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- https://ufdc.ufl.edu/UF00099103/00001
## Material Information- Title:
- Selected Piagetian tasks and the acquisition of the fraction concept in remedial students
- Creator:
- Dees, Roberta Lea, 1938- (
*Dissertant*) Kantowski, Mary Grace (*Reviewer*) Bengston, John K. (*Reviewer*) Bernard, Donald H. (*Reviewer*) Lewis, Arthur J. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1980
- Copyright Date:
- 1980
- Language:
- English
- Physical Description:
- xi, 276 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Circles ( jstor )
Educational research ( jstor ) Fractions ( jstor ) High school students ( jstor ) Learning ( jstor ) Mathematics ( jstor ) Mathematics education ( jstor ) Rectangles ( jstor ) Schools ( jstor ) Students ( jstor ) Arithmetic -- Study and teaching (Primary) ( lcsh ) Cognition in children ( lcsh ) Curriculum and Instruction thesis Ph. D Dissertations, Academic -- Curriculum and Instruction -- UF Fractions ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt ) - Spatial Coverage:
- United States -- Florida -- Gainesville
## Notes- Abstract:
- This clinical study was designed to answer two questions: 1, is there a relationship between the acquisition of cognitive structures, as exemplified in Piaget-type' tasks, and the acquisition of the fraction concept; and 2, is there any difference between the concrete or manipulable and the pictorial or written modes of presentation in assessing students' knowledge? Three instruments were developed. The first was a set of tasks, similar to those used by Piaget to test for the cognitive structures thought to be related to the concept of fraction: conservation of number, seriation, classification, class inclusion, conservation of distance, and conservation of area. Tasks were prepared in concrete or manipulable and pictorial forms. The other two instruments were fractions tests, one concrete or manipulable, and one written, containing parallel sections on. the concept of fraction (discrete, number line, and area models) and equivalence and comparison of fractions. The written test also included addition and subtraction of fractions with like denominators. A pilot study was conducted with four students at Gainesville High School, Gainesville, Florida, in summer, 1979. The main study was done in spring, 1980, with 10 girls and 15 boys in the tenth, eleventh, and twelfth grades (median age 16 years), who were enrolled in the compensatory mathematics classes in Eastside High School, Gainesville, Florida. Tests were administered individually; interviews were recorded. The tasks were administered first. The two fractions tests were given on the next available day, 12 students taking the concrete form first and 13 taking the written test first. In general, students scored very low. No students were successful on conservation of area tasks; 8% were successful on classification tasks. The best scores were 56%, conservation of distance; 44%, seriation; and 36%, conservation of number. No student passed all sections of either fractions test. Three students passed both forms on concept of fractions, discrete model. On the concrete form, scores were better on the discrete and area models of the concept of fraction (39% and 56%) respectively, being the average percentage of students correct per item in those sections) than on the number line model (average of 14% correct per item) . Performance was poor on the equivalent fractions section (average of 19% correct per item); no student could do the comparison of fractions task. On the written test, results were similar except on equivalent fractions: 2 students (8%) passed the section, and 7 other students (28%) answered 3 out of 4 items correctly, apparently using a reducing algorithm. Six students could add and subtract fractions, but were incorrect on many items related to the concept of fraction. To answer the two main questions, data were examined using Walbesser contingency tables. No strong trends were evident, but there were some patterns. Students who could conserve number performed more satisfactorily on the discrete model of fraction. Students performed better on the concrete form of the class inclusion task than on the pictorial (56 to 8%). Students taking the concrete form of the fractions test first were more successful on the written test than those who took the written test first, indicating that learning may have occurred during the administration of the concrete test.
- Thesis:
- Thesis (Ph. D.)--University of Florida, 1980.
- Bibliography:
- Includes bibliographic references (leaves 268-275).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Roberta Lea Dees.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 000099957 ( AlephBibNum )
07240220 ( OCLC ) AAL5417 ( NOTIS )
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SELECTED PIAGETIAN TASKS AND THE ACQUISITION OF THE FRACTION CONCEPT IN REMEDIAL STUDENTS BY ROBERTA LEA DEES A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1980 Copyright 1980 by Roberta Lea Dees ACKNOWLEDGMENTS I wish to acknowledge the inspiration of Jean Piaget, whose productive life ended in September, 1980. I wish to thank my committee members: Dr. Mary Grace Kantowski, who understood what I wanted to do; Dr. John K. Bengston, without whose assistance and support I could never have done this; Dr. Donald H. Bernard; Dr. John F. Gregory; and Dr. ArthurJ. Lewis. I am grateful to the other Dees women, Jennifer, Sarah, and Suzanna, who always had faith in me. I also wish to thank my friend, Sidney Bertisch, for his help and encouragement. TABLE OF CONTENTS PAGE ACKNOWLEDGMENTS iii LIST OF TABLES IN TEXT vii LIST OF FIGURES viii ABSTRACT ix CHAPTER ONE INTRODUCTION 1 TWO RESEARCH REVIEW AND RATIONALE 7 The Content: Fractions 8 Fraction Hierarchies 8 Interpretations of Rational Numbers 15 The Learner 24 Piaget's Theory 25 Disadvantaged Students 41 The Interaction of the Learner and the Content 42 Piaget's Fractions 43 Assessment of Students' Knowledge of Fractions 51 Diagnostic and Prescriptive Teaching 56 Clinical Study 64 Related Piagetian Research 65 Concrete versus Abstract Modes of Presentation 85 Rationale 93 The Student's Notion of Fraction 94 Concrete or Manipulable versus Pictorial or Written Presentations 102 Clinical Methodology Used 103 Question 1 104 Question 2 105 THREE PILOT STUDY 106 Subjects 106 Instruments 106 Piaget-type Tasks 108 Fractions Tests 110 Procedure 112 Findings and Discussion Piaget-type Tasks Concrete Fractions Test Written Fractions Test General Observations Resulting Modifications Tasks Instrument Fractions Tests FOUR MAIN STUDY Subjects Instruments Tasks Fractions Tests Procedure Testing Scoring Planned Data Examination FIVE FINDINGS OF MAIN STUDY AND DISCUSSION Findings Overall Data Possible Relationships, Questions ] and 2 Qualitative Data from Student Protc Discussion Overall Data Possible Relationships, Questions I and 2 Qualitative Data from Student Protc Limitations of the Study SIX IMPLICATIONS FOR RESEARCH AND FOR TEACHING Implications for Future Research Question 1 Question 2 Other Issues Suggested Research Implications for Teaching D TASKS (REVISED) E CONCRETE FRACTIONS TEST (REVISED) F WRITTEN FRACTIONS TEST (REVISED) G TASK RESULTS H CONCRETE FRACTIONS TEST RESULTS I WRITTEN FRACTIONS TEST RESULTS REFERENCE NOTES REFERENCE LIST BIOGRAPHICAL SKETCH TABLES IN TEXT Page 1. Sample of Results of Lankford Study 58 2. Success in Area Subtasks 124 3. Percentage of Students Successful on Tasks 143 4. Success on Subtasks of the Tasks 145 5. Success on Concrete and Pictorial Forms of Tasks 146 6. Percentage of Students Successful on Sections of Fractions Tests 147 7. Success on Items of the Concrete Fractions Test 148 8. Success on Items of the Written Fractions Test 150 9. Percentage of Students Successful on Sections of Fractions tests by Test Sequence, Concrete-Written and Written-Concrete 151 10. Average Percentage of Students Correct Per Item by Model of Fraction 165 FIGURES 1. Walbesser Contingency Table 2. Task I and Fractions Section A 3. Task V and Fractions Section B 4. Task IV and Fractions Sections A, B, and C 5. Task IV and Fractions Section A 6. Conservation of Number, Concrete and Pictorial 7. Seriation, Concrete and Pictorial 8. Classification, Concrete and Pictorial 9. Class Inclusion, Concrete and Pictorial 10. Conservation of Distance, Concrete and Pictorial 11. Conservation of Area, Concrete and Pictorial 12. Concept of Fraction (Discrete Model), Concrete and Written Forms 13. Concept of Fraction (Area Model), Concrete and Written 14. Equivalent Fractions, Concrete and Written 15. Frequency Distribution of Scores on Classifica- tion, Subtask A and Subtask B viii Page 152 154 155 155 156 157 157 157 158 158 158 159 160 160 179 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SELECTED PIAGETIAN TASKS AND THE ACQUISITION OF THE FRACTION CONCEPT IN REMEDIAL STUDENTS By Roberta Lea Dees December, 1980 Chairperson: Mary Grace Kantowski Major Department: Curriculum and Instruction This clinical study was designed to answer two ques- tions: 1, is there a relationship between the acquisition of cognitive structures, as exemplified in Piaget-type tasks, and the acquisition of the fraction concept; and 2, is there any difference between the concrete or manipulable and the pictorial or written modes of presentation in as- sessing students' knowledge? Three instruments were developed. The first was a set of tasks, similar to those used by Piaget to test for the cognitive structures thought to be related to the concept of fraction: conservation of number, seriation, classifica- tion, class inclusion, conservation of distance, and conser- vation of area. Tasks were prepared in concrete or mani- pulable and pictorial forms. The other two instruments were fractions tests, one concrete or manipulable, and one written, containing parallel sections on the concept of fraction (discrete, num- ber line, and area models) and equivalence and comparison of fractions. The written test also included addition and sub- traction of fractions with like denominators. A pilot study was conducted with four students at Gainesville High School, Gainesville, Florida, in summer, 1979. The main study was done in spring, 1980, with 10 girls and 15 boys in the tenth, eleventh, and twelfth grades (me- dian age 16 years), who were enrolled in the compensatory mathematics classes in Eastside High School, Gainesville, Florida. Tests were administered individually; interviews were recorded. The tasks were administered first. The two frac- tions tests were given on the next available day, 12 students taking the concrete form first and 13 taking the written test first. In general, students scored very low. No students were successful on conservation of area tasks; 8% were successful on classification tasks. The best scores were 56%, conservation of distance; 44%, seriation; and 36%, conservation of number. No student passed all sections of either fractions test. Three students passed both forms on concept of fractions, discrete model. On the concrete form, scores were better on the discrete and area models of the concept of fraction (39% and 56%, respectively, being the average percentage of students correct per item in those sections) than on the number line model (average of 14% correct per item). Performance was poor on the equivalent fractions section (average of 19% correct per item); no student could do the comparison of fractions task. On the written test, results were similar except on equivalent fractions: 2 students (8%) passed the section, and 7 other students (28%) answered 3 out of 4 items correctly, apparently using a reducing algorithm. Six stu- dents could add and subtract fractions, but were incorrect on many items related to the concept of fraction. To answer the two main questions, data were examined using Walbesser contingency tables. No strong trends were evident, but there were some patterns. Students who could conserve number performed more satisfactorily on the dis- crete model of fraction. Students performed better on the concrete form of the class inclusion task than on the pic- torial (56 to 8%). Students taking the concrete form of the fractions test first were more successful on the written test than those who took the written test first, indicating that learning may have occurred during the administration of the concrete test. CHAPTER ONE INTRODUCTION Students who enter secondary school have all received some instruction in basic mathematical concepts and skills. In spite of having received instruction, there exists a group of secondary school students who apparently have not learned these concepts and skills. The majority of students are able to master operations with whole numbers, except possibly long division. The introduction of fractions signals the beginning of a dramatic separation of students into those who succeed in mathematics and those who do not. Many frustrated educators have seen the advent of the hand-held calculator as salva- tion for those students who cannot operate with fractions; they say that students will ho longer be blocked from further progress in mathematics because they can not remem- ber when to "invert," etc. Others feel that real progress will still be missing unless students have a basic under- standing of the concept of fraction, on which the concepts of decimals and percentages are logically based. Those responsible for educating the masses realize that there is a lower end to every distribution; a normed test will always have Stanines 1 and 2. But in recent years, there has been an attempt in many states to agree upon specific mathematics competencies for all students. These competencies would become the minimum achievement required 1 for graduation from high school. This idea represents a considerable advance over the old "grading on the curve" method, in which the person who did the "worst" automati- cally failed. Competency testing also has its imperfec- tions. Nevertheless, only after competencies were chosen and tests administered did educators become aware of the large number of students who had not mastered these mini- mum skills. The finding was consistent with the results of the 1972-73 mathematics assessment by the National Assessment of Educational Progress (NAEP). In many secondary and adult schools, compensatory or remedial instruction is scheduled for students who have not mastered basic skills. Often smaller classes are ar- ranged; teaching aides or assistants are sometimes provided to lower the student-teacher ratio even more. However, the instruction appears to consist of essentially the same strategies that did not produce competence in previous years of schooling. Prejudicing chances for success, some schools assign their beginning teachers to these compensatory classes, the more experienced teachers being given the brighter students and the college preparatory subjects. Compounding the dilemma still further is the 3 current shortage of mathematics teachers; teachers teaching "out of field" are often assigned to compensatory classes. It seems evident that it is not a simple task to compensate for the learning not yet achieved in eight, nine, or ten years of school; it will require experience and knowledge both of the subject matter to be taught and of how students learn. It will probably not be accomplished by repetitious drill or by more of the same instruction that was not successful in the past. Fifteen years of teaching in various school settings, producing some success with these challenging students, have given this investigator a basic belief, which is as follows: To remediate or compensate mathematical deficien- cies, first, one must start where the student is; secondly, the student usually needs a hands-on or manipulative approach to discover concepts for himself or herself. This study provides an opportunity to begin to see whether these ideas can be substantiated. One purpose of this exploratory study is to investigate why certain secondary school students have not mastered a specific portion of mathematics content, the fraction concept. What can be learned about students' understanding of fractions could eventually lead to more successful in- structional strategies; first, it is necessary to identify what they know. Knowledge of where the individual student is 4 with respect to a mathematical concept has not been gained from standardized tests. More sensitive, individual testing, such as that used in clinical studies, is needed to reveal the student's thinking about mathematics. The secondary school student has through the years accumulated some information and misinformation about operating with fractions. Some mathematics educators attempt to isolate and identify the errors, and then remediate them. But such a method may not be successful if the student does not have a firm understanding of the concept of fraction on which to base the operations. What is the necessary foundation? Two things are needed: better diagnosis of what a student knows now about mathematics, and knowledge of what basic structures are required for the student to be able to learn mathematics. In trying to meet the first need, better diagnosis, one is led to the methods of Jean Piaget, who pioneered in the use of the clinical interview to try to understand children's thinking. His results in turn lead to possible insights into the second need. For in Piaget's work with young children, there are described behaviors remarkably like those observed in secondary school students who were having trouble with fractions. For some reason the cognitive development of certain students has been delayed, so that they do not have the cognitive structures often assumed to be present in all secondary school students. The question arises: is this a coincidence, or are the two deficiencies related? Could their tardy cognitive development be the cause of some students' difficulties in learning mathematics? It is not surprising to find that students who are behind in one academic area are slow in something else as well. But if specific Piagetian concepts are found to be related to the specific mathematical concepts to be taught, the finding could be very helpful to those who diagnose student defi- ciencies and plan instruction. In this study an attempt is made to identify basic cognitive structures necessary for an understanding of the fraction concept. Another interest of some mathematics educators, especially those involved in elementary school mathematics, is the laboratory method of teaching mathematics. The method is not new; the resurgent interest can probably be traced to the late sixties and the Nuffield Project in England. The use of concrete or manipulable objects in the learning of mathematics is fairly well established in elementary schools, but is rarely seen in secondary schools. A second purpose of this study is to consider whether there may be any justification for the use of these materials in secondary schools. 