John Von Neumann’s quote often rings true, but I’m not sure that’s not a good thing.
I saw this quote on twitter a few months ago. I can’t remember where I read it first, but it may well have been courtesy of Dylan Wiliam.
It immediately resounded with me. I feel that I learned a lot of maths this way. And I got a first from Cambridge, surely it must be fine?
On the contrary, whilst I was lucky enough to muddle through regardless, I think that this approach is not effective for the majority of learners.
Greg Ashman makes a good case for not focussing too much only on axioms, but I don’t think (m)any teachers are doing this. In my experience of teaching and observing, the opposite is much more common and I think this is mostly caused by the ‘curse of knowledge’.
I’m new to teaching mixed-ability classes and I’m experiencing this a lot personally, particularly in the lower years. I recently taught an introduction to algebra to year 7. Even after several similar examples, some of the pupils were struggling to simplify d+d+d+d as 4d. They were either not sure what it should be at all, or would write it as d^4.
In my previous selective schools, this was assumed knowledge from primary school so I never even thought about it. In retrospect, I suspect that some pupils didn’t understand it but just got used to it.
When I encountered this misunderstanding, I was at first tempted to say something like “d+d+d+d… it’s just 4 d’s” Unfortunately, this falls down when you get to d×d×d×d, but I also think it falls into the Neumann trap of just asking learners to get used to things. Instead, I should have checked that my pupils understood that 7+7+7+7=4×7. This understanding is important to avoid other future problems: pupils incorrectly substituting into expressions such as 2d, or not knowing what to do when faced with equations such as 2d=7, both because they’re not aware that 2d means 2×d.
I’ve updated my lesson plan to including a check of that knowledge at the start of the topic. I hope that in future this will help pupils to understand how to simplify d+d+d+d, rather than just get used to the answer.