Mathematical Modelling and Applications

Part of the International Perspectives on the Teaching and Learning of Mathematical Modelling book series, this compendium of recent research in mathematics education is specific to the teaching and learning of mathematical modeling. Members of the International Community of Teachers of Mathematical Modelling contributed the content. While the articles in this volume cover all levels of education from the early years to tertiary education, the greatest focus is on the secondary level. Most articles here offer largely disappointing observations on the capability of secondary education teachers to present modeling. For example, in “Mathematical Modelling as a Professional Activity: Lessons for the Classroom”:

While mathematical modelling has been described as “the most important educational interface between mathematics and industry” (Li 2013, p. 51), there are indications, however, that it is not emphasized in current teaching practices at upper secondary school (e.g. the preface in Stillman et al. 2015) nor is the coordination between school and working life strong enough…

This volume suggests, by touching on it in at least five articles, that the lack of preparation of pre-service teachers is a root cause of this deficiency. For example, in the Conclusion to “Mathematical Modelling Strategies and Attitudes of Third Year Pre-service Teachers”:

The inquiry revealed that it was not only a very challenging task for the participants, but also it was indeed very difficult for them to link the 'world out there' (reality) to the mathematics of the classroom. The real dilemma was captured in their search to find appropriate mathematics (mathematisation) to solve the problem.

Regardless of where the fault lies, this is identified as a threat to our future engineers (“The Mathematical Modelling Competencies Required for Solving Engineering Statics Assignments”). Excessive linearity appears as a danger to our crop of future economists: “Students’ Overreliance on Linearity in Economic Applications: A State of the Art”. There are some suggestions that such inadequacies being detected can at least partly be blamed on shifting policies. In “Implementing Mathematical Modelling: The Challenge of Teacher Educating”:

The expectation that teachers will help school learners develop mathematical modelling skills has gained visibility in the United States with the adoption of Common Core State Standards of Mathematics… Arguably, one of the least understood expectations among the set of CCSSM Standards (Gould 2013), and one of the most conceptually demanding domains of knowledge to nurture due to its complexity (Meyers 1984), mathematics teacher educators face the challenge of preparing teachers to first develop an understanding of the intricacies of mathematical modelling and then helping them define ways to implement it effectively.

Beyond examining educator preparation in this area, some contemporarily relevant teaching techniques get examined as in “Enabling Anticipation Through Visualisation in Mathematising Real-World Problems in a Flipped Classroom”:

Despite mixed results on effectiveness in secondary mathematics classrooms, on balance, the flipped classroom model appears to have potential that is worthy of further development and research as a means to leverage more time in the classroom for engaging in richer experiences that give more than lip service to mathematical modelling activity…

When not focused on teachers or techniques, students themselves are investigated. Source material includes detailed transcripts of conversations with students and case studies of individual students’ work as in “Long-Term Development of How Students Interpret a Model: Complementarity of Contexts and Mathematics”. This is one of the multiple studies here touching on and suggesting open-ended Fermi problems such as this one linked to a supporting video:

There are about 60 million people in the U. K.

About how many school teachers do we need?

Further on the subject from the Conclusions in “Design and Implementation of a Tool for Analysing Student Products When They Solve Fermi Problems”:

On the basis of our analytical results, we can state that Fermi problems require students to elaborate models with a high level of detail.

This international content comes from, among other places, Holland, Brazil, France, Spain, Chile, Germany, and Japan. From Japan there are positive results to report in “Experimental lessons with the Night-Time problem for Japanese 10th grade students…” This Night-Time Problem is a fairly open-ended problem in determining night length at a given latitude from a globe and current conditions with intriguing reuse possibilities. Also ready to adapt as a classroom capsule is “the design of a teaching sequence based on an archaeological context — the ruins of a Roman theatre discovered in Badalona (Catalonia) — implemented with 12–14-year-old students”. This has a multidisciplinary dimension including history, etc. A similar blend of areas occurs between history of mathematics and modeling in the ready lecture that is “The Velocity Concept: The History of Its Modelling Development”. Additional ideas ready to implement include exiting the classroom for engaging math walks and building a simple apparatus for L'Hôpital's pulley problem, one of early calculus problems with links to physics.

Tom Schulte is a mathematics instructor living in Mandeville, Louisiana.