Skip to main content
SpringerLink
Log in

The mathematical background of proving processes in discrete optimization—exemplification with Research Situations for the Classroom

  • Original Paper
  • Published:
ZDM – Mathematics Education Aims and scope Submit manuscript

Abstract

Discrete mathematics brings interesting problems for teaching and learning proof, with accessible objects such as integers (arithmetic), graphs (modeling, order) or polyominoes (geometry). Many problems that are still open can be explained to a large public. The objects can be manipulated by simple dynamic operations (removing, adding, ‘gluing’, contracting, splitting, decomposing, etc.). All these operations can be seen as tools for proving. In this paper we particularly explore the field of ‘discrete optimization’. A theoretical background is defined by taking two main axes into account, namely, the epistemological analysis of discrete problems studied by contemporary researchers in discrete optimization and the design of adidactical situations for classrooms in the frame of the Theory of Didactical Situations. Two problems coming from ongoing research in discrete optimization (the Pentamino Exclusion and the Eight Queens problems) are developed and transposed for the classroom. They underscore the learning potentialities of discrete mathematics and epistemological obstacles concerning proving processes. They emphasize the understanding of a necessary condition and a sufficient condition and problematize the difference between optimal and optimum. They provide proofs involving partitioning strategies, greedy algorithms but also primal–dual methods leading to the concept of duality. The way such problems can be implemented in the classroom is described in a collaborative study by mathematicians and mathematics education researchers (Maths à Modeler Research Federation) through the Research Situations for the Classroom project.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

Notes

  1. Discrete optimization is a branch of optimization in applied mathematics and computer science that deals mainly with problems where one has to construct an optimal solution from a finite (or countable) number of possibilities.

  2. That is, “nondeterministic polynomial-time complete” (See Garey & Johnson, 1979).

  3. Their proofs are distinct, whereas in continuous optimization problems, their resolutions are often put together in an asymptotic argument.

  4. We can point out that the word duality has the same meaning in geometry, algebra and combinatorics: it is linked to the notions of sphere covering, generating set and transversal on the one hand, and sphere packing, independent set and matching on the other hand.

  5. Packing and covering problems are existence problems for which we know effective procedures to get feasible solutions (see all the presented techniques for Pexistence).

  6. The word heuristic is often used in mathematics education as Pólya (1954) defined it. In problem-solving, it includes, for instance, analogy, generalization, induction, decomposing and recombining, working backward. We will keep in mind the definition of heuristic coming from operations research, which is non-contradictory with that of Pólya: “In general, for a given problem, a heuristic procedure is a collection of rules or steps that guide one to a solution that may or may not be the best (optimal) solution” (Laguna & Marti, 2013).

  7. Sometimes the heuristic gives a minimal or a maximal solution (as described in Fireworks, Sect. 6).

References

  • Alcock, L., & Simpson, A. (2009). Ideas from mathematics education. An introduction for mathematicians. The Higher Education Academy.

    Book  Google Scholar 

  • Arsac, G. & Mante, M. (2007). Les pratiques du problème ouvert. SCEREN-CRDP.

  • Artigue, M. (2014). Didactic engineering in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education. Springer.

    Google Scholar 

  • Artigue, M., & Blomhøj, M. (2013). Conceptualizing inquiry-based education in mathematics. Zdm–mathematics Education, 45(6), 797–810.

    Article  Google Scholar 

  • Balacheff, N. (2010). Bridging knowing and proving in math: A didactical perspective. In G. Brousseau (Ed.), The theory of didactical situations in mathematics. Kluwer Academic Publishers. 1997.

    Google Scholar 

  • Bezzel, M. (1848). Schachfreund. Berliner Schachzeitung, 3,363.

  • Bondy, J. A., & Murty, J. S. R. (2008). Graph theory. Springer.

    Book  Google Scholar 

  • Brousseau, G. (1997). Theory of the Didactical Situations in Mathematics. Kluwer Academic Publishers.

  • Campbell, P. J. (1977). Gauss and the eight queens problem: A study in miniature of the propagation of historical error. Historia Mathematica, 4, 397–404.

    Article  Google Scholar 

  • Coffin, F., Dupraz, M., Gravier, S., Manin, S., & Payan, C. (2006). Maths à Modeler: Situations-recherche pour l’enseignement des mathématiques auprès d’enfants présentant des troubles psychopathologiques. In N. Bednarz & C. Mary (Eds.), L’enseignement des mathématiques face aux défis de l’école et des communautés. Editions du CRP.

    Google Scholar 

  • Colipan, X. (2014). Etude didactique des situations de recherche pour la classe concernant des jeux combinatoires de type Nim. PhD Thesis, Grenoble University. Retrieved from https://tel.archives-ouvertes.fr/tel-01121726. Accessed on 12 Jul 2022.

  • Deloustal-Jorrand, V. (2004). L’implication mathématique: étude épistémologique et didactique. PhD Thesis, Université J. Fourier, Grenoble. Retrieved from https://hal.archives-ouvertes.fr/hal-00990938. Accessed on 12 Jul 2022.

