March 2006

Whenever a discussion comes up about students using calculators for mathematics in schools, there are some standard points that are usually made. Examples are given of how being able to mental arithmetic is an unecessary skills these days and examples are give on real life applications of mental arithmetic.

I am interested though in the special case of student who go on to study a subject such as engineering. If you want to be an engineer you obviously need to be able to do certain types of calculations, however you will also in practice always have access to a calculator.

The problem however is that trying to understand certain topics in engineering without knowing your times tables is like trying to read a French novel and having to look up every other word as you go in a dictionary. You may understand it all it in the end, but the likelihood is that you’ll miss lots of the subtleties and things worth appreciating about the novel. You probably won’t enjoy the experience as much either.

I remember trying to teach some engineering students who were stuck on a problem. When I delved down, I discovered that what was causing them problems was figuring out what the square root of twenty five was. All their mental energy was concentrated on working this out and making it difficult for them to actually think about the problem or concepts at a higher level. Not being able to do basic mental arithmetic was making it almost impossible for them to be do more complicated mathematics.

So I believe that anybody wanting to understand mathematics at a higher level, does need to put in the time doing routine practice of lower level mathematics questions. It's like practising your scales when you are learning to play a musical instrument. It's not what you're trying to learn in the end, but it is something that almost everybody needs to do along the way in order to become good at what they really are trying to learn.

We need to make this mechanical practice as simple and easy as possible so that we can concentrate on teaching engineering students how mathematics is something that helps you describe and predict interesting patterns and that by doing this you can solve problems. There was an article by Derek Raine recently in MSOR Connections that echoes to a large extent how I feel.

This all relates at a wider level to the idea that as you learn new skills you pass through three stages:

**Level 1 (Following)**Where learn to follow a set of rules or procedures**Level 2 (Detaching)**Where you learn where the rules you've learned at Level 1 breakdown**Level 3 (Fluent)**Where you don't actually need the rules any longer.

(Interestingly Aikido has levels called shu-ha-ri which correspond to these).

You can think of Level 1 as a form of scaffolding. It can be hard for people at Level 3 to see that people usually need to go through Level 1 and 2 first and it's obviously at getting to Level 1 where traditional-style formal education of some sort is usually extremely useful and where constructivist ideas are least useful. We want to get people to Level 3, but we should not dismiss the type of learning needed for most people to get Level 1 as a prequisite.