6 In a specific portion of mathematics content, the concept of fraction, this study will explore the following two questions: Question 1. Is a particular level of cognitive development, as indicated by performance on certain Piaget-type tasks, prerequisite to the student's acquisi- tion of the fraction concept? Question 2. Does the mode of presentation of mathema- tics content, concrete or symbolic, make a difference in students' performances? Or are students who are successful at a task in one format always able to perform the same task when it is presented in the other manner? CHAPTER TWO RESEARCH REVIEW AND RATIONALE In designing instruction on fractions, an educator must consider the goal, or what the student is to learn about fractions, and the present status, or what the student already knows about fractions. The teacher's instructional plan is the strategy that is expected to effect movement from the present status toward the goal. Mathematics educators have studied the instructional process from both ends. In this section, the goal, the mathematics content to be taught, will be considered first. Pertinent research about the student's acquisition of the fraction concept will be reviewed. Next, attention will be given to the learner's competence with respect to the goal. The congitive development theory of Jean Piaget will be presented as the theoretical framework for describing the development and knowledge of the learner with respect to fractions. The third section will consider the interaction between the learner and the mathematics content. Related aspects to be discussed include Piaget's study of how children understand fractions and educators' attempts to assess the learners' present knowledge of fractions. Current research trends to be reviewed include diagnostic and prescriptive studies; studies of students disadvantaged in mathematics; studies employing a clinical method; and studies which attempt to apply Piagetian theory to education. The ways in which students respond to the concrete and symbolic modes of presentation of mathematics content will also be examined. The Content: Fractions In considering mathematics content, not everyone agrees that the understanding of, and computation with, fractions are worthy goals. In an article on the metric system and mathematics curriculum, for example, Sawada and Sigurdson (1976) suggest that common fractions should be studied only at the conceptual level, and that decimal numeration should receive major attention. Fraction Hierarchies Those researchers who do select fractions as mathematics content to be taught often perform task analyses after the work of Gagne (Gagne, Mayor, Garstens, and Paradise, 1962). A specific concept or skill is analyzed by its subconcepts or subskills. A learning hierarchy, or a network of par- tially ordered subconcepts or subskills, is developed on the basis of logical relationships (Gagne, Note 1). The assumption is that if the hierarchy is valid, it gives a sequence, perhaps optimal, for teaching the component parts of the concept of skill. However, it appears that these "expert" generated learning hierarchies are not equivalent to student generated learning hierarchies with the same terminal behavior (Walbesser and Eisenberg, 1972). A task analysis technology, described by Resnick (1976) and Walbesser and Eisenberg (1972), has been developed to test the validity of a hierarchy with students, to find out whether the various hypothesized dependencies of the hierar- chy are supported. Greeno (1976) is concerned with showing how psychologi- cal theories might be used in formulating instructional objectives. He has attempted to identify the "cognitive objectives" needed to produce the desired outcome behaviors. Presenting his work as a serious proposal about what the goals of instruction are, he says, It may be that when we see what kinds of cognitive structures are needed to perform criterion tasks, we will conclude that something important is missing; but if that is the case, it also will be important to identify a more adequate set of criterion tasks in order to ensure that instruction is promoting the structures we think are important. (p. 124) Adding fractions was Greeno's first example. He constructed a "procedural representation," or a flow chart, for adding fractions. Recognizing that finding equivalent fractions is necessary both before and after the actual addition, he looked at three different models, or procedures, for finding equivalent fractions. The first is based on "spatial processing" of a region (or an area model). The second is a "set-theoretic" (or discrete) model. And the third is simply an algorithm, "operating directly on numeri- cal representations" (p. 133). Had he extended his reason- ing one step further, he might have wondered what 10 understanding of area a child would need to be able to use the spatial processing model, or what concept of number, to use the discrete or algorithmic model. Uprichard and Phillips sought to generate, then vali- date, a hypothesized hierarchy for adding (1977) and sub- tracting fractions (Note 2). The authors intended to give consideration "to both psychological and content (discipline) factors" in identifying and hierarchically ordering tasks. The procedures for both studies were essentially the same. Fraction addition and subtraction problems were divided into two levels, those with like and unlike denominators. Within each level, classes were identified by the nature of the denominators, prime or composite, and the nature of the relationship between the two denominators. Further, there were sum or difference categories, depending on various renamings required. Both studies were done with students in grades four through eight; the majority were in the fifth and sixth grades. Items were compared by two methods, the Walbesser (Walbesser & Eisenberg, 1972) contingency table, with ratio levels of acceptability as determined by Phillips (1972); and pattern analysis, after Rimoldi and Grib (1960). The end results were two lists of problems, in order of ascending difficulty. Conclusions were that problems of certain types should be taught before problems of other types, based on the assumption that those missed most often, and therefore, by definition, the hardest must depnd on the easier problems as prerequisites. Examination of these dependencies yields examples of concepts which could, logically, seem prerequisite to others, but which have a reversed order of difficulty for students. The following example is taken from the sub- traction study: It was found that tasks involving whole number sums greater than one ( such as 5 1/6) were more difficult than those involving mixed numbers (such as 1 1/6 2/6) (Uprichard and Phillips, Note 2, p. 10). Per- haps students were performing a rote algorithm on 1 1/6 (denominator times whole number plus numerator), rather than realizing that 1 can be renamed as 6/6 and 5 as 30/6. The authors say in summary that the results support the notion that both epistemological and psychological factors be considered when developing teaching sequences in mathematics. Some of the implications above would not necessarily be derived from logical analysis alone. Also, in interpreting the results of this study one must be conscientious of the limitations of indirect validation procedures. For example, confounding variables such as prior educational experience of subjects and errors of measurement must be considered. (p. 11) Students older than their subjects will have had even more experiences in school, and the partial learning they bring to a taskmay function as a confounding variable. The rote application of a poorly understood, or poorly remem- bered, algorithm is an example of this. In discussing the Uprichard and Phillips work, Underhill (Note 3) remarked that a hierarchy may be valid for original learning, but not necessarily for remediation when instruction has already been given. There may be some "subskill retention hierarchies" that could be omitted in remediation. As pointed out by Kieren in his review (1979) of the addition study, Uprichard and Phillips's analysis treated fractions as symbols to be manipulated according to formal algorithms (a very limited view). Kieren further suggested that such studies needed to have a sound epistemological basis from which to work, and that clinical evidence needed to be given to support the statistical analyses. It is knownwhich problems were missed, but it is not know why they were missed. Novillis (1976) studied more basic subconcepts of the fraction concept, with subjects in grades four, five and six. Each subconcept was depicted by a model that had as its unit either a geometric region (the part-whole model); a set (the part-group model); or a unit segment of a number line (which the author considered a specific form of the part-whole model). The investigator constructed a hierarchy of dependent subconcepts of the fraction concept and designed a fraction concept paper-and-pencil test of 16 subtests, one for each of the subconcepts. Most of the subtests contained one item each of the following types: a) given a fraction, the student was asked to choose the correct model. b) given a model, the student was asked to choose the correct fraction. c) given a model, the student was asked to select another model for the same fraction. d) given four models, the student was asked to choose the one that did not depict the same fraction as the others. To validate the hierarchy, Novillis (1976) used a category system equivalent to Walbesser's contingency table, and analyzed the individual dependency relationships using ratios developed by Gagne et al. (1962) and Walbesser (Note 4). Support was found for 18 of the 23 dependencies in the hierarchy. The author concluded that certain subconcepts were prerequisites to others. The main dependencies are given below, with the subconcept on the left being prerequisite to the subconcept on the right. Lower order subconcept Higher order subconcept associating fractions with associating a fraction part-whole and part-group with a point on a number models line associating a fraction using a fraction in a com- with a part-whole model prison situation invol- or with a part-group ving the respective model model associating a fraction associating a fraction with with a part-whole model the respective model where or with a part-group the number of parts was a model multiple of the denominator and the parts were arranged in an array that suggested the denominator associating a fraction associating a fraction with a part-whole model with the respective or with a part-group model having noncongruent model having congruent parts, where (in the case parts of part-whole models), the parts.were equal in area. (Novillis, 1976, p. 143) The author noted that the study was exploratory but inferred that elementary school students were not exposed to a sufficient variety of instances of the fraction concept or negative instances (cases where it is not valid) to per- mit generalization of the concept. Because they are relevant to the present study, two of Novillis's examples are given here: Many students can associate the fraction 1/5 with a set of five objects, one of which is shaded, but most cannot associate the fraction 1/5 with a set of ten objects, two of which are''shaded, even when the objects are arranged in an array that clearly indicates that one out ofevery five is shaded. . . If two rectangular regions have been separated into five parts such that in one case the parts are congruent and in the other case the parts are neither congruent nor equal in areas, and in each case one of the parts is shaded, then many students associate the fraction 1/5 with each of these regions and indicate that 1/5 of each region is shaded. (p. 143) Since the instrument for the validation was a written test, the intriguing question of why they missed the items cannot be answered. In the case of the second example, was the difficulty due to their concept of area? A clinical study, in which individual students could have been observed and interviewed, might have yielded further information. In a discussion of directions for research, Lesh (1975a) suggested that mathematics educators should apply Piagetian techniques and theory to rational numbers, and referred to Kieren's (1975) paper as "a first step in the direction of a Piagetian analysis of the concept of rational numbers" (Lesh, 1975a, p. 15). The work by Kieren seems to be motivated by curriculum development more than by the theory of Piaget. However, of available published work, it is closest in focus to the present study. Therefore it will he:discussed at length. Interpretations of Rational Numbers Kieren was concerned about the different possible interpretations of fractions, particularly the "algebraic" aspects of operations on rational numbers, which are usually not presented when fractions are introduced, and which sometimes get lost. He attempted to show the connection between the mathematical, cognitive and instructional foundations of rational numbers in the following way: He named seven different interpretations of rational numbers. For each of these interpretations, he stated the mathematical structures emphasized. Then he listed a set of related cognitive structures and a set of instructional structures (or sequences of necessary experiences). It is not always clear whether he was summarizing existing educational practices or whether he was making recommendations for instructional sequences. Kieren suggested these seven interpretations of rational numbers: 1. Rational numbers as fractions. This is his label for the most common interpretation of rational numbers, the symbols used in computation. In this interpretation, the associated mathematical structure is a set of procedures (or algorithms) for manipulating the symbols. Kieren gave very little attention to the other two kinds of structure for this interpretation: The corresponding cognitive structure is a set of skills. It is not necessary under this interpretation to assume any other structures underlying the skills. The pre- requisites for these skills would be skills in-computation with whole numbers and not developed concepts of part-whole relation- ships or proportionality. The major instructional strategy is diagnosis and remediation both based on elaborate task analysis. (Kieren, 1975, p. 107) It seems doubtful that the student could in fact organize and memorize these skills (the 160 different addi- tion types he mentioned, for example) if there not some other cognitive structures on which to anchor the skills. Concerning the instructional strategy, even though Kieren did say the interpretations were not independent (p. 103), this passage might still lead one to believe that this narrow, symbolic interpretation is to be readily found in classrooms. Actually it is highly unlikely that a teacher would present the algorithms for the first time without some attempt to give meaning to the processes by appealing to one or more of the other interpretations, or to some concrete device. 