  • Dijkstra, E. W. (1959). A note on two problems in connexion with graphs. Numerische Mathematik, 1(1), 269–271.

    Article  Google Scholar 

  • Dissa, S. (2020). Entre arithmétique et géométrie discrète—Une étude épistémologique et didactique autour du théorème de Bézout et du théorème de Pick. PhD Thesis, Grenoble Alps University.

  • Dorbec, P. (2007). Empilements et recouvrements. Ph.D. Thesis, Grenoble University. Retrieved from: http://tel.archives-ouvertes.fr/tel-00181722/fr/. Accessed on 12 Jul 2022.

  • Gandit, M., Modeste, S., Gravier, S., Balicco, M.-P. (2014). Les caisses de dynamite, un atelier de recherche Maths à Modeler. Atelier Journées de Chamonix ‘Jouer ou apprendre’, 49–58. https://halshs.archives-ouvertes.fr/halshs-02021307. Accessed on 12 Jul 2022.

  • Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-Completeness. Freeman.

    Google Scholar 

  • Giroud, N. (2011). Etude de la démarche expérimentale dans les situations de recherche pour la classe. PhD Thesis, Grenoble University. https://hal.archives-ouvertes.fr/tel-00649159/. Accessed on 12 Jul 2022.

  • Godot, K. (2005). Situations recherche et jeux mathématiques pour la formation et la vulgarisation. PhD Thesis, Grenoble University. https://tel.archives-ouvertes.fr/tel-00102171. Accessed on 12 Jul 2022.

  • Goemans, M. X., & Williamson, D. P. (1996). The primal-dual method for approximation algorithms and its application to network design problems. In D. S. Hochbaum (Ed.), Approximation algorithms for NP-hard problems (pp. 144–191). PWS Publishing Co.

    Google Scholar 

  • Goldin, G. (2010). Problem solving heuristics, affect, and discrete mathematics: A representational discussion. In B. Sriraman & L. English (Eds.), Theories of mathematics education: Seeking new frontiers (pp. 241–250). Springer.

    Chapter  Google Scholar 

  • Golomb, S. W. (1966). Polyominoes. Allen & Unwin.

    Google Scholar 

  • Gomory, R. (1960). Solving linear programming problems in integers. In R. E. Bellman & M. Hall (Eds.), Combinatorial analysis (pp. 211–216). AMS.

    Chapter  Google Scholar 

  • Gravier, S., Moncel, J., & Payan, C. (2007). A generalization of the pentomino exclusion problem: Dislocation of graphs. Discrete Mathematics, 307(3–5), 435–444.

    Article  Google Scholar 

  • Grenier, D. & Payan, C. (1999). Discrete mathematics in relation to learning and teaching proof and modelling. In I. Schwank (Ed.), Proceedings of CERME 1, Vol. 1 (pp. 143–155). Osnabrück.

  • Grenier, D., & Payan, C. (1998). Spécificités de la preuve et de la modélisation en mathématiques discrètes. Recherches En Didactique Des Mathématiques, 18(1), 59–100.

    Google Scholar 

  • Grenier, D., & Payan, C. (2003). Situations de recherche en “classe”, essai de caractérisation et proposition de modélisation. In V. Durand-Guerrier & C. Tisseron (Eds.), Actes du Séminaire National de didactique des mathématiques (pp. 189–205). IREM de Paris 7.

    Google Scholar 

  • Gravier, S., & Payan, C. (2001). On the pentomino exclusion problem. Discrete and Computational Geometry, 26, 375–386.

    Article  Google Scholar 

  • Hanna, G., & De Villiers, M. (Eds.). (2012). Proof and proving in mathematics education. Springer.

    Google Scholar 

  • Hanna, G., Jahnke, H. N., & Pulte, H. (Eds.). (2010). Explanation and proof in mathematics: Philosophical and educational perspectives. Springer.

    Google Scholar 

  • Hart, E., & Sandefur, J. (Eds.). (2018). Teaching and learning discrete mathematics worldwide: Curriculum and research. Springer.

    Google Scholar 

  • Heinze, A. Anderson, I. & Reiss, K. (Eds.)(2004). Proof and discrete mathematics. ZDM–Mathematics Education, 36(2&3).

  • Kartak, V., & Kabarisova, A. (2019). An integer programming approach to the irregular polyomino tiling problem. In I. Bykadorov, V. Strusevich, & T. Tchemisova (Eds.), MOTOR 2019. Communications in computer and information science (Vol. 1090, pp. 235–243). Cham: Springer.

    Google Scholar 

  • Laguna, M., & Marti, R. (2013). Heuristics. In S. I. Gass & M. C. Fu (Eds.), Encyclopedia of operations research and management science. Springer.

    Google Scholar 

  • Lampert, M. (1990). When the problem is not the question and the solution is not the answer. Mathematical Knowing and Teaching American Educational Research Journal, 27(1), 29–63.

