Challenging students: Real-world Mathematics

Each year the International Mathematical Modeling Challenge (IM2C) gives students an opportunity to team up and work collaboratively on a complex modelling task which connects the mathematical learning they’re experiencing in class with a real-world situation. Once they receive the task, teams have just five days to write and submit a report.

In this article, Ross Turner – who leads Australia’s involvement in the IM2C – unpacks the details of the problem presented to participating students this year.

The 2020 problem for the International Mathematical Modeling Challenge (IM2C) asked teams to review data on goods to be offered during a ‘flash sale’ in order to identify which sale items would likely be most popular and the store layout factors that might affect damage risks during a potentially frenzied sale event.

Teams had to develop and use a model that would predict damage to goods in order to recommend optimal product placement and department locations for a given store layout. Teams were also asked to create and evaluate a new and better floor plan for the flash sale scenario, and to write a one-page letter to the store manager presenting and supporting their findings.

Australian teams entering this year’s IM2C chose five consecutive days in late March to work on the ‘flash sale’ problem. Despite the pandemic-related disruption to schools here in Australia towards the end of Term 1, 102 teams submitted a report to the judging panel before the closure of submissions.

The judges identified 14 teams as ‘national finalists’ that had produced reports strong enough for very close consideration and, from these, six team reports were given higher awards, including the two entries that became Australia’s entries in the international phase of the IM2C. Those teams were from North Sydney Boys High School (in New South Wales) and Caulfield Grammar School (in Victoria). The full list of national finalists and teams receiving higher awards can be found here.

As in previous years, students from all levels of secondary schooling from Year 7 to Year 12 participated in teams, and produced some very impressive work. The proportion of boys participating was slightly higher than for girls. One or more teams from each Australian state or territory submitted entries.

Observations from the Australian judging panel

The Australian judging panel focused on the following modelling elements as they judged the team entries:

Problem definition

The team’s efforts at defining the actual problem they intended to solve as they interpreted the problem statement, and how they interpreted the mathematical opportunities. This might have included identifying any assumptions they would make to clarify and simplify the situation and to focus their efforts.

Model formulation

Teams needed to go beyond a purely qualitative analysis of the main variables (for example, damage risk and item popularity) as well as matters such as logistics, and marketing. To make it through to final judging, teams had to show some degree of mathematical formulation of at least part of the problem situation.

Data on several variables was provided in the problem statement, and additional data was sourced by some teams, on such matters as annual sales and sales revenue.

Models for ‘product desirability’ (mathematising related to part 1b of the question statement) were developed by many teams. These used information about one or more of product price, size of discount, customer rating, quantity available, and product brand (all given as part of the problem statement).

The reasoning and justification used is key to how the approach was evaluated by the judging panel. The way in which different kinds of variables, measured in different units and scales, were mathematically combined into an index of desirability (or similar) was a critical factor in evaluation of the model proposed. Many teams combined quite different data types without sufficient justification.

A central issue in evaluating team reports was determining how effectively different component models were combined into a well-argued and quantitatively defined decision-making model for placement of products, both in the given shop layout, and in an alternative layout. This was the thrust of part 2a of the problem statement.

Mathematical processing

Teams were required to show how the model(s) they had developed would inform the assignment of items for sale to the available display and sales locations (related to parts 2a, b and c of the question statement). The main question here is whether the models developed were applied in analysing the given store layout to devise a plan for placement of products in the store.

The culmination of the modelling exercise was for teams to identify shortcomings in their application of the model(s) to the given floor plan, and to devise a modified floor plan, to explain how their model(s) would be used to design and populate the floor plan, and to explain the ways in which this floor plan would be superior (part 2d of the problem statement).

Model evaluation

This can be one of the most challenging and critical aspects of good modelling practice. In relation to model evaluation, the following matters were particularly relevant:

  • Were arguments advanced about the shortcomings of the given floor plan?
  • Were mathematically supported arguments about the superiority of the alternative floor plan proposed?
  • Was there any evidence of sensitivity analysis? That is, would changing any of the variable used have an effect on the team’s solution?

Report quality

The final set of assessment criteria focus on the quality of the report itself. The following are important elements of that:

  • Have teams produced a readable report, or is it more like a school assignment?
  • Is the essential technical material presented in the body of the report, with the remainder and additional details in an appendix?
  • Does the summary give a good overview of approach, method and results?
  • Does the letter to the store manager communicate effectively with explanation, to the store manager?

If your students have not yet had a go at this modelling problem, they might like to try it. Strengthening the connections between the mathematical knowledge students are acquiring through their maths class work is an outcome we would all see as highly desirable. Working with a mathematical modelling perspective on problems that arise in the ‘real world’ provides an excellent way of strengthening those connections. In addition, the team collaboration provided by the IM2C approach provides another suite of tangible benefits to our students.

Teachers can download a guide from the IM2C website, which provides example problems and practice activities. To see how teams from across the globe fared in the international judging round, the results are now available on the International Mathematical Modeling Challenge website.