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Modelling
in Maths300

 

Charles Lovitt

How important is the concept of mathematical modelling to student learning?
It could be argued that the true power of mathematics lies in its ability to model the real world. Today's weather forecast, for example, came from a mathematical model 'inside a computer'.

Students should arguably understand and appreciate the whole process of mathematical modelling - not just parts of it. Many traditional textbook questions do not fulfill this important requirement because, while they use real world contexts, the true purpose of the questions is to practise particular algorithmic skills rather than to engage students in the whole mathematical modelling process. As a result, the link to the context is lost.

Two typical textbook examples are:

  1. The cost of renting a car is given by the formula C = 60 + 0.35K.
    • If K = 270, find C.
    • If C = 200, find k.
    The purpose of the first part of this question is to practise substitution; the purpose of the second part is to practise solving linear equations. What is clear however is that students learn little about the car rental industry. In other words the choice of a so-called real context is spurious.
  2. Sean is employed to mow the lawns in the local council gardens. He can mow 0.5 hectares per hour - how long will it take to mow the 3.5 hectares of the council lawns?
    The purpose of this question is to practise division of fractions and decimals. The student learns nothing about the role of councils in caring for their gardens, or the role of mathematics in that process. Again, the link to context is spurious.
This could and should be different in mathematics lessons and resources if we are to improve student learning. A better approach is to focus on the modelling process itself and in so doing students learn about the hugely important role that mathematics plays in solving real world problems.

There are many schematic versions of the modelling process. I particularly like this diagram which is a simple and elegant representation of the classic mathematical modelling process.

In the diagram, the scenario starts in the real world with some meaningful context. For example, a person is hiring a rental car. It will cost them $60 a day, including 100 km and 15c per km for every kilometres over 100 km.

Arrow 1 represents the process of creating a mathematical model or equation of this scenario, that is, going from the real scenario into the world of mathematics.

Mathematical Modelling Diagram

For this problem the mathematician develops the formula C = 60 + 0.15 (K - 100) where C is the cost (in $) and K is the number of kilometres travelled (and K>100). This formula is the mathematical model.

Arrow 2 represents the process of solving problems in the mathematical world using this formula. For example, typical textbook exercises might ask:

  • If K = 348, find C.
  • If C = 100, find K.
Using the context, the first of these questions asks, If I travel 348 km, what will it cost? - and the second asks, If I have $100, how far can I travel?. Answering the first question involves using substitution, while answering the second involves solving a linear equation; students have usually been drilled in both of these procedures. The danger here is that by focusing only on the 'drilled algorithms' the connection to the context can easily be lost.

Arrow 3 therefore represents the process of translating the mathematical answer back to the original scenario.

  • The answer to the first question was C = 97.2. This translates back to... it will cost you $97.20 to travel 348 km.
  • The answer to the second question meanwhile is K = 366.6666. This means that... for $100 you can travel 366 km.
Finally, Arrow 4 represents the process of checking that the answer(s) makes sense. For example, suppose the mathematician wrote the original equation as C = 60 + 15 (K - 100), the answer to the first question would be C = 3780. What!, says the customer incredulously, $3,780 dollars!. This leads the mathematician to check the model which exposes the missing decimal point. The point of this is that the stages above are cyclic - the model is always being continuously checked and if necessary refined.

In our mathematics courses we spend most of our time and energy on 'Arrow 2', i.e. practising the algorithms, and not enough on the whole mathematical modelling process. There are several lessons in Maths300 that highlight either the whole process or aspects other than 'Arrow 2'. For example, Speed Graphs (Lesson 147) develops a personal mathematical model of students walking and jogging, and highlights the predictive power of the model which students can then check physically.

Another great lesson is Baby in the Car (Lesson 111). In this lesson the major outcome is understanding the context. It is through the modelling process that the mathematics content (surface area to volume ratios) develops this understanding. The context is the important aspect, while the mathematics content is merely a vehicle to get there. There are other lessons with a similar structure to this such as Radioactivity (Lesson 7) and Chocolate Chip Cookies (Lesson 58).

Yet another lesson is Backtracking (Lesson 19). This lesson is quite remarkable in that it enables students to handle very complex linear equations with ease. The success of this lesson lies primarily in the fact that the energy of the lesson is in 'Arrow 1' which reinforces the connection between the mathematical model and the context that it represents, rather than a particular technique, algorithm or procedure which is 'Arrow 2'.

I have often had cause to reflect on why this lesson is so successful. I think it is because all the effort is in students telling me what happened to the number - once they see the algebra as a story, the rest (finding the starting number) becomes almost trivial. There is absolutely no mention of technical words like flowcharts, inverse operations, opposites etc. - in fact the introduction of such words (at this early stage) seems to be a definite negative.

Another forthcoming lesson which is successful for the same reason is Goods and Bads. The point of these two lessons is that success can be greatly supported by keeping the connection to the context ('Arrow 1') alive.

 

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