Didactical issues at the interface of mathematics and computer science
... These tools represent a potentially invaluable resource for the study of mathematical argument, but their application to mathematics has only just begun Corneli et al., 2019). Again, I am unaware of any application of these resources in mathematics education: there is a growing body of work applying digital tools to mathematics education (for example, Modeste, 2016;Durand-Guerrier et al., 2019), but not the tools specific to argumentation. They represent an as yet untapped resource, of considerable scale and importance. ...
The traditional view of evidence in mathematics is that evidence is just proof and proof is just derivation. There are good reasons for thinking that this view should be rejected: it misrepresents both historical and current mathematical practice. Nonetheless, evidence, proof, and derivation are closely intertwined. This paper seeks to tease these concepts apart. It emphasizes the role of argumentation as a context shared by evidence, proofs, and derivations. The utility of argumentation theory, in general, and argumentation schemes, in particular, as a methodology for the study of mathematical practice is thereby demonstrated. Argumentation schemes represent an almost untapped resource for mathematics education. Notably, they provide a consistent treatment of rigorous and non-rigorous argumentation, thereby working to exhibit the continuity of reasoning in mathematics with reasoning in other areas. Moreover, since argumentation schemes are a comparatively mature methodology, there is a substantial body of existing work to draw upon, including some increasingly sophisticated software tools. Such tools have significant potential for the analysis and evaluation of mathematical argumentation. The first four sections of the paper address the relationships of evidence to proof, proof to derivation, argument to proof, and argument to evidence, respectively. The final section directly addresses some of the educational implications of an argumentation scheme account of mathematical reasoning.
This text introduces the first of two volumes following the DEMIMES (Didactics and Epistémology of Mathematics and their Interactions for Mediation and Post-secondary Teaching) thematic school held in Autrans (France) from April 4 to 7, 2022.
Ce texte introduit le premier des deux volumes qui font suite à la tenue de l’école thématique DEMIMES (Didactique et Épistémologie des Mathématiques et leurs Interactions pour la Médiation et l'Enseignement Supérieur) qui s’est tenue à Autrans (France) du 4 au 7 avril 2022.
Interdisciplinarity (ID) represents nowadays a complex, multi-dimensional and timely
challenge in STEM and, more in general, in STEAM education. If on one side ID is at the core of the most urgent societal issues, in schools and universities disciplines are almost exclusively taught separately and rigid boundaries are created. After a long period of good practices, researchers are more and more perceiving the need to develop theoretical frameworks to rigorously define what ID is and to recognize if and how interdisciplinary knowledge and skills can be developed in teaching. In this paper, four theoretically-oriented studies are presented as outcomes of the IDENTITIES Erasmus+ project aimed to develop an approach to design interdisciplinary teaching modules for pre-service teacher education.
We compare the pencil/paper proofs and formal proof of two traditional proofs in high school geometry. We highlight the fact that slightly different formulations or proofs can lead to difficulties in the formalization. We discuss the challenges and impact on both mathematical teaching and on the design of AI tools for mathematical education.
The role of logic in mathematics education has been widely discussed from the seventies and eighties during the “modern maths period” till now, and remains still a rather controversial issue in the international community. Nevertheless, the relevance of discrete mathematics and algorithmic thinking for the development of heuristic and logical competences is both one of the main points of the program of Tamás Varga, and of some didactic teams in France. In this paper, we first present the semantic perspective in mathematics education and the role of logic in the Hungarian tradition. Then, we present insights on the role of research problems in the French tradition. Finely, we raise some didactical issues in algorithmic thinking at the interface of mathematics and computer science. Subject Classification: 97E30
Understanding learners' understanding is a key requirement for an efficient design of teaching situations and learning environments, be they digital or not. This keynote outlines the modeling framework cK¢ (conception, knowing, concept) created with the objective to respond to this requirement, with the additional ambition to build a bridge between research in mathematics education and research in educational technology. After an introduction of the rationale of cK¢, some illustrations are presented. Then follow comments on cK¢ and learning. The conclusion evokes key research issues raised by the use of this modeling framework.
