive one whole pancake and three-fourths (see Figure 31). Similar sharing was done by Jonah s group. Figure 31. Example of the Preserved-Pieces Strategy 157

171 11. Mark-all strategy: In this strategy all the discrete objects will be partitioned to obtain the desired number of pieces. However, at the time of sharing, only the objects that needed cutting will be cut. The pictorial representation that Dave and Carol gave to accompany Task 4 on Activity 12 indicated that they visually partitioned each pancake into four pieces but in assigning the actual share they gave each person one whole plus three-fourths of the remaining three pancakes. Each person would get. 12. Distribution strategy: For this strategy, all the discrete objects will be partitioned and cut. The pieces will then be distributed to each person until finished. Alton s group applied this strategy in sharing the pancakes in Task 4 Activity 12. Each pancake was cut in fourths where each person was given one fourth of each pancake eventually resulting in each person getting the equivalent to seventh-fourths. The members of this group recognized that that they could use the preserved-pieces strategy but opted to use this method instead to make it interesting. Research Question #3: How do physical and real world representations aid in the development of students understanding of fractions? Paramount to the first tier of Herscovics and Bergeron s (1988) model of understanding is the physical manipulation of objects that aid in laying the foundation for the constituent parts of the model. Throughout the study, these concrete objects formed the basic building blocks for the conceptual understanding of fractions. As mentioned in the previous chapters, varying manipulatives were integrated in the task-based activities. These included two-sided colored chips or counters, fraction circles, fraction triangles, fraction squares, Fraction Stack Bar, Fraction Balance and fraction stencils. From the onset of the study, the students were presented with a personal manipulative bag containing a ruler, erasers, flexible wires (similar to pipe cleaners) and two pencils. The students were free to use any of manipulatives in their bag, any of the ones mentioned above and those assigned during the specific activities to aid them with the assigned tasks. These include numerous task-based cut-outs, measuring cups, plastic containers for liquid and the liquid itself. As Steffe and D Ambrosio s (1995) hypothetical learning trajectory indicated, major significance was placed on the presentation of information, the situation of the scenarios and the support given to the learners during the process of constructing 158

172 knowledge. Lesh s translational model (Lesh, Post & Behr, 1987) depicts the connections among different external representations including real world situations and manipulative aids. This translational model reflects the ideology that the understanding of fraction is reflected in the students ability to represent fraction ideas in multiple ways, including the ability to make connections among the different representations. As the students worked with several manipulative models, they were advised to consider how these models were alike and different. They worked individually, in small groups as well as interacted with the teacher-researcher in a whole class setting as they discussed fraction ideas and worked with the various physical referents. Except for a number of the RNP activities which specified which manipulative to use, the students were not just limited to one type of manipulatives but were encouraged to use the ones best suited to aid in the completion of the task(s) at hand. Even though an attempt was made by the teacher-researcher to use the fraction circles, fraction triangles and fraction squares/rectangles to illustrate fractions, the students often resorted to using only the fraction circles and rectangles to model their fractions. This was also noted during the paper folding exercises. The students had the opportunity to work with paper circles, paper triangles or paper rectangles. Only Josh, Dave and Jay took up the challenge of folding the paper triangles to yield the given fractions. The paper rectangle became the most popular as it was easier to bend in shape (Ben). Karla remarked: It is easy to use the fraction triangles but it is hard to draw them and show equal parts for the fifths and sixths. For a number of the activities, the students worked with both discrete and continuous analogs. Even after working with both continuous and discrete model for the duration of the study, it was noted that 85% of the participants chose to draw a continuous diagram to illustrate a fraction in response to Task 7b in Activity 12. The remaining 15% opted for a symbolic representation. Left on their own, most of the students displayed an affinity to working with continuous rather than discrete objects. As Carol put it: For this one [showing the rectangular-shaped paper] there is only one thing for the whole but more than one thing here [showing the set of counters] stand for the whole. 159

173 In Activities 5, 6 and 7, the students worked extensively with discrete models mainly because they were required to. They exhibited relative ease when working with these models. Completing discrete wholes in Activity 7 posed quite a challenge to a number of the students. The main problem, however, was not in the type of analogs (continuous or discrete) used but in the students interpretation and understanding of the numerator and denominator. Despite the difficulty that they faced to ensure equality of the parts when dividing a circle, the circle seemed to be the most popular of all the continuous wholes introduced during the study. Although no specific attempt was made to eliminate the use of physical referents on problem-solving tasks, it was observed that for the most part, the sixth graders treated problems written in symbolic form differently from those connected to physical referents, partly because the first set of activities never stressed the quantification of fractions or used fraction symbols or words. Activity 11 presented a number of problem-solving tasks including Poither and Sawada (1983) Cake problem, Lesson 6 Student Page C and Lesson 22 Student Page A (Level 2) from RNP. These activities asked the students to use chips or draw pictures to help them solve the problem. Majority of the students ignored the instructions and proceeded to solve the problems without the use of any physical referents. For example, on item 5 from Lesson 22 (shown in Figure 32), Marla applied the division algorithm to arrive at the solution for the problem. She divided 18 by nine to get two and then used the two to multiply the five to yield the desired answer 10. She continued to use the same algorithm for the remaining problems on the task sheet. Her method typified the method used by most of the students. Carol, Karla and Claudia were the only students who attempted to use some form of diagram to illustrate their solution to that problem. Todd s mother cut a cake into 18 pieces. She is going to take 5/9 of the cake to a party. How many pieces of cake will she take to the party? Figure 32. Item 5 RNP Lesson 22 Student Page A - Level 2 160

