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.

What does it take to reach grade 5?

The availability of transcripts from the June GCSE EdExcel maths papers provides an extraordinarily rich resource to try to make sense of students’ thinking in engaging with the questions. I spent a rewarding couple of hours with a head of maths looking at questions in the ‘crossover’ i.e. where they appear on both the foundation and higher papers. That would imply that these are all targeted at grade 4 or grade 5. One very clear and immediate conclusion to be drawn is that it is easier to get a 4 if you take the foundation level exam. Comparing higher and foundation responses to the harder crossover questions, a 4 could be achieved without success in these but a 5 was not achieved even with success on some of the crossover questions at higher. This will need a more detailed analysis, but anecdotally it seemed very clear. This HoM’s school achieved outstanding results at grade 4 with a strategy of erring on the side of entry at foundation.

Continue reading What does it take to reach grade 5?

Research Based Pedagogic Tasks RBPTs

This has been a long time coming, but I intend to restart sharing thinking about work I’ve been doing in projects around the world! I am part of a Sheffield Hallam, British Council project for the Indian Institute of Science Education and Research in Pune. The idea is to work with teachers of undergraduate science and maths to develop tasks requiring problem solving skills in settings of genuine benefit. The problem with maths, is that the problems are frequently unrealistic or too simplistic. One of our co-tutors wrote to ask my thoughts on this … Continue reading Research Based Pedagogic Tasks RBPTs

WOW Maths: Indian Maths Scheme

Publishers are feverishly producing complete new sets of text books for English schools using consultants and academics from Shanghai and from Singapore. This comes on the back of a Nuffield Foundation report suggesting that the poor quality of text books is a key issue for schools in England. The overwhelming demand to teach to the test is now accepted, with even Ofsted saying there was; “too much teaching concentrated on the acquisition of disparate skills that enabled pupils to pass tests and examinations but did not equip them for the next stage of education, work and life.” So, in a major crisis of confidence for English maths educators, we look to the successful TIMMS nations. But what are the key elements of the Shanghai Maths ‘Mastery’ and Singapore Maths? Well certainly in Singapore they are keen to assert that this is a reworking of what was best in maths education in the UK from the early days of the ATM with Dienes, Gattegno and Cusienaire. Continue reading WOW Maths: Indian Maths Scheme

HP Prime in the Palace of Westminster

Next week Tim Peake is going to the international space station. This is an important moment in UK space. The UK space agency has organised a whole raft of school activities, events, resources and so on to coincide with his mission. One of these is the Space to Earth Challenge. The essential idea here is that pupils will find wacky and amusing ways to travel the distance that Tim will travel in returning to Earth and track their progress in doing this. I was commissioned by the project and HP to write a set of maths activities to develop the mathematical thinking involved in the parallel science and PE activities that had been developed. Continue reading HP Prime in the Palace of Westminster

Technology and Maths Exploration

20151008_110555 - Copy (450x800)It is always good to get the chance to actually teach real kids in a real school. I have always said that if you are going to teach students about linear functions, it would be a crime to do it without any technology. So, up comes the topic and in I go with the technology. We have organised a class set of HP Prime handhelds for the school and my job was to get students started using them, so they would be sufficiently familiar at the start of the linear functions topic. Continue reading Technology and Maths Exploration

HP Prime Instant Set Up and Update

I had the opportunity to visit the very beautiful Swiss city of Zurich last week to run a session for Swiss senior high school maths teachers. It is always very interesting to see the differences in the way that mathematics teaching and indeed schools are in different countries. Firstly, the school is for ages 16-19 and has roughly 3000 students in a city centre environment. Striking interior architecture, excellent catering facilities, a range of fascinating teaching spaces and a very well equipped presentation room were all very impressive (and that’s not to mention the three floors of underground heated car parking!) Continue reading HP Prime Instant Set Up and Update

Talking Maths in the Esoteric Domain: HP Prime Wireless

At the ATM London Branch conference on Saturday, Kate Gladstone-Smith from Langdon Park School in East London, presented her research into the nature of communication she had observed in maths classrooms and how this differed according to the set, the students were in. (Anyone not from the UK will need to know that in English schools teachers decide in advance how well students will do with a subject and place them in ‘top’ and ‘bottom’ sets (i.e. class/teacher groups) accordingly). Continue reading Talking Maths in the Esoteric Domain: HP Prime Wireless

Wireless Prime has arrived!

I took delivery of a box of new Primes with the wireless kit last week. This is really exciting. From a pedagogic point of view, it seems to me that the big move is to generate genuine classroom dialogue, supported by serious technology. The Prime solution gives you enough machines for a class, in a box you can easily hold in one hand. You give them out to your students. They turn them on. You launch the connectivity software on your PC and that’s it. Everyone is connected. Continue reading Wireless Prime has arrived!