Boris Koichu

Cognitive Development of Proof

This chapter traces the long-term cognitive development of mathematical proof from the young child to the frontiers of research. It uses a framework building from perception and action, through proof by embodied actions and classifications, geometric proof and operational proof in arithmetic and algebra, to the formal set-theoretic definition and formal deduction. In each context, proof develops over the long-term from the recognition and description of observed properties and the links between them, the selection of specific properties that can be used as definitions from which other properties may be deduced, to the construction of ‘crystalline concepts’ whose properties are a consequence of the context. These include Platonic objects in geometry, symbols having relationships in arithmetic and algebra and formal axiomatic systems whose properties are determined by their definitions.

The effect of promoting heuristic literacy on the mathematical aptitude of middle-school students

Heuristic literacy – an individuals capacity to use heuristic vocabulary in discourse and to apply the selected heuristics to solution of routine and non-routine mathematical tasks – was indirectly promoted in a controlled five-month classroom experiment with Israeli 8th grade students (N = 92). The experiment achieved a moderate mean effect size, which is in line with some previous research on heuristics. The novel result of the study is that those students of the experimental group who were below sample average at the beginning of the experiment benefited from the heuristically-oriented intervention significantly more than the rest of the students. It is argued here that this is, in part, due to communicational aspects of the intervention.

When Do Gifted High School Students Use Geometry to Solve Geometry Problems

This article describes the following phenomenon: Gifted high school students trained in solving Olympiad-style mathematics problems experienced conflict between their conceptions of effectiveness and elegance (the EEC). This phenomenon was observed while analyzing clinical task-based interviews that were conducted with three members of the Israeli team participating in the International Mathematics Olympiad. We illustrate how the conflict between the students’ conceptions of effectiveness and elegance is reflected in their geometrical problem solving, and analyze didactical and epistemological roots of the phenomenon.

If not, what yes?

This article presents an instructional approach to constructing discovery-oriented activities. The cornerstone of the approach is a systematically asked question ‘If a mathematical statement under consideration is plausible, but wrong anyway, how can one fix it?’ or, in brief, ‘If not, what yes?’ The approach is illustrated with examples from calculus and geometry. It is argued that the ‘If not, what yes?’ approach facilitates conjecturing and proving, constructing meaningful examples and counterexamples and has a potential for creating learning situations, in which responsibility for achieving desirable mathematical results is devolved from an instructor to the learners.

On the relationships between (relatively) advanced mathematical knowledge and (relatively) advanced problem-solving behaviours

This article discusses an issue of inserting mathematical knowledge within the problem-solving processes. Relatively advanced mathematical knowledge is defined in terms of three mathematical worlds; relatively advanced problem-solving behaviours are defined in terms of taxonomies of proof schemes and heuristic behaviours. The relationships between mathematical knowledge and problem-solving behaviours are analysed in the contexts of solving an insight geometry problem, posing algebraic problems and calculus exploration. A particularly knowledgeable and skilled university student was involved in all the episodes. The presented examples substantiate the claim that advanced mathematical knowledge and advanced problem-solving behaviours do not always support each other. More advanced behaviours were observed when the student worked within her conceptual-embodied mathematical world, and less advanced ones when she worked within her symbolic and formal-axiomatic worlds. Alternative explanations of the findings are discussed. It seems that the most comprehensive explanation is in terms of the Principle of Intellectual Parsimony. Implications for further research are drawn.

What Do High School Teachers Mean by Saying “I Pose My Own Problems”?

The aim of this chapter was to identify mathematics teachers’ conceptions of the notion of “problem posing.” The data were collected from a web-based survey, from about 150 high school mathematics teachers, followed by eight semi-structured interviews. An unexpected finding shows that more than 50% of the teachers see themselves as problem posers for their teaching. This finding is not in line with the literature, which gives the impression that not many mathematics teachers are active problem posers. In addition, we identified four types of teachers’ conceptions for “problem posing.” We found that the teachers tended to explain what problem posing meant to them in ways that would embrace their own practices. Our findings imply that most of the mathematics teachers are result-oriented—as opposed to being process-oriented—when they talk about problem posing. Moreover, many teachers who pose problems doubt the ability of their students to do so and consider problem-posing tasks inappropriate for their classrooms.

