Geoff Tennant - Promoting access to mathematics for all
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14/9/14: some thoughts on the learning and teaching of mathematics

We've just finished hosting a major East African mathematics education conference, bringing delegates from Tanzania, Kenya, Uganda, Rwanda, with visitors also from Botswana, Zimbabwe, South Africa, Italy, Sweden, France and Canada (think that's everyone!)  Great pleasure to meet so many wonderful people as we look to explore what we can to support mathematics learning in this region, lots of ideas on how we can continue to network - including PAMC, this may now get off the ground, watch this space!

So, not surprising that I've been thinking a bit about the learning and teaching of mathematics recently, more so than normal.  I was very struck by one of the presenters who videotapes his undergraduate lectures and then uploads them to a website so that students can watch them again - or for the first time if the missed the lecture.  One student wrote to him to thank him for doing this, and then continued to say this:

After I watched your lectures through 3 or 4 times I
realised that you ask really simple questions

On the face of it this makes no sense - surely if the lecturer is asking simple questions then that should be immediately obvious?  But I would want to suggest that this makes a very profound point about mathematics: once the penny drops things become clear, until that point things seem impenetrable.  I wonder how many people have had the experience when at school of going into a classroom after a mathematics class for an older group has happened, and to look in wonder and mystification at what was left behind on the blackboard?  In my experience this continues up through undergraduate mathematics, in any one year having no real idea what was going on in the year above until we got there.  As far as I can make out, this 'secret garden' aspect of the subject does not apply in quite the same way in other disciplines, but am very happy to be shown to be wrong on this point!

A few related points:

If you're doing mathematics right then you're doing a simpler
version of mathematics than if you're doing it wrong

One example of this is when I was tutoring a boy for an independent senior school entrance exam (non-calculator), and he was faced with a particularly hideous simplification of a fraction to do, with a large number of fractions within fractions to sort out.  However, if the simplification was done correctly, then a huge amount of cancellation was possible, so the arithmetic involved was actually straightforward.  One may consider that this makes it a good question in that it is testing only one skill, that of simplifying fractions.  The problem here is that if you start to go wrong then the cancellations will not appear so you end up doing all kinds of long multiplications which the examiners had not intended.  Maybe these questions should come with time limits?

Most time spent studying mathematics is undertaken by
people who do not understand what is going on

This is a similar point to that above.  As long as things are making sense then mathematics can be done with relatively little effort.  Once things make no sense any more then the going gets very tough going.  So, I got my way through my undergraduate degree through a large amount of memorisation of theorems and proofs, spending hours before tutorials trying with limited success to work through the example sheets provided.  Stephen Hawking, so the story goes, spent 1/2 hour before tutorials producing totally perfect solutions which he then put in the bin as he was leaving, simply because he couldn't see the point in keeping them.

An awful lot can be achieved with a small amount of content knowledge

Every now and again I wage a small war against standards of numeracy when I go shopping and calculate the cost of my purchases in less time than it takes the shop assistant to reach for a calculator.  I then get treated like some kind of genius - but please, all I'm doing are basic calculations I learnt at primary school!

To take a different example, consider the following multiple choice question, a solution to which is given at the bottom of this post:

The probability that the next person you see will have an above average number of arms is:
A:    0
B:    nearly 0
C:    1/2
D:    nearly 1
E:    1

You might like to consider this problem before reading on.

I've used this problem many times, mostly with trainee teachers, and found it appropriately challenging, with almost everybody getting it wrong initially, discussing it, eventually getting to the correct answer with a level of 'Oh, yeah!' going on.  Except for one occasion when I threw it out to a 12 year old boy whose family I met whilst on holiday, who subverbalised for about 10 seconds before getting the answer right with flawless reasoning.  I make no apology for the fact that my jaw dropped open in amazement!

