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]]>In terms of methods, it’s a common situation, right? : finding the balance between conceptual understanding and an efficient method that “works” as in it yields a correct solution must of the time.

What I was thinking here is, what’s the benefit of drawing the grid vs just doing the calculations when doing e.g 34×5. And then maybe 35×15.? If we are trying to reinforce understanding, better to write out all the calculations than use the “method” of boxes. I agree, the diagonal boxes is different again and often gets even more fraught.

Final thought, why type of students are doing box method in Yr10 vs those doing formal long multiplication?

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]]>PS When I left the school her daughter presented me with a big cardboard box. It was warm. All I could think of was that the stupid child had given me a puppy! In fact she’d got up at five a.m. to make me my own tray of date slice! ]]>

Let me start by explaining why I say that this is conceptual. It is because it links to a topic that we only just touch on in ordinary A’level maths (there’s a bit more in FM) but which is crucial in degree level maths: vectors. Having taught vectors for many years in universities, I would have been overjoyed had any of my students met this kind of exercise when studying transformations. To see the circle of ideas between points, vectors, translations, and transformations makes so much more of mathematics connect together.

The second thing to say is that I don’t see the dichotomy that you allude to between procedural and conceptual approaches to mathematics. Understanding and capability go hand-in-hand and it varies from person to person and from topic to topic as to which is in the lead. There are some topics where it is possible to have a sufficient understanding and overview before one starts learning procedures, but there are others where actually one should become au fait with the methodology in order to gain a conceptual understanding. See that the method *works* often leads to insight as to what’s going on underneath.

It continually frustrates me as an educator to see this attitude of “I won’t try until I fully understand every step”. This leads to the “Where do I begin?” problem when attempting more intricate problems: the student hasn’t developed the habit of just starting. They understand that to solve a complicated problem they need to break it down, but don’t realise that one often doesn’t break it down at the outset but rather chips off piece after piece until it all falls apart.

A related frustration is when I’m explaining a method to a student then they don’t *write things down* as we’re going along. They expect to be able to hold the entire thing in their head and then reproduce it when I’m gone. That’s not how mathematicians work! Part of the point of developing mathematics was to avoid having to think too hard at any one stage!

Anyway, there’s my initial thoughts. To sum up, I think this is a very good method for teaching scaling (the term “enlargement” annoys me!) to any ability of student.

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