174 The ease at which these students solved these problem solving tasks on fractions surprised the researcher. In my teaching experiences over the years I have seen eight- and nine-graders baffled over problem solving questions such as the one shown in Figure 34. The participants in this study were able to transfer knowledge gained from previous activities to the problem solving tasks. For example, Richy reasoned that if 1/3 of the box of gumdrops has seven gumdrops and there are three thirds in a whole then the box has 21 gumdrops, cause 3 7 = 21 (see Item 2 Lesson 6 Student Page C RNP Level 2). Karla: 21 gumdrops are needed because you have to get 2/3 more so you add 1 more group of 7 then in order to get 3/3 you add 1 more group of 7 to equal 21. [She used x s to illustrate her solution when asked to verify.] Chrissy, one of the few who did some form of pictorial representation, arrived at the solution by drawing a circle, cut the circle in three equal parts, assigned seven to each part then add the three seven s to get 21. Alton drew seven small shaded circles to represent one group of one-third and then drew two more sets of seven circles to obtain 21. No form of teacher intervention was necessary as the students were simply working individually and verifying their results among themselves. During the problem-solving sessions, the teacher-researcher main task was to ensure that the students were on-task. When asked about the impact of the real world representation tasks, the students had positive things to say. Here are a few of the comments. Polly: They helped me to understand fraction better because they are used in examples and stories. Chrissy: Marla: Alton: Brian: Bob: I am able to look at fractions in different ways. I am better able to understand fractions. They seem to let working with fractions become easier. This shows that we use fractions everyday without realizing it. When we grow up we will be having fractions everywhere we go. I love the sharing activities. Fractions seem to be fun. The way the fractions lessons are done are really cool. I know I will always remember how to work with fractions. 161

175 Mary: Paul: They help to show that fractions are interesting. They teach me how to use fractions. Carol: I can use what I learn to divide a cake or pie equally if someone asks me. In ordering fractions, a number of the students conjured pictures of slices of a pizza or a rectangular chocolate bar. Polly and Jay placed the fractions within a real world setting before arriving at a solution. The class was ordering fractions. Polly: I picture them being pies, and I tried to find the bigger slice. Jay: I don t think so: 2/7 is bigger. It s like sharing a pizza. You get more to a slice when you have to share for seven people instead of nine. You re going to have two bigger sevenths than two ninths. The Fraction Breakfast gave the students the opportunity to practice their sharing abilities in real ways. Some of the students credited this activity as one of their reasons for understanding fractions a whole lot more. Jay: The Fraction Breakfast shows that learning fraction can be fun. Josh:. Mary: It shows how fractions are important in our everyday lives without realizing it I will never look at fractions the same way again. They will have more meaning to me. SUMMARY The equality of the parts in a whole plays a critical role in the conceptual understanding of fractions. Besides being cognizant of that fact, a learner should possess the capability to divide given wholes (continuous or discrete) into equal parts. To achieve this end, various partitioning strategies were observed and documented in the literature. Poither and Sawada (1983), Charles and Nason (2000) and Lamon (1996) have shared a number of partitioning strategies they observed while working with young children. Some of the strategies are common among the prominent literature mentioned above, with slight variation. This study used the partitioning strategies reported by Charles and Nason (2000) and Lamon (1996). Activities 11and 12 were purposefully designed to examine the students partitioning behavior and their strategy for solving 162

176 problems related to the real world. The students responses to these activities revealed that they were able to partition the given objects in a variety of ways. Their partitioning skills were honed during the tasks for Activities 1 and 4. The students expressed the need to make other mathematics topic relevant by designing the tasks similar to the problem solving tasks used during the study. They enjoyed interacting with each other. 163

177 CHAPTER 6 SUMMARY, CONCLUSION, DISCUSSION AND IMPLICATIONS Introduction This study that was done with the set of sixth graders from a private school in southeast Florida was a whole class exploratory study designed to examine these students understanding of fractions as they participated in task-based activities during an eight week teaching sequence. These activities included tasks depicting real world representations and those requiring the use of physical referents. The students understanding of fraction was analyzed and assessed using the Herscovics and Bergeron s (1988) model of understanding. Mathematical understanding is achieved when the learner is capable of making connections between ideas, facts, or procedures (Hiebert & Carpenter, 1992). In reference to the current study, an understanding of the part-whole and quotient subconstructs of a fraction does not necessarily imply that the students are able to make the connection between each. Throughout this study, a careful analysis was undertaken to reveal the connections, if any, the students made while engaging in the process-oriented activities. The analyses and interpretations of the behaviors of the students were made from the written task sheets and the video- and audio-tapes collected during the study. These were augmented with the teacher-researcher s journal and input from a peer reviewer. The function of this chapter is primarily to discuss and summarize the findings unraveled during the teaching sequence, and to outline the implications of the study. The summary deriving from the study will be discussed under the major themes: the nature of the student understanding of fractions, the students partitioning strategies and physical and real world representations. The limitations of the study were already discussed in Chapter

178 The Nature of the Students Understanding of Fractions It was noted during the study that very few students initially had a formal definition of a fraction although they were able to give a symbolic representation. The few who attempted a definition hinted towards the part-whole concept of the fraction. By the end of the study, the majority of the students viewed the fraction from a part-whole perspective. The other subconstructs of the fraction were possibly ignored due to the emphasis that was placed on the part-whole relationship for the first five activities done during the study. With the exception of the ratio subconstruct, the other three subconstructs (measure, operator and quotient) as outlined by Kieren (1976) were included in some of the activities but were not accentuated in a verbal or written form as much as the part-whole relationship. The students worked with the number line and measuring cups (measure subconstruct) in at least three activities. The quotient subconstruct was referred at least two activities. For example, in Activity 11 the students were asked to share nine cookies equally among four persons. Another problem solving 3 task required them to find how many children went on a trip if of the children from a 5 class of 25 students went on the trip (Activity 11). Findings related to the study also indicated that the students possessed a clearer idea of the distinction between the top number (numerator) and the bottom number (denominator) of the fraction. Herscovics and Bergeron s (1988) model was used as the lens through which the students understanding of fractions was observed. This model was developed by Herscovics and Bergeron in an attempt to provide an epistemological analysis of the various conceptual schemata taught at the elementary level (Herscovics and Bergeron, 1988, p. 15). It is a two-tiered model; the first tier describes the understanding of the preliminary physical concepts while the second tier describes the understanding of the mathematical concept itself. Each tier has three levels or constituent parts which do not necessarily occur in a linear fashion. Due to the physical connotation as well as the mathematical phenomenon that is attached to the fraction concept, this model of understanding is well suited to be used as the analysis tool through which this subset of the rational number concept was observed. 165

