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.
Well, mathematical modelling now has a serious place in key stage 4. So, get yourselves ready. Do not look through the document looking for any coherence, though. You won’t find it. This is another pot-pouri. Some very odd things like frequent reference to mechanics as an example of mathematical modeling, even though it is not taught as that at A level. More a collection of known models being applied. The real trick is to get students to develop their own models critically and develop methods for validation. That’s why real engineering projects go through more than one development iteration and A level mechanics problems do not. My Pizza Project article develops the model creating phase here. Also, Venn diagrams are back for probability problems, but set theory is not. Vectors in different format are back, but matrices are not. My favourite is the explicit teaching of Roman numerals (up to 100 in year 4 and then, I kid you not, up to 1000 in year 5).
Mr Gove was on question time yesterday. We cannot learn to be creative until we have a through grounding in the facts and techniques needed, he says. Everyone agrees with this (including our good selves). The trouble is, that no-one questions which facts and techniques and for why. The secretary of state himself quotes long division as an example. But could he tell us which creative mathematics is opened up by being able to do long division. Certainly, at A level we can divide polynomials this way, and for sure, unpicking the process to see how it works, provides deep insights into the power of place value, but, as a technique to be learned, it is just a pandering to an imagined perfect past. The trouble for us, is that everyone … the man from industry, the children’s author and all of the politicians agree with him. I say get kids to memorise Pascal’s/The Chinese triangle and chant their squares, and cubes. That would genuinely help them engage with maths creatively. But 11×12? Why? Is old money making a comeback?
So, there’s a big opportunity. Some interesting if oddly chosen hard maths, that modelling word and even ‘proof’ is in there too. An absence of levels is a major blessing. But, ordinary kids have to be able to do this. Escalante got all of his students to AP calculus with ganas. We must be able to do this too. The stakes have been raised.
Two weeks ago, I had a very nice trip down to Cape Town. It is a very beautiful city indeed. However, I did a series of sessions mixing HP Graphing Calculators, GeoGebra and Data Streaming to groups of maths teachers, trainee maths teachers and undergraduate engineering students at the Cape Penninsula Technical University and the University of the Western Cape. South Africa, in the post apartheid era has been trying to bring all of its education systems up to the level of the former elite schools. As you can imagine, this is a tough task, although the government’s commitment is clear having one of the highest proportional spends on education anywhere in the world. The universities I visited are excellent examples of that move to change and I was delighted to work with really enthusiastic students and teachers. Continue reading Cape Town Maths
The ATM/MA london Branch was treated to on of Peter Ransom’s barnstorming performaces last Saturday. A big message that we share with our PGCE students is that teaching is a performance art and ensuring that your lessons have a good dose of theatre will bring students in to your message. Well, Peter brings avery big dose of theatre. Right down to the brilliant stand-up touches … is he really going to drop the cannon ball? Well, yes, naturally. We got through transformational geometry, force functions in suspension bridge chains, cannon ball stacking sequences and the destructive impact of cannon balls by linear and quadratic scaling. So, no messing maths. Please come back soon to see the photos … and come to our next session which will be 10:30 Saturday 24th March (King’s College London, the Franklin Wilkins Building on Stamford Street, SE1, just down from the IMAX cinema), which is the Danny Brown maths Workshops. Read all about it at Danny’s site: www.makemaths.com
To any teachers out there who come across my blog, we have a competition:
WIN 30 HP40GS calculators and a data streaming kit (worth £2000)
We are setting up an exemplar STEM teaching room for teachers. We need top quality activities to do in the room, which will be premiered at the Olympia BETT show. We need a detailed exposition of the activity together with any supporting materials (e.g. worksheets etc.) In entering, you give us permission to use the activity (suitably credited) in the STEM room and in a booklet which we will produce with all of the entries and which will be available free to teachers. We would like activities using HP39/40GS or 50G graphing calculators, but any other software which runs under windows (and therefore on the new HP windows 7 slate) would do fine. The activity must be an application which merges mathematics with science or engineering. Send one zipped folder of files with your full contact details to: firstname.lastname@example.org. Closing date is Friday 9th of December.
So, why do we teach students maths in school? How tempting it is, to say “because it’s useful”. Well, I defy anyone to respond to this post by finding a single example taking from a school maths text book, in which something happens that (a) could be described as useful and (b) happens in the manner that it might do if someone were actually doing it. Continue reading Applied Maths