I am working with a small handful of (thankfully grateful) year 10s during registration time at the moment. We are concentrating on complementing topics covered in maths lessons, to try and help them cope better during class time. Currently the topic is proportion.
We began by talking about enlarging shapes, and how when a square is enlarged, how can we recognise the new shape? Will it still be a square? What about when we enlarge circles? This in turn led to a more general idea of proportion, and with it, the difficulties in spotting an enlargement of a typical rectangle, or a triangle. The bigger shapes are still rectangles and triangles respectively, but not necessarily enlargements. The process feels more complicated, because it is. Anyway, I digress.
Over the last few weeks we’ve covered enlargements and ratios. The students found both topics a little challenging at first, but picked up the ideas, and I managed to find some methods that worked well with them to systematically find answers to fairly diverse questions such as “If I have x money, and the total is split into this ratio, how much have you got?” and “x money is divded into this ratio, how much do you have, how much do I have?”, and “x money is split such that I have this much and you have that much, what is the ratio?”.
Each question was being answered confidently eventually, albeit quite slowly. I could live with that. But what bothered me was that all the students kept making silly mistakes. Not with the method, they had a method they understood and could use, but still they’d come up with the wrong answer. Each of them, at different times, kept getting something wrong – but still, the methods were always perfect. So what was happening? They would talk me through their thinking, and the things they were saying they were doing to the numbers were correct. Those descriptions would yield perfect answers. However the students were simply doing the calculations badly.
It became clear that this was not an issue of mathematical understanding, it was an inability to calculate. The more I explored this, the more saddened I became. These students, as with so many I’ve come across in my career, had no recall of any multiplication table at all. Not even the twos. As such they relied entirely on guestimating, which frankly only works even marginally well if you already have a good understanding of your times tables anyway! Explaining that the two times table only ever produces even numbers, and demonstrating it with dots seemed like a revelation to them. Explaining further that *all* times tables of even numbers only contain even numbers, and linking that back to the two times table, again seemed like some kind of dark secret nobody was supposed to know about. We then looked at the 9 times table, and logically deduced that each sum 9 x n must be close to 10 x n (we didn’t use algebra, just to be clear!), and must be less than 10 x n. Therefore it’s quite easy to figure out what the first digit of the number will be. For 9 x 4, it must be 30 something, and 9 x 6 must be fifty something etc.
Again, light bulb moments. One student exclaimed joyfully (after further exploration of the 9 times table I should add) “I can do it! I know my 9 times table for the first time!”. I should have felt happy for her, but still didn’t. Performance, afterall, is not learning. If she remembers next time I see her, which will likely only occur if she practices fairly regularly, then maybe I’ll share her confidence. I hope she does.