179 Although there were evidences that these students had previous formal instructions in fraction, none of the students were operating at an advanced stage of formalization. This was evident from the findings related to the pretest where most of the students were placed in category 2 revealing the idea that the students mostly had a limited idea of the fraction concept. Jay, Polly and Marla frequently quantified the partwhole relationship before it was required but still exhibited signs of misconception in the understanding of the physical concepts associated with the fraction. The students satisfied the criterion for intuitive understanding during the pretest and later activities by recognizing parts that were partitioned. This included the recognition of equality of parts even though the parts were transformed. During the teaching sequence, the students participated in twelve activities designed and developed to investigate the remaining five levels or constituent parts of the Herscovics and Bergeron s (1988) model of understanding. The students exhibited signs of possessing logico-physical procedural understanding as they partitioned continuous and discrete wholes the main criterion for this level of understanding. The main hitch came when the students were dividing the circle into equal parts of three. Ashley drew two vertical lines to complete her partition but was later challenged by the students as to the equality of the parts. Marla shared with the students her method of ensuring that the parts were equal. This method became the standard strategy used by the students when partitioning a circle into three parts throughout the study. Partitioning the equilateral triangle did prove to be challenging to some of the students but they were able to negotiate and practice until they were satisfied the parts were equal. When partitioning discrete wholes, the students experienced little difficulty when the number of objects to be partitioned is a multiple of the number of partitions. They used the division algorithm or dealing strategy (Davis & Hunting, 1990; Davis & Pitkethly, 1990). When asked to partition 12 circular chips into 5 parts, all of the students were momentarily stumped as they were not sure of what to do with the remaining two chips. Jay offered a solution that made sense to them so the students proceeded to divide the remaining two cookies into five parts each then dealt one part of each circle to the five groups. They also used varying unique and familiar techniques to ensure equality of the parts. It was noted that information gained from a previous unit in geometry was referenced while checking for 166

180 equal parts. The geometry unit was done before the fraction unit solely because of the reorganization of the time to facilitate the study. The logico-physical abstraction level of understanding is the last component in the first tier of Herscovics and Bergeron s (1988) model of understanding. At this level the relationship between the part and whole is emphasized. As soon as the students were clear on the meaning of the term part-whole relationship it became evident that the sixth graders possessed logico-physical abstraction understanding as they successfully satisfied the five criteria related to this level. The findings of the first four activities of the teaching sequence indicate that the students understood the physical concepts that underlie the fraction. These activities aimed at looking at the unquantified part-whole relationship via partitioning, repartitioning and determining part-whole relationships. The use of fraction language at this level was ignored and not encouraged. The first two activities showed that the students were capable of reconstituting the whole from its parts, were aware of the equivalence of the part-whole relationship in spite of a variation in the physical attributes of the whole, they recognized the equivalence of the part-whole relationship regardless of the physical transformations of the whole making note of the relationship between the size of the parts and the number of equal shares. The most challenging criterion to be satisfied was encountered when the students were given the opportunity to repartition already partitioned continuous whole. Repartitioning discrete whole would result in the simple task of regrouping where the original partition would be lost. The art of regrouping was already evident while assessing for logico-physical procedural understanding and consequently tasks involving discrete wholes were eliminated from the tasks comprising this activity. The participants were challenged by the tasks of repartitioning already partitioned wholes where they could not apply the halving algorithm thus the first of the three tasks were accomplished with ease. When asked to repartition a three-part chocolate bar so four persons could get an equal share, the students considered the task to be impossible. After much discussion and negotiation, seven different configurations were evident. The most popular configuration that emerged yielded twelve equivalent parts the number of parts analogous to the lowest common multiple of 3 and 4. The remaining tasks in this activity were completed with 167

181 relative ease as the students used the knowledge gleaned from this task to do their repartitioning. Five activities were used to assess the students understanding of the emerging mathematical concept as it related to the fraction concept. The quantification of the fraction, verbally and written, formed the core of understanding for the second tier of Herscovics and Bergeron s (1988) model of understanding. At the logico-mathematical constituent part, the students were able to orally express the fractions associated with a number of concretely depicted part-whole relationships and to make concrete representations of part-whole relationships from orally given fractions. This was done with relative ease with the greatest challenge occurring when the students had to rename non-unit fractions to unit fractions. It was evident from the responses that the sixth graders demonstrated logico-mathematical procedural understanding of unit and non-unit fractions. One of the misconceptions that emerged from the activities had to do with the identification of the whole if the whole was part of a circle. The initial inclination of the students was to complete the circle figuring that the whole must be a circle. It took much discussion before some of the students became convinced that John s patio (see Assessment Task Activity 5 in Appendix B) represented a whole although it had a semi-circular shape. By and large, the students possessed explicit logico-mathematical procedures that they could use to relate to the underlying preliminary physical concepts and were capable of using these procedures appropriately. The use of physical referents at this point was downplayed although the students had the opportunity to reference them if necessary. The art of paper folding was introduced and used in the second tier of the model of understanding. Due to the prevailing use of the fraction symbol in previous activities by the students, the logico-mathematical abstraction and formal understanding were dealt with simultaneously. One of the prerequisites for formal understanding is the mastery of the conventional fraction symbol. With the aid of a RNP activity, the students had the opportunity to view the relationship that existed among three different representations of a fraction - fraction name, fraction symbol and a pictorial representation. There were able to complete this task with very little challenge. The tasks at the logico-mathematical abstraction level paralleled those encountered at the logico-physical abstraction level with 168

182 the exception that at the second tier constituent part, the use of the fraction language was permissible and evaluated. The possession of logico-mathematical abstraction understanding as it relates to the fraction concept indicates that the learner is capable of reconstituting the whole from its parts, ordering fractions according to their size and generating and quantifying equivalent fractions. It was obvious that some of the students possessed this type of understanding even before they did the tasks designed to assess such understanding. Activities 7 10 solidified this notion and also gave proof to the fact that the participants of this study, despite the hurdles they encountered in completing some of the tasks, did possess this type of understanding. The students worked to reconstitute continuous as well as discrete wholes. Activity 7 provided the most challenge as their understanding of the numerator and the denominator was highlighted. Two sessions of Activity 7, two assessment tasks and individual interviews were held with the aim of clarifying and verifying the students ability to reconstitute discrete wholes given parts of the whole. The major deterrent was the students insistency to inadvertently group in twos despite the number of parts mentioned in the numerator. Formal understanding was evident as the students were able to generate equivalent fractions and order fractions without the aid of physical referents. On tasks that required the use of the fraction as an operator, the students displayed a level of formal understanding that was not expected by the researcher. The students solved real world representation problems without any form of teacher intervention and without little use of manipulatives even when instructed to do so. The Students Partitioning Strategies One of the aims of this study was to reveal the partitioning strategies that the sixth graders employed while completing various tasks throughout the duration of the study. These strategies were categorized according to Charles and Nason (2000) and Lamon (1996) classification of partitioning strategies. Depending on the task at hand, each child opted to use one or more of the partitioning strategies that were outlined by the researchers above. Throughout the study the students were given the opportunity to partition continuous and discrete wholes. They partitioned regular geometric shapes such as the circle, parallelogram, triangle and rectangle. They partitioned cutout shapes of stars, the letter L, heart shapes and other irregular geometric shapes. They partitioned 169

