## Mathematical Modelling

Mathematical modelling is pretty well defined under MEI and can be found in Edexcel in the Stats work. There is a flow diagram which I will attempt to reproduce (eventually; you can see it within the OCR/MEI spec, such as P35 of here); meanwhile, in words:

In principle:

Make some assumptions relating to an identified topic for modelling.

Develop a mathematical model based on those assumptions

Develop the analysis to produce some results

Go compare those results against reality

Review your assumptions [= Refine the model]

In practice, there is a phase when the assumptions are made pretty fierce so that any analysis can take place within the scope of the modeller. For exam purposes (back when there was A-level coursework) one used to force students to make at least one cycle of adjusting assumptions.

Here’s an extended example showing the eventual thinking:

I want to model some tennis, since I’m under pressure to do some modelling and it s the summer and we are supposed to know about projectiles and this is projectiles I’m playing with in the week. It sounds like I’ll be allowed to go play tennis and call it maths, so good idea. Topic identified.

Projectiles at school means the ball is a particle, there is no air resistance, no spin, no Coriolis effect. Assumptions tick.

Oh. now what do I do? If I’m supposed to do any experiment, then (I do Physics) I need something repeatable.... I know, it’ll have to be a serve. Okay, I’ll model tennis serves.

Modelling: I can find the official dimensions of a tennis court. I can write a model for an ace serve. On the way I’ll have to make assumptions about my tennis, like how hard I hit it.

Cycle one: I discovered that my first guesses at speed of hitting the ball don’t get over the net. Apparently I must hit it harder than .....m/s from 2.4 m in the air immediately over the baseline or it won’t clear the net. Good question then would be [1] what speeds and angles permit a valid serve (it is ‘in’ play)?

Cycle two: Assumptions now include me hitting it hard enough. I went and hit some balls to discover a range of values at which I can hit the ball. Discovered that hitting it harder makes accuracy deteriorate and don’t know what to do with that. Possible area for exploration. Discovered (from looking at the work of others) that the sort of ball that will go past me (i.e. an ace in the sense I can’t react well enough to hit a response) is going faster than I can hit it. This then defines temporarily what I mean by ‘a better player’. Discovered that the precision with which I need to hit the ball, just above the net, is a problem and I’d like to think that’s a Stats problem not a Mechanics one.

Cycle Three: Thought about good players, prompted by Wimbledon coming soon. They hit it hard, like ... m/s. My modelling says they can’t get the ball to land at all ‘in’. Why is that? Sir says I should look at my assumptions (I lost two evenings of work trying to find my mistake). Eventually i realise that the pros make the ball land in because they spin the ball. Sir says there is evidence that air resistance is the greater effect (greater than gravity). Don’t get this.

Get the idea? For exam purposes the concept is achieved if the student explores all the routes on the flow chart (down one side, up the middle, down the same side as before) with no huge errors in the maths and uses maths of an appropriate standard or higher. From the point of higher learning, most students became interested in the problem outside/beyond the maths and so genuinely were modelling in the sense that the meaning of the results was not an academic-only issue. Usually two-thirds to three-quarters would be sufficiently stimulated that research went on outside lessons a they tried to find out what they wanted to know. Which is quite different from putting in extra hours bevcause of exams or deadlines. Students faced with a second investigation generally were far more efficient in their use of time but just as interested in the results.

I did this for Mechanics modules 1, 2 & 4 and for Stats 1 & 2 over perhaps ten years. I found myself in the position of demonstrating ideas to teachers at different schools (representing the Board) several days a year until the AQA decided that coursework was not working (succeeding in causing learning to occur) any more. In terms of learning (High-Performance Learning is a current buzz-phrase) it was brilliant, and the department I was a member of often looked at bringing investigations back into the curriculum even while they were excluded from exam content. The drive for results made that happen rarely, which i always thought to be a shame. perhaps we failed to express ourselves in the right vocabulary to garner support; it is probably a selling problem s much as an academic one.

The practical problems of handling coursework were many.

1.  Maths teachers in general are reluctant to do anything practical - it is well outside their comfort zone. I tried to encourage this to happen and those that came to experience a presentation went away less uncomfortable - but then they were not an unbiased sample in the first place.