2. Rational numbers as equivalence classes of fractions. A rational number is defined as a set of ordered pairs of integers. In mathematical structure, the rational numbers, together with the operations of addition and multiplication, constitute an ordered field. The principle underlying concept needed, according to Kieren, is that of an ordered pair of numbers. He sees three phases: perceiving a real situation and its coordinate parts in order, being able to represent these coordinates symbolically, and associating the symbols again with a coordinate reality. Kieren continues: With rational numbers the child must learn to identify part-whole situations, learn verbal and numerical codes for these, and learn to correctly identify a code (fraction) with a part-whole setting. As a cognitive capstone of this ordered pair concept set, the child must realize that a part-whole setting can be seen in a set of equivalent ways, and that the various fractions which represent the elements of this set are equivalent. (Kieren, 1975, p. 109) But logically, it is not necessary to understand anything about part-whole situations to use the equivalence class concept of rational numbers. For example, a rational number (a, b) can be said to be equivalent to (1, 2) if (a 1) / (b 2) = 1/2. Alternately, one could give a geometric meaning of equivalence. Referring to Kieren's (1975) graph (p. 108), partially reproduced to (,,1) left, one could say that the rational number (a, b) is equivalent to (1, 2) if (a,b) <0," lies on a line through (0, 0) and (1, 2). Kieren suggests that the proper instructional strategy for this conceptual development is exposure to a wide variety of part-whole settings. He mentions four settings: state-state (static comparison between a set and one of its subsets), state-operator (divide 3 cookies among 5 persons), operator-state (use 5 of a dozen eggs) and operator-operator (cut a pie in eighths, serve 5). The student must also understand that these ordered pairs are numbers. This understanding must include the relationship of this new set of numbers to whole numbers, and a "notion of operations consistent with the fractional and equivalence notions" (p. 110). This second notion, Kieren feels, de- pends on the ability to partition both discrete and continu- ous quantities. Examples he gives are: dividing 15 plants among 5 pots, dividing a rope into 5 equal pieces, and dividing some crackers among 4 people. In concluding this section, he says that in this interpretation, the child must be able to assign a pair of numbers to a part-whole situation. This, of course, entails the ability to logically handle the part-whole relationship in both the discrete and continuous cases. The ability to handle class inclusion may be very important in the former case, while partitioning plays a role in the latter. (Kieren, 1975, p. 110) 3. Rational numbers as ratio numbers. An example of a ratio number is the number x, where x is to 1 as 1 is to 8. This interpretation leans heavily on the previous one, as it depends on ordered pairs and operations proceeding from equivalence classes. This interpretation is a sophisticated one and Kieren does not expect the child to be able to deal with it until the proportionality schema is developed, probably not until later adolescence. 4. Rational numbers as operators or mappings. In this interpretation, 2/3 is an operator which maps 3 onto 2, yielding a line segment 2/3 as long as the original. A finite analog would be giving 2 boxes of crayons to every 3 children. (Thus 6 children would need 4 boxes, etc.; equivalence can be seen in operators in this way.) Of the operations, multiplication and division can each be thought of as one operator following another, and are easier than addition and subtraction. Kieren says that three cognitive structures are critical to this interpretation. One is the notion of proportion. However, he says, the rational number notions in this interpre- tation can be developed as concrete generali- zations about a large number of concrete situations. Thus, these notions from the point of view of the child can be considered preproportional. It should also be noted that the fraction notion in this interpreta- tion is based on the quantitative comparison of two sets or two objects; hence, part- whole or class inclusion notions are not cen- tral to the interpretation. (Kieren, 1975, p. 115) The child must also have a structure of composition (one operator followed by another) and be able to replace these transformations by their product. The third structure is that of properties, parti- cularly those of inverse and identity, and the underlying reversibility notion. Instructional strategies would include work with simi- lar figures, which Kieren calls "preproportional," and ex- change games with finite sets. 5. Rational numbers as elements of a quotient field. The rational number x is a solution to an equation of the form ax = b, where a and b are integers. Field axioms are assumed. This interpretation relates rationals to abstract algebraic systems, and "is not closely related to the natural thought of the child" (Kieren, 1975, p. 121). Be- cause it requires formal reasoning, this interpretation will not be detailed further. However, Kieren says that the more primitive cognitive structure underlying the quotient concept is partitioning: if there are 6 pizzas for 5 children, what is an equal share for each? His simpler illustration is this: Here are 20 letters to be divided evenly in 5 mailboxes. This problem can be solved by distribution of the letters one at a time into the mailboxes, like dealing out cards (Kieren, 1975, p. 121). 6. Rational numbers as measures. Rational numbers are points on the number line. Addition is the simple laying of two vectors end-to-end and reading the result. This interpretation gives an intuitive notion of order. Kieren gives the cognitive structures that seem particularly important: The first is the notion of a unit and its arbitrary division. The child must realize that one can partition the unit into any number of congruent parts. Second, the child must be able to conceptualize part- whole relationships in this context and recognize equivalent settings arising from partitioning of the unit (1/2 = 3/6). Third, the child must develop the concept of an order relation. This involves both the ability to order physical reality and the ability to use correctly the symbolic order statements. Underlying these structures are more general structures, conservation of length and substance, and a general notion of ordinal number. (p. 125) Instructional activities are suggested by both forms of division, measurement and partitioning. Equivalences can be shown with rods or paper strips of different colors. 7. Rational numbers as decimal fractions. In this interpretation, rational numbers are those which can be expressed as either terminating or repeating decimals. The operations are extended from those for whole numbers, making computation simple. In division, a remainder is not needed. Teaching from this viewpoint would not provide pre-experience for the rational expressions of algebra. The cognitive structures necessary are similar to those for measurement. However, the child must be able "to generalize in the symbolic domain" (Kieren, 1975, p. 126). Also, one out of six parts, or 1/6, is a natural extension of counting; saying "about .16" is not. There- fore measuring and estimating are critical. Estimating, he says, involves a general notion of unit and the ability to think hypothetically. Instructional activities would include any work with the numeration system, and operations with whole numbers. Metric system measurement and money also provide natural models for decimal fractions. And estimating length is a pre-decimal fraction activity. Kieren says that "the processes of seriating and comparing are of paramount importance as is the whole notion of order" (p. 127). Having described these interpretations of fractions, Kieren makes the point that all should be considered in the curriculum. Given these interpretations, he says, a curriculum developer-instructional designer "can then ascertain the necessary cognitive structures for meeting the objectives and develop sequences of instructional activities which contribute to the growth of these structures" (Kieren, 1975, p. 128). He says further that a researcher who asks, "How does the child know rational numbers?" must go through a similar process. He can study selected interpretations in more detail and identify what he believes to be the most important cognitive structures. Settings can then be developed or used which allow one to see the extent to which a child has such struc- tures. The growth of such structures can then be studied developmentally. Alterna- tively, the importance of such structures can be tested. Here, one would test the effect of having or not having some structure on attaining some rational number objectives. (p. 128) As mentioned elsewhere, students' learning does not always proceed logically, or according to researchers' expectations. Therefore, to ascertain these necessary cognitive struc- tures as Kieren suggests may require in-depth study of students and their learning. Kieren then summarizes the "conglomerate picture of rationals," including some work that has been done in developing hierarchies of skills, and suggests curriculum research. He further suggests clinical research such as that of Inhelder and Piaget (1969) on the growth of logical thinking, saying, Some aspects and behaviors of rational number will be impossible to study in their "natural state." They will undoubtedly be colored by instructional experience. (p. 140) As already observed, secondary students will have had many such instructional experiences, which may confound the study of their concept of fraction. Kieren also says that "it would seem that conservation of area and length might be related to continuous partitive division" (Kieren, 1975, p. 141). This idea will be dis- cussed further in following sections. The Learner Two aspects of the learner will be considered: first, what is known about the learner's cognitive structure, as described in the cognitive development theory of Piaget; and secondly, what is known about the remedial student. Not all scholars agree with everything Piaget says. In fact, according to Flavell (1963), the system has an extraordinary penchant for eliciting critical reactions in whoever reads it. Piaget has done and said so much in a busy lifetime that foci for possible contention and disagreement abound. More than that, he has consistently done and said things that run so counter to accepted practice as to make for an immediate critical reaction in his reader, almost as though he had deliberately set out to provoke it. (p. 405) Flavell also disagrees with certain parts of the theory, but concludes that Piaget's work "is of considerable value and importance, with a very great deal to contribute to present understanding and future study in the are of human development" (p. 405). Piaget's theory of cognitive development is not a theory of education, but of knowing, which may or may not be related to the knowledge purveyed in schools. Piaget has left educational implications and applications to others (Sigel, 1978, p. xvii). However, the following discussion of his theory will indicate that the cognitive structures he describes are of importance to school learning. Piaget's Theory Piaget is an epistemologist. He studies the nature of knowledge; he is concerned with finding out how the ability to know develops. He looks for commonalities in children's knowing that do not depend on what school they attend, their emotional state or other factors (important though they may be to the overall condition of the child). During decades of study on hundreds of children, Piaget concluded that there were definite levels of cognitive development which were invariant in the sequence in which they emerged. Unfortunately, this idea gives rise to the first of many common misinterpretations of Piaget's ideas. An example is the following: The research of Piaget, et al. suggest that all students by about the age of 12, should be able to correctly use an external frame of reference to properly predict water level, pendulum position, etc. (Dockweiler, 1980, p. 214) A review of the work cited (Piaget and Inhelder, 1956) fails to turn up the suggestion by Piaget and Inhelder that any student should do anything at a particular age. Piaget does not view cognitive development level as age dependent. Flavell emphasizes Piaget's position on the stage- age question: Piaget readily admits that all manner of variables may affect the chronological age at which a given stage of functioning is dominant in a given child: intelligence, previous experience, the culture in which the child lives, etc. For this reason, he cautions against an overliteral identifi- cation of stage with age and asserts that his own finTings give rough estimates at best of the mean ages at which various stages are achieved in the cultural milieu from which his subjects are drawn. . Of course not all individuals need achieve the final states of development. . Piaget has also for a long time freely conceded that not all "normal" adults, even within one culture, end up at a common genetic level; adults show adult thought only in those content areas in which they have been socialized. (Flavell, 1963, p. 20) The present study is not focused on the chronological age at which a student has reached a stage; all of the subjects are "behind" Piaget's children. Attention is given instead to whether the student has reached a stage, and whether having reached it has anything to do with enabling the acquisition of mathematics concepts. In this discussion, the major developmental stages themselves will be called "periods," in accordance with Piaget's stated preferences (Flavell, 1963, p. 85); the word "stage" will refer to subdivisions with the period (except where reference is made to authors who use the former nomenclature). The first period, called the sensorimotor period, lasts from birth to about two years of age. The last one, the formal operational period, in which an individual becomes able to think about thoughts and reason about reasoning, has been found by Piaget to be completed at about age 15. These periods at the two ends of the developmental scale were not exhibited by the students in this study. Therefore, attention will be paid only to the middle period, called by Flavell "the period of preparation for and organization of concrete operations" (Flavell, 1963, p. 86). The first of two major subperiods is that of preopera- tional representations, and the second is that of concrete operations. In the preoperational subperiod, found by Piaget to last roughly from 2 to 7 years of age, the child is learning to use language as representation of thought. His under- standing of space increases to include such concepts as more and less, larger and smaller, before and after. He learns to discriminate differences in objects, colors, etc. Yet, in the preoperational child, perception is a stronger influence than reason. During the concrete operational subperiod, about 7 to 11 years in Piaget's findings, the child acquires a conser- vation schema. She can classify objects on the basis of a common characteristic. She learns to seriate, or put things in order from smallest to largest or vice versa. The following are some examples of Piaget-type tasks and how children react in each of the two subperiods. Conservation of number. In Piaget's theoretical analysis, the concept of number is derived from "a synthe- sis of class inclusion and seriation" (Sinclair, 1971, p. 152). Piaget's own volume, The Child's Conception of Number (1965), includes conservation of quantity, one-to- one correspondence, logical classification and order relations, each of which were given at least one chapter. In spite of its complexity, conservation of number has been chosen to present first, because it can make a vivid illustration of what Piaget means by conservation. What will be presented here is a simplistic version, with emphasis on the tasks themselves, based on work by Copeland (1979), Formanek and Gurian (1976), and Lesh (1975b). A child is shown two rows of beads displayed as follows: 0 0 0 0 0 0 0 0 0 0 The child agrees that there are the same number of beads in each row. If one row is now spread out, like this, 0 0 0 0 0 0 0 0 0 0 the child of 5 or 6 years may think that there are more beads in the bottom set, because the row is longer. Even if he counts each row, he may still be influenced by what 29 he sees, the length of the rows, in making judgment about which set contains more beads. He would be said to have the ability to conserve number if, in this case, he could realize that the number of beads remained constant even when the beads were rearranged. There are three stages into which children's responses can be divided: I: Says that the second row contains more beads. Pressed for a reason, says, "Because I can tell by looking," etc. II. Is transitional. Appears to conserve, but is not sure; counts to see whether the rows are equal in number. III. Realizes that rearranging does not change the number. Asked for a reason, replies, "Because you didn't put any more or take any away." These three stages are typical of the sequences Piaget finds in other conservation tasks (quantity, length, area, volume, mass). The stages are summarized by Flavell (1963): I, no conservation; II, conflict between conservation and nonconservation, with perception and logic alternately getting the upper hand; and III, a stable and logically certain conservation. (pp. 312-313) Flavell also says that, in this task, "a genuine concept of cardinal number is by no means guaranteed by the ability to mouth appropriate numerical terminology [or count] in the presence of objects" (p. 313). 