183 lines and liquids and at the end of the study had the opportunity to share items of food equally among themselves. Twelve distinct partitioning strategies (see Chapter 5) were observed. The three most popular strategies were the regrouping strategy identified by Charles and Nason (2000), preserved-pieces strategy and mark-all strategy identified by Lamon (1996). Physical and Real World Representations The manipulatives or physical referents used during this study were carefully selected and ranged from cutouts of figures to graduated containers for assessing students partitioning strategies. Each manipulative was categorized as either a continuous or discrete whole depending on the context in which it was placed. For example, one circle could be used as a continuous whole and a set of circles used as a set of discrete objects. Based on the findings of the study the students showed a preference toward working with the continuous whole than the discrete whole although some research literature purported that children develop an understanding of discrete quantities before a comparable understanding of continuous quantities (Hiebert & Tonnessen, 1978; Poither & Sawada, 1983). Although discrete models were used by the students, majority of the students exhibited a preference in using continuous models as forms of reference for given fractions. The participants of the study have expressed in verbal and written form their appreciation for working with fraction problems that represent real world situations. Significance of the Study A study of students construction of fractions as they participate in problemsolving activities and their subsequent understanding of this very important middle school topic is significant for a number of reasons: 1. It adds to the growing scholarly documents on students sense-making of fractions and how this affects further operations on fractions. 2. It provides real-time activities and application where students work with discrete and continuous quantities interchangeably. 3. It provides support for several instructional and curricular strategies. 170

184 4. It provides teachers with an alternative curriculum that addresses not only prerequisite knowledge and skills which support the acquisition of fraction concepts but also the learners intuitive knowledge base. 5. The findings and interpretation of the study can inform classroom practice and pedagogy. 6. The inquiry method that was used during the tasks and interviews will foster an important transformation in the students thinking of fractions. CONCLUSIONS AND DISCUSSION This research project was designed to probe a set of sixth graders sense-making or understanding of fractions as they worked with physical referents and real representations. At the beginning of the study, students such as Brian, Alton, Kisha, Dahlia, Claudia, Ben and Carol s understanding of fraction appeared to be a bit unstable. However, by the end of study, these students along with their classmates seemed to make the qualitative leap of sophistication (Lamon, 1999) towards a formal understanding of fractions. Kamii, Lewis and Kirkland (2001) made the distinction between physical and logicomathematical knowledge which coincides with Herscovics and Bergeron s (1988) model of understanding. Physical knowledge refers to the knowledge of objects that exist in the external world while logicomathematical knowledge consists of the mental relationships, which each learner creates from within. The responses to the tasks done in this study revealed that the participants logicomathematical knowledge was constructed through their own thinking. The physical referents provided the launching pad for such critical thinking. According to Behr, Harel, Post and Lesh (1992) semantic and mathematical analysis of the part-whole, quotient and operator subconstructs of rational number concepts there exists a one-for-one matching between a sequence of manipulations of physical objects and a sequence of manipulations of mathematical symbols. This matching reveals that children s understanding occurs from the manipulation of objects. In traditional classrooms, fractions are taught algorithmically with little attempt to ground them in a meaningful contextual basis. More often than not, they do not have a deep understanding of the mathematics behind them. Consequently, when the given 171

185 problems do not fit neatly into the structure within which the algorithm was taught, even competent students can have difficulty (Bulgar, 2003; Davis & Maher, 1990). Table 13. Classification of Students Difficulties CLASSES DIFFICULTIES PARTITIONING Inability to produce odd partitioning especially in circular wholes. Unreliable strategies to ensure equality of the parts Repartitioning already partitioned whole RECONSTITUTION Assuming that the whole must be a circle if the part is a sector of the circle ORDER Reliance on the whole number language (e.g since 4 > 2 then ¼ > ½ QUANTIFICATION Identifying unit fractions from its equivalent fraction Table 13 highlights the major challenges that one or more students encountered during the study. Research (Behr, Wachsmuth, Post & Lesh, 1984; Lamon, 1999, Mack, 1995) has shown that students whole number language often interfered in their ability to quantify a fraction appropriately. This is especially noted when students are comparing unit fractions such as 1/5 and 1/7. Some students figured that 1/7 is larger than 1/5 because seven is bigger than five. Biddlecomb (2002) commented that this overgeneralization was not only evident in young children but also in college algebra students who made similar errors in claiming that 1/(x + 1) is bigger than 1/x because (x + 1) is larger than x (p. 167). It is important for the teacher to recognize the fraction knowledge that the students bring into the classroom (Mack, 1995). Children at an early age are familiar with sharing their cookie in half with someone else. However, as they get older more sophisticated partitioning methods become more evident. Poither and Sawada (1983) 172

186 noted the difficulties students faced when dividing into odd number of parts. As students are encouraged to discuss the part-whole relationship using different shapes and sizes they realize that it is easier to represent certain fraction using a particular shape. This was evident during this study where the participants frequently used the rectangle to represent fractions with odd number denominators rather than the well-used circle. Very few students in the middle grades (grades 5 9) are able to partition into any number of given parts whether mentally or physically (Smith, 2002). Consequently, significant attention should be placed on partitioning tasks at this level or an earlier grade level as partitioning forms the basic building block for the fraction concept. One danger in emphasizing the part-whole relationship in the construction of a fraction is the possible reinforcement of fraction as two numbers or not a number at all instead of the fraction as a single number (Kieren, 1988). It is also possible that over dependence on the part-whole subconstruct can thwart the development of the fraction as other subconstructs. In addition, sometimes the learners conception of the part-whole meaning of fractions interferes with their ability to show fractions on a number line (Bright, Behr, Post, & Wachsmuth, 1988). Children often consider the given segment of number line as the whole as was evident in the pretest. Critical to the students sense-making or understanding of fractions is the knowledge that the same fraction always represents the same relative amount regardless of the variation in physical attributes and/or transformation of the parts in the whole as highlighted in Activity 3. Different sizes and shapes of objects should be used to aid students to come to this realization. Repartitioning already partitioned whole provides it own set of challenges to students. Repartitioning tasks should be aimed at allowing the students to discover the lowest common multiple or a common multiple of the denominator of original fraction and the denominator of the new required fraction part. For example, when repartitioning a rectangle already partitioned in thirds, into fourths, the strategies used should lead to common multiples such as 12 and 24. The concepts of equivalence and addition and subtraction of fractions arise in repartitioning activities. Manipulatives have been at the forefront of recent educational reforms of mathematics teaching and learning. Yeatts (1991, p. 7) suggested that manipulatives assist students in bridging the gap from their own concrete sensory environment to the 173

