I’ve been thinking a lot recently about “formal methods”, e.g. Column Addition or Long Multiplication.

As secondary maths teachers we don’t pay much attention to these. I assess their ability to carry out the procedure and have sometimes attempted to teach long multiplication explicitly. But more often than not it is assumed that these methods have been learnt at primary school have been pretty well-practised and are therefore secure.

We have been doing a lot of number work with Year 7 so far this term including exploring in some detail the Laws of Arithmetic.

Here is an example of the type of question we have been looking at:

**127 + 54 + 73**

Most of my students’ first instinct with a question like this is to draw up a nice column addition and solve it. Would you agree that most students do this?

I want them to look at the *structure* of the numbers first. To realise that addition is commutative and that it’s much easier if we add the 127 to the 73 first because that’s a number bond to 200.

I am having 2 issues with this approach:

- My students are not convinced that this is a valid exercise. Basically they think it is contrived (which, of course all exercises are). They think I have chosen numbers to make this work, it will only work for those certain numbers and basically they feel like I have tricked them.
- Some students perceive the message that the formal methods they spent hours practising at primary school are now not the way we do things at secondary school. I have said that this is not the case, they are
**not** the devil’s work, and that those methods still have a place. But we need to have smarter ways to work. The fact that we have spent many lessons not solving calculations using formal methods leads me to believe that they are confused about what I want them to do.

Fundamentally I want my students to rely less on the authority of the teacher and rely more on their own understanding. I want them to see the structure in everything they do, to be certain that what they have done is valid because the mathematics tells them it is so, not because I tell them that they are right.

How should we bridge the gap between informal methods which require (and develop) depth of understanding and formal methods which are efficient and accurate?

I have an idea using manipulatives. I am not sure if it will work, but if I get chance to try it with the class, I will write about it again later.

In the meantime, does this resonate with anyone else? How do you bridge the gap? I’d love to read your comments.