La lettre de la Preuve


ISSN 1292-8763

Juillet/Août 2000


Aczel J. C. (2000) The Evaluation of a Computer Program for Learning Logic: The Role of Students' Formal Reasoning Strategies in Visualising Proofs. CALRG Technical report 192. Computers and Learning Research Group, Institute of Educational Technology, The Open University, Milton Keynes, UK

Breton P., Gauthier G. (2000) Histoire des théories de l'argumentation. Coll. Repères. Paris : La décourverte

Miyazaki M. (2000) Levels of proof in lower secondary school mathematics. Educational Studies in Mathematics 41(1) 47-68.

 All the following papers are published in Deduction Systems for Mathematics Education,
 proceedings of a Workshop at CADE-17, edited by Erica Melis (see a quick presentation below)

Bundy A., Moore J.D., Zinn C. (2000) An Intelligent Tutoring System for Induction Proofs Abridged Project Description, pp. 4-13

Dahn B. I. (2000) Knowledge Management in the TRIAL-SOLUTION Project (First Considerations). pp. 14-19

Farmer M. W. (2000) A Proposal for the Development ofan Interactive Mathematics Laboratory for Mathematics Education. pp. 20-25

Fiedler A. (2000) Determining and Structuring the Content of Presentations of Proofs in Natural Language. pp. 44-51

Lowe H., Dobbin A. (2000) Towards a Proof Assistant forNovice Formal Methodists. pp. 52 sqq

Melis E. (2000) The `Interactive Textbook' Project. pp. 26-34

Sommer R., Rozenfeld M., Ravaglia R. (2000) A Proof Environment forTeaching Mathematics. pp. 35-43




The complete text from all the following references from the Jape Project
 can be downloaded from the project site, just click on the icon

Aczel J. C., Fung P., Bornat R., Oliver M., O'Shea T., Sufrin B. (1999) "Using computers to learn logic: undergraduates' experiences" in Cumming, G., Okamoto, T. & Gomez, L. (eds.) Advanced Research in Computers and Communications in Education: Proceedings of the 7th International Conference on Computers in Education, Amsterdam, IOS Press

Aczel J. C., Fung P., Bornat R., Oliver M., O'Shea T., Sufrin B. (1999) "Influences of Software Design on Formal Reasoning", in Brewster, S., Cawsey, A. & Cockton, G. (Eds.) Proceedings of IFIP TC.13 International Conference on Human-Computer Interaction INTERACT '99, Vol. 2, pp. 3-4, Swindon, UK, British Computer Society, ISBN 1-902505-19-0

Aczel J. C., Oliver M., Fung P. (1999) "Progressive Refinement of Formal Reasoning Strategies using Software for Learning Logic", Proceedings of the 20th anniversary conference of the Computers and Learning Research Group, The Open University, June 1999

Aczel J. C., Fung P. (1999) "Evaluation of a software tool for supporting formal reasoning", Proceedings of the day conference of the Student Research Centre, Feb 1999, Institute of Educational Technology,

Aczel J. C., Fung P., Bornat R., Oliver M., O'Shea T., Sufrin B. (1999) "Undergraduates' Computer-Assisted Formal Reasoning", Presentation to the day conference of the British Society for Research into Learning Mathematics, Feb 1999

Aczel J. C., Fung P., Bornat R., Oliver M., O'Shea T., Sufrin B. (1999) "Computer Science Undergraduates Learning Logic Using a Proof Editor: Work in Progress", Proceedings of the Psychology of Programming Interest Group, University of Leeds

Bornat R., Sufrin B.A. (1999) "Animating formal proof at the surface: the Jape proof calculator", The Computer Journal, 43(3), 177-192



Barbin E. (1996) Quels conceptions épistémologiques de la démonstration pour quels apprentissages ? in : L'enseignement des mathématiques : des repères entre savoirs, programmes et pratiques (pp.195-210). Pont-à-Mousson : Topiques éditions

Davis E. (1893) On the teaching of elementary geometry. Bulletin of the New York Mathematical Society 3, 8-14

Jones P. (1944) Early american geometry. Mathematics Teacher 37, 3-11

Keigwin H. (1892) The old and new methods in geometry. Educational Review. 4, 182-184.

Kitcher P. (1977) On the uses of rigorous proof. Science 196, pp. 782-783 [A review of Lakatos' Proofs and refutations]

Kuntz G. (1996) De l'intelligence artificielle aux fiches-méthodes. in : L'enseignement des mathématiques : des repères entre savoirs, programmes et pratiques (pp.55-68). Pont-à-Mousson : Topiques éditions

Legrand M. (1996) Débat scientifique en cours de mathématiques et spécificité de l'analyse. in : L'enseignement des mathématiques : des repères entre savoirs, programmes et pratiques (pp.171-192). Pont-à-Mousson : Topiques éditions

Nunn T. (1925) The sequence of theorems in school geometry. Mathematics Teacher 18, 321-332

Richards E. (1892) Old and new methods in elementary geometry. Educational Review. 3, 31-39.

Ryan J. (1928) Two methods of teaching geometry: syllabus vs. textbook. Mathematics Teacher 21, 31-36

Shutts G. (1892) Old and new methods in geometry. Educational Review. 3, 264-266.

Stroup P. (1926) When is a proof not a proof. Mathematics Teacher 19, 499-505


Proof and perception IV

Michael Otte


All mathematical knowledge we might have seems to fall into two categories.

1. Knowledge that certain mathematical claims follow from certain other mathematical claims, or bodies of claims.
2. Knowledge of the consistency of certain mathematical claims, or bodies of claims.

