After visiting Bath and listening to Alex Bellos’s talk I was led to muse particularly on the cognitive processes used by the two girls in the abacus clip. I had been aware for some time that generally [and I know that in general one shouldn’t generalise] the best mathematics pupils are not the best at visio-spatial tasks. I shall give three examples from by own experience that exemplify what I mean; the fact that all three are from a non-WASP background may or may not be relevant and is up for discussion later.
The examinations referred to are those used by many British independent schools for transfer at 13+. They are written to a set of syllabuses based on the National Curriculum and are administered by the Independent Schools Examinations Board. The core suite of exams is called Common Entrance and those used to assess the award of scholarships are called the Common Academic Scholarship. A fair number of schools set their own entrance papers, and even more their own scholarship exams, but generally within the bounds of the Common Entrance syllabuses.
The first example relates to a boy who we shall call child A. He was studying at the time for a scholarship to a top Public School and we were working through various old papers to get the hang of the house style. The algebraic and computational tasks he whistled through but when we came to a question about trains travelling to and from Edinburgh he just couldn’t see it. The use of this expression is deliberate; I tried all sorts of ways of explaining it using algebra and ratios, diagrams, at one point we even had pencil cases as trains, with his classmates moving them back and forth [His contemporaries included B and C who both gained a full scholarship to another top Public School] Even with their help, A could not visualise the task in his mind and so couldn’t assertain where or how to start the question. We had to agree in the end that there was going to be the odd question that was just not his thing but that there would be plenty that he could tackle and do well. I should add that a couple of months later he sat the real paper and was awarded a scholarship, to add to his music scholarship he had already won; we are not talking about a stupid child here!
More recently I taught a boy called D from an FSU. His algebraic and computational skills were some two years ahead of his English contemporaries but he had studied little or no graphical work and whilst doing a recent Common Entrance paper, I discovered he had never even seen a net, let alone stuck one together. So we set to and started with some pyramids, then a cube and a cuboid followed by several other prisms. His lack of experience of this sort of learning was evident and several attempts ended up as a ball of paper and glue in the bin, projected there with the accompaniment of a variety of words which we agreed I probably didn’t want translated. It was very noticeable during this exercise that he was not able to relate the flat net to the eventual 3D solid and that, other than the occasional Slavic outburst, said very little. As with A, he simply couldn’t visualise the task in his head and so could not predict how the shape was going to develop. Finally he, another boy and myself made three square-based pyramids with a vertical edge. Here again, he could not see initially that it was going to form a pyramid nor, once we had finished, that they could fit together to form a cube. He studied the finished shape for a while then suddenly said [for best effect, please use stereotypical Russian accent here], “Ah, is why formula for pyramid is one third.” Essentially, back with algebra he was on safe ground, and prepared to contribute. We also looked at the UKMT JMC papers; he had done Kangaroo papers back in his home country. Here, as to some extent with A and also B, who was considered by his peers to be an automaton, he struggled with the questions requiring deductive reasoning but those involving computation or formulaic algorithms he whizzed through. He got top marks in his transfer exams and I expect him to gain an A* in his iGCSE a year early.
The last example relates to Common Academic Scholarship exams. Over the last few years this paper has seen a noticeable shift towards more visio-spatial questions. Certainly five years ago one would not have expected to see a graph plotted and certainly no transformations. A recent exam appeared to feature more of them than ever. One of my top mathematicians sitting the paper E was, as with the first two boys, very strong algebraically and computationally. However, ask him to draw something and not only would he be a slow starter, his work would be sluggish, messy and inaccurate. He gained his scholarship to a good Public School and came top in the Maths paper, despite it not playing to his strengths.
So what do these three have in common? If we deal with the obvious first, all of them are boys and all of them are from non-British backgrounds. A and E were both raised in the UK but are from Sino- and Indo-asian families. D only came to England recently and was raised by his family in his homeland. All three cultures emphasis the importance of computational prowess as a merit-worthy skill like the Japanese with their abacus clubs and competitions. Deductive reasoning and the graphical representation of processes are, however, less promoted; the English curriculum features much of the latter area and the focus of testing at scholarship level for independent schools features the former in particular. Does this then mean that boys, especially with a background of another culture, are disadvantaged either in accessing the National Curriculum or in attempting the hardest scholarships to top Independent schools? Certainly the shift in the Common Academic Scholarship seems to favour the girls I teach who generally prefer questions requiring neat, careful construction rather than all hands to the pump full on algebra tasks.
My best two pupils in the last decade have actually been both from a WASP background. F was, like B, considered by his classmates to be in another league from them, and was almost autistic in his single mindedness to tackle questions. He had been home tutored for some time, with clearly a focus on computational processes, but he, like the other boys mentioned so far, found the trickier deductive elements of JMC beyond him and only achieved a Gold Certificate. B went one better and took the JMO, just gaining a bronze medal. C not only took the JMO, but scored significantly more than B and received a medal and extra mentoring. He was not as good a pure mathematician as any of the other boys I have mentioned here but he was far more flexible in his approach to problems. A, B, D, E and F were all well-programmed calculating machines: give them a question to compute be it numerical or algebraic and they all knew how to work it out and would do so with relentless precision. Require then to think laterally or even worse draw their solution and they were, relatively speaking, sunk. C on the other hand would struggle with the hardest algebra but could tackle the most fiendish logic problem and construct or plot as well as any girl I have taught. So which would be the better mathematician: the machine or the thinker? Perhaps I have just been unlucky not to meet a pupil with a foot in both camps but given that I have been teaching for over twenty years so will have taught in that time well over a thousand pupils, and often the top sets, they can’t be that common. It is worth pointing out at this stage that the lists of top awards in the JMO are consistently populated by Sino-Asian names but, with no direct experience, I cannot comment on where they were brought up and how much of their success is to due calculating and how much to thinking.
Finally back to the two girls, using different parts of their brain at the same time to carry out two simultaneous tasks. Clearly they are the best of the best but are able to compute, visualise and also be flexible [given that they don’t know what word the other girl will come up with] all at the same time. It doesn’t seem unreasonable to expect to have found more pupils that can at least do all three not at the same time. What does that mean for how and what I teach particularly to my best pupils to encourage more ability in their weaker areas? Food for thought as I consider my syllabuses for the year ahead.