Monday, 2 May 2011

Modelling at the core

The process of applying mathematics to solving a real-life problem can be analysed in terms of the four steps of the modelling process:

1. Set up a mathematical model of the problem.

2. Find the solution to the mathematical problem.

3. Interpret the mathematical solution back in the real world context.

4. Check the solution against the constraints and realities of the original real life problem.

Of course, in practice, when we are solving problems we do not necessarily go through these steps precisely, one at a time in sequence. We often go backwards and forwards between the real world and the world of mathematical symbols. But these four elements will always be there when we are applying mathematics to a real-life context.

Here's a simple illustration.

Anglian Water is calculating how much our monthly direct debit for water and sewage should be. We are currently paying £31 a month. We will pay this for a further two months, before the new monthly payment begins. Their estimate for charges for the next 12 months is in total £564.96. And we carry over a debit of £56.36 from the last six months.

What should be the new monthly charge? That’s the real-life problem.

Step 1 is to set up the mathematical model, which is done as follows:

Add the estimate for 12 months charges (£564.96) to the existing debit (£56.36), subtract the expected credit for the next two months (£31 multiplied by 2), and divide the result by 12. This gives the following calculation:

[564.96 + 56.36 – (31 × 2)] ÷ 12.

Step 2 is to solve the mathematical problem: in other words, do the calculation:

[564.96 + 56.36 – (31 × 2)] ÷ 12

= [621.32 – 62] ÷ 12

= 559.31 ÷ 12

= 46.6091667 (on a calculator).

So, the mathematical solution is 46.6091667.

Step 3 is to interpret this back in the real world. In this case the number read from the calculator represents a monthly payment of £46.6091667.

Step 4 is to check this against reality. Clearly, they can’t set up a monthly debit for £46.6091667. £46.61 looks a possibility. But, in fact, the company’s policy is to round the result to the nearest pound, which is reasonable enough given that the original data was based on an estimate obtained from recent meter readings, which would not justify any greater degree of accuracy. So, the new monthly payment will be £47.

All four steps in this process are important and involve significant reasoning. But the way some politicians and journalists talk you would think that the only step that matters is Step 2, doing the calculation! As if this step alone is the real mathematics. The interesting thing here is that Step 2 is the one part of the process that can be handed over to an inexpensive piece of electronic equipment (i.e. a calculator) – whereas all the other steps require genuine mathematical reasoning.

When Michael Gove talks about teaching the ‘core arithmetical functions’ in primary mathematics, I fear that he is thinking only of Step 2 (done without a calculator): which is, of course, the one step of the process that in practice any sane person would do on a calculator. My argument is that the whole process – all four steps – constitutes what should be at the core of learning about number operations in the primary school.

Knowing what calculation to do is in practice actually more important than being able to do the calculation. Being able to interpret the answer on a calculator is crucial mathematical thinking. Checking your solution against the constraints of reality is genuine mathematics.

1 comment:

  1. Dear Derek Haylock

    I attended your course 'Language and Low Attainment in Mathematics' - at the University of East Anglia in 2003, since then I have been involved in helping pupils develop better skills with number and arithmetic. What you said back then is still very much the case in schools today for this group of children. You may recall we had a chat about Dr Catherine Stern. Subsequent to this I went to New York to attend a Stern training course and then a year later set up a small company to bring Stern back to the UK.

    The range of manipulatives and her teaching approach is a wonderful way for young children to learn about number and arithmetic because it follows a child's natural stages of learning and development namely the sensory-motor period. Further, deeply embedded in the Stern pedagogy is the stimulation of cognitive functions. Deficits in processing systems are a major cause as to lack of progress. I am pleased to report our findings, that once a child begins to follow the Stern programme and concrete teaching, progress and cognitive development is evident on one single term.

    The factors associated with low attainment, as you suggest, are dealt with and overcome through the visual, auditory and kinaesthetic input (VAK) and small-step progression in delivery, which in itself is natural differentiation.

    The latest government figures indicate there are almost 2 million children with SEN or a disability, many would have difficulties with maths. I believe Stern's approach has its place in Early Years and Foundation level maths enabling children to put the crucial building blocks of arithmetic in place, covering all four steps in a simple logical manner. So my question to you is would you be interested in finding out, or refreshing your knowledge about this programme? If so, I would be happy to demonstrate its effectiveness. I also note from the books you have published, you have made references to Stern and have used Stern blocks in illustrations of various concepts.

    I do hope, as someone who is still very passionate about mathematics; who recognised Stern's work back then, that you are curious to find out more and I look forward to you getting in touch.

    Best wishes
    Vikki Horner - Numeracy Advisor-SEN
    Maths Extra 01747 861 503