What is mathematics?

Yesterday Sasha Borovik sent me a link to a document entitled Making it all add up: Business priorities for numeracy and maths by the Confederation of British Industry.

Sasha drew my attention to the fact that this document contains a definition of mathematics. In fact it contains two definitions:

What is numeracy?

Confidence with the handling of numbers, general mathematical awareness and its application in practical contexts.

What is mathematics?

A developing language for description, deduction, verification and calculation – more a set of tools than a specific learned skill.

The main problem with the definition of numeracy is that all the confidence and general awareness in the world is useless if you lack accuracy, of which no mention is made.

Here is a true story I have told many times. After we announced a policy of forbidding calculators in most exams, the students (who tended to regard a calculator as a security blanket in those days) became extremely worried. I tried to reassure them that my first-year probability exam would contain no hard sums. At a certain point, one candidate needed to multiply 10 by 11. So on the rough working page, he wrote 10 down eleven times and added them up, obtaining the correct answer. Was that student numerate? I think not. If the answer had been wrong, would he have noticed?

On to the definition of mathematics. I would say it is good as far as it goes. But again they have left out the most important thing. Here is Richard Feynmann’s comment:

… mathematics is not just another language … it is a language plus logic.

Finally, with reference to mathematics as a set of tools, there is some resonance with Tim Gowers’ “two cultures of mathematics”, where he distinguishes theorem-based mathematics such as algebraic geometry from technique-based mathematics such as graph theory, a typical tool in the latter case being the probabilistic method. I suspect, from the document, that (as Sasha suggested to me) the CBI have taken a much more limited definition of “tools”, and would only admit tools which solve problems outside mathematics, and in particular, problems in industry (however defined).

The document itself gives clear evidence of a change in the role of the CBI, who now sub-title themselves “The voice of business”. Once, business and industry were seen as somewhat opposed; now, we have so little manufacturing industry that any kind of business can call itself an industry, e.g. “the financial services industry”, “the leisure industry”. The change hasn’t made them more humble, rather the opposite; they clearly feel they have a right to dictate to schools and universities what kind of graduates they should produce.

This wouldn’t matter so much but for some serious evidence of lack of thought in the document itself.

There should be an opt-in for students who achieve a good benchmark standard in maths at GCSE … to study at least an AS level in maths after 16.

Since this option already exists, this sentence can’t mean what it says. It seems that the writer would like this to be compulsory, or at least to be expected (so that pupils would have to make a deliberate decision to opt out).

The latest CBI surveys show a fifth (18%) of employers …

If that standard of estimation (and of language) is good enough for British business, then I despair.

P.S. I showed the document to a group of graduate students yesterday. The cover shows people holding large arithmetic signs up. One student remarked, “The person with the minus sign will have trouble making it all add up.”

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About Peter Cameron

I count all the things that need to be counted.
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3 Responses to What is mathematics?

  1. Anonymous says:

    They really do seem to be using “opt-in” to mean something very different from what I mean by it. For example, “Students who fail to achieve … should be opted-in to courses to help them to do so.”

  2. Ross Templeman says:

    The story about the student that had to resort to adding up the tens isn’t as surprising to me as one might hope.

    When my cousin was in year 6 about 2 years ago I was asked by my aunt to help my young relative with some of her mathematics homework. I noticed pretty quickly that basic mental arithmetic seemed to be an underlying weakness.

    As such I asked my cousin to recite her 5 times table for me. To this she replied “5, 10, 15, 20, 25, 30, 35, 40, 46, 50, 55, 60” in a single breath. I then asked her what seven fives were, to which she gave the reply “um…er…60?”. She wasn’t trying to be funny, it was simply the case that at junior school she has been taught to memorise her multiplication tables by the sequence of ‘answers’, a useless method for developing skill in basic arithmetical reasoning.

    Being a mathematics graduate myself (as fate would have it Professor Cameron you actually taught me Probability 1 at Queen Mary in 2004) I couldn’t stand for this and re-taught her using the “old fashioned” method of reciting “two fours are eight” etc followed by random questions. She is much improved as a result, but I cannot fathom why the school hadn’t done this themselves.

  3. I definitely saw one of my students with “2+1=3” displayed on their calculator screen during an linear algebra exam. Performing elementary row operations didn’t appear to pose a problem, but this evidently did!

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