187 more abstract levels of mathematics. However, as Kamii, Lewis, and Kirland (2001, p. 31) noted: the mathematics we want children to learn does not exist in manipulatives. It develops as children think, and manipulatives are useful or useless depending on the quality of thinking they stimulate. Ball (1992) pointed out that working with manipulatives do not automatically create mathematical knowledge. In the context of the teaching and learning of fractions, Smith (2002) placed the burden of the proper use of fraction manipulatives on the shoulders of mathematics teachers. He noted that fraction manipulatives that are skillfully used by teachers can support children s developing knowledge of fractions, but only when these materials are seen as representations of many, many examples of divided quantities. Implications for Teaching and Future Research From their constructions of students fraction schemes, D Ambrosio and Mewborn (1994) concluded from their findings that developing a conceptual understanding of fractions takes time (p. 160). Even after eight weeks of intense investigations and observations of students performance on fraction activities, the surface of this vital middle school topic is barely scratched. Considering the number of topics to be covered in the middle school mathematics curriculum, hardly any mathematics teacher can realistically spend so much time on only the basic aspects of the topic. Operations on fractions must be done before seventh grade. I am suggesting that fewer topics be taught at the sixth grade level so that ample time can be spent in the teaching and learning of fractions. This will minimize the time spent on fraction instruction in seventh and eighth grade giving way for the introduction of the topics eliminated from the sixth grade curriculum. The preliminary partitioning activities should be done in earlier grades (3 5) thus setting the pace for the operations and applications of fractions in sixth grade. Based on her studies on young children s informal knowledge of rational numbers, Mack (1990, 2000) recommended that mathematics teachers seek to develop a strand of rational number knowledge based on partitioning, as a way to build upon students informal knowledge. As previously mentioned, the research literature is replete with studies that suggest that the knowledge of partitioning is necessary for having a conceptual understanding of fraction (Armstrong & Bezuk, 1995; Behr, 174

188 Harel, Post, & Lesh, 1992; Mack, 2000). This study has shown how the partitioning activities that the students were engaged in provided the base for understanding equivalent fractions and the concept of the fraction as an operator. Fraction tasks should be designed to elicit a variety of strategies and representations from the learners. The results of the study indicate that with adequate instruction and meaningful engaging activities, students are capable of solving problems relating to fair sharing and the order and equivalence of fractions. Fraction words can be used instead of fraction symbols in the initial conceptualization of the fraction (Sáenz-Ludlow, 1994). As a matter of fact, research (Saxe, Taylor, McIntosh & Gearhart, 2005) shows that students knowledge of conventional fractional symbolization can develop somewhat independently from partwhole relations. Students are capable of developing a deeper understanding of mathematical symbols by relating symbolic representations to physical referents that are meaningful to them (Ball, 1993; Mack, 1995). Implications from this study support the pedagogy of using physical referents to aid in building middle school students understanding and foundations of fraction concepts. Much emphasis should be placed on the unit fraction prior to the introduction to non-unit fractions. Continuous and discrete wholes should be used with similar intensity. Students should be provided with numerous experiences with manipulative materials. Further research is needed to elaborate how the activities undertaken during the study will impact the students ability to add, subtract, multiply and divide fractions. A more detailed study is necessary to investigate the measure and operator subscontructs. Also, considering the prevalence of the acquired knowledge from the unit of geometry that was taught before the fraction unit, an investigation could be launched as to the effects, if any, that a geometry unit would have on the teaching and learning of fractions. CONCLUDING REMARKS This research does not, by any means, provide an exhaustive study on the understanding of basic fraction concepts. The findings may confirm the role that partitioning and problem solving tasks play in the development of students understanding of fractions but there may be other mathematical ideas that play a similar critical role. Ongoing research and classroom practice will aid in identifying some of 175

189 these ideas. In the meantime, mathematics teachers are continuously faced with the responsibility of creating tasks that meaningful and engaging tasks that will foster higher order thinking skills in students. This responsibility becomes especially crucial and challenging as US students mathematics scores on standardized and international assessment tests are relatively low when compared with scores from other countries. Certain mathematics topics have contributed to this demise, the fraction concept being at the forefront. For the past 25 years there has been concern for the nature of students understanding of fractional numbers. This study attempts to investigate this phenomenon with the hope that the findings will provide a framework from which teachers can begin to create and model task-based activities that will prod the children towards a formal understanding of the fraction concept. There is much to be done. This study is just one of the springboards to achieve this goal. 176

190 APPENDIX A PreTest Fraction Name: Multiple-Choice Questions 1. Which rectangle is not divided into four equal parts? 2. Which shows 4 3 of the picture shaded? A. B. C. D. 177

191 3. The figure below shows that part of a pizza has been eaten. What part of the pizza is still there? A. 3 8 B. 3 5 C. 5 8 D What fraction of the circle is shaded? A. 1 Between 0 and 4 B. 1 1 Between and 4 2 C. 1 3 Between and 2 4 D. 3 Between and

192 5. Which picture shows that 5 2 is equivalent to 10 4? 6. Which of these fractions is smallest? A. 1 6 B. 2 3 C Students in Mrs. Johnson s class were asked to tell why 5 4 is greater than 3 2. Whose reason is the best? A. Kelly said, Because 4 is greater than 2. D. B. Keri said, Because 5 is larger than C. Kim said, Because 5 4 is closer than 3 2 to 1. D. Kevin said, Because is more than Free-Response Questions General Information 1. (a) What is your definition of a fraction? (b) How would you show a friend what a fraction is? How would you show this same fraction in another way? 179

