Mathematical modelling and discrete mathematics: opportunities for modern mathematics teaching

Abstract

Discrete mathematics and mathematical modelling, along with the educational discourse surrounding these, have many connections. However, ways that the educational discourse on discrete mathematics can benefit from the inclusion of examples of mathematical modelling and the accompanying discussion are currently under-researched. In this paper, we elaborate on the educational potential of examples of mathematical modelling based on the usage of methods from discrete mathematics, with a focus on secondary education. We first describe vertex-edge graphs as possible topics of discrete mathematics that are accessible at school level within modelling lessons. Secondly, in the context of a case study, we describe modelling activities with students at the end of lower-secondary education, using a classical problem of discrete mathematics originating from the Königsberg bridge problem. The students’ solution processes for this optimisation problem based on graph theory are described. Their approaches are examined referring to the phases of the modelling cycle, using the method of qualitative content analysis. We studied in particular the extent to which students use concepts related to vertex-edge graphs in specific sub-phases of the modelling process. The analysis allows the required sub-competences of modelling to be identified and the connection of these competences with discrete mathematics to be worked out. On the basis of this analysis, educational opportunities of teaching discrete mathematics and mathematical modelling are assessed. Overall, we point out the possibilities and opportunities for using examples from the field of discrete mathematics to acquire modelling competences and to foster the linkage of mathematical modelling and discrete mathematics at school level.

Teaching and learning discrete mathematics

“…In England, students focusing on mathematics can take a special track in which they have extensive exposure to discrete mathematics. In many countries, it appears that the opportunities for dealing with discrete mathematics in schools, especially when it goes beyond combinatorics, are often only seen on the level of optional recreational mathematics (Colipan & Liendo, 2022;Gravier & Ouvrier-Buffet, 2022;Greefrath et al, 2022).…”

“…English (1991English ( , 1993 reports on young children's strategies as they engage with combinatorial problems. As another example, students can naturally `invent' graph theory to solve a problem (Ferrarello & Mammana, 2018;Greefrath et al, 2022;van den Heuvel & Krabbendam, 1991). We can only wonder how much better would the students' work be if they already knew some graph theory or had previous experience with counting or recursive problems?…”

“…In this way, discrete examples could also be used to counter the difficulties in understanding nonlinear multiplicative problems (Tillema & Gatza, 2016). Greefrath et al (2022) deal with students thinking about real connections using graph theory, which the students came up with on their own. So, engaging with this context can promote modeling competence (Greefrath & Vorhölter, 2016).…”

“…However, there are only a few studies in this area so far (Ferrarello & Mammana, 2018;Ferrarello et al, 2022;Medová et al, 2019). Greefrath et al (2022) shows how, because of the ease of accessibility of graph theory, students can develop models related to their world and address issues of optimization.…”

Mathematical Modelling: A Retrospective Overview

“…These studies focused on the effects of practices based on MM skills on students' conceptual learning (Blomhøj & Jensen, 2003;Cetinkaya, Kertil, Erbaş, Korkmaz, Alacacı & Cakıroğlu, 2016;Çavuş-Erdem & Gürbüz, 2018;Galbraith, 2012;Greefrath et al, 2022;Haines & Crouch, 2010;Kaiser & Sriraman, 2006;Kertil, Erbas & Cetinkaya, 2017;Lesh & Lehrer, 2003;Maaβ, 2006;Michelsen, 2006;Verschaffel, Greer & De Corte, 2002;Xu et al, 2022). In addition to these, success based on MM competencies (Bukova-Güzel & Uğurel, 2010;Hıdıroğlu, Tekin-Dede, Kula-Ünver & Bukova-Güzel, 2017;Kandemir, 2011;Kaya & Keşan, 2022;Sokolowski, 2015), high-level thinking skills and problem-solving (Didis, Erbas, Cetinkaya, Cakıroğlu & Alacacı, 2016;English, Fox & Watters, 2005;Kim & Kim, 2010;Lesh & Harel, 2003;Niss & Blum, 2020;Niss & Højgaard, 2019;Schukajlow, Kolter & Blum, 2015) creativity, awareness, performance, and motivation (Akgün, Çiltaş, Deniz, Çiftçi & Işık, 2013;Bora & Ahmed, 2019;Bukova-Güzel, 2011;Daher, 2021;Lu & Kaiser, 2021;Novak, Daday & McDaniel, 2018;Sriraman, 2009;Wessels, 2014;Zbiek & Conner, 2006 ) and other topics were also discussed.…”