Mathematics and numeracy... but I don't know where to start.
Being functional mathematically doesn’t just mean being able to work out your gas bill or check your change in the shop, valuable though these skills are. It means a willingness to try things out. There is something about mathematics problems that often provokes one of two responses from our learners:
1. I can do this because I can see what it is I need to do.
2. I can’t do this because I don’t know where to start.
And the mathematics student’s intuitive sense of inverse proportionality, summed up in the phrase “the shorter the question is, the harder it is to solve” is a reflection of the second of these.
Show that √ 2 is irrational.
Why do some polygons tessellate and others don’t?
What’s the best mobile phone package for me?
As maths teachers we need to teach our learners to accept a third option:
3. I can’t see exactly what I need to do, but I am going to try some things anyway.
Option 3 is what real mathematicians do. Did Andrew Wiles know all the steps of his proof for Fermat’s Last Theorem when he started out? Do you know the financial benefits of different phone packages without research?
Inside the classroom, mathematics can sometimes seem to be a trick played on learners – the teacher knows how to do it and the learners have to guess what’s in their mind. Showing that √ 2 is irrational is easy, once you know that way to do it is to assume the opposite is true. Maybe the first person to figure it out immediately realised that proof by contradiction was going to bear fruit, but I doubt it.
Let’s take the question “Why do some polygons tessellate and others don’t?”
Where to start? What does tessellate mean? What is a polygon? Can I draw some? Can I make some? Can I make tentative conjectures? Am I going to be mortified if my initial conjectures are false? What if I don’t find the answer – is all my work wasted?
Part of the anxiety that many people feel when given a maths problem is that there is no middle ground. You either get it right or you get it wrong. If you get it wrong then your work is wasted and you can define yourself as “useless” at maths.
Perhaps we can encourage our learners to honour the middle ground. You might not find out why some polygons do tessellate but you might be able to say why some don’t. Or you might find sets of polygons that tessellate but are not sure whether you’ve missed any. To teach our learners to be mathematically functional we have to help them develop the courage to go up blind alleys, to try things out no matter how trivial they may seem, to accept that making mistakes is how we learn not to make them in the future and (Andrew Wiles is a case in point) that we seldom get things right first time.