Showing papers in "Educational Studies in Mathematics in 1989"

Finding and Using Patterns in Linear Generalising Problems.

Abstract: Linear generalising problems are questions which require students to observe and use a linear pattern of the formf(n)=an+b withb≠0 This study reports responses of students aged between 9 and 13 to these questions, documenting the mathematical models that they select, the strategies used in implementing them and the explanations they give Substantial inconsistency of choice of model is observed; students who began a question correctly frequently adopted a simpler but incorrect model for more difficult parts of the question Students who had undertaken a course in problem solving implicitly used a linear model more frequently and consistently and their explanations more often related the spatial patterns and the number patterns They seemed to understand the relationship between the data and the generalising rule more completely

Journal writing and mathematics instruction

Abstract: In this article we discuss the educational value of engaging mathematics students in a specific form of writing to learn—the keeping of a journal throughout a mathematics course. As the result of an analysis which comprises both conceptual and empirical components, we suggest that journals have the potential to contribute to mathematics instruction in several ways. As the students write the journals, they can be encouraged to express and reflect upon their feelings, knowledge, processes and beliefs about mathematics, and consequently grow along each of these dimensions. By reading their students' journals, teachers may receive a wealth of information about their students and the course, and consequently improve their teaching. Finally, journals can create a new form of dialogue between the teacher and each student, thus allowing for more individualized instruction and a supportive classroom atmosphere.

Constructivism, Mathematics and Mathematics Education.

Abstract: Learning theories such as behaviourism, Piagetian theories and cognitive psychology, have been dominant influences in education this century. This article discusses and supports the recent claim that Constructivism is an alternative paradigm, that has rich and significant consequences for mathematics education. In the United States there is a growing body of published research that claims to demonstrate the distinct nature of the implications of this view. There are, however, many critics who maintain that this is not the case, and that the research is within the current paradigm of cognitive psychology. The nature and tone of the dispute certainly at times appears to describe a paradigm shift in the Kuhnian model. In an attempt to analyse the meaning of Constructivism as a learning theory, and its implications for mathematics education, the use of the term by the intuitionist philosophers of mathematics is compared and contrasted. In particular, it is proposed that Constructivism in learning theory does not bring with it the same ontological commitment as the Intuitionists' use of the term, and that it is in fact a relativist thesis. Some of the potential consequences for the teaching of mathematics of a relativist view of mathematical knowledge are discussed here.

Technology and Mathematics Education: A Survey of Recent Developments and Important Problems.

Abstract: This paper, prepared for the survey lecture of theme group T2 at the Sixth International Congress on Mathematical Education in Budapest, gives an oveview and analysis of recent progress in applying electronic information technology to creation of new environments for intellectual work in mathematics. The paper is divided into six major sections considering the impact of: 1. Numerical computation; 2. Graphic computation; 3. Symbolic computation; 4. Multiple Representations of Information; 5. Programming and Connections of Computer Science and Mathematics Curricula; 6. Artificial Intelligence and Machine Tutors.

Preservice elementary teachers' explicit beliefs about multiplication and division

Abstract: This study, conducted in the United States, was designed to assess the extent to which the beliefs, “multiplication always makes bigger” and “division always makes smaller,” are explicitly held by preservice elementary teachers. Paper and pencil instruments were administered to 136 preservice elementary teachers in a large university in the southeastern United States. Responses on the instruments and in interviews provide insight into the sources of preservice teachers' beliefs and into the relationship among preservice teachers' beliefs, computational skills, and performance on word problems.

Mathematics achievement of girls and boys in grade eight: results from twenty countries

Abstract: Over the past two decades researchers have shown considerable interest in the relationship between the gender and the mathematics achievement of children in the upper grades of the elementary schools. Many studies to date have shown that by age 13 boys are significantly superior to girls in both their mathematical performance and their attitudes toward mathematics (Aiken, 1976; Backman, 1972; Benbow and Stanley, 1980; Maccoby and Jacklin, 1974; Mullis, 1975; Suydam and Riedesel, 1969), and that the male advantage is especially pronounced among high-scoring exceptionallygifted students, with boys outnumbering girls 13 to I (Benbow and Stanley, 1983). It should be stressed that some research has also shown that gender-related differences in achievement vary considerably both within and among countries (Schildkamp-Kuindiger, 1982), and are influenced by the educational environment and by psychosocial processes (Burton, 1986; Fennema, 1985; Leder, 1986; Walden and Walkerdine, 1986). In a most recent overview of cognitive gender differences, Feingold (1988) looked at standardized tests administered between the years 1960 and 1983, and found that gender differences declined over the years surveyed. The purpose of the present analysis is to pursue the question of gender differences in mathematics achievement from a cross-cultural perspective, on the basis of the data of a large-scale international study of mathematics. The breadth of this study makes it possible to gain new information about the scope and the variation of gender differences.

