I was invited speak at the Maths Mastery conference in London last month. My essential brief was to engage with using technology to support mathematics learners. I have increasingly wanted to take a wide view of things so I interpreted technology very liberally. Having trained teachers in using graphing calculators for both TI and HP for many years, I am well versed in the language of multiple representations. The possibility is there to see a function represented graphically, symbolically and as a table of values and to swap quickly between them and to see how each representation gave you different insights into the nature of the function.

For example, a linear function has a particular algebraic form, it has a straight line graph and a table of values with a common difference. We exploited all of these in our Pizza project, showing that the natural tendency when watching change over time (the declining temperature of a cooling pizza) is to look for a linear change. The difference is roughly equal over the minute intervals we used, for the 10 minute length of the experiment. This is forcefully confirmed visually when a real time graph being drawn is very nearly a straight line. So, we feel empowered to hypthesise a linear function which symbolically can be used to calculate extrapolations. It is these that undermine our initial thoughts (put time = 24 hours into the function and we quickly see there is something wrong). Then we can go back to the graph and change the axes to see the nature of the slight curve and look again at the nature of the differences from equal differences, which themselves have a pattern.

It is clear that this notion of multiple representations runs throughout mathematics mastery. Having run ATM branch for such a long time it is good that the Singaporeans who kicked the mastery thing off did fully acknowledge that all they were doing was recovering the work of the founders of the ATM. The ATM started as the association of teaching aids in mathematics. The teaching aids of Cattegno, Dienes, Cuisenaire et al. had largely been removed from school classrooms, especially in secondary schools, but are now making a welcome return. The physical manipulative is a powerful representation. Converted to a picture of itself it is a diagram and both of these represent some number or calculation or more. Teachers show pictures of things and assume they are the thing. A picture of a chocolate cake is not a chocolate cake. (Ask Magrit for more on this and let the NCETM know). A graph is not a function, nor is the symbolic representation. Developing mathematicians need to learn the art of switching views. So, teachers need to give them opportunities to do so.

So, we have computer technology, manipulative technology and I finished with human technology. The learner experiencing the mathematics within themselves. I started with the classic maths gym where everyone holds their arms in the shape of different graphs. I do linear functions varying a and b in f(x)=ax+b (after some errors, everyone knows what the a and the b do) and then quadratics f(x)=ax²+bx+c (here everyone knows what the a and the c do. But what does the b do?) It is always good to find out what you don’t know. Feeling it within yourself is however a powerful experience. More dramatic (but in truth I only got enough time to say it), is to solve puzzles as a human team. The frogs puzzle and the tower of Hanoi (correctly the tower of Brahma) where a team each play the part of one of the pieces. No communication of any kind is allowed. So, you have to feel your own part in the process. This yields insights of a qualitatively different type than is possible doing the whole thing yourself. Teams have done this in the Mayor’s Fund’s count on us challenge (that I run for them) and become so good we had to abandon it. We now get teams to compete as the counters in a game of Hex on a 4 by 4 boards (drawn with huge hexagons on the floor). They still find this nicely hard. Try it.

So, take a wide view. Mastery is rooted in multiple representations (and in the ATM), but the technology that can be used to represent them are many and varied, as are the representations themselves.