I believe that the mathematical community (appropriately dened) is facing a great challenge to re-evaluate the role of proof in light of the power of current com- puter systems, of modern mathematical computing packages and of the growing capacity to data-mine on the internet. Add to that the enormous complexity of many modern mathematical results such as the Poincar e conjecture, Fermat's last theorem, and the classication of nite simple groups. With great challenges come great opportunities. I intend to touch upon the current challenges and opportu- nities for the learning and doing of mathematics. As the prospects for inductive mathematics blossom, the need to ensure that the role of proof is properly founded remains undiminished.
This is a tutorial on finite automata. We present the standard material on
determinization and minimization, as well as an account of the equivalence of
finite automata and monadic second-order logic. We conclude with an
introduction to the syntactic monoid, and as an application give a proof of the
equivalence of first-order definability and aperiodicity.
A Source Book for the History of Mathematics, but one which offers a different perspective by focusing on algorithms. With the development of computing has come an awakening of interest in algorithms. Often neglegted by historians and modern scientists, more concerned with the nature of concepts, algorithmic procedures turn out to have been instrumental in the development of fundamental ideas: practice led to theory just as much as the other way round. The purpose of this book is to offer a historical background to contemporary algorithmic practice. Each chapter centres around a theme, more or less in chronological order, and the story is told through the reading of over 200 original texts, faithfully reproduced. This provides an opportunity for the reader to sit alongside such mathematicians as Archimedes, Omar Khayyam, Newton, Euler and Gauss as they explain their techniques. The book ends with an account of the development of the modern concept of algorithm.
In this article, we come back to the seminal role of epistemology in didactics of sciences and particularly in mathematics. We defend that the epistemological research on the interactions between mathematics and informatics is necessary to feed didactical research on today’s mathematics learning and teaching situations, impacted by the development of informatics. We develop some examples to support this idea and propose some perspectives to attack this issue.
Aimed at any serious programmer or computer science student, the new second edition of _Introduction to Algorithms_ builds on the tradition of the original with a truly magisterial guide to the world of algorithms. Clearly presented, mathematically rigorous, and yet approachable even for the maths- averse, this title sets a high standard for a textbook and reference to the best algorithms for solving a wide range of computing problems. With sample problems and mathematical proofs demonstrating the correctness of each algorithm, this book is ideal as a textbook for classroom study, but its reach doesn't end there. The authors do a fine job at explaining each algorithm. (Reference sections on basic mathematical notation will help readers bridge the gap, but it will help to have some maths background to appreciate the full achievement of this handsome hardcover volume.) Every algorithm is presented in pseudo-code, which can be implemented in any computer language, including C/C++ and Java. This ecumenical approach is one of the book's strengths. When it comes to sorting and common data structures, from basic linked list to trees (including binary trees, red-black and B-trees), this title really shines with clear diagrams that show algorithms in operation. Even if you glance over the mathematical notation here, you can definitely benefit from this text in other ways. The book moves forward with more advanced algorithms that implement strategies for solving more complicated problems (including dynamic programming techniques, greedy algorithms, and amortised analysis). Algorithms for graphing problems (used in such real-world business problems as optimising flight schedules or flow through pipelines) come next. In each case, the authors provide the best from current research in each topic, along with sample solutions. This text closes with a grab bag of useful algorithms including matrix operations and linear programming, evaluating polynomials and the well-known Fast Fourier Transformation (FFT) (useful in signal processing and engineering). Final sections on "NP-complete" problems, like the well-known traveloling salesmen problem, show off that while not all problems have a demonstrably final and best answer, algorithms that generate acceptable approximate solutions can still be used to generate useful, real-world answers. Throughout this text, the authors anchor their discussion of algorithms with current examples drawn from molecular biology (like the Human Genome project), business, and engineering. Each section ends with short discussions of related historical material often discussing original research in each area of algorithms. In all, they argue successfully that algorithms are a "technology" just like hardware and software that can be used to write better software that does more with better performance. Along with classic books on algorithms (like Donald Knuth's three-volume set, _The Art of Computer Programming_), this title sets a new standard for compiling the best research in algorithms. For any experienced developer, regardless of their chosen language, this text deserves a close look for extending the range and performance of real-world software. _--Richard Dragan_
In mathematics education, it is often said that mathematical statements are necessarily either true or false. It is also well
known that this idea presents a great deal of difficulty for many students. Many authors as well as researchers in psychology
and mathematics education emphasize the difference between common sense and mathematical logic. In this paper, we provide
both epistemological and didactic arguments to reconsider this point of view, taking into account the distinction made in
logic between truth and validity on one hand, and syntax and semantics on the other. In the first part, we provide epistemological
arguments showing that a central concern for logicians working with a semantic approach has been finding an appropriate distance
between common sense and their formal systems. In the second part, we turn from these epistemological considerations to a
didactic analysis. Supported by empirical results, we argue for the relevance of the distinction and the relationship between
truth and validity in mathematical proof for mathematics education.