30 Seriation. Seriation is the act of putting things in order. Its beginnings are in the child's broad discrimi- nations between big and little. In one version of the seriation task, a child is given about 10 sticks of different lengths and is asked to put them in order. In stage I, a child can order two, or maybe three, sticks at a time, but there is no overall scheme. In stage III, the child has a plan, and methodically selects the longest (or shortest), then the next longest (or next shortest), etc., and completes the series efficiently. If some sticks are introduced as having been "forgotten," the stage III child can insert them with no problem (Copeland, 1979, p. 96). The stage II child can usually form the series by "trial and look." as Copeland calls it (p. 96). But a plan or system is noticeably absent. In fact, the child may end up with two or three unconnected subseries, as Lesh (1975b, p. 97) shows: Even if such a child can complete the series, he may still be unable to insert a "forgotten" stick, Lesh says (p. 97). Copeland says this child "considers the series already built to be complete and feels no need to insert the addi- tional sticks" (Copeland, 1979, p. 94). In Piaget's language, the preoperatory levels, stages I and II, "lack coordination in that subjects can put two or three elements in order at a time but cannot put all the elements in order. The operator level sees a general (reversible and transitive) coordination linking these specific actions into a whole" (Piaget, 1976, pp. 300-301). One source of difficulty in the seriation task lies in the tendency of some stage I and II children to make pairs. It may not be simply that they can only attend to two at a time (that is, they can consider a < b, but not a < b and b < c simultaneously). Another factor may be, Piaget says, that the conceptualization on which the cognizance is based, which starts from the results of the act, is not only incomplete but often incorrect as well, because the child's pre- conceived ideas influence his reading of the situation--that is, he sees what he thinks he ought to see. (1976, p. 300) In this context, once a stage I child picks up two sticks and orders them, she may continue making pairs in that fashion, disregarding the original instructions, because she hears the directions she thinks she ought to hear. Classification. In a simple classification task, a child is given a collection of objects or pictures and asked to put "the ones that are alike" together. Flavell's (1963) discussion of children's responses will be abbre- viated. In stage I, the child tends to organize classifiable material, not into a hierarchy of classes and subclasses founded on similarities and differences among objects, but into what the authors [Piaget and Inhelder] term "figural collections" [like pictures]. . It is a relatively planless, step-by-step affair in which the sorting criterion is constantly shifting as new objects accrue to the collection. . Partly in consequence of this inch- by-inch procedure bereft of a general plan, the collection finally achieved is not a logical class at all but a complex figure (hence figural collection). The figure may be a meaningful object, e.g., the child decides (often post hoc) that this aggregation of objects is "a house." Or instead, it may simply be a more or less meaningless configuration. . Frequently, at least part of the child's collection is founded on a similarity-of-attribures basis. What often happens is that the child begins by putting similar objects together, as though a genuine classification were in progress, and then "spoils" it by incor- porating his "class" into a nonclass, con- figurational whole. (Flavell, 1963, pp. 304-305) Flavell says that the stage I child may also begin by putting squares together, but fails to include all the squares or contaminates his collection with nonsquares. This is an illustration of his inability to differentiate, and hence coordinate "class comprehension (the sum of qualities which define membership in a logical class) and class extension (the sum total of objects which possess these criterial qualities)" (p. 305). He explains: In a genuine classification, these two pro- perties must always be in strict correspon- dence: the definition of the classification basis determines precisely which objects must constitute its extension, and the nature of the objects in a given collection places tight constraints on the definition of the class they together form. But for the young child, there seems to be no such strict correspondence. (p. 305) It is noted by Flavell that these gaps in the child's understanding may be hidden. "The child's ability to bandy about classification-relevant phrases (e.g., 'dogs are animals,' "some of these are red,' etc.) either under ordinary questioning or in spontaneous discourse, is likely to be a most unreliable guide" (Flavell, 1963, p. 306). A stage II child can form nonfigural collections on the basis of similarity of attributes. He can generally assign every object in the display to one or another group. Still troublesome are groups or collections with only one member, or, worse yet, no members. Copeland reports Inhelder and Piaget's (1969) findings: "The concept of the singular class is not operational until eight or nine years of age, and the empty or null class is not operational until ten to eleven years of age" (Copeland, 1979, p. 69). Stage III does not occur, for Piaget, until the child has mastered class inclusion. This will be treated below as a separate task. Class inclusion. The important aspects of class inclu- sion can be exemplified by these two tasks, taken from Flavell (1963, pp. 307-309). In one, the child is to be tested on the notion of "some" and "all" (a reflection of the understanding of class comprehension and extension, discussed above). A series of objects is shown, such as the following collection: Red Blue Blue Blue The questions asked take two forms: a) Are all the blue ones circles? or Are all the squares red? etc. b) Are all the circles blue? or Are all the red ones squares? etc. Being able to answer questions like those in a does not guarantee that the child can answer questions like those in b. In the second experiment, the child is shown a set of flowers with a large subclass of primroses and a few other (various) flowers. It is first established that the child understands that the primroses are flowers. Then questions are asked on the "quantification of inclusion" (Flavell, 1963. p. 308): 1) If I took away all the primroses, would there still be flowers left? 2) If I took away all the flowers, would there still be primroses left? 3) Are there more primroses or more flowers? Strangely enough, some children can answer questions 1 and 2 correctly and still "fail" question 3. In Piaget's interpretation, if B is the set of flowers and A is the subset of primroses, The child can recognize that A and A' comprise B when he focuses attention on the whole B (thus, he can perform B =-- + A'), btE "losesT B (and the fact that A = B A') when he isolates A as a comparison term. With B momentarily inaccessible as an object of thought, the child cannot do other than compare A with its complement A'. (Flavell, 1963, p. 309) CFlavell's emphasis] Conservation of distance. Distance and length are not the same thing. Length is the measure of something which takes up space (one-dimensional) and distance is space (one-dimensional) which can be filled up with something. If movement is involved, the situation is complicated further, according to Piaget, Inhelder and Szeminska (1960): Questions about the strips of paper . . may be asked in terms of "static" length or in terms of distances travelled. The answer is not always the same in both cases and the two languages should not be confused. (p. 106) The conservation task to be discussed below, adapted from Formanek and Gurian (1976, pp. 32-34) concerns the linear space between two points. Two small toys, such as cowboys or soldiers, are placed about 50 cm apart. The child is asked if the toys seem to be "close together" or "far apart." (Either one is satisfactory; this establishes a frame of reference.) Then a low screen, or barrier, is placed about midway between the toys, as if it were a fence separating them. The child is then asked whether they are still as close together or as far apart, depending on the child's first reply. The screen is then replaced, first with a larger screen, high enough to hide the two toys from each other, then with an obviously three- dimensional object, like a block of wood. Each time the child is asked to make a judgment about whether the distance has changed and why. In stage I, children are thrown off by the partition and no longer seem to be able to consider the total distance between the two toys; they will only look at the distance each toy is from the screen. A stage II child can consider the total distance, but the distance seems less, because the obstruction is taking up space. For them, distance is empty space. Children in stage III realize that the obstruction is irrelevant to the distance between the two toys; they state confidently that they are just as far apart because they haven't moved. Conservation of area. Logically, adding one more dimension would tend to complicate matters. The "farm" task, adapted from the version given by Piaget, Inhelder and Szeminska (1960, pp. 262-273), will illustrate some of the complexities involved in considering area. The child is shown two rectangular sheets of green cardboard and told that they represent fields of grass. It is established that they are the same size, by putting one on top of the other, if necessary. Then a tiny model of a cow is placed on each field and the child is asked whether both cows have the same amount of grass to eat. Thus, the frame of reference is established. Then the investigator begins to change things. Two identical "barns" are added, one to each field, and again the child is asked whether the cows have the same amount of grass. According to the authors, every child says that they have (Piaget, Inhelder and Szeminska, 1960, p. 263). A second barn is then introduced into each field, but in a different arrangement: in one, the barn is juxtaposed to the previous one; in the other, the second barn is placed elsewhere in the field, not near the first barn. The child is asked the same question; if it is answered correctly, a third barn is added (in a row in one field, spread out in the other), then a fourth barn, and so on. The authors found results analogous to the previous ones: During stage I we find it difficult to pursue the enquiry, but at stage IIA children are ob- viously interested, yet they refuse to admit that the remaining areas are equal, often at the very first pair of houses. Here there is no trace of operational composition, and judgment is based entirely on perceptual appearances. At level IIB we find a complete range of intermediate responses: up to a certain number of houses the subject admits the remaining meadowlands are equal; beyond that number the perceptual configurations are too different. Here there is intuitive articulation in varying degrees, but not operational composition. At stage III, however, . children recognize that the remainders are always equal, relying on an operational handling of the problem which convinces them of the necessity of their reasoning. (Piaget, Inhelder and Szeminska, 1960, pp. 263-264) Variations in the above procedure produced some surprises. In the discussion above, it is not mentioned where each of the first two barns was placed on each field. In the experiment, they were placed identically. Yet in other experiments, it turned out that if one of the barns was placed in a corner, and the other in the center of the other field, the remaining space did not seem equal to all children (p. 263). The authors used rectangular "bricks" to represent the barns. They relate one example in which the bricks were first placed in identical positions, and the child identified as GAR agreed that the amount of green was the same. Then; investigators And like this (one in the centre of Bi with the length of the brick parallel with the length of the meadow, another at one end of B2 and laid breadth- ways)? [GAR] No, there's more green left here (B2). Investigator] Why? LGAA Because there's all this left (free space). (p. 264) Bi C Bz 2 39 This situation is reminiscent of some optical illusions in which the orientation changes the appearance of a length. After carrying out other experiments, the authors noted further that the conservation of "space remaining" did not necessarily occur simultaneously with conservation of "space taken up" (Piaget, Inhelder and Szeminska, 1960, p. 286), and that once an area (or plane surface) had been cut, its area might not seem the same to some children, even when it was put back together (p. 295). Other features of the theory. There are many other experiments Piaget has done which would be illustrative of his theory and his method. These six tasks were chosen because of their relationship to the present study. Some of the other relevant features of Piaget's theory, taken primarily from Flavell (1963) and Travers (1977), are discussed below. The child develops a "schema," an organization of ideas or behaviors, a structure in the intellect which enables the child to understand. New information that is found is "assimilated," or added into the existing schema. The process of assimilation entails adding knowledge or behavior consistent with actions already organized within the schema. Later, as the child acts on the environment, the child changes the schema or builds a new one to accommodate new behaviors in response to new situations. "Accommodation" is this building of new schemata or modifying of old schemata to adapt to new situations. A child adapts to the environment by an interplay of assimilation and accommodation. There are definite stages of cognitive development, invariant in sequence. Each stage is the foundation for the next stage. To go from one stage to the next, the child needs to mature chronologically, and also needs experience with the environment. Further, there has to be a problem that the child wants to solve. The child is not satisfied with the solution produced by the present stage of development; Piaget calls this a state of "disequilibrium." When the child finds a new solution to the problem at a higher cognitive level, equilibrium is restored. This process is called equilibrationn." Thus, cognitive development results from the child's interests and drives in interaction with the environment. Conjectures could be made about what might happen when some of these requirements for development are not met. For example, a child might have matured chronologically without having had the experiences which induce development. The particular environment may not have presented problems that the child wanted to solve. Or the child's interests may have been in art