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This study describes a teacher preparation program in mathematics and science and explores what impact the reformed curricula and teaching methods instituted in the program have on prospective teachers' understanding of rational numbers and integers. The study pursues obtaining in-depth insights regarding prospective teachers' concept development. The pretest and the posttest consisted of tests for computational skills, number sense, and conceptual understanding. Results suggest a relationship between a reformed mathematical environment and enhanced student achievement. (KHR)

This study investigates the development of 5th grade children's understanding of quotient and the classroom norms and practices that constrain or enable that understanding. It reports not only how the children's understandings develop, but also why and under what conditions they develop. The results of this study indicate that children progressed from treating fractions as exclusively part-whole to having at least two parallel conceptions: part-whole and fair share while at the same time beginning to conceive of the division operation as generalizable to any pair of whole numbers. Some children began to see the operation as generalizable to division by fraction. The general learning trajectory of the class is discussed. (KHR)

This paper reports on a study of seventh grade students (N=4) who attend a rural K-12 school. Students participated in a 5-week teaching experiment designed to build on their existing knowledge of the unit concept and extend it to the rational number operations of multiplication and division. Data collected comes in the form of videotapes, audiotapes, researcher journal, and students' written work. Other data consist of individual student interviews conducted prior to and at the conclusion of the teaching experiment. Analysis of the results reveals that students' concepts of unit are enriched by participating in the teaching experiment. Among the conclusions drawn were that students developed a flexible concept of unit, modeling provided continuity between conceptual domains, equipartitioning remained a persistent difficulty, sustained focus on the measuring unit is hard to achieve, development of models is impeded by students' selection and use of measurement units, unitizing skills endure and are extendible, and models can inform procedural methods and/or provide alternative solution methods. (DDR)

Over the past five years, Pesek (Simoneaux), Gray and Golding have been actively involved in the Louisiana Systemic Initiative Program (LaSIP) and the Louisiana Collaborative for Excellence in the Preparation of Teachers (LACEPT) grants through Southeastern Louisiana University. Through these grants teachers from the region are inserviced on implementing the directives and philosophy of the Mathematics Association of America's (MAA) "A Call for Change" and the National Council of Teachers of Mathematics' (NCTM) "Curriculum and Evaluation Standards for School Mathematics", and preparation courses for elementary majors are being redesigned. This paper reports some tasks being implemented across the mathematics content and methods courses in one curriculum strand, namely rational numbers. Elementary education majors at Southeastern are required to take 12 hours in mathematics (algebra, probability and statistics, number sense, and geometry), three hours in mathematics education, and are certified for grades 1-8. (Contains 24 references.) (Author/ASK)

Many have argued for the importance of numeracy, yet little is known about the opportunities for numeracy available to people in their daily lives. In this study, characteristics of rational numbers in magazines written for children, teenagers, and adults were analyzed and compared. Analysis indicated that difficult mathematical concepts that appear in the media such as fractions, percents, and averages are much more prevalent in adults' magazines than in those written for children and teenagers. Adults are often presented with rational numbers that are related to each other. Numbers in teenagers' texts do not appear to provide a transition to those found in adults' texts, despite the fact that through formal schooling teenagers have encountered all the mathematical concepts that are frequently found in adults' texts. Implications for preparing students for the numeracy demands of everyday life are discussed. An appendix contains the coding scheme used in the study. Contains 27 references. (Author/MKR)

When teachers possess detailed knowledge about children's thinking and problem solving, it can profoundly affect their knowledge of their students and their planning for instruction. Reported is a project designed to demonstrate the feasibility of redesigning courses for prospective teachers in mathematics and science by incorporating into the courses research results from cognitive science and by focusing on the development of pedagogical content knowledge. After the background of the project is presented, the section on the project's implementation describes the preparation stages and the three courses offered in four sections. Section 1: Cognitive Seminar for Teacher Preparation Project, describes the organization of knowledge, how it will be presented in the classroom, and how student learning and attitudes towards learning will be assessed for the three courses being designed. Section 2 describes the mathematics course entitled "Calculus for Middle School Teachers," including course development activities, evaluation data, and implications for curriculum development. Section 3 describes the mathematics course entitled "Mathematics Course for Elementary/Middle School Teachers: Rational Numbers, Proportional Reasoning, Probability, Statistics," including course planning, instruction, information about students, evaluation of student affect and knowledge measures, and a discussion of instructor impressions of students. Section 4 describes the biology course entitled "Process and Inquiry in Life Science," including an overview of the course, the materials and methods used during teaching, results of science process skills and affective attitudes of experimental and comparison groups, and discussion and conclusions from the results. Appendices including pertinent documents with respect to activities used in lessons taught, evaluation instruments for knowledge and attitude measures, data gathered in the study, and reports made at the Psychology of Mathematics Education are given. (MDH)