It is widely attested that university students face considerable difficulties with reasoning in analysis, especially when dealing with statements involving two different quantifiers. We focus in this paper on a specific mistake which appears in proofs where one applies twice or more a statement of the kind “for all X, there exists Y such that R(X, Y)”, and forgets that in that case, a priori, “Y depends on X”. We analyse this mistake from both a logical and mathematical point of view, and study it through two inquiries, an historical one and a didactic one. We show that mathematics teachers emphasise the importance of the dependence rule in order to avoid this kind of mistake, while natural deduction in predicate calculus provides a logical framework to analyse and control the use of quantifiers. We show that the relevance of this dependence rule depends heavily on the context: nearly without interest in geometry, but fundamental in analysis or linear algebra. As a consequence, mathematical knowledge is a key to correct reasoning, so that there is a large distance between beginners' and experts' abilities regarding control of validity, that, to be shortened, probably requires more than a syntactic rule or informal advice.
Résumé
Les difficultés de manipulation, par les étudiants, des énoncés contenant deux quantificateurs différents, rencontrés dans de nombreux raisonnements en analyse, sont bien attestées. Nous nous intéressons plus spécialement dans cet article à une erreur qui apparaît dans certaines preuves lorsque l'on applique deux fois ou plus un énoncé de la forme “pour tout X, il existe Y tel que R(X,Y)” et que l'on oublie que dans un tel cas, a priori, “Y dépend de X”. Nous analysons cette erreur d'un point de vue logique et d'un point de vue mathématique, puis nous l'étudions à travers deux enquêtes, l'une historique et l'autre didactique. Nous montrons que les professeurs de mathématiques soulignent l'importance de la règle de dépendance pour éviter ce type d'erreur, tandis que la déduction naturelle dans le calcul des prédicats fournit un cadre de référence logique pour analyser et contrôler l'usage des quantificateurs. Nous montrons que la pertinence de la règle de dépendance dépend fortement du contexte: pratiquement sans intérêt en géométrie, elle est tout à fait fondamentale en analyse et en algèbre linéaire. De ce fait, les connaissances mathématiques sont la clé d'un raisonnement correct, si bien qu'il y a une grande distance entre le débutant et l'expert concernant le contrôle de la validité, que quelques règles syntaxiques ou quelques conseils informels ne permettent vraisemblablement pas de réduire.
We discuss what constitutes knowledge in pure mathemat- ics and how new advances are made and communicated. We describe the impact of computer algebra systems, automated theorem provers, programs designed to generate examples, mathematical databases, and theory formation programs on the body of knowledge in pure mathemat- ics. We discuss to what extent the output from certain programs can be considered a discovery in pure mathematics. This enables us to assess the state of the art with respect to Newell and Simon's prediction that a computer would discover and prove an important mathematical theorem.
