Geoff Tennant - Promoting access to mathematics for all
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3/5/15: maths and real life contexts

One of the wonderful things about being involved in education at any level, certainly higher education, is learning alongside our students.  This comes about in a number of ways.  One is learning from their experiences, so acting as dissertation supervisor has given all kinds of insights into education across East Africa.  Another is learning whilst preparing for teaching sessions, finding new articles to refresh ideas, trying to keep up to date with what is being said about mathematics education across the world.

So one of the things I address with the mathematicians is real life contexts.  An article I have found particularly helpful is written by my former boss at Leicester Professor Janet Ainley:

Ainley, J. (2012). Developing purposeful mathematical thinking: a curious tale of apple trees. PNA, 6(3), 85–103.

In this article Janet argues that there are two apparently contradictory reasons for using real life contexts in mathematics.  One is to motivate / stimulate / make things easier.  So, for example, there are very good reasons for learning basic arithmetic, with hundreds and thousands of applications which give relevance as times tables and other arithmetic tools are thoroughly learnt.

But the other reason is to apply mathematics which has been learnt to new contexts - so, for example, a standard ending to a set of exercises on Pythagoras is to ask questions about lengths of ladders and diagonals of television sets, where students need to draw their own diagrams and work out for themselves what Pythagoras question is being asked before going on to answer it.  In other words, real life contexts make the mathematics more difficult.

(Parenthetical thought: a related question here is, how real life are real life mathematics questions?  Two reasons for being a bit sceptical here.  One is that it can be very difficult to see why an answer to a question should be useful.  I think I'm quoting Janet Ainley correctly from her inaugural professorial lecture in citing the question, "A beetle has 6 legs, how many legs do 8 beetles have?"  As she asked, why should this be interesting?  Are we proposing to make them all bootees?  The other is that the assumptions being made can be, under examination, untenable.  An example here is the question, "169 people are waiting for a lift which can carry 14 people at a time.  How many trips does the lift make?"  To get this question correct we need to engage with the context sufficiently to see that not only do we need to do a division but also that, whatever the remainder, we must necessarily round up to the next whole number.  But engage with the context any further than this, in wondering whether people will squeeze in beyond 14 to a journey, whether people will use the stairs, whether more people will join the queue before all 169 have been transported, and you get the question wrong.  So, you need to engage with the context so far and no further.  End of parenthetical thought.)

Another way of thinking about real life contexts is that mathematics can throw light on the contexts, and also contexts can throw light on mathematics.  So, restricting examples to money, we can do all kinds of calculations as we are shopping, budgeting, etc. which are genuinely useful (unlike calculating number of legs of beetles).  But also, money can provide contexts which help throw light on areas of mathematics.  So, for example, if I'm wanting to explain why 2 / 0.4 = 5, I might relate the sum to the question, "How many lots of 40p do I need to collect to get UKP2?"  But let me be clear here, if I really did want to know the answer to this question, I would use informal methods, I would not do the sum 2 / 0.4.  What we have here is very much a one way street, the context helps to explain the mathematics, the mathematics does not give us a helpful way to solve this problem.  Implying that it does is one of the things which can give mathematics a bad name - take an easy idea and make it difficult, oh yes!

Also, borrowing an idea from Professor Malcolm Swan, before rearranging the formula p = nc - e, I might spend some time exploring the situation whereby I charge 20 people UKP30 each to attend a course, with expenses of UKP150 giving me a profit of UKP450.  Again, the context helps throw light on the mathematics, I wouldn't want to be making any claims for the opposite way round.

But here in East Africa there's a problem.  Each of Tanzania, Kenya and Uganda have shillings as their currency, referred to informally in Kenya as 'bob' - speaking as someone just old enough to remember shillings in the UK, this seems really strange!  In Tanzania there are about 3000 shillings to UKP1, with the minimum unit of currency that I've seen 50 shillings.  Confusingly there are about 140 Kenyan shillings to a pound and 4500 Ugandan shillings.  I assume (but don't know for a fact) that in colonial days the shillings were the same value as the UK shilling, or 5 pence, the divergence since then telling us, I suppose, something about the respective strength of the economies since then.  So, this blows my 2 / 0.4 question out of the water - although could substitute "How many bricks of length 40cm. do I need to go along a 2m. length of wall?"  Although I have seen questions in textbooks which require children to do calculations with cents (100 in a shilling) some decades after this went out of date in the real world.

But if I want to use Malcolm Swan's question about courses here, I have, as far as I can make out, a three way choice.  I can continue to use pounds or dollars, which may well be unfamiliar and promote the idea that mathematics is an alien, western concept.  Or I can simply substitute shilling for pound, which makes the prices ridiculously small.  Or I can add 3 0s to the sums of money, which runs the danger of obscuring the point I am trying to make.  None of these options appeals to me but I can't think of any others.  Any thoughts on this point very gratefully received!

But there is one final point which needs to be made.  Relating mathematics to contexts necessitates using words.  This is an area which is much explored in the UK, but here in East Africa almost all learners in secondary school, and many in primary school, are learning in a language other than their mother tongue.  Before we get to the mathematics we have to work through the language which can be a major barrier, a major reason for getting answers wrong.  My favourite example here is, "What must I add to 13 to get the answer 20?"  Please, don't let's conclude that, if a child gives the answer 33, she can't do subtraction!  Far from motivating and making things easier, I'm actually demotivating and making things more difficult.

So, where does this leave my mathematics students at the University?  Better able to analyse situations they find themselves in, I hope.  Better able to disentangle situations and diagnose where problems have occurred.  Clearer about what they're doing with children and why.  But, alas, without any easy solutions to some pretty intractable problems.

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