Overcoming a Pitfall of Circularity in Research on Problem Solving by Mathematically Gifted Schoolchildren

A considerable portion of research on mathematical giftedness seeks to compare between problem-solving experiences of gifted and average-ability schoolchildren. In some comparative studies, either quantitative or qualitative, some of the identified differences can be (implicitly) embedded in the study design. In light of the evaluation criteria adopted from research on general intellectual abilities and problem-solving competences, such studies bear a danger of falling into the pitfall of circularity. The goal of this article is to discuss three ways of overcoming this pitfall. The discussion converges to methodological implications for evaluating past research and conducting further research on problem solving by mathematically gifted schoolchildren.RésuméUne bonne partie de la recherche sur le talent mathématique vise à comparer les expériences de résolution de problèmes chez les enfants doués et les enfants d’habileté moyenne à l’école. Dans certaines études comparatives, qu’elles soient quantitatives ou qualitatives, au moins une partie des différences relevées sont (implicitement) liées à la conception même de la recherche. À la lumière des critères d’évaluation adoptés dans les études sur les habileté intellectuelles générales et sur la capacité de résoudre le problèmes, il appert que ces études courent le risque de tomber dans le panneau de la circularité. Le but de cet article est de présenter trois façons d’éviter ce risque. l’article se penche ensuite sur certaines implications méthodologiques permettant d’évaluer les recherches antérieure sur le sujet et d’ouvrir de nouvelles perspectives de recherche sur la résolution de problèmes chez les enfants particulièrement doués pour les mathématiques.

'good research' conducted by talented high school students: the case of sci-tech

In this paper, we present SciTech, the summer international research program for talented high school students, organized in the Technion — Israel Institute of Technology [1] [1] The first author of this paper is proud of founding SciTech in 1992 when he was the head of the Harry and Lou Stern Family Science and Technology Youth Center at the Technion. Information on the various activities of the Youth Center aimed at promotion of excellence and interest in mathematics, science and technology of high school students can be found in the website of the Technion (http://www.mechina.technion.ac.il/en/index.html) . Our major thesis is that while taking part in scientific projects under supervision by the Technion research stuff, the high school participants in SciTech are given the opportunity to work as researchers, and not just as learners. We begin with a brief discussion of the term ‘good research.’ This is followed by a description of SciTech: we outline how students are accepted to the program and what they do in it. The main part of this paper is devoted to discussion of some, not necessarily the most successful, mathematical projects, which we had the pleasure to supervise. We conclude the paper with general observations concerning the projects, based on our experience as SciTech mentors.

3-D Dynamic Geometry: Ceva's Theorem in Space

This is a snapshot about a student’s discovery. Usually, dynamic geometry software is used in investigations of 2-dimensional problems. In this snapshot we want to describe how Geometry Inventor (1994) was used by a high school student as a trigger to the discovery of a three-dimensional extension of the classical Ceva’s theorem. Reading the paper of Noble, Nemirovsky, Wright and Tierney (2001) we found that the discovery of the young student therein can be explained in the framework of multiple learning environments. In their paper, they use the word ‘‘environment’’ to describe a configuration of tools such as manipulative materials or computer software, as well as tasks, expectations, and conventions. They propose perspectives of learning in which students connect experiences in different environments through the development of family resemblance across their experiences. In the following story we show a student’s discovery in a computer-learning environment, which enabled him to connect the experiences gained there in other environments.

Four (algorithms) in one (bag): an integrative framework of knowledge for teaching the standard algorithms of the basic arithmetic operations

ABSTRACT In this article we present an integrative framework of knowledge for teaching the standard algorithms of the four basic arithmetic operations. The framework is based on a mathematical analysis of the algorithms, a connectionist perspective on teaching mathematics and an analogy with previous frameworks of knowledge for teaching arithmetic operations with rational numbers. In order to evaluate the potential applicability of the framework to task design, it was used for the design of mathematical learning tasks for teachers. The article includes examples of the tasks, their theoretical analysis, and empirical evidence of the sensitivity of the tasks to variations in teachers’ knowledge of the subject. This evidence is based on a study of 46 primary school teachers. The article concludes with remarks on the applicability of the framework to research and practice, highlighting its potential to encourage teaching the four algorithms with an emphasis on conceptual understanding.