or some other endeavor which did not induce the conflict, or disequilibrium, necessary for cognitive growth. In these cases, the expected structures may not have developed. Such a situation does not preclude further growth. Piaget's theory does not put a ceiling on development at any age. Therefore, the theory is compatible with the possibility that children of intellectually deprived environments may not yet have achieved the cognitive development of which they are capable. The next section will focus on these disadvantaged students. Disadvantaged Students Pikaart and Wilson (1972) "examined the research on the slow learner in mathematics and found it lacking" (p. 41). The meager research that is available, they say, parallels the development of the idea that intelligence is quantifiable. IQ scores are of little use, they say. A more fruitful approach . is to consider specific learning aptitudes of slow learners and to adapt instruction to take account of these individual differences. (p. 42) In Suydam's (1971) summary of research on teaching mathematics to disadvantaged pupils, she notes that the summary does not contain many studies done with students in the secondary school. One of the reasons she gives for this is that there are not as many slow learners or low achievers or otherwise disadvantaged students still enrolled in mathematics courses in the secondary school. The process of selection or tracking precludes most students in any of the subsets of the disadvantaged from going beyond a general mathematics course. (p. 3) To "enrolled in mathematics course," she might have added "enrolled in school." With compulsory attendance over at around age 16, many who have not been successful by then drop out. The studies concerning disadvantaged students that are listed by Suydam (1971) usually focus on comparing different teaching methods, and will be mentioned later. Compensatory and remedial programs have proliferated; still mathematics education researchers interested in secondary school mathematics have devoted the bulk of their resources to studying the students who are in the college bound track, taking courses in algebra and geometry. It is hoped that this study of disadvantaged secondary students will be a start in the direction suggested by Pikaart and Wilson (1972). The child's development is left now for a consideration of the student's school learning, as the learner interacts with mathematics content. Interaction of the Learner and the Content The discussion of what happens when the learner interacts with mathematics content must be limited for this review. The topics chosen can be explained by first summarizing the previous two sections. First, efforts to study the content, the fraction concept and operations with fractions, were discussed. Included were calls for research to find the underlying cognitive structures of fractions. Secondly, in looking at the learner, relevant aspects of cognitive development theory were described. The lack of research on students who have difficulty learning mathematics was mentioned. This section, on the interaction of the learner with the fractions content, will relate the preceding sections. For example, in spite of his disavowal of educational objectives in general, Piaget did consider what are almost pre-fraction concepts in some detail. This work on fractions will be reviewed. Next will be a description of assess- ment efforts aimed at finding out what students in general know about fractions, and then of diagnostic and prescrip- tive studies, concerned with why the individual student has not learned fractions and what might be done about it. Attention will also be given to clinical studies, which often include detailed observation of interactions between learner and mathematics content. The neo-Piagetian research will be included. Last will be a discussion of the concrete-versus-symbolic modes of presentation of mathe- matics content in attempts to assess students' knowledge. Piaget's Fractions Much of Piaget's work has been done with small children, so he has not given much attention to fractions. He has, however, considered "Subdivision of Areas and the Concept of Fractions" as Chapter 12 of The Child's Conception of Geometry (Piaget, Inhelder and Szeminska, 1960). He describes work with children whose ages range from 4 to around 7 years. He is not studying "fractions" as they are normally taught in school, however. For example, a child is asked to cut a cake, to "divide it up so that the man and the woman will both have the same amount of cake to eat" (Piaget, Inhelder and Szeminska, 1960, p. 304). The child does not have to know either the notation "1/2" or the words "one half" to be able to perform the task. When Piaget writes about "their idea of a fraction" (p. 310), he seems to be talking about the children's idea of partitioning, a basic component of, or perhaps even a pre- concept to, the idea of a fraction. (The fact that the work is a translation may add to the confusion.) The procedure was as follows: The child was expected to use a wooden knife to divide a circular cake made of modelling clay equally between two dolls. After the division was performed, the child was asked whether, if the pieces were put back together, it would be equal to the original whole. Those children who could divide the cake into halves were then asked to divide the cake between three dolls, and so on, up to six dolls. The youngest children often cut two pieces of arbitrary sizes for the two dolls, leaving the remainder of the cake (either ignoring it or pushing it aside). When pressed by the interviewer as to what was to be done with the remainder, a child might refuse to discuss it (p. 305) or even try to hide the leftover part (Piaget, Inhelder, and Szeminska, 1960, p. 306). At this stage the child was concerned neither with equality of shares, nor with exhausting the whole. Some children also seemed to think that two pieces required two cuts. More advanced children could correct their mistake, having made two cuts, by subdividing the remainder and parceling out more cake to the dolls, so that the cake was exhausted, at least, whether equally subdivided or not. In trying to comprehend the children's behavior, Piaget suggests that the half-to-whole, and generally, part-to-whole, relationship can be understood by the child perceptually. But, he says, it is a far cry from such perceptual or sensori-motor part-whole relations to operational subdivision. There are syste- matic difficulties in understanding part- whole relations on the plane of verbal thinking. . When we used phrases like "a part of my bunch of flowers is yellow," or "half of this bunch is yellow," etc., we found that even children of nine or ten thought of the whole bunch as yellow because they thought of the part (or half) as something absolute rather than as being necessarily relative both to the other part (or half) and to the whole. Typical replies were these: "What's a half?-- Something you've cut off.--What about the other half?--The other is gone." Obviously the half that is cut off and thought of as a thing apart without reference either to the whole or to the other half echoes the little pieces which are cut off in actual fact by children of two to four. S. Quite early on children elaborate means of dealing with reality at the level of action, and even at the level of con- crete operations, but these solutions still need to be re-worked at the level of verbal thinking by means of formal schemata. (Piaget, Inhelder, and Szeminska, 1960, p. 308) Thus, when a half is cut off, it may become to the child an entity on its own, with no further reference to the whole of which it was a part. Piaget refers to his earlier work on the part-whole relation, when he was studying the child's conception of number: Thus, when shown a large number of brown beads alongside two white beads, all these beads being made of wood, the child under seven could not understand that there were more wooden beads than brown beads for he persisted in forgetting about the collection as a whole when concentrating on the brown beads and therefore came to the conclusion: "There are more brown beads than wooden beads because there are only two white beads." (Piaget, Inhelder, and Szeminska, 1960, p. 308) In the partitioning task, again, subdivision must be reconstructed in thought, but with reference to a concrete situation. It cannot be assumed that a child who is able to physically partition an object can verbalize the actions. In analyzing the actions of the smallest children, mentioned earlier, Piaget says that the most striking thing is the presence of a part-part, rather than a part-whole, relationship. For them, "the relation between parts is one of juxtaposition and not of a nesting series" (p. 309). The child ignores the quantitative aspect, that two halves are equal, for example, and also the relation of the part to the whole, "from which it may be parted in fact but to which it still relates in thought" (Piaget, Inhelder, and Szeminska, 1960, p. 309). Piaget's analysis of the fraction concept continues: The notion of a fraction depends on two fundamental relations: the relation of part to whole (which is intensive and logical) and the relation of part to part, where the sizes of all other parts of a single whole are compared to that of the first part (a relation which is extensive or metric). (p. 309) He describes the necessary components of the notion of a fraction as follows: 1. The child must see the whole as composed of separable elements, i.e., divisible. Very young children, he says, see the whole as an inviolable object and refuse to cut it. Later, the children are prepared to cut it, but then the act of cutting it may make the object lose its wholeness. 2. A fraction implies a determinate number of parts. Children who do not realize that the number of shares should correspond to the number of recipients begin by randomly breaking off pieces. 3. The subdivision must be exhaustive, i.e., there must be no remainder. There has been mention already of children who refuse to share out the remainder, apparently satisfied that when they have made up the two parts they were asked for, anything left over is neither part nor whole and has nothing to do with the two real parts: these alone are real because these alone go to make up their idea of a fraction. (Piaget, Inhelder, & Szeminska, 1960, p. 310) 4. There is a fixed relationship between the number of subdivisions and the number of intersections, or cuts to be made. 5. The individual parts must be equal. 6. The parts themselves have a dual character: they are parts, but they can also be wholes, and thus are sub- ject to being subdivided further. This is the understand- ing necessary for finding fourths by halving halves. 7. The sum of the parts equals the original whole. Somehow, cutting the cake changes it for some children. A subject identified as SOM thinks there is more in two half-cakes than there is in one whole (p. 327). Subject GIS says that they are the same, but: asked.to choose between a whole cake and two halves, chooses the whole, saying, "I get more to eat this way" (p. 320). In a measured understatement, Piaget says, "We see how paradoxical are these replies" (p. 329). Some of the conditions for understanding the fraction concept may seem obvious, but Piaget has discovered their necessity by seeing their absence in the thinking of children. He further says that these seven conditions must still be part of a general structure to be operational. There must be an anticipatory schema: children must be able to anticipate the solution before they can solve the prob- lem. That is, they must plan ahead where all the cuts will be before they make the first cut. In the absence of such a plan, successive fragmentation of the cake is made. Piaget emphasizes the complexity of the task: The subdivision of an area . is fraught with considerable difficulty for young children and its complications compare in every respect with those pertaining to logical subdivision of the nesting of partial classes within an inclusive class. (Piaget, Inhelder and Szeminska, 1960, p. 333) Piaget defines the substages in the subdivision of areas according to whether the seven conditions are met by the children, and whether they can halve, trisect, quarter, etc. (trisecting being more difficult than quartering, since quartering can be done with two successive dichotomies). He concludes the chapter with the following summary: The facts studied in this chapter show not merely that there is a clear parallel between the subdivision of continuous areas and that of logical classes, but also that notions of fractions and even of halves depend on a qualitative or intensive substructure. Before parts can be equated in conformity with the extensive characteristics of fractions, they must first be constructed as integral parts of a whole which can be de-composed and also re-assembled. Once that notion of part has been constructed it is comparatively easy to equate the several parts. Therefore, while the elaboration of operations of subdivision is a lengthy process, the concept of a fraction follows closely on that of a part. For parts which are subordinated to the whole can also be related to one another, and when this has been achieved, the notion of a fraction is complete. (Piaget, Inhelder and Szeminska, 1960, pp. 334-335) Although Piaget says that "the notion of a fraction is complete," it must be noted that his discussion has dealt basically with one interpretation of fraction, that of subdividing continuous substances. He studied primarily one medium, clay, which is three-dimensional, though he referred to the task as "subdividing areas." Piaget did try some different shapes and some plane figures, finding it easier, for example, for the children to trisect a rectangle than a circle, and the longer the rectangle, the easier. Some of the children were given a "sausage" of modeling clay and it was found to be the easiest solid to trisect, presumably because it was like an elongated rectangle (p. 319). Piaget's work has given valuable insight into some of the components necessary to a child's concept of fraction. But he did not treat a linear model or a discrete model. He considered only unit fractions, and then with small denominators. He did not look at equivalence, or comparisons between different fractions. And certainly it was not his purpose to study how children learn about 51 fractions as ratios or quotients or how they come to per- form mathematical operations with fractions. These other aspects need to be given in-depth study also. The next topic is students' general knowledge of frac- tions as taught in school, and as evidenced by assessment tools. Assessment of Students' Knowledge of Fractions Students have all received instruction in fractions by the end of the sixth grade, so it is appropriate to ask what understandings and skills they carry with them into junior high and high school. In view of most elementary school mathematics pro- grams today, Carpenter, Coburn, Reys, and Wilson (1978) say, "13-year-olds should be thoroughly operational with fractions" (p. 34). However, in their summary of the NAEP mathematics assessments, they say that overall results on fraction concept tasks were low. Of all three groups, 13- year-olds, 17-year-olds, and adults, no more than about two thirds responded correctly to an exercise that dealt with fraction concepts (p. 34). Some of the NAEP results were reviewed earlier (Carpen- ter, Coburn, Reys, & Wilson, 1976), when only two exercises were released. The first was: 1/2 + 1/3 = (p. 137) Only 42% of 13-year-olds and 66% of 17-year-olds were successful in solving this exercise. Of various incorrect responses, the most common was obtained by adding both the numerators and the denominators (30% and 16%, for 13 and 17-year-olds, respectively). In speculating about this and other errors, the authors say the results suggest "that students are not viewing the fractions as representing quan- tities but see them as four separate whole numbers to be combined in some fashion or other" (Carpenter, Coburn, Reys, & Wilson, 1976, p. 138). The multiplication exercise was: 1/2 x 1/4 = (p. 137) The students performed better on this exercise, getting 62% and 74% correct answers. The incorrect responses did not show a pattern. The authors noted that these results were consistent with data from various state assessments and other research (p. 139). The later, more complete report (Carpenter, Coburn, Reys, & Wilson, 1978) describes the testing of the concept of fraction: Asked what fractional part of a small set of marbles was blue, 65% of 13-year-olds answered correctly (p. 37). The three older groups were given this