scispace - formally typeset
Search or ask a question

Pupils' needs for conviction and explanation within the context of dynamic geometry.

01 Jan 1998-
TL;DR: The research attempted to determine whether pupils were convinced about explored geometric statements and their level of conviction, and established whether pupils exhibited an independent desire for why the result, they obtained, is true, within the context of dynamic geometry.
Abstract: Recent literature on mathematics education, and more especially on the teaching and learning of geometry, indicates a need for further investigations into the possibility of devising new strategies, or even developing present methods, in order to avert what might seem to be a "problem" in mathematics education. Most educators hIld textbooks, it would seem, do not address the need (function and meaning) of proof at all, or those that do, only address it from the limited perspective that the only function of proof is verification. The theoretical part of this study, therefore, analyzed the various functions of proof: in order to identify possible alternate ways of presenting proof meaningfully to pupils. This work further attempted to build on existing research and tested these ideas in a teaching environment. This was done in order to evaluate the feasibility of introducing "proof' as a means of explanation rather than only verification, within the context of dynamic geometry. Pupils, who had not been exposed to proof as yet, were interviewed and their responses were analyzed. The research focused on a few aspects. It attempted to determine whether pupils were convinced about explored geometric statements and their level of conviction. It also attempted to establish whether pupils exhibited an independent desire for why the result, they obtained, is true and if they did, could they construct an explanation, albeit a guided one, on their own. Several useful implications have evolved from this work and may be able to influence, both the teaching and learning, of geometry in school. Perhaps the suggestions may be useful to pre-service and in-service educators.
Citations
More filters
Journal ArticleDOI
TL;DR: In this article, the authors present an analytic framework to describe and analyze students' answers to proof problems and verify the usefulness of learning in dynamic geometry computer environments to improve students' proof skills.
Abstract: As a key objective, secondary school mathematics teachers seek to improve the proof skills of students. In this paper we present an analytic framework to describe and analyze students' answers to proof problems. We employ this framework to investigate ways in which dynamic geometry software can be used to improve students' understanding of the nature of mathematical proof and to improve their proof skills. We present the results of two case studies where secondary school students worked with Cabri-Geometre to solve geometry problems structured in a teaching unit. The teaching unit had the aims of: i) Teaching geometric concepts and properties, and ii) helping students to improve their con- ception of the nature of mathematical proof and to improve their proof skills. By applying the framework defined here, we analyze students' answers to proof problems, observe the types of justifications produced, and verify the usefulness of learning in dynamic geometry computer environments to improve students' proof skills.

218 citations


Cites background from "Pupils' needs for conviction and ex..."

  • ...…Balacheff, 1988a and b; Bell, 1976a and b; Harel and Sowder, 1996; Sowder and Harel, 1998), and descriptions of students’ beliefs when deciding whether they are convinced by an argument about the truth of a statement, or not (De Villiers, 1991; Harel and Sowder, 1996; Sowder and Harel, 1998)....

    [...]

Journal ArticleDOI
TL;DR: In this article, a broad descriptive account of some activities that the author has designed using Sketchpad to develop teachers' understanding of other functions of proof than just the traditional function of "verification" is given.
Abstract: This paper gives a broad descriptive account of some activities that the author has designed using Sketchpad to develop teachers’ understanding of other functions of proof than just the traditional function of ‘verification’. These other functions of proof illustrated here are those of explanation, discovery and systematization (in the context of defining and classifying some quadrilaterals). A solid theoretical rationale is provided for dealing with these other functions in teaching by analysing actual mathematical practice where verification is not always the most important function. The activities are designed according to the so-called ‘reconstructive’ approach, and are structured more or less in accordance with the Van Hiele theory of learning geometry.

98 citations

Journal ArticleDOI
25 Apr 2008-Zdm
TL;DR: In this paper, the condition of transparency refers to the intricate dilemma in the teaching of mathematics about how and how much to focus on various aspects of proof and how to work with proof without a focus on it.
Abstract: The condition of transparency refers to the intricate dilemma in the teaching of mathematics about how and how much to focus on various aspects of proof and how and how much to work with proof without a focus on it. This dilemma is illuminated from a theoretical point of view as well as from teacher and student perspectives. The data consist of university students’ survey responses, transcripts of interviews with mathematicians and students as well as protocols of the observations of lectures, textbooks and other instructional material. The article shows that the combination of a socio-cultural perspective, Lave and Wenger’s and Wenger’s social practice theories and theories about proof offers a fresh framework for studies concerning the teaching and learning of proof.

51 citations

Journal ArticleDOI
TL;DR: For example, the authors argues that mathematical proofs that explain why some mathematical theorem holds and proofs that merely prove that the theorem holds without revealing the reason why it holds can be seen as explanations.
Abstract: Unlike explanation in science, explanation in mathematics has received relatively scant attention from philosophers. Whereas there are canonical examples of scientific explanations (as well as canonical examples of nonexplanations, such as �the flagpole,� �the eclipse,� and �the barometer�), there are few (if any) examples that have become widely accepted as exhibiting the distinction between mathematical proofs that explain why some mathematical theorem holds and proofs that merely prove that the theorem holds without revealing the reason why it holds. This essay offers some examples of proofs that mathematicians have considered explanatory (or not), and it argues that these examples suggest a particular account of explanation in mathematics (at least, of those explanations consisting of proofs). The essay compares its account to Steiner's and Kitcher's. Among the topics that arise are proofs that exploit symmetries, mathematical coincidences, brute-force proofs, simplicity in mathematics, merely clever proofs, and proofs that unify what other proofs treat as separate cases.

