Archive for September, 2010

Advice

September 30, 2010

I was going to start this post with “The nice thing about working in maths education is…” but I realise that what I’m going to go on to say suggests that working in maths ed can be a double edged sword. The nice part of it is that people ask me for advice and for my opinion on their children’s maths education. The negative part of it is that people ask me for advice and for my opinion on their children’s maths education!

This post was prompted by a good friend of mine asking for advice on her daughter’s maths development after a particularly tricky homework question on division. I felt a little bit out of my depth – daughter is Key Stage 1 and my experience is Key Stage 3, 4 and 5, but I have done a little reading on primary maths and spent enough time listening to the wisdom of my primary colleagues that I felt able to give a little advice. I decided the most useful thing I could suggest was for mother and daughter to explore maths in everyday contexts as much as possible to help to develop a strong sense of number and a confidence at solving problems. I know my friend enjoys baking, so I suggested that one useful activity was baking together, with a chance to discuss halving and doubling quantities, sharing out the biscuits when they’re made, and so on. It may not solve every worry my friend has, but it certainly won’t do any harm!

Another foray into primary maths for me occured when I visited my niece recently. She is almost 5, and has just started school. I recently bought a book for her at a car boot sale – the Ladybird book of Shapes. We spent a happy half hour reading it together, and then drew some shapes and coloured them in. Last time I visited, we went out as a family on a long journey, and we played a game together in the car.

I’m thinking of a shape. Ask me some questions and see if you can guess my shape.

After a few goes where I modelled the sort of questions she might ask (Does it have straight sides? How many sides does it have? Does it have curved sides? Are all the sides the same length?) she came up with the following:

I’m thinking of a shape with one straight side!

It took me ages to suggest that it would also have a curved side and that she was in fact thinking of a semicircle! It’s not often that I am mathematically out-thought by someone a sixth of my age.

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Mind your language, again.

September 20, 2010

You may remember that way back in May I wrote about the language we use in maths. Back then, I was thinking mainly about the importance of using unambiguous language, but I’ve been thinking about language again, particularly words which mean something specific in mathematics but also have an English meaning in everyday conversation. (I don’t know enough about languages other than English to know whether maths-specific vocab is shared with a more general usage, but I’m interested in examples people have to share, in any language.)

The word that got me thinking about this is a very simple one, “or”. I was editing an article for secondary school students about tree diagrams, written for us by one of our NRICH summer students this year, and part of the article dealt with “the probability of A or B”. I felt the need to highlight the mathematical usage of “or” which means A or B or both, because many students on meeting such a statement might assume it to be an exclusive or which doesn’t include the “both” option. It reminded me of a sign I saw outside a cafe at the weekend:

Toasted Teacakes or Crumpets with regular Tea or Filter Coffee for just £2.25

This common English usage of the word “or” strongly hints at an exclusive or; if I ordered teacakes AND crumpets, with tea AND coffee, I’d expect the bill to come to £4.50 rather than £2.25.

There are many other examples of words which mean something unexpected when used in a mathematical context – for example the word “expect”. (In case you can’t tell, a lot of my work this month has been thinking about probability problems!) I realised I was throwing in the word “expect” without ever giving a mathematical definition of expectation. Does it make sense to someone with a layman’s understanding of the word that we “expect” to take two flips of a coin to get a Head? Or do we need to take the time to explain that half the time we get it on the first flip, a quarter of the time on the second flip, and eighth of the time on the third flip, and so on, which averages out to two flips?

Part of the problem is familiarity. Those of us who are conditioned into the world of mathematics can slip easily between our subject’s specialist vocabulary and the same words used in a different sense. But whenever we are working with those who have not yet mastered the vocabulary, we may need to be explicit in making the distinction between the words in their common usage and the specific mathematical meaning we are attaching to them.

More probability – maths with cards.

September 6, 2010

I’ve been thinking about a card game/trick shared by a colleague. Take a pack of cards with JQK removed, and deal them out in a long line of 40 cards. Place a counter on one of the first six cards, and then move forward that number of cards – for example, if you place your counter on a 3, you’d move 3 cards forward. Continue until you can’t move any further without going off the end of the line.

Then place another counter on another of the first six cards, and do the same.

If you try it, you may be surprised that quite often, you end up on the same card. Trying to explain why is quite fun. (Analysing the probabilities would be horrendous, but when I set this as a problem with students I’m planning to make it an experimental rather than theoretical probability task.)

My unanswered question is this though; is it possible to order such a pack of cards in such a way that each of the first six cards will take you to a DIFFERENT finishing point? We think we have a convincing argument why this is impossible… but maybe we’ve missed something and it is possible, or maybe you have a really neat way of thinking about this – if so, please share it!