193 2. Do you like learning about fractions? Why? Why not? 3. Why do you think we need to learn about fractions in school? Fractions 4. Where should you put 5 3 on the number line? Why? What does the fraction five-eighths look like? Using a diagram, show this fraction in two different ways? 6. How many fourths are in a whole? Draw a diagram to show this. 7. Think carefully about the following question. Write a complete answer. You may use drawings, words, and numbers to explain your answer. Be sure to show all your work. Jose ate 2 1 of a pizza. Ella ate 2 1 of another pizza. Jose said that he ate more pizza than Ella, but Ella said they both ate the same amount. Use words and pictures to show that Jose could be right. 180

194 8. 6 persons shared 3 pizzas. How much was each person s share? Why do you think so? 9. Here are some cookies: Suppose you eat only the dark ones. How much of the cookies would you eat? Why do think so? If the cookies are rearranged as follows: How much of the cookies would you eat now if you eat the dark ones? Why do you think so? 10. The shaded part of each strip below shows a fraction. A. This fraction strip shows 6 3. B. 181

195 What fraction does this fraction strip show? C. What fraction does this fraction strip show? What do the fractions shown in A, B, and C have in common? Fill in the fraction strips below to show two different fractions that are equivalent to the ones shown in A, B, and C. 11. The circles below represent two pies of the same size one for you and one for your friend. you eat this much your friend eat this much Did you eat as much pie as your friend? Why do you think so? 182

196 12. Bert s father cuts a cake into 8 pieces. He is going to take 3-fourths of the cake to the party. How many pieces of cake will he take with him? A. Draw a picture below to solve the problem. B. The number of pieces of cake taken to the party is: 13. is the unit. You have two pieces this size. What fraction is this? 14. Draw in the box a set of circles 3-fourths as many circles as you see here: 15. What fraction of the circle is part c? 183

197 16. How many fifths are shaded? 17. Circle is 3-fourths of some length. Draw the whole length below. Explain why it is the whole. 184

198 APPENDIX B ACTIVITIES FOR TEACHING SEQUENCE Investigating Fractions: ACTIVITY 1 Task 1: You are given a set of shapes a circle, rectangle, triangle, a parallelogram, and the letter L. a. Partition the circle into three parts. b. Partition the triangle into six parts. c. Partition the rectangle into eight parts. d. Partition the parallelogram into four parts. e. Partition the letter L into five parts. Task 2: You are given twelve (12) chips. a. Partition the set of chips into four parts. b. Partition the set of chips into six parts. c. Partition the set of chips into three parts. d. Partition the set of chips into five parts. Task 3: You are given three (3) lines. a. Partition the first line into two parts. b. Partition the second line into five parts. c. Partition the third line into eight parts. Task 4: You are given one cup of water. a. Partition the cup of water into two shares. b. Partition the cup of water into three shares. c. Partition the cup of water into five shares. 185

199 Individual Assessment Task Activity 1 1. Here are four circles each divided into three parts. Decide which circles are partitioned and which are not. Put a check mark ( ) beside the ones that you believe are partitioned. a. b. c. d. Describe your method of verifying that the circles are partitioned. You may use the letter written beside the circle to identify the circle. 186

200 2. The rectangle below represents a yummy cornbread. You want to share the cornbread equally with five of your friends and yourself. Show two different ways in which in you could divide the cornbread equally among the six of you. a. b. 3. Romy has given you 8 cookies all of the same size to share equally for three persons. In the space provided below, describe and model how you would share the cookies. 187

201 Activity 2 Individual Tasks Task 1. Assemble the parts of the circle to form a whole circle. How many parts make up the whole? Make a model of the whole with its parts in the space below. Task 2. You are each given one part (yellow) of a circle. How many pieces the same size do you believe are needed to complete the whole circle? Group Tasks Discuss and find the solution to these tasks. Task 3. You are given 5 parts of a triangle. How many more pieces of the same size are needed to complete the whole triangle? Task 4. You are given 1 part (green) of circle. How many pieces of the same size are needed to complete the whole circle? Task 5. You are given 8 parts of a whole square. How many more pieces of the same size are needed to complete the whole square? Task 6. You are given 1 part of a cup of water. Predict how many more similar parts will give you a whole cup of water. 188

202 Group Assessment Task Activity 2 Instructions: How many more pieces of the shaded parts are needed to complete the whole object? For each one, share the method or strategy that you use to complete the whole. You may write your solution beside the figure. 1. Rectangle 2. Circle 189

203 Instructions: Look carefully at the diagram below. How many more parts are needed to complete the whole object? For each one, share the method or strategy that you use to complete the whole. You may write your solution beside the figure. 3. Circle 4. Parallelogram 5. Circle 6. Rectangle 190

204 Activity 3 (Shapes for the Activity are found at the end of Appendix B) 1. Phil, John, Jane and Tom each have a rectangular bar of chocolate of different sizes. The rectangles represent the size of the chocolate bars and the shaded part represents how much of each chocolate was eaten. Did each child eat as much of his/her chocolate bar as the other? Discuss in your group. 2. Kris bought a small pizza from Pizza Hut while Mitch bought a small pizza from Papa John. When the pizzas came, they found out that Kris pizza is smaller than Mitch s is. The circles below represent the pizza. The shaded part represents how much of each pizza was eaten. Did Kris eat as much of his pizza as Mitch? Discuss in your group? 3. Look carefully at the shaded parts of the wholes in the following figures. Is the part-whole relationship the same? If so, what is it? If not, why? Discuss in your group. 4. Is the part-whole relationship the same in these squares? If so, what is it? If not, why? Discuss in your group. 5. a) Look at the set of circles and the set of diamonds. Is the part-whole relationship the same in both set? If so, what is it? If not, why? Discuss in your group? b) Compare the larger set of circles with the smaller set of circles. Is the partwhole relationship the same in both set? If so, what is it? If not, why? Discuss in your group? 6. Are the shaded parts in the following squares the same? What is the part-whole relationship? Discuss in your group? 7. The circles on the given sheet of paper represent a set of cookies. The dark ones represent the ones that are eaten. Is the same share of cookie eaten in each set? Discuss in your group? Explain your reasoning. Group Assessment Task Activity 3 Shapes are shown at the end Appendix B Using all the figures you have been working with including some new ones, sort them according to their part-whole relationship. 191