problem: There are 13 boys and 15 girls in a group. What fractional part of the group is boys? (p. 38) Described as "very disappointing," the results were 20%, 36%, and 25% correct answers, for 13-year-olds, 17-year-olds, 53 and adults (Carpenter, Coburn, Reys, & Wilson, 1978, p. 38). The authors commented that the cause of the errors cannot be determined from the data. Perhaps there is a problem with the language "fractional part" that would contribute to the-"I don't know" re- sponses. But the committed errors must be due to a lack of mastery of fraction con- cepts and their application to problem con- texts. (p. 38) In a multiple choice exercise, two common fractions less than 1 were given; respondents were asked to select another fraction between them. Correct answers were given by 56% and 83% of 13 and 17-year-olds (p. 39). However, when given six very common fractions less than 1 and asked to write them in order from smallest to largest, "no age group could perform this task adequately" (p. 39). Asked which fraction was the greatest of 2/3, 3/4, 4/5, and 5/8, 26% of 13-year-olds and 49% of 17-year-olds an- swered correctly. The authors say, The strongest distractor for both the 13-year-olds and the 17-year-olds was 2/3. The exercise clearly shows that 13-year-olds are not yet operational with fractions. (p. 40) In a survey intended to find out whether deficiencies in fractions skills were due to current instructional pro- grams, Ginther, Ng, and Begle (1976) went to "the most ad- vantaged schools" in their area and tested about 1,500 eighth graders. The students were in intact classes iden- tified as average by their teachers. A battery of fractions tests was designed, to include the cognitive levels of 54 computation, comprehension, and application. In the compu- tation section, the second easiest problem (the second high- est percentage correct) is given for each of the operations, with the percentage of students correct: 1/6 3/4 5/8 x 32 7/8 5/16 + 5/8 2/5 63% 58% 65% 40% correct (Ginther, Ng, & Begle, 1976, pp. 3-4) The authors were apparently not concerned by these low percentages, commenting in the conclusion that the students had a reasonable understanding of the fraction concept (p. 9). They did, however, decry the students' lack of under- standing of structure. In the comprehension section, items were intended to be answered very easily by students who understood the structure of the rational number system. The following are examples, presumably still the second easiest: 2 x[= 1 . 42% correct (p. 4) I I 1 1 I I I I I 0 A 1 A is 1/2 1/3 1/4 3/8 5/8 62% correct (p. 5) The fifth easiest diagram question was as follows; There is a drawing on the left. Part of the drawing is shaded. The drawing suggests a fractional number. You are to choose the fraction on the right which names the same fractional number as the shaded part of the drawing. Circle the letter in front of your answer choice. 1/6 3/6 5/6 7/6 None of these 88% correct (Ginther, Ng, & Begle, 1976, p. 6) The following example from the applications section was the fourth easiest in its subsection: A girl weighs 64 1/2 pounds. Her brother weighs 1/2 as much as she weighs. How many pounds does he weigh? 54% correct (p. 7) These results were included here to illustrate that even in the "most advantaged" schools, many students do not have a thorough understanding of fractions. The authors concluded that the poor understanding of the structure of the rational number system was due to poor instructional programs, and that until elementary and junior high school teachers could teach fractions in a more meaningful way, much of the work on fractions should be postponed to secon- dary school (p. 9). An alternative explanation is that the students may not have reached the level of cognitive deveop- ment necessary to profit from the instruction. Efforts to help the individual student will be discussed next. Diagnostic and Prescriptive Teaching Diagnostic and prescriptive teaching is not new, but is emerging as an important area in mathematics education. The State of Florida has recently passed a law requiring early childhood teachers to use diagnostic and prescriptive techniques when teaching the basic communication skills. In the teaching of mathematics at all levels, the techniques seem especially appropriate. The pioneers in using diagnosis and prescription in the teaching of mathematics were Brownell, Brueckner and Grossnickle, who did extensive work in the field beginning in the twenties and working through the forties. Interest in that effort waned during and after the war, but the preoccupation in the late sixties with disadvantaged students and the current emphasis on ensuring that minimal competencies are mastered has caused a rebirth of interest in the field. The term "diagnosis" refers to knowing not just that the student missed the problem, but why (in the sense of "what type of error was made?"). "Prescription" means the assignment of instruction specifically designed to correct that type of error. This method of teaching has been des- cribed as shooting with a rifle, rather than with a shotgun (Glennon and Wilson, 1972, p. 283). There have been some attempts to find the causes of "discalculia," or mathematical disability (Farnham-Diggory, 1978), including studies of brain damage (Luriya, 1968) and of the hemispheres of the brain (Davidson, Note 5). Concen- trating more on psychology than biology, Scandura (1970) reviewed research in "psychomathematics." He concluded that there are a large number of unspecified, but crucial, "ideal competencies which underlie mathematical behavior. These need to be identified. . There is also the urgent need to consider how the inherent capacities of learners and their previously acquired knowledge interact with new input to produce mathematical learning and performance. (p. 95) These urgent needs might best be met through in-depth obser- vations of individual students and their learning, as is done in clinical studies. In the meantime, many diagnosticians have taken the pragmatic viewpoint: they would like to know how students learn mathematics, but meanwhile, they try to find out specifically what students are doing wrong and to correct or remediate those errors. Glennon and Wilson (1972) wrote a state-of-the-art paper for the 35th National Council of Teachers of Mathematics (NCTM) Yearbook, The Slow Learner. They defined diagnostic- prescriptive teaching as "a careful effort to reteach success- fully what was not well taught or not well learned during the initial teaching" (p. 283). They suggested the interview technique perfected by Brownell (Brownell & Chazal, 1935) for finding out what students were doing wrong. Lankford has used individual diagnostic interviews to survey the computational errors of seventh graders as they worked problems involving whole numbers and fractions. He tested 176 students in six intact seventh grade classes. In the interviews he directed students to "say out loud" their thinking as they computed (1974, p. 26). The percen- tages correct on the fraction exercises are not surprising, in view of the national assessment data; in general "the performance was much below that with whole numbers" (Lank- ford, 1972, p. 30). A sampling taken from that article (pp. 20-30) follows: Table 1 Sample of Results of Lankford Study Percentage of Exercise Attempted Exercises Correct 3/4 + 5/2 47 3/4 1/2 58 2/3 x 3/5 63 9/10 3/10 41 Which is larger, 2/3 x 5 or 1 x 5? 61 It should be noted that in the last exercise cited above, there were two choices; students could have been correct 50% of the time by chance. In fact the interviews showed, Lankford said, that sometimes students gave the correct answer for the wrong reason. 59 The main findings, of course, were students' thinking patterns. In addition of fractions, for example, out of 97 incorrect answers, 62 were found by adding the numerators and also adding the denominators; 10, by adding the numera- tors and taking the larger denominator; and 6, by adding the numerators but multiplying the denominators (p. 30). These errors might have been predicted by experienced teachers, but Lankford says, "relatively large whole numbers were a 'surprise' as when 3/4 + 5/2 = 86 . and 3/4 1/2 = 22" (p. 31). Students were adding or subtracting the numerators and denominators; the surprise lies in the manner in which the results were written. Apparently either a fraction did not have meaning as a small number for these students, or the students did not connect the meaning of a fraction with computations done on paper. Another error demonstrates the lack of understanding of the meaning of a fraction: 3/8 + 7/8 = 11/15 (p. 31) The answer was derived from 3 + 8 = 11 and 7 + 8 = 15; the procedure may have been a persevering pattern from the column addition of whole numbers. And to change 3/4 to an equivalent fraction, one student reasoned, "4 times 1 equals 4 and 1 + 3 is 4, so 4/4" (p. 31). The conclusion that 3/4 = 4/4 again indicates that the student did not understand the concept of fraction, 60 or did not connect the concept with the computation. Some students even stated that "2/3 is greater than 1" (Lankford, 1972, p. 34). In concluding, Lankford gives pointers in the use of the diagnostic interview, suggesting that teachers can learn how well instruction has been imparted by using this method with their own students. Glennon and Wilson (1972) also recommend Brownell's models of ideographicallyy oriented procedures,"which they feel are effective techniques for both diagnostic and pre- scriptive teaching. They give credit to both Brownell and Piaget for their contributions to the development and use of idiosyncratic procedures in mathematics education, but cite as the more easily understood and readily used the work of Brownell (p. 308). Even with Brownell's and Piaget's clinical procedures, especially frustrating and challenging are those students called "disadvantaged" or "slow learners" or "low achievers." Many teachers feel that if a formula could be found to enable their learning, all students would benefit from the formula. The only logical way this idea could be in 61 error is for slower students to actually learn in a quali- tatively different manner from the more successful students' manner. What has been discovered, if anything, about this possibility? In Suydam's (1971) summary of research on teaching mathematics to disadvantaged students, cited earlier, she lists the following as one of the statements that can be implied from the research: The mathematical characteristics which distinguish disadvantaged from advantaged pupils appear to exist in degree rather than kind. That is, disadvantaged and advantaged pupils have similar abilities and skills, but differ in depth or level of attainment. (p. 13) It is an assumption of this study that the above statement is true, and that what is learned about the learning of disadvantaged students will help the advantaged students as well. Suydam also found that "active physical involvement with manipulative materials, which is believed to be important for all children, may be even more so for the disadvantaged" (p. 13). However, as she noted earlier, "little research has been done on this specific topic with specific sets of disadvantaged pupils" (p. 5). She concludes, Groups of disadvantaged pupils are not all disadvantaged in the same way. There is as much need to individualize instruction for disadvantaged students as for other groups of students. (p. 13) Currently many compensatory and remedial instructional programs aimed at teaching basic skills do not take these individual differences into account. There is an older study which was designed to address the problems of these students in a substantive way. Al- though the students were not in secondary school, but in the upper elementary grades, the spirit and method of the study and the questions asked make its review appropriate. The purpose of the study, reported by Small, Avila, Holtan and Kidd (1966), was to "explore factors related to low achievement and underachievement in mathematics education and to determine if there are individual levels of abilities in abstractive thought with respect to mathematics concepts" (p. 4). This pilot study was an attempt to identify charac- teristics of low achievers and underachievers in mathematics in grades 4, 5 and 6, in hopes of finding new approaches to remediation, thereby making it possible to intervene in the processes which often lead to failures and dropouts. Low achievers were defined as students of average IQ whose average percentile scores on all sections of a standardized achievement test were at least two deciles below their present grade placement level. Underachievers were students of average IQ whose nonmathematics scores were equal to or above their grade placement, but whose 63 mathematics computation and concepts scores were two or more deciles below their nonmathematics percentile averages.. Small et al. (1966) used a case study approach with 12 underachievers and 11 low achievers. Each student was tested individually on two concpets, place value and linear measurement. There were three levels of questions on each subtest: concrete (the test material was a physical model which could be manipulated by the subject); semi-concrete or pictorial (materials used were photographs of real ob- jects); and abstract (questions were asked verbally or sym- bolically). The report included affective results. First, there was no consistent pattern on levels of abstraction; the ability to operate on the different levels is an individual problem and must be identified for each student. Secondly, both the low achievers and the underachievers seemed to experience more emotional adjustment problems than did the typical student population. The underachieving stu- dent was most often a child with a large amount of anxiety and a relatively unharmonious home in which high achievement was considered importatn. The low achieving student probably needed a comprehensive compensatory program at school. Several recommendations were made for underachievers, basically aimed at reducing their anxiety. The authors recommended a diagnosis and remediation plan involving levels of abstraction, for testing by other researchers. The Small et al. (1966) study serves both to introduce the general field of clinical studies and to focus attention on the concrete-versus-abstract question. Clinical Study In the study discussed above, a "case study" approach was used with 23 subjects. Tests were administered indivi- dually. (The testing instruments are given, but the report is brief and details of the diagnostic interviews are not available.) No hypotheses were being tested; rather, the researchers were searching for factors which might be used to form hypotheses concerning low achievers and under- achievers. In many clinical studies the interview, as developed by Piaget, is used as the basic technique to gain informa- tion about children's thinking. This approach may seem unscientific to some researchers trained in standardized testing, for, as Flavell (1963) says, no two children will ever receive exactly the same experimental treatment. Even though the initial questions may be uniform, in the course of this rapid sequence, the experimenter uses all the insight and ability at his command to understand what the child says or does and to adapt his own behavior in terms of this understanding. (p. 28) The same individual attention used in diagnosis needs to be used in studying the interaction of the learner with instruction, as in "teaching experiments." According to Steffe, teaching experiments share these characteristics: 65 They are usually long term interventions, with a small number of students. Researchers study how children learn, or the "dynamic passage from lack of knowledge to knowledge present" (Steffe, Note 6). This microscopic attention to individual students is expected to yield much information, in contrast to tradi- tional paper-and-pencil standardized testing, where "there is no way of knowing exactly what respondents were thinking" (Carpenter, Coburn, Reys, and Wilson, 1976, p. 137). In mathematics education research, according to Kilpatrick, we not only want to know that certain people do better at certain things; we also want to know their characteristics, and what interaction is occurring. These things, he says, can not be learned from statistical analyses. Neither can anything be learned without sensitivity. A suggested approach is,"Let me look very intensively at a small number of people and see what is happening" (Note 7). This is the approach of a clinical study. The next topic will be a cursory look at how others have interpreted Piaget's works and the resulting impact on mathematics education. Related Piagetian Research The many efforts to make sense of, and subsequently, to make use of Piaget's voluminous output can be roughly categorized as follows: 1) validation (or invalidation) studies, where attempts are made to replicate his experiments; 2) "training," or learning, studies, where experi- menters test to see whether children can be taught the cognitive structures Piaget has described (classification, seriation, etc.); 3) applications or extensions of his theory and/or his methods to other situations, to education in particular. (In a sense, of course, group 2 is a subset of group 3.) There will be no attempt here to give a comprehensive review of this work. The earlier works can be located in Flavell's (1963) definitive book on Piaget's work, and Lovell (1971a) has reviewed "twenty-five years of Piaget research in intellectual growth as it pertains to the learning of mathematics" (p. 2). Some general comments will be made, including mention of a few relevant studies, with the primary attention given to the third group. 