This view leads to the impression that mathematical knowledge is simply logical knowledge. Today, however, our logical investigations rely on mathematics and because of this, logic appears as a chapter of mathematics. Therefore in the end, mathematics appears as discipline which is supposed to justify itself. Such a "naturalism" is rather unusual. The core of this circularity seems to come down to the following. Ever since Euclid and until the 19th century, as a rule, axiomatics was used as an instrument for dealing with the complexities of some realm of intended applications (think of Euclid's axiomatization of geometry and of Newton's axiomatization of mechanics). Ever since Hamilton and Boole, and particularly after Hilbert and Gödel, axiomatics began to be used as an instrument for dealing with the complexities of mathematics itself. It became a means of mathematizing or modeling mathematical theories themselves!


To



Deduction Systems
for Mathematics Education
a Workshop at CADE-17

Erica Melis

TSG 12
Proof and Proving in Mathematics Education

Chief organiser
Paolo Boero

 The purpose of this workshop, in the  application area of automated and inter-active theorem proving, is to establish more communication between current education projects in the CADE community, to exchange ideas and opinions, and to make available the experience of education systems from other AI-communities. The submission of project descriptions were explicitly encourage.

Editor contact :

With contributions from...
M. G. Bartolini Bussi J. Bolite Frant, M. Rabello de Castro M. de Villiers N. Douek M. A. Gravina D. Grenier K. Harada, E. Gallou-Dumiel, N. Nohda L. Healy C. A. Maher, R. D. Kiczek G.W. LandmanF. Olivero P. R. Richard G. Roulet Y. Sekiguchi  

Renaissance de la démonstration
dans l'enseignement des mathématiques aux Etats-Unis ?


Eric Knuth

"The principal difficulty in the mathematics is the length of inferences and compass of thought, requisite to the forming of any conclusion. And, perhaps, our progress in natural philosophy is chiefly retarded by the want of proper experiments and phaenomena, which are often discovered by chance, and cannot always be found, when requisite, even by the most diligent and prudent enquiry. As moral philosophy seems hitherto to have received less improvement than either geometry or physics, we may conclude, that, if there be any difference in this respect among these sciences, the difficulties, which obstruct the progress of the former, require superior care and capacity to be surmounted."

Hume, 1748, An Enquiry Concerning Human Understanding

 Cette contribution d'Eric Knuth au thème  de la lettre de mai-juin, est maintenant  disponible en langue française.

La preuve, langue et culure

Prueba, lingua y cultura

Proof, language and culture

Comment on

"The rebirth of proof in school mathematics in the United States" by Eric Knuth
(published in the Proof Newsletter, May/June 2000)


A. Keith Austin

A part of the proof website is now devoted to a repository of material which could help to inform research about the state of the teaching of proof in mathematics in different countries. All contributions are welcome...

See also La lettre de la Preuve, septembre/octobre 1999, concerning issues related to proof and culture.

Eric Knuth in his short paper reports that the majority teachers feel proof is not appropriate for all students. Keith Austin suggests, in his reaction, a possible explanation for this ...

Any further reactions are welcome, contact the Proof Newsletter editor.

Web Archives

Visualisation in the Software Development Process

Principal Investigator

Dr. Pat Fung

 The aim of this project was to evaluate and refine a visualisation tool (Jape) intended to support the formal  reasoning of computer science students; and in so doing to increase understanding of the effectiveness of such tools in supporting training in the software development process, and in particular the cognitive processes and factors which affect visualisation software's ease of use.
  The particular tool under investigation allows users to manipulate proofs on screen, using a mouse. An educator can control which logic Jape uses, how proofs are presented, and what actions the user can take. The implementation studied here - ItL Jape - is pre-loaded with propositional and predicate logic, with Natural Deduction rules and with Fitch boxes. When users apply a rule to a line of the proof, the software shows the effect on the proof.
  Three empirical studies were carried out. In the first study, four subjects were videotaped as they used the tool Jape to assist them in constructing proofs as part of their course in formal logic. In the second study, usage data was collected from a year's cohort of computer science undergraduates. In the third study, ten subjects were videotaped using the program in task-based interviews.
  A quantitative analysis of tests, surveys and logfiles suggests that subjects' backgrounds appeared to have little effect either on how much the program was used, or on how much progress was made with the conjectures in the program. However, on average, the more a subject used Jape the higher his or her score in course outcome measures. Moreover, progress in Jape was a significant factor in course performance, even when significant background variables such as gender, degree course, and prior programming experience are taken into account. This suggests that visualisation tools can be appropriate for different user groups.
  Evidence from observation and interviews suggests that the main advantages of ItL Jape for many users are that it allowed them to consider many more examples than would be possible on paper, it encouraged experimentation with different routes to a proof, and it challenged inaccurate and forward-fixated reasoning.
  In order to explore the role of visualisation in this process, a model of users' knowledge in terms of conjectured proof strategies was developed. Rule-specific strategies that enable users to decide what rule should be applied in given situations are distinguished from global strategies that help users to make and debug plans. Users are categorised by their prior knowledge of the rules. This model provides insights into the reasons for success and failure in learning from Jape, explains the differences in users' proving behaviour on paper and using the visualisation tool ItL Jape, and enables predictions about interface refinements that would enhance learning.


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How to publish Cabri figures
on the Web?

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Editeur : Nicolas Balacheff
English Editor :
Virginia Warfield, Editor en Castellano : Patricio Herbst

Advisory Board : Paolo Boero, Daniel Chazan, Raymond Duval, Gila Hanna, Guershon Harel,
Celia Hoyles, Maria-Alessandra Mariotti, Michael Otte,
Yasuhiro Sekiguchi, Michael de Villiers

La lettre de la Preuve


ISSN 1292-8763