# The Mastery of Multiple Representation

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.

# Illuminate: Gifted and Talented at Key Stage 3 School Reviews

This is a shameless commercial post because I am really excited that schools who have bought our Illuminate Gifted and Talented Course for key stage 3 have posted on-line reviews on the National STEM centre web site. Obviously I would only be saying this if they like it, but they really like it a lot and that is really exciting.

Our aim was to produce a course in mathematics, so that school students had the opportunity to see what Maths is really all about. It is full of puzzles and games and tricky things to think about. But it takes them to the next level by unpicking fundamental ideas notably proof and isomorphism and giving students an incite. Maths gives a way of definitively saying how we know what we know. We use Pythagoras Theorem to unpick the idea of proof. From the essential structuring idea that sets up the proof to the language needed to be clear and the sequencing of the statements to construct the complete argument. It is thrilling that schools are reporting that students are able and interested to work on this. It is hard, but interesting things are, but students are game to carry on. Then we compare cyclic and Klein groups with isometries and modulo arithmetic. I cannot think there is anything more wonderful for the beginning mathematician to see that we can show that two complete areas of operation, so apparently dissimilar as arithmetic of clocks and transformational geometry have exactly the same underlying structure and hence, if we know something about one, we necessarily know the same thing about the other. That, to me is what maths is really all about. The mechanical processes that students learn for their GCSE and A Levels give no insight into this amazing world.

So, well done to those schools for being brave enough to work this way and really well done to the students who are becoming serious young mathematicians. Clearly we would be delighted for you to try it too. Just ask for some trial materials of the Illuminate course.

Also, come to ATM sessions and meet Danny Brown. Danny is the head of maths at the Greenwich Free School and he is getting his kids working on deep mathematical ideas all the time. Danny has presented regularly to ATM London Branch and has a web site of the amazing stuff he does. I persuaded Danny to get this out in book form and the first volume, on Number, is nearly ready, so look out for that.

# Assessment

I had an interesting conversation with a former maths teacher who was telling me how much she disliked ‘investigations’. She said that you could never tell whether a student had done the work themselves or if their Dad had done it for them. It was clear to me that steering the conversation round to wondering about the difference between investigating mathematically and submitting GCSE coursework, wasn’t going to get me anywhere, so I had to nod and force a polite smile. On one level it was deeply depressing how pleased maths teachers were when GCSE coursework was abandoned for maths exams. On the other, the whole process had been so discredited … Continue reading Assessment

# Maths Events

I was asked to run a session for PGCE students using a kit of parts we make called Maths-for-a-Day. Basically, we took the content of the kits I had produced for the shopping centre events I organised during Maths Year 2000 and packaged them up in a box suitable for a school maths event. I asked for 6 volunteers from the group to staff the activities and the remainder were punters. Continue reading Maths Events

# Maths Puzzles: Sustaining Activity

At higher level GCSE, it is possible to get a grade B having got 60% of the paper wrong. Since this is the benchmark for moving on to an A level course, there could be a concern that students could decide, say to avoid learning algebra and concentrate on geometry and statistics in order to get the B (or vice versa). In the main, the questions that they choose will have one or two steps at most to a solution, or if more are needed then guidance will be offered in the form of question structuring. In these circumstances, more extended A level questions, where the mathematics required may cover more than one area, would prove a significant culture shock. Continue reading Maths Puzzles: Sustaining Activity