205 Shapes are at the end of Appendix B Activity 4 Task 1 (a) 1. Partition the circle into four parts. 2. Let s assume that the circle represents a pie. Share this pie evenly among eight people. Describe in words what you did. 3. You had already divided the pie into eight equal pieces and then realize that you have to share the pie for eight more people that is, for sixteen people? Show the partition on your circle and describe what you did. 4. What if you have to share the same pie with 32 people? Describe what you would do. Task 1 (b) 1. Divide the rectangle into 6 equal parts. Describe what you did to arrive at your solution. 2. Divide this same rectangle into 12 equal parts. What did you do? Task 2(a) Partition this rectangle into four parts. [You could let the rectangle represent a 3 bar chocolate and you have to share it evenly with four people.] Describe what you would do to obtain four equal shares. Task 2(b) Romy ordered a pizza from Pizza Hut for you. When the pizza came it was divided into 3 equal slices. There are four of you. How would you divide this pizza into 4 equal shares? Use the circle to show. Task 3 (a) This circle represents a circular birthday cake that is already divided into equal slices. Show by shading the parts, how much one person would get if only four persons share the cake. Explain your reasoning. Task 3 (b) This birthday cake is already divided into five equal pieces. Show how you would partition this cake so that 3 people will get an equal share. Explain what you have done. 192

206 Task 3 (c) Divide this rectangular pizza from Dominoes (already sliced in 7 equal parts) into 3 equal shares. Show by shading one person s share. Explain your reasoning. Task 3 (d) This rectangular cake is sliced into 8 pieces. How would you share this cake with 2 persons so that each one will get the same amount? Shade the portion for 1 person. Task 3 (e) This L represents a flat candy cane already striped into equal shares. Repartition this candy to share with four people. 193

207 Assessment Task Activity 4 Task 1. STEPS: Here is a parallelogram that needs to be divided into 12 equal pieces. List the steps that you would use to do this. Follow your steps on the drawing. Task 2: John and his two friends have just won a giant cookie from the Cookie Factory for their art work. When the cookie arrived they found out that 194

208 the cookie is partitioned only for two persons and there are three of them. Using the circle below, how can John divide this cookie so each of the three persons get an equal amount of the cookie? Shade the amount that one person would get. Describe your steps. STEPS: 195

209 Task 3: This chocolate bar is divided into equal pieces. Four children have decided to share the bar equally among them. On the rectangle below, shade one child s share. 196

210 Activity 5 TASK 1: INSTRUCTION: Look at the diagrams that I will be showing you. Tell what the part-whole relationship is? Decide what it is; write it down on the paper. a. b. c. d. e. f. g. h. i. j. k. l. m. n. o.. TASK 2: INSTRUCTION: Look at the diagrams that I will be showing you. Tell what the part-whole relationship is? Decide what it is; write it down on the paper. a. b. c. d. e. f.. TASK 3: INSTRUCTION: Look at the diagrams that I will be showing you. Tell what the unit fraction is for each of them? Decide what it is; write it down on the paper. a. b. c. d. e. f. g. h. i. TASK 4: Listen to the unit fractions that I will be saying. Represent these fractions whichever way you want to. You can do your drawing in the space below or on the blank paper you are given. 197

211 1. Look on the diagram. Show a half. Assessment Task Activity 5 2. Work on these problems in your group. a. Here is picture of a pizza with one piece removed. The piece is of the whole pizza. (say the word) b. Here is a picture of a candy bar which someone has started to cut into pieces. The small piece is of the whole candy bar. (say the word) Draw lines to finish cutting the candy bar into equal parts. c. John has a patio that looks like this. Draw on John s patio to show it divided into two equal-sized parts. Each part is of John s patio. (say the word) Mary said John s patio is really one-half (not a whole.) What would you say to Mary? 3. Listen to the fractions that I will be saying. Draw diagrams to illustrate these fractions. 198

212 4. Gina is given these cookies. She ate some (shown by the shaded ones.) Gina ate of her cookies. (say the word) Say and write the unit fraction for the cookies that Gina ate. 199

213 Activity 6 TASK 1: Use your paper to fold, shade and name the following fractions ,,, Task 2: The students completed Lesson 5 (Student Pages A, B, C, D, E) of RNP Level one. Discrete Tasks for Activity 6 Task 1: Write the word name and the symbol name for the fraction shown below. Task 2: Write the word name and the symbol name for the fraction shown below. Task 3: Write the word name and the symbol name for the fraction shown below. Task: 3: Draw a picture of the fraction two-thirds ( 3 2 ). 200

214 Activity 7 TASK 1: Answer the following: a. How many one-fourths would be needed to make a whole? Draw a picture to show your answer. b. How many one-sevenths would be needed to make a whole? c. This shape represents 5 1 of whole. How many of them are needed to complete the whole. d. Sydney divides his chocolate bar into 6 1 pieces. How many pieces did he get? DO NOT WORK ON TASK 2 UNTIL YOU ARE TOLD TO DO SO!!! Task 2: a. Show three-fourths with chips. Use 12 chips in all. Draw a picture of your display. b. Show two-sixths with chips. Use 6 chips in all. Draw a picture of your display. c. Describe the steps you would take to show 7 3. d. Work in your groups to complete the worksheet. 201

215 Assessment Task Activity 7 (Task 1) 1. How many fifths are in a whole? 1 2. John ordered a large pizza from Papa John. He found out that each slice is of 12 the whole pizza. How many slices of pizza did he get? 3. Show two-fifths with chips. Use 10 chips in all. Draw a picture of your display. 4. Show two-sevenths with chips. Use 7 chips in all. 5. Show 8 5 with chips. Use 16 chips in all. 202

216 Assessment Task Activity 7 (Task 2) 1. How many one-ninths are needed to make a whole? 2. How many 13 1 are needed to make whole? 3. You may use the chips to help you to do these problems. a. Fold the square sheet of paper into eight equal parts. Shade three-fourths. b. Fold the rectangle sheet of paper into nine equal parts. Shade 3 1 of the parts. c. This is 4 1 of whole. Draw the whole. X X d. X X X X is 9 2 of a unit. Draw the unit. X X e. is 8 3 of a whole. Draw the unit. 203