This document reports on a small-group teaching experiment whose goal was to understand how fourth- and sixth-grade children develop concepts of common and decimal fractions. Both the Grade 6 and the Grade 4/5 children were taught common and decimal fractions through discussion and by using manipulatives, beginning with basic concepts of common fractions. Results showed: (1) common fractions were initially interpreted as parts of a whole, using region models; (2) Grade 6 students seemed to find manipulatives helpful, but Grade 4 students used manipulatives less and seemed more interested in completing as many questions as they could without recourse to manipulatives; (3) students were able to relate common and decimal fractions; (4) most students could develop fraction and decimal concepts, learn the associated operations meaningfully, and perform satisfactorily on end-of-unit tests; and (5) teachers felt they had learned much from their involvement in the project, including: the need for clarification of roles of researcher and teacher, importance of student interviews, usefulness of journals and student-constructed questions, and judicious use of worksheets. Appendices contain reports of lessons in Grade 4 and Grade 6, project evaluation by teachers and researchers, pre- and post-test questions, interview questions and transcripts, and worksheets. Contains 14 references. (MKR)

This paper describes the Rational Number Project (RNP), teaching experiments concerned with the teaching and learning of fractions among 4th and 5th grade students. Interviews with 4th grade students who used the RNP curriculum and with students who used a traditional curriculum were conducted by RNP staff as well as classroom teachers. This paper provides teachers with examples of students' conceptual and procedural thinking to give them a picture of what it means for students to exhibit number sense with fractions. Focusing on children's thinking and the effect of the use of manipulative models on children's thinking is suggested as one way of teaching fractions. (KHR)

The development of proportional reasoning is a major focus of the middle grades curriculum. The challenge for educators is to find contexts that engage students and that facilitate the study of proportional reasoning. This chapter explores proportional thinking with students in grades 3-8 by using a number of books in which the underlying stories center on proportional thinking. Results indicate that literature provides an effective vehicle for engaging middle school students in mathematics. Middle grades students became engaged in doing mathematics as a result of reading books, and the contexts seem to offer insight into ways in which to engage in mathematics. (KHR)

This chapter presents a yearlong teaching experiment involving a 4th grade class in a New Jersey school that focused on fractions. The 25 children worked in pairs or in small groups, then came together as a whole class for sharing in larger discussions. The development of children's thinking is explored through analyzing videotapes of class sessions, studying the children's written work and researcher's field notes, and listening to students' conversations as they discussed their ideas with one another. Results indicate that students built important understanding of fractions by modeling, drawing pictures, and eventually developing the notation for their ideas. (KHR)

This study was undertaken to gain insights into children's understanding of rational numbers as quantities; that is, the extent to which they associate a size with a fraction like 2/3. Eight children in an experimental group in DeKalb, Illinois, chosen to reflect the range from low to high ability, were observed during 30 weeks of experimental instruction during grades 4 and 5. A classroom-sized group of 34 middle-ability children in grades 4 and 5 in Minneapolis simultaneously took part in the same teaching experiment, providing children with manipulative-oriented instruction. Seven interview assessments, each preceded by about 4 weeks of instruction, were videotaped with each DeKalb child and with eight Minneapolis children. Written tests were also given. Data from the three fifth-grade assessments are included in this report. The three tasks are described, and children's reactions are reported in detail. They had varying success on the tasks. It appeared that three knowledge structures are essential for the development of a quantitative understanding of rational number: estimation, fraction equivalence, and rational-number order. These structures appeared to develop somewhat independently, but need to be coordinated for success with rational number situations. (MNS)

This volume presents a partial description of the lessons in the preliminary version of the PLATO fractions curriculum. Each lesson has three parts: review, new material, and a student-selected option. Students receive immediate feedback from the computer as they progress through each lesson. Five groups of lessons are described: meaning of fractions, mixed numbers, equivalent fractions, addition and subtraction (like denominators), and addition and subtraction (unlike denominators). An outline and flowchart is presented for each group. For each lesson within a group, a statement of purpose, a brief description, and sample computer displays are provided. (SD)

A 20-task learning hierarchy for subtraction of fractions was deductively derived using Gagne's task analysis. To test this analysis empirically, composite items were written for each level and administered to students in grades 3-6. Test results were analyzed by the Walbesser Technique and Pattern Analysis; the acceptance levels developed by Phillips and Kane were used as criteria. These analyses yielded ratios for consistency, adequacy, and completeness which were below the acceptance level. The hierarchy was then revised to maximize these ratios, and an index of agreement of .85 was obtained between the expected pattern of responses and the observed level. The investigators note that: (1) results may have been affected by the fact that very few students were able to answer correctly items from the upper levels of the hierarchy; and (2) the study should be replicated using students with a broader range of abilities and ages. (SD)

