Gisèle Lemoyne

Addition and multiplication: Problem-solving and interpretation of relevant data

This study constitutes a first step towards the elaboration of a method for intervening in the processes of solving concrete mathematical problems, based on the processing of expressions often included in the statement of addition and multiplication problems presented to second-cycle elementary students (ages 9 to 12). Forty-eight students took part in the experiment: the 29 subjects of the experimental group were divided into small groups of about 5 subjects in each and had to perform various learning exercises. The aim of these exercises was essentially to improve skills in analyzing and processing certain expressions frequently included in the statements of addition and multiplication problems. The effect of these learning exercises on problem-solving was evaluated by a problem-solving pre-test and post-test presented to the subjects in both groups. The learning exercises helped significantly to improve the problem-solving performance of subjects in the experimental group. Analysis of the verbal productions of the subjects in this group during the learning exercises revealed certain factors related to the effectiveness of the method of intervention proposed in this research.

Coordination of Knowledge of Numeration and Arithmetic Operations on First Grade Students

The aim of this study was to develop a better understanding of the processes involved in the construction of the oral and written symbolic systems of numbers and to grasp their role in the elaboration of the modeling function of numbers. Tasks related to both spoken and written number sequenceS and to addition and subtraction problems were given to six first graders. Analysis of the childrens behavior permits discussion of certain theoretical questions raised by the research results of the construction of number sequence, contributes to a better understanding of the relationship between numeration and arithmetic operations and finally, in light of these results, permits us to propose ideas for teaching.

Connaissances utilisées par des elèves de 8 à 12 ans dans la formulation de problèmes arithmétiques concrets

RésuméDes énoncés de problèmes arithméthiques concrets rédigés par des élèves de 8 à 12 ans sont éxaminés aux fins: a) de dégager les apports respectifs des connaissances linguistiques, sémantiques, procédurales et des schémas de connaissance dans la formulation de ces problèmes; b) d’identifier les étapes de développement des modèles mentaux qui président à la réalisation de cette activité. Les résultats de cette recherche montrent que les modèles mentaux de problèmes chez ces élèves ne semblent pas différer des modèles théoriques actuels construits pour expliquer l’activité de résolution de problèmes (Mayer, 1983; Kintsch & Greeno, 1985). Des schémas de connaissance semblent en effet présider à la formulation de problèmes; des connaissances sémantiques orientent également le choix des nombres. Les contributions des connaissances linguistiques et procédurales sont par ailleurs moins évidentes. Enfin, l’analyse des résultats conduit à la formulation de certaines hypothèses sur le développement de modèles mentaux de problèmes arithmétiques concrets.AbstractThis study analyses children development of semantic, linguistic, procedural and schematic knowledge in the context of writing arithmetic word problems. 139 children aged between 8 and 12 years old were presented with a task which consisted in writing arithmetic word problems, according to some contraints: words, questions or measures to include in their problems; type of problems to write. Results show the relevance of actual theoritical models of problem solving (Mayer, 1983; Kintsch & Greeno, 1985). Schematic knowledge seem indeed more important than other knowledge in the process of writing arithmetic word problems; semantic knowledge are also used to choose relevant numbers or measures; the roles of linguistic and procedural knowledge seem less evident. Finally, some hypotheses related with the development of mental models of arithmetic word problems are formulated.

UNE INTRODUCTION NON CLASSIQUE AUX ALGORITHMES D'ADDITION ET DE SOUSTRACTION *

The aim of our study was to analyze a didactical sequence for the teaching of addition and subtraction procedures and algorithms. In the conception of that sequence, we have taken into account diagnostic and repair models for procedural bugs in addition and subtraction algorithms, as well as learning and teaching methods for multi digit additions and subtractions. The didactical sequence included situations involving many of the characteristics associated with procedural bugs; however, when the children encountered those situations they had many conceptual tools to detect their mistakes and correct them, giving that way a meaning to the actions made in addition and subtraction procedures and algorithms. Our teaching activities were submitted to second grade school children (7–8 years old). The didactical interactions and the procedures used by children in problem solving activities were analyzed in order to get a better understanding of the interaction between numbers, numeration and operations knowledge which are involved in the construction of addition and subtraction procedures and algorithms and to relate childrens knowledge acquisition to the didactical situations.

What Concepts are and How Concepts are Formed

With respect to the question of how mathematical concepts are formed by children, we can draw upon several theoretical positions, going from the most general point of view, relative to the universal subject, to the most specific point of view, relative to the didactic subject. In this chapter I will rely on the theory of conceptual fields, whose originator, G. Vergnaud, describes it thus: “The theory of conceptual fields is a cognitivist theory, which sets out to provide a coherent framework and some basic principles for the study of the development and learning of complex competencies, notably those which arise in the sciences and technologies” (Vergnaud, 1991, p. 135) I shall try first to show the necessity of that theory for the didactics of mathematics in relation to that of genetic epistemology; in the second section I shall elaborate on what the theory of conceptual fields contributes, with the notion of scheme-algorithm, to questions on the cognitive sources of observable errors in written calculations.