Isn’t it amazing how quickly and easily adolescents can learn the words of a song, and yet they struggle to remember mathematical formulae? Memory in mathematics is quite peculiar. In common with learning the words of a song, once you get started, it can be surprising how much comes back, but, as with songs, it may be the later bits that are harder to recall.
How are mathematical ideas committed to memory? Caleb Gattegno suggested that memorising something costs energy. You have to invest effort, and the process is not always reliable. Even when it can be relied upon, it may not actually inform learners’ actions. I recall a class in South Africa who could complete the chant “vertically opposite angles …”, but could not recognise pairs of angles as being ‘opposite’. By contrast, images (which usually have enactive, iconic and symbolic components – thank you Bruner!) come for free, according to Gattegno. Well, almost for free. As long as there is an emotional or dispositional component, as well as cognitive and enactive components. Having ‘seen’ something, whether simply visually, or even having appreciated and comprehended something, is not sufficient in itself to lodge in retrievable memory. The learner has to have some commitment, some disposition to engage.
I would like to develop these ideas in a sequence of blogs working on different mathematical topics. Take for example trigonometric identities. They are all based on a few fundamental relationships:
sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
cos(A + B) = cos(A)cos(B) – sin(A)sin(B)
Pythagoras (more about this another time)
sin(–A) = –sin(A); cos(–A) = cos(A); tan(A) = sin(A) ÷ cos(A)
sin(A – B) and cos(A – B) can be worked out from these, as can tan(A + B) and tan(A – B). If enough work is done using these to deduce other relations, they too will be lodged not just in memory, but will be available as actions to enact whenever needed.
I think that Gattegno used the term awareness to refer to ‘that which enables an action to be enacted’. So if you ‘have the awareness’ of the expansions of sin(A + B) and cos(A + B) as actions fully integrated into your functioning, then whenever you see A + B in the context of trigonometry, those expansions will become available to you to enact.
Consequently, I for one would begin the A-Level sequence on trigonometry by displaying those relationships on a poster in the classroom. I would tell learners that the poster was coming down in, say, two weeks, by which time they would be expected to have memorised them so fully that they know them forwards and backwards, both as expansions and as contractions.
During the two weeks I would get them to use those relationships to deduce a whole raft of others. This would serve to develop their facility with algebraic manipulation, which is well known to be a weakness for many learners on entry to university. It would also help them internalise those relationships, and appreciate how much can be worked out from them. For example, I would get them to complexify those relationships so that they become sensitised to recognise them in a variety of forms such as
sin(2A), sin(3A), …, sin(2A + 3B), … sin(A + B + C).
I would emphasise that none of these other relationships need be memorised as they have the facility to re-develop them from the original expansions. Then I would invite them to create complex expansions and challenge each other to recognise what had been expanded, such as:
sin(x + y)cos(x – y) + cos(x + y)sin(x – y) and then using these to deduce sin(A) + sin(B) as ‘half-angle formulae’.
One of the things I have picked up from Chinese research into so-called ‘Shanghai pedagogy’ is that it can be very effective to spend time on recognising complex versions of simple relationships, or simple relationships embedded in complex settings, because this enriches learners’ appreciation of the relationships and augments their example spaces.
Only when the posters came down would I show them how those two relationships are nothing more than the coordinate calculations for the points on a unit circle centred at the origin, making angles of A, B and A + B with the positive x-axis. The idea is to provide connections, which augment their comprehension, but only after they have internalised the expansions. The reason is that starting with the coordinate calculations is likely to focus their attention on the coordinates, and the slightly complicated calculations, leaving them with an impression of complexity and potential confusion.
For many concepts, it is useful to see them emerge in context so that their significance can be appreciated, but for some concepts it is more useful to gain facility with their use first, so as to be able to appreciate their significance before dwelling on how they arise.
Pedagogically it all has to do with making choices, which in my view is an opportunity for the teacher to experience a fleeting moment of freedom … but more about that anon.
All the best,
John Mason worked for the Open University Mathematics Department, where he designed two of the mathematics summer schools, contributed to numerous courses, and then helped form and run the Centre for Mathematics Education. He has written numerous books and booklets, as well as research articles and book chapters, the best known being ‘Thinking Mathematically’ (1982, 2010).