An investigation of young children's understanding of multiplication

Abstract: The issue of how young children learn arithmetic has generated considerable research that focuses upon the child as an active “meaning maker” in the process of learning mathematics. Close observation of the children's actions and utterances have led to formulations that provide a description of children's cognitive development with respect to counting (Gelman, 1972; Schaeffer, Eggleston and Scott, 1974; Steffe, Von Glasserfeld, Richards and Cobb, 1983) and to addition and subtraction (Carpenter, Moser and Romberg, 1982; Fuson, 1984; Baroody, 1984) This paper presents an analysis of children's solution strategies for multiplication tasks. Consideration is also given to the difficulties experienced by unsuccessful children.

The arithmetic connection

Abstract: From test and interview data obtained during an investigation into Grade 10 students' conceptions of algebraic generalisation and justification, we have extracted evidence of the extent to which these students have coordinated the “worlds” of arithmetic and algebra, and can move freely between them. The data show more dissociation than we expected, even among students who were successful at standard algebraic tasks. Conceiving algebra as “generalised arithmetic” may obscure the many genuine obstacles that the learner has to overcome in moving from fluent performance in arithmetic to fluent performance in algebra while achieving and maintaining a smooth coordination of both modes of action.

Un processus d'apprentissage du concept d'aire de surface plane

Abstract: On construit un processus d'apprentissage du concept d'aire de surface plane, pour des eleves de 9–12 ans Il s'appuie sur les hypotheses suivantes: - developper le concept d'aire, en tant que grandeur, aide les eleves a etablir des relations entre les cadres geometrique et numerique - une identification precoce entre les grandeurs et les nombres amene les eleves a faire des confusions entre longueurs et aires

Adolescents' ability to communicate spatial information: Analyzing and effecting students' performance

Abstract: This study presents evidence which suggests that even though students in grades 6 through 8 are familiar with a variety of types of representation modes, they have great difficulty in successfully communicating spatial information. A Building Description Task, which consists ofa building made up of ten small cubes taped together and a set of instructions, was presented to a sample of middle school children. They were asked to “help your friend to know what your building looks like.” Students' attempts were classified by representation mode (verbal, mixed, graphic) and analyzed by grade and by sex. In addition, the effect of instruction in spatial visualization activities on students' preference for representation mode and rate of success on the task was assessed. The findings were examined relative to the practical teaching implications, to individual differences in spatial visualization ability and to design of spatial tests.

Providing a computer based framework for algebraic thinking

Abstract: This paper describes and presents the findings of a study which aimed to trace the development of pupils' use and understanding of algebraic ideas within a Logo programming context relating this to their use and understanding of similar ideas within a non-computational context. The research consisted predominantly of a three year longitudinal case study of four pairs of pupils (aged 11–14) programming in Logo during their normal school mathematics lessons. The data included video recordings of all the case study pulils' Logo sessions, and individually presented Logo and algebra structured interviews. The overall conclusion of this research is that Logo experience does enhance pupils' understanding of algebraic ideas, but the links which pupils make between Logo and algebra depend very much on the nature and extent of their Logo experience.

Solving Structured Geometric Tasks on the Computer: The Role of Feedback in Generating Strategies.

Abstract: This paper describes a year-long study of the way a group of six 12-year-old children went about solving a special class of geometric problems, using a computer and a limited set of Logo programming tools. We examined children's solution strategies from the perspective of whether the strategies led to insights about the underlying mathematical relations. It was found that the “feedback” aspect of the computer graphics screen encouraged solution strategies that were qualitative and driven by visual data, rather than being based on explicit or implicit relations. These visually based strategies did not lead to the kind of mathematization of problem-situations that we as mathematics educators would like to see and, in fact, seemed to be a barrier to the development of higher levels of geometric thinking.

Sex-related differences on objective tests among undergraduates

Abstract: Objective tests, such as multiple-choice and relationship analysis questions, are often used in mathematics to assess certain ranges of skills and abilities. This article examines the performance of men and women students in mathematics at university level and shows that there are significant differences between the sexes.

