Guershon Harel

Problem Solving, Modeling, and Local Conceptual Development

The research reported here describes similarities and differences between (a) modeling cycles that students typically go through during 60–90 min solutions to a class of problems thast we refer to as model-eliciting activities, and (b) stages of development that students typically go through during the “natural” development of constructs (conceptual systems, cognitive structures) that cognitive psychologists consider to be relevant to these specific problems. Examples of relevant constructs include those that underlie children’s developing ways of thinking about fractions, ratios, rates, proportions, or other elementary, but deep mathematical ideas. Results show that, when problem solvers go through an iterative sequence of testing and revising cycles to develop productive models (or ways of thinking) about a given problem solving situation, and when the conceptual systems that are needed are similar to those that underlie important constructs in the school mathematics curriculum, then these modeling cycles often appear to be local or situated versions of the general stages of development that developmental psychologists and mathematics educators have observed over time periods of several years for the relevant mathematics constructs. Furthermore, the processes that contribute to local conceptual development in model-eliciting activities are similar in many respects to the processes that contribute to general cognitive development. Applying principles from developmental psychology to problem solving—and vice versa—is a relatively new phenomenon in mathematics education (Lesh & MATHEMATICAL THINKING AND LEARNING, 5(2&3), 157–189 Copyright

Advanced Mathematical-Thinking at Any Age: Its Nature and Its Development

This article argues that advanced mathematical thinking, usually conceived as thinking in advanced mathematics, might profitably be viewed as advanced thinking in mathematics (advanced mathematical-thinking). Hence, advanced mathematical-thinking can properly be viewed as potentially starting in elementary school. The definition of mathematical thinking entails considering the epistemological and didactical obstacles to a particular way of thinking. The interplay between ways of thinking and ways of understanding gives a contrast between the two, to make clearer the broader view of mathematical thinking and to suggest implications for instructional practices. The latter are summarized with a description of the DNR system (Duality, Necessity, and Repeated Reasoning). Certain common assumptions about instruction are criticized (in an effort to be provocative) by suggesting that they can interfere with growth in mathematical thinking.

What is Mathematics? A Pedagogical Answer to a Philosophical Question

1 The framework presented here is part of the DNR Project, supported, in part, by the National Science Foundation (REC 0310128). Opinions expressed are those of the author and not necessarily those of the Foundation.

The role of isomorphisms in mathematical cognition

Recognizing and exploiting structural relationships between situations differing in surface features is an inherent part of mathematical cognition. Laboratory-based experimental studies have shown that subjects generally show little awareness of such relationships when presented with isomorphic problems. However, these findings should be interpreted in the context of unmotivated participants performing abstract tasks over a short period with minimal opportunity for development of stnlctural awareness. Another collection of studies has demonstrated that people often fail to apply mathematics principles to situations outside the classroom, to which they are at least potentially applicable; these findings reflect a major shortcoming in mathematics education. We recommend that awareness of structure, including specifically the recognition of isomorphisms, should be nurtured in children as part of the general development of expertise in constructing representational acts. A balanced view of the goals of mathematics education encompasses both the need to teach mathematics so that its applicability to many contexts is recognized, and a recognition of the importance and power of mathematics as desituated cognition.

Case studies of mathematics majors’ proof understanding, production, and appreciation