- A V Aho
- R Sethi
- J D Ullman
Aho AV, Sethi R, Ullman JD (1986) Compilers, Principles, Techniques. Addison
Wesley
Arora S, Barak B (2009) Computational Complexity: A Modern Approach. Cambridge University Press
Introduction à l'approche écologique du didactique -L'écologie des organisations mathématiques et didactiques
- M Artaud
Artaud M (1998) Introduction à l'approche écologique du didactique -L'écologie
des organisations mathématiques et didactiques. In: Actes de la IXème école d'été
de didactique des mathématiques, ARDM & IUFM, Caen, pp 101-139
Experimental approaches to theoretical thinking: artefacts and proofs
- F Arzarello
- Mgb Bussi
- Ayl Leung
- M A Mariotti
- I Stevenson
Arzarello F, Bussi MGB, Leung AYL, Mariotti MA, Stevenson I (2012) Experimental approaches to theoretical thinking: artefacts and proofs. In: Proof and proving
in mathematics education, Springer, pp 97-143
L'informatique et son enseignement dans l'enseignement scolaire général français: enjeux de pouvoir et de savoirs
- G L Baron
- É Bruillard
Baron GL, Bruillard É (2011) L'informatique et son enseignement dans l'enseignement scolaire général français: enjeux de pouvoir et de savoirs. In: Recherches et
expertises pour l'enseignement scientifique, vol 1, De Boeck Supérieur, pp 79-90
Logiques classiques et non classiques : essai sur les fondements de la logique
- Da Costa
Da Costa NCA (1997) Logiques classiques et non classiques : essai sur les fondements de la logique. Masson, Paris
Douady R (1986) Jeux de cadres et dialectique outil-objet. Recherches en didactique
des mathématiques 7/2:5-31
Spécificité de la preuve et de la modélisation en mathématiques discrètes
- D Grenier
- C Payan
Grenier D, Payan C (1998) Spécificité de la preuve et de la modélisation en mathématiques discrètes. Recherches en didactique des mathématiques 18(2):59-100
Logique, automates, informatique
- P Gribomont
- D Ribbens
- P Wolper
Gribomont P, Ribbens D, Wolper P (2000) Logique, automates, informatique. In:
Beets F, Gillet E (eds) Logique En Perspective: Mélanges Offerts à Paul Gochet,
Ousia, pp 545-577
Enseignement des sciences mathématiques : Commission de réflexion sur l'enseignement des mathématiques
- J Hopcroft
- R Motwani
- J Ullman
Hopcroft J, Motwani R, Ullman J (2007) Introduction to Automata Theory, Languages, and Computation, 3rd edn. Addison-Wesley
Howson AG, Kahane JP (eds) (1986) The Influence of computers and informatics
on mathematics and its teaching, international commission on mathematical instruction edn. ICMI study series, Cambridge University Press
Kahane JP (2002) Enseignement des sciences mathématiques : Commission de réflexion sur l'enseignement des mathématiques : Rapport au ministre de l'éducation nationale, cndp edn. Odile Jacob, Paris
Recherche binaire et méthode de dichotomie, comparaison et enjeux didactiques à l'interface mathématiques -informatique
- A Meyer
- S Modeste
Meyer A, Modeste S (2018) Recherche binaire et méthode de dichotomie, comparaison et enjeux didactiques à l'interface mathématiques -informatique. In:
proceedings of EMF, Paris, France, to appear
Enseigner l'algorithme pour quoi ? Quelles nouvelles questions pour les mathématiques ? Quels apports pour l'apprentissage de la preuve
- S Modeste
Modeste S (2012) Enseigner l'algorithme pour quoi ? Quelles nouvelles questions
pour les mathématiques ? Quels apports pour l'apprentissage de la preuve ? PhD
thesis, Université de Grenoble
Modelling algorithmic thinking : the fundamental notion of problem
- S Modeste
Modeste S (2013) Modelling algorithmic thinking : the fundamental notion of problem. In: proceedings of CERME 8, Antalya (Turkey)
L'expérimentation en mathématiques
- D Perrin
Perrin D (2007) L'expérimentation en mathématiques. Petit x 73:6-34
Theories of programming languages
- W V Quine
Quine WV (1950) Methods of Logic. Harvard University Press
Reynolds JC (1998) Theories of programming languages. Cambridge University
Press
Sedgewick R, Flajolet P (2013) An Introduction to the Analysis of Algorithms. Pearson Education
Corps Et Modèles: Essai Sur l'Histoire de l'Algèbre Réelle
- H Sinaceur
Sinaceur H (1991a) Corps Et Modèles: Essai Sur l'Histoire de l'Algèbre Réelle.
Vrin
Sinaceur H (1991b) Logique : mathématique ordinaire ou épistémologie effective ?
In: Hommage à Jean-Toussaint Desanti, Trans-Europ-Repress
The concept of truth in the languages of the deductive sciences. Prace Towarzystwa Naukowego Warszawskiego
- A Tarski
Tarski A (1933) The concept of truth in the languages of the deductive sciences. Prace Towarzystwa Naukowego Warszawskiego, Wydzial III Nauk
Matematyczno-Fizycznych 34(13):172-198, english translation in Tarski (1983)