1) The validation studies generally support Piaget's theory, although variations are reported. Lovell (1971a) summarizes a group of these: By and large the stages in the development of the structures, proposed by Piaget, are found but there are differences. The age range for the elaboration of a particular structure is considerable even in children of comparable background and ability as judged by teachers or by test results. (p. 5) 67 Lovell states further that the situation, the actual apparatus used, and the previous experiences of the children are all variables affecting their behaviors. "It is now clear that the tasks are subtle, that the relevant ideas have to be carefully devised and that analysis has to be thoughtfully considered" (Lovell, 1971a, p.6). 2) Flavell (1963) reviews 20 training or learning studies (pp. 370-378) which pertain to the teaching of the various cognitive structures. Results were mixed; only a few reported significant differences between the trained groups and the control groups. Flavell comments, Probably the most certain conclusion is that it can be a surprisingly difficult undertaking to manufacture Piagetian concepts in the laboratory. Almost all the training methods reported impress one as sound and reasonable and well-suited to the educative job at hand. And yet most of them have had remarkably little success in producing cognitive change. It is not easy to convey the sense of disbelief that creeps over one in reading these experiments. (p. 377) Just as they are difficult to induce, the conservation concepts are difficult to extinguish when actually once acquired, he says. The one study he reported in which the training group clearly outperformed the control group was one by Smedslund (1961), in which the keynote of the training procedure was the induction of cognitive conflict and the absence of external reinforcement. Piaget's response to these efforts is usually amuse- ment. In the first place, he does not understand why edu- cators want to accelerate what he considers the child's natural development. Even assuming that such acceleration is a worthwhile goal, he is skeptical. Whenever he is told that someone has succeeded in teaching operational struc- tures, there are three questions he asks. First, is the learning lasting, two weeks, a month later? "If a structure develops spontaneously, once it has reached a state of equilibrium, it is lasting; it will continue throughout the child's entire life" (Piaget, 1964, p. 184). And when the learning is achieved by external reinforcement, he asks, what are the conditions necessary for it to be lasting? Secondly, how much generalization is possible? "You can always ask whether this is an isolated piece in the midst of the child's mental life, or if it is really a dynamic structure which can lead to generalizations" (p. 184). The third question is, "What was the operational level of the subject before the experience and what more complex structures has this learning succeeded in achieving" (p. 184)? We must see, he says, which spontaneous operations were present at the outset and what operational level has now been achieved after the learning experience. 69 Recent training studies by mathematics educators have included those by Coxford (1970), Johnson (1975), Kurtz and Karplus (1979), Lesh (1975b), and Silver (1976). Some re- port successful training and some do not. 3) The unsettled questions just mentioned bear on the present section. In his article on psychology and mathema- tics education, Shulman (1970) says that Piaget's charac- terizations of number-related concepts have helped shape our ideas of what children of different ages might learn meaningfully. This has thus influenced some current concep- tions of readiness: To determine whether a child is ready to learn a particular concept of principle, one analyzes the structure of that to be taught and compares it with what is already known about the cogni- tive structure of the child of that age. If the two structures are consonant, the new con- cept or principle can be taught; if they are dissonant, it cannot. One must then, if the dissonance is substantial, wait for further ma- turation to take place. (p. 42) If the degree of dissonance is small, Shulman says, Piaget's theory does not recommend, but neither precludes, training procedures aimed at achieving the desired state of readiness. Brainerd (1978) disagrees entirely with Piaget's model of learning. Since he assaults major theses, not trivial details, his arguments will be mentioned. He first takes issue with the notion that concepts will arise na- turally and need not be trained. Brainerd's is a typical oversimplification of Piaget's "notion," which actually includes as requirements for this "natural" development not only chronological maturation, but also an appro- priate set of experiences, providing for disequilibrium and subsequent, higher-level equilibration (Copeland, 1979; Flavell, 1963). Further, Brainerd says that those Piagetians who do training experiments insist that the training be as natural as possible and include opportunities for self-discovery. Brainerd maintains that there is not a continuum from artificial to natural, and that there is no evidence that natural is better (Brainerd, 1978, pp. 83-84). The same original sources, in this case Piaget's theory as stated by his co-workers, can yield different interpretations. Another person, reading the same quotations Brainerd has selected (pp. 69-78), might summarize them using the phrase "relevant to the child," for example, instead of the word "natural." (This interpretation would render irrelevant Brainerd's admitted digression on Rousseau (pp. 79-84), subtitled "Is Mother Nature Always Right?") If Brainerd's recommended methods of teaching, or training, are worthwhile, whether natural or not, then of course they should be used. For example, he mentions "correction training," in which verbal feedback from the experimenter is accompanied by "a tangible reward (e.g., candy or a token) following correct responses" (p. 86). This method of teaching 71 is not recommended by some psychologists. Not only may the reinforced behavior be extinguished when the reinforcements are removed, but also, extrinsic rewards may actually de- crease the intrinsic value of the learning activity for the subject, thus doing more harm than good (Levine & Fasnacht, 1974, p. 820). Other types of training Brainerd mentions as success- ful are "rule learning" and "conformity training." In rule learning, as the name implies, the students are taught a rule or rules "which may subsequently be used to generate correct responses on a concept test" (Brainerd, 1978, p. 87). In conformity training, children who missed the concept questions on protests are grouped with children who answered the pretest questions correctly. Asked to arrive at "con- sensual answers," the conservers apparently convinced the nonconservers. Brainerd says that "79% of the pretest nonconservers learned all five concepts. . All improve- ments were stable across a 1-week interval" (p. 88). One must accept the statement that the 79% gave correct res- ponses; Piaget would want to wait more than a week to see whether the children had "learned all five concepts." In further critique, Brainerd selects three predictions he says the Piagetian theory makes. First, learning inter- acts with children's knowledge of to-be-trained concepts. But, Brainerd says, few learning theories would not say that. Secondly, preoperational children cannot learn con- crete operations concepts. Brainerd says that this has been disproved (Brainerd, 1978, p. 105). And thirdly, concepts belonging to different stages must be learned in a certain order. Brainerd says that this is a trivial outcome; the way the stages are set up, each stage includes the concepts of the previous stage (pp. 100-101). He concludes that "although we may need a readiness perspective on concept learning, Piaget's approach does not seem to be it" (p. 105). The basic thrust of this study concerns the possibili- ty of improving our knowledge of how students learn, or fail to learn, mathematics, the fraction concept in particular. The value of Piaget's theory in this effort, if any, will not be that it is correct and aesthetically satisfying in every detail, but that it adds to our knowledge of how students learn or fail to learn, that it enriches our diag- noses of students; difficulties, and possibly, that it, eventually, inspires more successful teaching techniques. Consequently, the first and third predictions, which Brainerd finds insignificant, are not weak points in this context; the ideas might prove to be valuable to an educa- tor attempting to sequence instruction for the student's maximum success. 73 In trying to refute the second prediction ascribed to Piagetian theory, Brainerd again violates Piaget's assumptions. In setting the stage for the studies that he says prove that preoperational children can learn concrete operational concepts, he states, "preschoolers should be almost completely untrainable. . in a sample of 3- to 4-year-olds . it should be safe to assume that concrete- operational mental structures are not present" (p. 96). As mentioned previously, Piaget's theory does not say what should be, but describes what has been observed. The mental structures of a child are developed individually and may not be congruent with those of his age group. It seems evident from Piaget's experiments that it is not "safe to assume" anything about a child's thinking. Brainerd cites a train- ing study on number and length conservation with 4-year-olds, saying that there was clear evidence of transfer. The same subjects passed roughly 41% of the items on the mass and liquid quantity post- tests. (Brainerd, 1978, p. 100) With the item format not available, it is not convincing that 41% correct answers represents clear evidence. In the other experiments mentioned, retention was again tested only one week after training. If Brainerd's objection is correct, however, and Piagetian concepts can be trained, and if certain Piagetian concepts are found to be related to mathematical concepts, then the path is obvious: students should be trained in Piagetian concepts before, or in conjunction with, their mathematical instruction. Certainly many mathematics educators have seemed to heed Piaget's (1973) invitation: If mathematics teachers would only take the trouble to learn about the "natural" psycho- genetic development of the logico-mathematical operations, they would see that there exists a much greater similarity than one would expect between the principal operations spontaneously employed by the child and the notions they attempt to instill into him abstractly. (p. 18) Piaget optimistically says that one can anticipate a great future for coopera- tion between psychologists and mathematicians in working out a truly modern method for teaching the new mathematics. This would con- sist in speaking to the child in his own lan- guage before imposing on him another ready-made and over-abstract one, and, above all, in inducing him to rediscover as much as he can rather than simply making him listen and repeat.(p. 19) Lovell has called for studies which give other than pass or fail responses, and suggests that more emphasis should be placed on careful observation of the schemes which lead to correct solutions. He says that such studies are likely to throw light on the nature of the schemes (in respect of mathema- tical ideas) available to normal as compared with dull and disadvantaged pupils. . The classical Piagetian structural model must be supplemented. (Lovell, 1975, p. 187) Carpenter expressed the research need as follows: What is essential is the construction of good measures of children's thinking and the iden- tification of specific relationships between performance on those measures and the learning of particular mathematical concepts. (p. 76) Several studies have used Piaget's cognitive structures as measures of children's thinking and have attempted to relate them to mathematics learning. Those most pertinent to the study of fractions will be discussed. Hiebert and Tonnessen (1978) wanted to extend Piaget's analysis of fractions in continuous situations to other physical interpretations. They decided to replicate the experiments with continuous models and to investigate whether Piaget's analysis applied equally well to a discrete model of fractions. Nine children, 5 to 8 years old, were given three tasks in videotaped interviews. They were asked to divide a quantity of material equally among a number of stuffed animals so that the material was used up. In the area task, a circular "pie" of clay was used; in the length task, a piece of licorice, and in the set/subset task, penny candy (four times as many candies as animals). Two children had tasks dealing with halves, three with thirds, three with fourths, and one with fifths. Six of the nine children succeeded in discrete (set/ subset) tasks; only two succeeded in both length and area tasks. The explanations offered by the authors are that discrete quantity tasks do not require well-developed anticipatory schemes, while continuous quantity tasks do. Discrete tasks were solvable by number strategies (e.g., counting); length and area tasks first required a sub- division into equal pieces. Concerning the developmental sequence, the authors said that in area representation, some children were successful with halves and fourths, but not with thirds. In the length task, the level of difficulty corresponded with the number of parts. And in the set/subset task, no order-of-difficulty sequence was observed. The pre- dominant one-to-one partitioning strategy was used with equal success for all fractional numbers. Hiebert and Tonnessen (1978) conclude that the Piagetian conceptual analysis of fraction is adequate to describe the children's strategies in the length and area tasks, but not in the set/subset task. Nothing inherent in the task forces the child to use the part-whole approach, since the task can be solved by simpler strategies (counting and one- to one partitioning). Further, they say that meaningful comparison of the discrete and continuous interpretations of fractions was not possible. They did not find generalizable identifying criteria that define a complete part-whole fraction concept across all physical interpretations. "It appears that further theoretical work involving a conceptual analysis of fraction must include psychological, as well as logical, analyses if this comparison is to be meaningful" (Hiebert and Tonnessen, 1978, p. 378). In upper elementary school, the ratio interpretation of fractions is important. But if, as Piaget has suggested (Lovell, 1971a, p. 8) and Lovell and others have confirmed (Lovell and Butterworth, 1966), proportional reasoning is not available to children until they reach the period of formal operations, then how can they understand fractions as ratios and solve proportions? Steffe and Parr (1968) investigated the success with fractions of fourth, fifth and sixth graders who had been exposed to two curricula, one using fractions as ratios, the other, as quotients, or fractional numbers. Among the authors' conclusions were the following statements: Children solve many proportionalities presented to them in the form of pictorial data by visual inspection both in the case of ratio and fractional situations. Whenever the pictorial data, which display the proportionalities, are not conducive to solution by visual inspection, the proportionalities become exceedingly diffi- cult for fourth, fifth and sixth grade children to solve, except for the high ability sixth graders. (p. 26) The authors raise this question, in implications for further research: Is it possible to construct a "readiness test" for the study of ratio and fractions in the elementary school? Such a test may have its foundation in the psychological theory of Piaget. (Steffe & Parr, 1968, p. 26) Efforts are being made to use Piaget-type tasks in classroom diagnosis. Johnson (1980) suggests that ele- mentary teachers can use such tasks in diagnostic inter- views. Information thus gathered, along with that obtained through traditional means, "allow the teacher to develop a program based on the diagnosed strengths and weaknesses of the child" (p. 146). A set of 18 tasks are described. The two tasks that are relevant to this study are Task 17, "Meaning of a fraction," and Task 18, "Concept of a fraction." Task 18 is an abbreviation of the cake-cuttings of Piaget, discussed in Chapter 2. However, Johnson's di- rections do not seem to be complete enough for a teacher's guide. The teacher may not know how to interpret it when a child cuts off two small slices for the two dolls, leaving a large portion of cake (perhaps trying to get rid of it under the table). Some sample expected answers could be provided, along with some criteria for deciding which answers exhibit what sort of understanding. 