AbstractProof understanding, production, and appreciation (PUPA) are important parts of a mathematician’s repertoire. Many university students in the US, however, have difficulty with proof. One intent of this study was to examine the development of such students’ PUPAs and possibly to identify significant influences on that development, through interviews with the students throughout their studies in mathematics. Three case studies show a great variance in the development of the students’ proof skills. Some students come to university with excellent PUPAs and continue to thrive in a proof environment. Others enter university with poor PUPAs and unfortunately graduate without a significant change in their proof skills and attitudes. Still others come with poor proof skills but do show some growth during their undergraduate mathematics programs. Results of teaching experiments suggest that making proofs tangible is a means of helping those with poor PUPAs to grow in their proof understandings and abilities.Sommaire exécutifTrois études de cas sur des étudiants qui se spécialisent en mathématiques dans une université américaine font ressortir des différences saisissantes pour ce qui est d’un aspect important des mathématiques: les compétences concernant la compréhension, la production et l’évaluation des preuves (PUPA). À partir d’entrevues réalisées avec 36 étudiants au cours de la seconde moitié de leur formation en mathématiques à l’université, ces études de cas illustrent les différences frappantes qui existent quant aux preuves chez les étudiants qui terminent leur formation universitaire en mathématiques. « Ann », par exemple, n’a presque pas évolué dans la compréhension, la production et l’évaluation des preuves et reconnaît sa faiblesse dans ce domaine, mais doute de pouvoir être en mesure de produire une preuve mathématique acceptable même au terme de sa formation. « Ben », quant à lui, possédait déjà des connaissances enviables en matière de preuves au moment où il est entré à l’université, c’est pourquoi on peut présumer que ces connaissances seront renforcées au moment où il terminera son programme. « Carla » est peut-être plus typique, car au début elle recourait surtout à des méthodes de justification simples, mais ses connaissances dans ce domaine ont évolué au cours de sa formation, et elle termine son programme avec une connaissance qui, si elle n’est pas parfaite, est sans doute acceptable. Les implications possibles qu’on peut dériver des entrevues sont, entre autres, les suivantes: •Les étudiants ne doivent pas croire que la production de preuves soit chose facile; ils doivent au contraire apprendre qu’une preuve est souvent le résultat d’un travail intellectuel considérable.•Une programmation délibérée visant la compréhension, la production et l’évaluation des preuves dans le curriculum de premier cycle sera nécessaire si on veut promouvoir cet aspect chez la majorité des étudiants.•Les institutions ont le devoir d’aider les étudiants dont la compréhension, la production et l’évaluation des preuves sont faibles au moment où ils entrent à l’université. Certaines expériences d’enseignement réalisées dans une autre université laissent supposer qu’il existe au moins quelques principes susceptibles de guider les enseignants en vue d’améliorer la compréhension, la production et l’évaluation des preuves chez leurs étudiants. L’un de ces principes, celui qui consiste à « rendre les preuves tangibles », dérive du « principe de nécessité » proposé par Harel en enseignement des mathématiques: pour que les étudiants puissent apprendre, il faut qu’ils soient en mesure de voir la nécessité intellectuelle de ce qu’on prétend leur enseigner (1998, 2001). « Rendre les preuves tangibles » signifie donc que les preuves présentées doivent, au yeux des étudiants, inclure des objets mathématiques familiers (aspect concret), être explicites quant à l’idée qui les sous-tend (aspect convainquant) et entraîner clairement la nécessité d’une justification des différentes étapes de la preuve (aspect essentiel).Les questions qui regardent l’environnement et la durabilité sont des exemples de nouveaux discours dont les caractéristiques épistémologiques générales indiquent qu’on s’éloigne des limites disciplinaires et de la segmentation des connaissances pour s’orienter vers la perméabilité des limites, les relations entre les choses et une meilleure compréhension de la dynamique qui régit l’interaction entre l’humanité et les écosystèmes. La thèse de l’article est que l’enseignement des technologies doit aider les jeunes d’une part à accepter la complexité, l’ambiguïté et l’incertitude liées à la notion de durabilité, et d’autre part à ne pas fermer prématurément leur esprit devant de nouvelles possibilités. Il est essentiel de les encourager à garder espoir et à faire preuve d’un certain optimisme si on veut qu’ils remettent en question les opinions reçues et qu’ils aient le courage d’explorer des points de vue divergents sur le monde. En insistant sur la nécessité d’une collaboration interdisciplinaire dans les curriculums technologiques, l’article prône l’intégration des idées provenant d’une nouvelle science émergeante, celle de la durabilité, aussi bien en enseignement des technologies que dans les programmes d’études commerciales.

Students’ understanding of proofs: a historical analysis and implications for the teaching of geometry and linear algebra☆

The process of observing and analyzing studentsO behaviors is interesting and complex but also unstable. It is unstable because it involves countless variables, many of which are uncontrollable. Despite this, what we learn from this process is useful, even essential, in designing and implementing mathematics curricula for both students and teachers. This presentation is about studentsO behaviors in relation to justi®cation and proof. Some of these behaviors are assumed to be due to faulty instruction in school; others seem to be unavoidable, in the sense that they are of human developmental nature. Analyzed from a historical perspective of mathematical development, these studentsO understandings of proof can be classi®ed into three categories: · Category 1: In this category, studentsO understandings of proof (viewed in relation to those of their instructors) seem to parallel the Greek conception of mathematics (viewed in relation to that of modern days). · Category 2: In this category, studentsO understandings of proof are reminiscent of the 16±17th century conception of mathematics. · Category 3: In this category, studentsO understandings of proof seem, to a large extent, to be a result of faulty instruction in the elementary and secondary schools. www.elsevier.com/locate/laa

On teacher education programmes in mathematics

Current teacher education programmes suffer from a lack of attention to the three crucial components of teachers’ knowledge: mathematics content, episte‐mology, and pedagogy. As a result, they cannot achieve the desired quality in teachers as was envisioned by the current mathematics education leadership. Teachers’ mathematics knowledge is far from being satisfactory even in terms of the standards for high‐school mathematics. The work on epistemology and pedagogy is detached from a personal experiential basis of teaching, and thus it is in conflict with the well established principle that knowledge construction (and this includes mathematics knowledge as well as knowledge of mathematics epistemology and pedagogy) is a product of personal, experiential problem‐solving activity. The effort of teacher education programmes must centre on these three components of teachers’ knowledge base. In particular, teachers’ knowledge of mathematics should be promoted and evaluated in terms of mathematics values, not spe...

Three Principles of Learning and Teaching Mathematics

I have touched upon certain aspects of each one of the LACSG recommendations. I have suggested that the focus on proofs should not begin in the first course in linear algebra, but should be emphasized throughout the mathematics curricula in all grade levels.

Higher Mathematics Education

This chapter pertains to higher mathematics education in various countries of the world. We try to present a review of significant and interesting research investigations and teaching experiences. The chapter starts with the description of some general features of mathematics education at university level that are more or less common in different countries. Following this, a number of research projects, both in calculus and linear algebra, on epistemological and cognitive aspects and curricular renovations are reported, and some nontraditional teaching methods that seem to be more successful than traditional teaching are presented. The chapter concludes with some summarising remarks and perspectives for the future drawn from the reported research projects and investigations on higher mathematics education.

Using geometric models and vector arithmetic to teach high‐school students basic notions in linear algebra

In [1] we discussed the difficulties students at the university level and the high‐school level have with the existing approaches of teaching linear algebra (for a review of these approaches see [2]) and offered an alternative approach. A summary of these difficulties and some of the elements of this approach are presented in this paper. In addition, we will (a) discuss the scientific and pedagogical importance of linear algebra; (b) show how vector arithmetic in solving geometric problems was used to introduce basic notions of linear algebra; (c) outline how the necessity principle was implemented in this approach.