An intelligent tutoring system approach to teaching primary mathematics

Abstract: This paper is concerned with recent developments in the construction of intelligent tutoring systems. The authors have developed an adaptive computer tutor for arithmetic, called Shopping on Mars; this activity is unique in that it supports informal methods of calculation. It involves carrying out shopping transactions within the context of an adventure game. The program runs on an Archimedes 310 microcomputer and is aimed at 8-to12-year-olds. Its main purpose is to help children develop an extensive repertoire of informal techniques for performing arithmetic in various everyday situations, and in particular, to learn to formulate shortcuts to problem solving. The tutor keeps a chronological record of how far individuals have progressed and of their observed difficulties, and it provides immediate feedback. Finally, the activity provides a rich, cooperative learning experience; the computer tutor acts as an intelligent participant in activities involving pairs of children. We describe the rationale behind Shopping on Mars, the design of its interface and instructional architecture, our strategies for intervention, and the special description language-based on direct manipulation-that we have evolved for child-computer communication. The results of some preliminary field trials are presented. We conclude by discussing the role of our activity in the classroom and the contribution which intelligent computer-based activities in general might make to the teaching of primary mathematics.

An exploratory approach to advanced mathematics

Abstract: There is little current use of more exploratory approaches to the teaching and learning of advanced mathematics. This article describes a simple exploratory teaching model, and reports the findings of a small-scale action-research project to implement and evaluate it. There are discussions of student perceptions and attitudes, and of student attainment. The principal findings are that there was no underachievement under the exploratory approach, and that student preferences for routinised mathematical activity and directed teaching style were strongly associated.

Tableur et calcul algebrique

Abstract: Cet article est un compte rendu de recherche concernant l'apprentissage a l'utilisation d'un tableur (MultiplanTM) avec des eleves de 15 ans dans un college francais. Cette recherche a un caractere exploratoire dans un domaine actuellement peu etudie.

Sur une situation d'éveil à la déduction en géométrie

Abstract: Nous nous interessons a l'argumentation des eleves de 10–11 ans et 12–13 ans, avant l'introduction scolaire de la demonstration. Nous nous limiterons ici aux arguments utilises par des eleves de ces deux niveaux scolaires pour formuler et valider deductivement une conjecture en geometrie. Nos hypotheses sont que les raisonnements deductifs peuvent etre developpes tres tot sur une categorie de problemes, ceux ou l'apprehension operatoire de la figure joue un role heuristique; les raisonnements sont alors fortement suggeres par l'apprehension operatoire de la figure. Nous decrivons et analysons les raisonnements des eleves sur un des problemes relevant de cette categorie. L'analyse des entretiens cliniques conduits avec 11 binomes montre la fecondite des hypotheses.

Accuracy and stability of cognitive structures and retention of geometric content

Abstract: This study sought to determine whether clear distinctions in cognitive structures which existed at the time of instruction between students classified as formal operational and concrete operational would persist over time and result in prototypical structures for each group. In addition, this study investigated the effect of the ability to produce an accurate cognitive structure on the retention of geometric content and the stability of cognitive structures.

Gender differences in the junior secondary (grade 7) mathematics curriculum in Hong Kong

Abstract: This article seeks to examine gender differences across the mathematics curriculum in the various topic areas and mathematical abilities. It is suggested that a profile of these differences, when compared with students at different grade levels or across curricula, would be more fruitful for classroom teachers and curriculum developers. Breakdown and discriminant function statistics reveal that boys performed better than girls in Geometry and excelled in comprehension, and solving routine and non-routine problems. Girls were better at mathematical manipulation problems.

Better mathematics for low attainers

Abstract: This article describes the work of the Low Attainers Mathematics Project which has produced the report Better Mathematics (Ahmed, 1987). Mention is made of the way in which the report was written. The article concludes with some remarks about bringing about large scale changes in teaching method through research.

Gestion des relations entre variables dans un environnement de programmation logo

Abstract: Notre propos est centre sur l'analyse des procedures de resolution mises en oeuvre par les eleves dans l'activite de programmation. Pour ce faire, nous proposons une methode d'analyse utilisant des criteres. L'observation de ces criteres sur des situations de programmation en Logo met en evidence les differents types de difficultes rencontrees par les eleves pour structurer et coordonner leurs procedures informatiques. Lorsqu'il y a codage par des variables, nous distinguons deux types de codages: analytique ou descriptif. L'analyse des consequences du choix entre ces deux types montre que le codage descriptif est une procedure de resolution plus simple, car elle evite d'avoir a exprimer une relation entre des variables et permet de decomposer un probleme complexe en plusieurs sous-problemes plus simples. Ce resultat concorde avec des observations faites en mathematiques dans des situations de mise en equations ou systeme d'equations.