NB: this post is more opinion led than most of mine!
On Tuesday evening I attended a webinar on the ARK Mathematics Mastery programme (@MathsMastery I found this a very interesting hour or so, but the one quote that I took away from it was the following
Symbols don’t mean anything unless you understand the concept.
They showed a picture similar to the one below, and commented that the symbol for ‘5’ means nothing unless you associate it with the concept of ‘five-ness’.
I’m constantly thinking the same thing when teaching something like expansion of brackets. When faced with the following question “Expand \( 3(2x -4) \)” I have had a fair number of students ask how you know if the second term is a plus or a minus – this just shows that sometimes I end up teaching the mechanics of a concept of method without ensuring there is solid understanding to build on. This is most definitely not a good thing – there is no point, in my opinion, in being able to complete the square of a quadratic for example without understanding why it is useful, and what it tells you about the quadratic. In my experience there are a significant number of undergraduate mathematics students who don’t realise that completing the square gives them important information about the turning point of a quadratic.
Having a concrete understanding of number and the number system is incredibly important. I see the overall goal of education to be “to prepare children for the world”. If by the time they leave school they don’t have a solid understanding of how numbers work and how to use them but can mindlessly expand a linear bracket because we have been told that it is on the scheme of work, have we really prepared students to deal with real life?!