Fourteen research reports related to mathematics education are abstracted and analyzed. Two of the reports deal with teacher education, two with problem solving, three with basic operations, and one each with learning disabled students, rational numbers, proportional reasoning, counting, teacher effectiveness, group cooperation on mathematical tasks, and verbal problem solving. Research related to mathematics education which was reported in RIE and CIJE between October and December 1979 is also listed. (MK)

This study focuses on basic mathematical skills mastered by middle school students. A test designed for the investigation was administered to 400 seventh- and eighth-grade pupils in three middle schools in Houston, Texas. The test, which focuses on fractions, was administered by the regular mathematics teacher during the mathematics class period and all students were given sufficient time to finish all items on the test. Among the results, the data indicated that while the renaming of fractions to higher terms and the renaming of an improper fraction to a mixed numeral were skills mastered by many of the pupils, more students have difficulty renaming a mixed numeral to an improper fraction. (MP)

Presented is a detailed discussion of the performance pattern of third- and sixth-grade pupils in the Los Angeles Unified School District (LAUSD) for the skill area of fractional numbers. The report begins with a brief and general introduction regarding minimum competencies and continues with tables showing the performance levels of non-English speaking/limited English speaking (NES/LES) and English/Bilingual students. The results discussed are those of a feasibility study conducted in fall 1978, which involved 3,835 students. Among the findings, the data revealed that all students in third grade have difficulty in differentiating between the concept of fraction and the concept of ratio when in a parts to whole context. It was also found that students at the sixth-grade level have difficulty adding and subtracting fractions or multiplying a whole or mixed number by a fraction. Performance patterns are identified for both groups in fractional number skills and suggestions for improving instruction are advanced. (MP)

This document provides an overview of a National Science Foundation (NSF) funded project, Rational Number Ideas and the Role of Representational Systems. The rational number project consists of interacting instructional, evaluation, and diagnostic/remedial components. General project goals are: (1) to describe the development of the progressively complex systems of relations and operations that children in grades two through eight use to make judgments involving rational numbers; and (2) to describe the role that various representational systems (e.g., pictures, manipulative materials, spoken language, written symbols) play in the acquisition and use of rational number concepts. The project aims to develop a psychological "map" focusing on several aspects of the learning process. The project is concerned not only with what children can do naturally, but also with what they can do accompanied by minimal guidance or following theory based instruction. A list of five studies currently planned as future activities for the rational number project staff is included at the conclusion of this document. (MP)

Some diverse fields in which the effects of mental representations have been noticed are first discussed, largely through illustrations of differing problem-solving experiences. Then the LAKOS Project (Logical Analysis of Cognitive Organizational Structures) at the University of Osnabruck is described. A central concern is the specification of a model of the representation and organization of knowledge in memory. A primary goal is to describe the cognitive structures of individuals so precisely that a machine can be made to simulate aspects of their behavior. In the remainder of Part A of this paper, two cases of inconsistent student behavior in rational number learning are documented. Details from interviews are included, followed by comments. In Part B, a computer model of the mental organization of knowledge is presented which was conceptualized to understand in detail some crucial aspects of cognitive functioning and of the origins of suboptimal behavior. A tentative conclusion based on the interviews and model is then projected. (MNS)

A two-year study was conducted with fourth-grade children in the context of extensive teaching experiments concerned with the learning of rational number concepts. Representational difficulties in using the number line model were investigated. While instruction in the second year attempted to resolve observed learning difficulties, the results of both years showed that children have considerable difficulty with number line representations which show an unreduced form of a given fraction. Explorations of the data suggest difficulty with partitioning and "unpartitioning" number line representations, with translations between various representational modes, and with coordinating symbolic and pictorial information on a number line (Author)

This report describes an investigation in England of some of the problems children experience with fractions. In particular, the restricted view that some children have of fractions is investigated. The methodology involved a large number of interviews with secondary school children (aged 12-14). The results of these interviews led to the design of teaching experiments which involved both small groups and whole classes with a pretest, immediate posttest, and delayed posttest. Results of the study address student understanding of models of fractions, the division aspect of a fraction, fractions as numbers, and equivalent fractions as well as the teaching module that was developed. Implications for the teaching of fractions include: (1) the learning of fractions would be facilitated if teachers developed their skills of listening to children and of encouraging them to talk about their interpretation of fractions and associated problems; (2) a greater emphasis could be placed on the division interpretation of fractions; (3) more attention needs to be given to the basic ideas of equivalence of fractions; (4) more attention needs to be given to the limitation of the "part of a whole" model of a fraction; and (5) more attention needs to be paid to the importance of the transition from the realm of counting numbers to that of rational numbers. Appendices include the interview and testing materials. (PK)