2. It was far too easy to ‘cheat’ in the sense of finding answers among ones peers. This was sometimes unavoidable and so I tried using pairs and encouraging exploring different tasks or aspects of the same task.

3. Those chasing answers soon found that, if they could make a model happen on a spreadsheet, they got many more answers. few had the capacity to write stuff that worked, so what tended to happen was that the class would produce one that they all used, or they’d find one from a previous year. This was more than ten years ago; now, one would simply find a model on the internet and use it. That rather ruins the idea from the Board’s viewpoint, though it could still be turned into good learning. Following that idea right now, suppose students needed to prove that their model gave appropriate answers (so they can then trust the results); suppose that the range of possible values for each specific case generated analysis and consequence... this would then be demonstration of using a mathematical tool sensibly. It would mean that what is being measured is not any ability to do modelling but more to find a test a model. I guess that is deemed not to be Maths and therefore not included. In the sense in which cross-curriculum work should be encouraged (or what is school for?) it seems to me to be a valid exercise whether or not it falls entirely within “maths at school”.

DJS 20130526

Exercise:

Look up the dimensions of a tennis court. Assume you hit the ball 2.4 metres above the ground from exactly over the baseline. Find a minimum speed that puts a ball over the net. Now find  a minimum when you hit the ball at a given angle. Find the particular speed that hits the ball horizontally.

Explore the special cases that result in the ball clearing the net and arriving in the last few cm of the service area (i.e. just ‘in’). Find the limiting values (show that there is an upper limit as I claim above).

Explore what sort of bounce a tennis ball gives in your circumstances and discover where the receiver needs to be.

Explore having the serve go to the other corner of the service box and the consequences. Smart students will develop their own spreadsheet to handle the calculations [This was, i think, one of the pragmatic reasons for dropping this element of the course, as it became too easy for students to find a model without any understanding occurring]. Using the model to develop analysis of problems in tennis and the way they are written up determines all sorts of learning taking place.

2.   Model what happens when you’re walking in the rain. Do you get wetter by running or by walking?

3.   Model a penalty kick at soccer. I claim that a goal-keeper can touch the bar easily; I claim that he can’t fall fast enough to reach either corner by the time you can kick the ball to a corner unless he moves before you kick it. I claim that you can’t kick the ball reliably to just inside the corner at that speed.

4.   Model a penalty flick at hockey. this is not the same as football because (i) the ball is smaller (ii) the goal is a lot smaller (iii) the goal-keeper is asymmetrical. Theories to test: I don’t think you can move the ball past the goalie inside their reaction time, but I also think there are some places the goalie simply cannot reach in that same time. A good investigation produces a contour map of where a goalie can reach within a given time. This then informs both players how to adjust their behaviour.

5.  Model a tennis serve assuming only air resistance matters, not gravity. Assume air resistance varies directly with speed. Carry out modelling to match exercise 1. Investigate when the choice of models is clear (I’d assume this one applies to the professionals, but what about club players?)

6.   Consider a serve in squash. The bounce of a ball matters significantly. Some serves offer returns that are nearly unplayable, so investigation should determine the viability of the various options.

This is my version of the MEI (OCR) modelling flow diagram: I cannot (see how to) make the diagram have arrows, so you must follow the numbering. Box 7 has input from boxes 6 and outputs to either 8 or 9.

1                                               Identify a problem
|

2                                Make assumptions to allow work to begin
Maths____________________|____________________Expt

3         Represent the problem             \   /                            Design an
in mathematical form                  |                              experiment
8                       |                        Review assumptions                |
4M   |                                    |                  4E             |
4M      Solve the mathematical              |                 Conduct an experiment
problem to produce                    |                                and derive
theoretical results                       |                          practical results
|
|                                  |
5                       |
|                                  |
Select information from                      |                        Give a theoretical
experience, experiment                |                         interpretation of
or observation
|                                  results
|
|                                  |
6                        |
|                                  |
Compare with                             |                            Compare with
theoretical results                            |                        experimental results

9                                                   Present findings

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