    Google Scholar 

  • Lockwood, E., Ellis, A., & Lynch, A. (2016). Mathematicians’ example-related activity when exploring and proving conjectures. IJRUME, 2(2), 165–196.

    Google Scholar 

  • Lucas, E. (1882). Récréation n°4: Le problème des huit reines au jeu des échecs. Récréations mathématiques (Vol. 1). Gauthier Villars.

    Google Scholar 

  • Maher, C. A., Powell, A. B., & Uptegrove, E. B. (Eds.). (2010). Combinatorics and reasoning. Representing, justifying and building isomorphisms. Springer.

    Google Scholar 

  • Mariotti, M. A., Durand-Guerrier, V., & Stylianides, G. J. (2018). Argumentation and proof. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven (Eds.), Developing research in mathematics education—Twenty years of communication, cooperation and collaboration in Europe (pp. 75–89). Taylor & Francis.

    Google Scholar 

  • Modeste S. (2012). Enseigner l’algorithme pour quoi ? Quelles nouvelles questions pour les mathématiques ? Quels apports pour l’apprentissage de la preuve ? PhD Thesis, Grenoble University. Retrieved from https://tel.archives-ouvertes.fr/tel-00783294. Accessed on 12 Jul 2022.

  • Ouvrier-Buffet, C. (2003). Construction de définitions / construction de concept : vers une situation fondamentale pour la construction de définitions en mathématiques. PhD Thesis, Grenoble University. Retrieved from https://tel.archives-ouvertes.fr/tel-00005515. Accessed on 12 Jul 2022.

  • Ouvrier-Buffet, C. (2009). Maths à Modeler: Research-situations for teaching mathematics. In E. Barbeau & P. Taylor (Eds.), ICMI Study 16, Challenging mathematics in and beyond the classroom (pp. 192–198). Springer.

    Google Scholar 

  • Ouvrier-Buffet, C. (2011). A mathematical experience involving defining processes: In-action definitions and zero-definitions. Educational Studies in Mathematics, 76(2), 165–182.

    Article  Google Scholar 

  • Ouvrier-Buffet, C. (2015). A Model of mathematicians' approach to the defining processes. In K. Krainer & N. Vondrová (Eds.), Proceedings of CERME9 (pp. 2214–2220). Charles University, Faculty of Education and ERME

  • Ouvrier-Buffet, C. (2020). Discrete mathematics teaching and learning. In S. Lerman (Ed.), Encyclopedia of mathematics education. Cham: Springer.

    Google Scholar 

  • Ouvrier-Buffet, C. (2021). Exploring proving processes 1130 in elementary schools—The potentialities of polyominoes and Research Situations for the Classroom. HAL  https://hal.archives-ouvertes.fr/hal-03499721. Accessed on 12 Jul 2022.

  • Ouvrier-Buffet, C., Alvès, M., & Acker, C. (2017). La chasse à la bête—une situation recherche pour la classe. Grand N, 100, 5–31.

    Google Scholar 

  • Polyá, G. (1954). Mathematics and plausible reasoning. Princeton University Press.

    Book  Google Scholar 

  • Rosenstein, J. G. (2018). The absence of discrete mathematics in primary and secondary education in the United States… and why that is counterproductive. In E. Hart & J. Sandefur (Eds.), Teaching and learning discrete mathematics worldwide: Curriculum and research (pp. 21–40). Springer.

    Chapter  Google Scholar 

  • Rosenstein, J. G., Franzblau, D. S., & Roberts, F. S. (Eds.). (1997). Discrete mathematics in the schools. DIMACS Series in Discrete Mathematics & Theoretical Computer Science (Vol. 36). AMS & NCTM.

    Google Scholar 

  • Simkin, M. (2021). The number of n-queens configurations. The Harvard Gazette. arXiv:2107.13460v2. https://doi.org/10.48550/arXiv.2107.13460

  • Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289–321.

    Google Scholar 

  • Stylianides, A. J. (2016). Proving in the elementary mathematics classroom. Oxford University Press.

    Book  Google Scholar 

  • Weber, K. (2011). Why and how mathematicians read proofs: An exploratory study. Educational Studies in Mathematics, 76(3), 329–344.

    Article  Google Scholar 

  • Welsh, D. J. A. (2010). Matroid theory. Courier Corporation.

    Google Scholar 

  • Yackel, E., & Cobb, P. (1996). Socio-mathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27(4), 458–477.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Fourier, FED Maths à Modeler, CNRS/Grenoble Alps University, Grenoble, France

    Sylvain Gravier

  2. Laboratoire de Didactique André Revuz, Paris-Est Créteil University, Paris, France

    Cécile Ouvrier-Buffet

Authors
  1. Sylvain Gravier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cécile Ouvrier-Buffet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cécile Ouvrier-Buffet.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gravier, S., Ouvrier-Buffet, C. The mathematical background of proving processes in discrete optimization—exemplification with Research Situations for the Classroom. ZDM Mathematics Education 54, 925–940 (2022). https://doi.org/10.1007/s11858-022-01400-3

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11858-022-01400-3

Keywords

Search

Navigation