217 Interview Tasks Activity 7 a. This is 4 1 of whole. Draw the whole. X X X b. X X X X is 7 2 of a unit. Draw the unit. X X X c. is 4 3 of a whole. Draw the unit. 204

218 Activity 8 TASK 1: Listen carefully to the questions that I will be giving. Give your answers to these questions. Which is larger? a. One-third or one-fifth? b. One-seventh or one-eleventh? Order these unit fractions from the smallest to the greatest , or Order these fractions from the largest to the smallest ,, and TASK 2: a. Tyra entered Papa John Pizza Factory. She saw 2 friends at one table and 3 friends at another table. Both groups have just been served a large pizza. Which group should she sit with so that she gets the most to eat? Explain your reasoning. b. Using the circles below, show Tyra s share at the table with the 2 friends and also at the table with the 3 friends. Table with 2 friends Table with 3 friends Tasks 3 & 4: Do the worksheets Lesson 6 Student Pages B and C (RNP Level One) Verification Task for Activity 8: Lesson 7 Student Pages A and B (RNP Level One) 205

219 Activity 9 Equivalent Fractions TASK 1: 1 Use your paper strips to fold and shade. Using the same strip, fold your paper into four 2 1 equal parts (1/4). How many fourths are shaded? What is the relationship between 2 2 and? Fold the same strip into eighths, how many eighths are shaded? Write the 4 1 relationship for the three fractions and write two more equivalent fractions of. 2 TASK 2: 1 Use your paper strips to fold and shade. Using the same strip, fold your paper into 4 eight equal parts (1/8). How many eighths are shaded? What is the relationship 1 2 between and? Fold the same strip into sixteenths, how many sixteenths are shaded? 4 8 Write the relationship for the three fractions and write two more equivalent fractions 1 of. 4 TASK 3: 1 Use your paper strips to fold and shade. Using the same strip, fold your paper into 4 twelve equal parts (1/12). How many twelfths are shaded? What is the relationship 1 3 between and12? Fold the same strip into twenty four equal pieces, how many twentyfourths are shaded? Write the relationship for the three fractions and write two more 4 1 equivalent fractions of

220 Activity 10 Students completed Rational Number Project Lesson 18 Student Page A Level One and Lesson 8 Student Pages A and B Level 2. Activity 11 TASK 1: Work on Lesson 22 Task sheet. TASK 2: You are invited to a birthday party where 10 girls are invited. The circular birthday cake is to be divided amongst the ten individuals. Using the given circle, show how you would divide the cake so each person gets the same amount. TASK 3: You are expected to share 9 cookies equally for 4 persons. Use a model to show how you would make your partition. 207

221 Activity 12 Task Sheet for Fraction Breakfast ADDING TASTE TO FRACTIONS GRADE 6 Name: Group #: Introduction Today you will be taking a delicious journey into the world of fractions. As you partake of the morning s breakfast that is provided for you, be mindful of the mathematics that is involved. Remember to share your insights with your group members. Expectations 1. There should be no sharing of food across groups. 2. The food should be shared equally amongst each group member. 3. Answer all questions on the worksheet. 4. Remember that you are being videotaped and audio taped. 5. Please speak clearly in your discussions so that the recording will be clear. Enjoy Your Meal!!! 208

222 FRACTION BREAKFAST WORKSHEET 1. You have 12 quiches to share equally among your group members. a. How many quiches did each person get? b. What fraction of the set of quiches did each person get? c. Describe in detail the strategy or method you used in determining the fraction of quiche each person will get. You may draw a picture to demonstrate what you did. d. Write two fractions that are equivalent to the fraction you wrote above. e. If you give one of your quiches to one of your group members, how many quiches will you now have? Write a fraction to represent the number of quiches that person will have? f. Write the fraction that represents the number of quiches for three group members? Describe how you get your answer. 209

223 2. Share the orange juice equally among your group members. The bottle holds 64 fluid ounces. Each cup holds 9 fluid ounces. a. What fraction of the orange juice does each person get? b. How much fluid ounces were served? c. What fraction of the juice was served? d. What fraction of the juice remains in the container? e. If you divide the remaining portion of the juice equally with your group members, how much fluid ounces will each of you get? f. How many cups full will you get from the remaining portion of juice? 3. Share the three tostados (flat bread) equally among yourselves. One tostado is a whole. a. What fraction of the tostados did you get? b. Describe your sharing method. 210

224 c. Did you get less than a whole, a whole or more than a whole? d. Write two fractions that are equivalent to the fraction you wrote above. 4. I have decided to give each group 7 pancakes (syrup is also provided) to share equally among the group members. a. Describe the way you choose to share the pancakes. You may draw pictures to show your partition. b. With your group members, discuss then write another way you believe the pancakes could be shared equally. c. If you do not want pancakes, how many pancakes will each member get if you divide the 7 pancakes equally with the other members of the group? 211

225 Extended Activity [Individual ON YOUR OWN] 5. 1 pint of milk is divided equally between 3 cups, all the same size. How much of the pint of milk will each cup hold? 6. A sausage has been sliced into 28 slices of the same size. How many slices would equal 7 3 of the sausages. Use picture to demonstrate your answer. Thoughts and Comments 7. (a) What is your definition of a fraction? (b) How would you show a friend what a fraction is? How would you show this same fraction in another way? 8. Do you like learning about fractions? Why? Why not? 212

226 9. Why do you think we need to learn about fractions in school? 10. How have the exercises you have been doing for the past weeks helped you in working with fractions? 11. You may add any comments pertaining what you have been doing for the past 6 weeks in fractions in the lines below. 213

227 INTERVIEW #1 (20 30 minutes) I am going to ask you some questions about the tasks that we previously did in class. I am very interested in how you come up with the answers, so it is important for you to tell me what you are thinking about. The interview will not be graded, so you do not have to worry about wrong answers. Are you ready? Listen to this story: Fay, Laura and Jim bought a bag of circular cookies. The bag contains 11 cookies. How should they share the cookies so each person gets the same amount? Here are some paper cookies that you can use to help you. Talk aloud as you solve the problem. Tell me what you are thinking. You can also make a picture to help you. Listen to this story: March 5, 2005 is Paul s birthday. He bought a square cake for his seven friends and him to eat. He wants to divide the cake into eight equal parts. Can you show Paul two different ways that he could share his cake? You can use the cut out squares to help you. Talk aloud as you solve the problem. 214

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