What makes it an interesting question is that the content knowledge needed to do it is very basic - in principle, once you've met the concept of probability and average - typically at lower secondary level - you're ready to do it.  But yet the reasoning skills here are quite complex which is what makes this a challenging question.  So my final point:

It is better to have mastery understanding of a small amount of
mathematics than a cursory understanding of a large amount

Many times back in the UK, speaking with people in some cases with very high levels of seniority in mathematics education, I expressed the view that most graduate mathematicians would be better off with a thorough understanding of what is deemed to be further mathematics A level than with a very thin understanding of what is actually taught in Universities.  I thought I was being highly contentious, but to my surprise, almost everybody to whom I have said that agrees with me.  Yet, as far as I can make out, across the world and across the age range, mathematics is being taught as a subject to be memorised, has no relevance to reality, and is devoid of any fun or enjoyment.  I sit here and weep.

I'm pleased to say that my own contribution, looking at how algebra can be introduced keeping strong links with number, went down very well.  So maybe I can make a small contribution here - although this coming week takes me back to my normal job of preparing documents for the Tanzanian Commission for Universities and other such jovialities.....

Solution to the problem: 
The probability that the next person you see will have an above average number of arms is:
A:    0
B:    nearly 0
C:    1/2
D:    nearly 1
E:    1

Assume that average here means mean average.  The overwhelming majority of the people in the world have 2 arms, a small number have 1 or 0, the average is therefore slightly less than 2.  In all likelihood the next person you meet will have 2 arms, which is an above average number of arms.  The answer is therefore D: nearly 1.

I'm aware that it's possible to pick holes in the logic, very happy to debate this if you'd like to comment!

6 Comments to 14/9/14: some thoughts on the learning and teaching of mathematics:

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Jenny Spence on 17 September 2014 12:54
Your question on the simplifying fractions brought to mind one of my major bugbears - use of calculators. When Chris was younger we used to get a mathematics calendar - it was great fun to do, with a problem per day and the answer was always the date. So you knew that any complicated multiplying/dividing etc was going to result in a whole number so using a calculator was pointless. But it was really hard work persuading him to ditch the calculator when multiplying out, for example, a long line of square roots (he would do it on his fairly basic calculator and of course end up with a non-whole number because the calculator lost accuracy, and he'd be slower than doing it without too). I can only presume he picked up using a calculator at school, because I certainly try longhand first so he didn't get it from me. And it is SO satisfying to cancel it all down and get ... 2. And he's still pretty rubbish at factorising quadratic equations - always goes for the formula first. Whatever are they teaching them these days!!!!!!!
Reply to comment

Geoff on 17 September 2014 22:47
Thank you, Jenny, agree with you about calculators. As an initial teacher trainer, the line I took was that calculators should be kept in the bag unless there was a clear reason to have them out, and I can think of 3 such reasons: to explicitly teach their use; when doing calculations manually would get in the way of what you're trying to learn (trigonometry would be a strong example of this); and as a learning tool - so nice activities around BODMAS can be set up, for example. Oftentimes topics like fractions, area, perimeter, ratio, proportion and many others can be opportunities to practise basic numeracy on the side. Maybe we need to start a campaign on this!
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Samuel Bengmark on 21 September 2014 03:00
Dear Geoff I enjoyed reading this post with some not so well known citations. ;-) The "arm-problem" was clever. Maybe there are also people with three or more arms, but without a doubt they must be fewer than pushing the average up to 2. So to continue in the same vein, is a more likely that the total number of arms in the world right now is even than odd?
Reply to comment

Geoff on 21 September 2014 10:06
Thank you, Samuel, am I allowed to say that you're the lecturer referred to at the beginning who records his lectures? Apart from the first quotation you are welcome to quote anything here as Tennant (2014). There are some very rare cases of conjoined twins which gives rise, depending on definition, to three and even four arms, but I'm almost certain that the argument above nevertheless holds. As to whether the total number of arms in the world is odd or even, I'm inclined to go for evens but am open to be persuaded otherwise! Thank you for contributing, Geoff
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Luwilo Sanga on 22 September 2014 00:33
Thank you Geoff, it is true that we had a very good moment during CANP4 2014. I real learned a lot from the EXPERTS. Please keep on helping us via to keep networking alive. Thanks for the hardwork.

Geoff on 22 September 2014 05:18
Great to have you with us, Luwilo, so pleased you found the whole experience so beneficial. Just the return to normality has gone well - and don't forget to submit something for PAMC!

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