79 Task 17 purports to test the student's understanding of the meaning of a fraction. The materials are two 4-inch by 8-inch rectangular regions. Here are the directions: Take the two regions and ask the child if they are the same size. The child should be allowed to place one on top of the other to verify. Now mark region A and region B as in the diagram below. A B Ask, "What is a fraction name for each part of region A?" "What is a fraction name for each part of region B?" Now point to a part of region A and ask if that part is the same size as one part of region B (pointing to a part of B). (Johnson, 1980, p. 164) There may be confounding factors in the above example, relevant to the tasks used in the present study. First, an optical illusion may be operating; the horizontal length of region B may appear to be greater than that of region A, when in fact they are the same. There is also the example in Piaget's study of conservation of area, reported earlier in this paper, where the different orientation of two iden- tical bricks changed a child's perception of the area re- maining in the field. While this situation is not exactly analogous, it casts doubt as to whether the child will see the horizontal parts of region B as equivalent to the vertical parts of region A. It seems to be assumed by Johnson that the child can conserve area. Piaget has reported protocols in which children have started with two rectangles exactly alike; having cut one into two or more parts, the experimenter asked whether there was as much room in each, the cut rectangle and the uncut rectangle. Several children maintained that there was more room in the rectangle which had not been cut (even when the experimenter put the cut pieces back together, right on top of the uncut rectangle) (Piaget, Inhelder and Szeminska, 1960, pp. 275-277). Could the markings on Johnson's rectangles function in the same way, to make the child think the area had changed? If the vertical marks changed region A, did they change A in the same way that the crossed marks changed B, if they changed B? Care also needs to be taken in the use of vocabulary. What is the interviewer's definition of "the same size?" And does it happen to be the same definition the child is using? A tall skinny man and a short fat man might have the same mass or perhaps the same volume (or possibly even both?), but one might not say that they are the same size. These considerations echo the comment of Lovell (1971a) He was, in turn, quoting Mayer (1961), who said that future teachers needed a "course which attempts to explore the profound aspects of the deceptively simple material they are going to teach" (Lovell, 1971a, p. 12). Certainly Task 17 was more complicated than it appeared on the surface. 81 Of all the models of fraction, the area model seems to be appealed to most often in schools. Therefore Taloumis's (1975) area study may bear on the teaching of the fraction concept. Taloumis wanted to standardize the reporting of abilities of primary school children in area conservation and area measurement. Also to be studied was the effect of test sequence on performance. Of the 168 children in grades 1 through 3, half did the area measurement tasks first, the other half, the area conservation tasks first. Tests were administered individually. There were three conservation tasks. In the first one, two rectangles (index cards) were shown. As the child watched, one of the index cards was cut on the diagonal. The halves were separated, rotated and rearranged into an isosceles triangle. The child was asked whether the two shapes (rectangle and new triangle) had the same amount of space. The second conservation task was the farm problem discussed earlier in this paper. In the third task, the congruence of two green "gardens" and the congruence of two brown "plots of ground" for flowers were established. The brown plots were placed in the gardens, and one of the plots, which was sectioned, was changed into successively longer rectangles. The child was asked whether each garden had the same amount of ground for flowers, or, if not, which one had more. 82 In the area measurement tasks, the child was to use as measuring devices 1-unit squares, 2-unit rectangles, and half-unit squares to compare two noncongruent shapes (the unions of rectangles). In the second task a triangle was to be compared with a polygonal shape. Taloumis found that the sequence of presentation did affect the performance on the second group of area tasks. The conclusion includes the following: If area conservation tasks are administered first, the scores on area measurement tasks are increased, and vice versa. The impli- cations for future researchers are: l)train- ing in area measurement may improve a child's performance in area conservation; 2) learning takes place across Piagetian tasks given in sequence. (Taloumis, 1975, p. 241) She concludes that Piaget's theory that the ability to measure quantities is dependent on acquired concepts of conservation does not appear to be completely tenable. Piaget's stand may not be be completely tenable. On the other hand, there might be a simple explanation for Taloumis's results: the two tasks are not all that different. In the first conservation task (Ci), for example, two plane figures are being compared. In the first measurement task (MI), two plane figures are also being compared, but with the assistance of some smaller increments of area (unit squares, etc.). Consider Piaget's work on area. In a conservation of area task, a child is being asked to compare the area of a rectangle with a second one which has been transformed into a pyramid. After asking the usual question about the amount of room in each shape, the interviewer says, "What if I covered it with cubes" (Piaget, Inhelder, & Szeminska, 1960, p. 281)? The child is then led to cover first one area, then the other, with the cubes, which serve exactly the same function as Taloumis's unit squares do in task M,. For Piaget, the tasks C, and MI are both conservation tasks. Therefore it is not at all surprising that they were found to be interdependent. When Piaget studies the measurement of area, the task is slightly different. He again asks the child to compare the areas of two polygonal regions, but using two separate techniques. With the first method, there are enough or nearly enough measuring cards to cover the area being measured. He wants to discover the age at which children will use the smaller cutouts as a middle term, or common measure. In the second method, the subject is presented a limited number of square unit cards which he must then move from one part of the surface being measured to another. The point then being observed is not simply that the child answers "equal" or "not equal," but whether the child realizes the transitivity of a common measuring term, a basic component of measurement (Piaget, Inhelder, & Szemin- ska, 1960, pp. 292-293). In explaining the dependence of measurement of area on conservation of area, Piaget mentions the "harder prob- lem," the conservation of completmentary areas, where the child must not only understand the space of "sites" which are occupied and those which are vacant, but also the re- ciprocal relation between the area within a perimeter and the area outside it (Piaget, Inhelder, & Szeminska, 1960, p. 291). A child may be able to comprehend the area of a thing which takes up space before the area of the "site," or space taken up. The analogous difference in one di- mension was mentioned in the discussion of conservation of distance. In addition to realizing the transitivity of a common measuring term, in order to measure area, the child must "understand composed congruence (i.e., that a number of sections taken together equal the whole which they cover)" (p. 294). Taloumis (1975) says further that the scores showed that significant learning took place during the assessment, and that there seemed to be transfer of learning in both directions (p. 241). This result is not incompatible with Piagetian theory. For children who were transitional, the testing situation may have provided the necessary cognitive conflict, or disequilibrium, to enable equilibration at a higher level with regard to the conservation of area. In 85 fact, the "keynotes" in Smedslund's (1961) training study, conflict with no feedback, were apparently present in Taloumis's assessment procedure. The explorations with concrete manipulatives may have also been helpful to the children in Taloumis's study. There is considerable interest in the use of manipulatives in instruction and, more recently, in diagnosis. Concrete Versus Abstract Modes of Presentation The mathematics education literature has for years in- cluded recommendations that concrete, manipulable materials be used in instruction (Lovell, 1971a; NCTM, 1954; Suydam, 1970; & Swart, 1974). Shulman (1970) says that "Piaget's emphasis upon action as a prerequisite to the internaliza- tion of cognitive operations has stimulated the focus upon direct manipulation of mathematically relevant materials in the early grades" (p. 42). Of course, as Piaget uses "action," internal cognitive operations are actions. In Piaget's concept, actions performed by the subject are the raw materials of all intellectual and perceptual adaptation (Flavell, 1963, p. 82). The infant performs overt, sensori- motor actions; with development, the intelligent actions become more internalized and covert. As internalization proceeds, cognitive actions become more and more schematic and abstract, broader in range, more what Piaget calls re- versible, and organized into systems whic--are structurally isomorphic to logico-algebraic systems. (Flavell, 1963, p. 82) [Flavell's emphasis] Flavell insists that despite the enormous differences between them, the abstract operations of mature, logical thought are as truly actions as are the sensorimotor adjustments of the infant (Flavell, 1963, p. 82). Piaget's notion of development, then, is active, interactive; think- ing and knowing are actions that one performs. Flavell also interprets certain of Piaget's beliefs about education: In teaching a child some general principle, one should parallel the developmental process if possible. The child should first work with the principle in a concrete and action-oriented context. Then the principle should be- come more internalized, with decreasing dependence on per- ceptual and motor supports (moving from objects to symbols of objects, from motor action to speech, etc.)(p. 82). It must be remembered that Piaget was not himself an educator; he provided a theoretical rationale for certain recommendations, but no practical instructions for teaching. Some mathematics educators have tried to apply strategies which would provide for active learning in the spirit of Piaget. They reason that children should be provided both concrete or manipulable objects and diagrams which could illustrate the mathematical concepts being taught symboli- cally. It is not clear that teachers or students always know what to do with these learning aids. Payne (1975) reviewed research on fractions done primarily at the University of Michigan. Most of the studies that compare different instructional sequences are not germane to this study, but some do relate to the question of mode of presentation. For example, Payne says that "where meaningful approaches to operations on frac- tions have been compared to mechanical or rule approaches, there appears to have been some advantage for the ones that were meaningful" (p. 149). Further, he says, "when there was an advantage favoring meaningful approaches, it was usually most evident on retention tests" (p. 149). "Meaningful" and "mechanical" were not always clearly de- fined; however, Green's (1970) study, according to Payne, had a logical development but relied heavily on physical representations (Payne, 1975, p. 150). Green investigated the effects of concrete materials (one inch paper squares) versus diagrams and an area model versus a "fractional part" model on fifth graders' learning the algorithm for multiplication of fractions. Since the study is not available in its entirety, excerpts of Green's summary, as quoted by Payne, will be given. Basically the approach using area was more effective, and the diagrams and manipulative materials were equally effective. Of further interest is Green's note: The failure in finding a fractional part of a set definitely points to the need to find a more effective way to teach this im- portant concept. Particular attention should be given to overcoming the difficulty chil- dren have with the "unit" idea, relating the model and the procedure for finding a frac- tional part of a set, and delaying the rule until there is understanding of the concept. (Payne, 1975, p. 153) Perhaps the difficulty alluded to is caused by the need to have logical class inclusion firmly in place for the understanding of a part-whole relationship (Kieren, 1975; Piaget, Inhelder, & Szeminska, 1960). Payne says that Green's results were better than those of similar studies. Green's approaches all involved visual models: either concrete materials that children manipulated or diagrams of regions. Since all her retention scores were almost 90% of posttest scores, Payne concludes that the use of visual materials in developing algo- rithms has a more important effect on retention than does a purely logical mathematical develop- ment. (Payne, 1975, p. 155) However, the use of manipulative materials did not seem to have the expected advantage in achievement. Payne says that evidently it is not a simple thing to relate a child's thought to his use of concrete materials or diagrams (p. 156) Kurtz and Karplus (1979) undertook a training study to see whether ninth and tenth grade prealgebra students could be taught to become proficient in proportional reason- ing. Manipulative materials were hypothesized to be more effective and to engender more favorable attitudes than paper and pencil activities alone. The authors' con- clusions were that proportional reasoning was taught suc- cessfully, that manipulative materials and paper and pencil activities provided equal cognitive gains, but that the manipulative version was considerably more popular than the paper and pencil version (Kurtz & Karplus, 1979, p. 397). Except for studies such as the above, the use of mani- pulatives in instruction has been primarily restricted to the elementary schools. An interesting result came from a study (Barnett & Eastman, 1978) of ways to train prospective ele- mentary teachers in the use of manipulatives in the class- room. Subjects either received demonstrations only (control group) or both demonstrations and "hands on" experience with the manipulatives (experimental group). On the test on the uses of manipulative materials, the authors found no signifi- cant difference between the groups. However, the experimen- tal group did better on the mathematics concept posttest. The authors suggest that a plausible explanation for this result may be that although subjects do not learn to "teach better" by actually using manipulatives, they may better learn the mathematics concepts involved. The results of several studies have suggested that many preservice elementary teachers have not reached the level of abstract operations, and hence they might need manipu- lative aids themselves in order to learn the mathematical concepts that they are expected to teach. (pp. 100-101) |