51 citations

Book
23 Jul 2008
TL;DR: The authors describe how students encounter proof in a community of mathematical practice at a mathematics department and how they are drawn to share mathematicians' views and knowledge of knowledge of the mathematical community.
Abstract: This thesis aims to describe how students encounter proof in a community of mathematical practice at a mathematics department and how they are drawn to share mathematicians’ views and knowledge of ...

46 citations

References
More filters
Journal ArticleDOI
TL;DR: This edition of the book should find a new generation of general readers and students who would like to know what mathematics is all about and will prove invaluable as a course text for a general mathematics appreciation course, one in which the student can combine an appreciation for the esthetics with some satisfying and revealing applications.
Abstract: Winner of the 1983 National Book Award! \"...a perfectly marvelous book about the Queen of Sciences, from which one will get a real feeling for what mathematicians do and who they are. The exposition is clear and full of wit and humor...\" - The New Yorker (1983 National Book Award edition) Mathematics has been a human activity for thousands of years. Yet only a few people from the vast population of users are professional mathematicians, who create, teach, foster, and apply it in a variety of situations. The authors of this book believe that it should be possible for these professional mathematicians to explain to non-professionals what they do, what they say they are doing, and why the world should support them at it. They also believe that mathematics should be taught to non-mathematics majors in such a way as to instill an appreciation of the power and beauty of mathematics. Many people from around the world have told the authors that they have done precisely that with the first edition and they have encouraged publication of this revised edition complete with exercises for helping students to demonstrate their understanding. This edition of the book should find a new generation of general readers and students who would like to know what mathematics is all about. It will prove invaluable as a course text for a general mathematics appreciation course, one in which the student can combine an appreciation for the esthetics with some satisfying and revealing applications. The text is ideal for 1) a GE course for Liberal Arts students 2) a Capstone course for perspective teachers 3) a writing course for mathematics teachers. A wealth of customizable online course materials for the book can be obtained from Elena Anne Marchisotto (elena.marchisotto@csun.edu) upon request.

1,157 citations

01 Jan 1990

318 citations


"Pupils' needs for conviction and ex..." refers background or result in this paper

  • ...It also takes cognisance of the levels expounded by the Van Hieles and also previous research conducted by other authors and researchers, including that of De Villiers (1990;1991) and Zack (1997 : 291-297)....

    [...]

  • ...For example, Hanna (as quoted in De Villiers, 1990 : 21) claims that "mathematical concepts and propositions are .. . conceived and formulated before proofs are put in place"....

    [...]

  • ...Although it is quite possible to achieve a high level of conviction that a conjecture holds true by experimentation, this does not provide a deeper understanding as to why the conjecture may be true (De Villiers,1990 : 19)....

    [...]

Journal ArticleDOI
TL;DR: In mathematical research, the purpose of proof is to convince as discussed by the authors, and the test of whether something is a proof is whether it convinces qualified judges. In the classroom, on the other hand, it is to explain.
Abstract: In mathematical research, the purpose of proof is to convince. The test of whether something is a proof is whether it convinces qualified judges. In the classroom, on the other hand, the purpose of proof is to explain. Enlightened use of proofs in the mathematics classroom aims to stimulate the students' understanding, not to meet abstract standards of “rigor” or “honesty.”

270 citations


"Pupils' needs for conviction and ex..." refers background in this paper

  • ...Besides, we do not fully understand the physical processes by which the computer works (Hersh, 1993 : 393)....

    [...]

  • ...This view of proof functions "as the last judgement, the final word before a problem is put to bed" (Hersh,1993 : 390)....

    [...]

  • ...100 years from now the map theorem will be, I think, an exercise in a first - year graduate course, provable in a couple of pages by means of the appropriate concepts, which will be completely familiar by then" (Hersh, 1993:393)....

    [...]

Journal ArticleDOI
TL;DR: In this paper, the authors analyse pupils' attempts to construct proofs and explanations in simple mathematical situations, to observe in what ways they differ from the mature mathematician's use of proof, and derive guidance about how best to foster pupils' development in this area.
Abstract: Viewed internationally, the proof aspect of mathematics is probably the one which shows the widest variation in approaches. The present French syllabus adopts an axiomatic treatment of geometry from the third secondary school year (age 14), Papy's Mathdmatique Moderne is axiomatic from the age of 12, early American developments based primary school number work on the laws of algebra. In England, proofs of geometrical theorems have been steadily disappearing from O-level syllabuses for thirty years, and 'it continues to be the policy of the SMP to argue the likelihood of a general result from particular cases'. (Preface to Book 5). Underlying this divergence in practice lies the tension between the awareness that deduction is essential to mathematics, and the fact that generally only the ablest school pupils have achieved understanding of it. The purpose of the work described in this paper is to analyse pupils' attempts to construct proofs and explanations in simple mathematical situations, to observe in what ways they differ from the mature mathematician's use of proof, and thus to derive guidance about how best to foster pupils' development in this area. In a previous paper (Bell, 1976), I have shown that pupils' attempts at making and establishing generalisations, and at supporting these with reasons, can be interpreted in terms of a number of identifiable stages of attainment which are loosely related to age. Two of these stages were fairly well-defined - Stage 1

261 citations


"Pupils' needs for conviction and ex..." refers background in this paper

  • ...(Bell, 1976 : 26) This function of proof is concerned with the logical organisation of propositions into a deductive system....

    [...]