The unconsciously incompetent

What happens when you build something on shaky foundations? It falls down.

Knowledge is a bit like that. Concept is built on prior concept giving rise to skills that can be practised and applied.

I was reminded about this in a maths lecture today at Edge Hill University on the subject of 'errors and misconceptions'. We were challenged to draw a picture or diagram of errors and misconceptions and how they relate to each other. I tried to draw a picture of a house standing on shaky foundations with each stone representing a concept - some of which were solid, some broken and some completely absent. Whatever your picture might be, the fact is that as maths teachers we can easily see the errors, but it's a bit more difficult to identiy the misconceptions.

An error could be a simple calculation mistake arising from being hasty or not checking the answer. Or it could point to an underlying misconception - a broken foundation - which could lead to many repeated errors.

I remember speaking with a friend when, both in our twenties, I realised he could not subtract even small numbers accurately. He would always be one out. For example 24-9 would be 16. This was because when counting back, he would always start by counting the initial number as 1, rather than 0 - if only he'd been shown number lines at primary school! I'm not quite sure how he got through 'A' levels and a degree - but he did. Anyway, after that conversation his misconception was fixed - the foundation was more sturdy than before.

So as teachers how can we correct misconceptions?

Firstly we need to avoid causing misconceptions ourselves. Have you ever heard a teacher say: "Five take away seven: you can't do that!" or ""Fives into three can't go!"? Have you ever said that yourself? O have you ever drawn triangles, squares or rectangles with all their bases horizontal and parallel with the bottom of the page? The lecturer today called this being 'unconciously incompetent' - a condition we should try to avoid. We need to be deliberate and precise in our language, using words that will enable future concepts to be easily built on what we have taught. In his report of 2008, Williams wrote: "It is often suggested that 'mathematics itself is a language' but it must not be overlooked that only by constructive dialogue in the medium of the English language in the classroom can logic and reasoning be fully developed". So Mathematics teachers need also to be masters of english so that they don't unintentionally teach misconceptions.

Secondly spot the misconception. If an the same error is repeatedly occuring in a student's work, that's a glaring clue. However some misconceptions are harder to find and one-to-one conversations with children - that 'holy grail' of 'Quality First Teaching' which all teachers aspire to where they can spend some time in purposeful dialogue with each child in their care at least once a week.

Thirdly use concrete examples, or models and images. Misconceptions are often because students don't get the abstract form of maths. They need to have to take it back to concrete examples - counters, teddy bears, things, whatever, or at at least use powerful models and images such as the number line. It's amazing how many uses teachers can find for chocolate when illustrating some mathematically...

I'd love to have examples of others experiences of errors and misconceptions - please comment below.

Photo: Broken Brick by Ternus on Flickr