Tag Archives: Materials

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

Count On Us Challenge: Congratulations to Kender School!

Kender School in Lewisham are the winners of the Mayor’s Fund for London’s Count On Us Challenge.

Val from the Maths Zone takes on Johnny Ball
Val takes on Johnny Ball!

Very well done to them and very well done to all of the schools who took part. The grand final took place yesterday at City Hall with 13 schools who had won their way through heats and semi finals to compete at solving 24®Game puzzle cards. Each card has four numbers; you can add, subtract, multiply or divide in any combination, but you must use all four numbers to make 24. Continue reading Count On Us Challenge: Congratulations to Kender School!

Financial Education: How does that work?

From September 2014 it will be compulsory for schools to teach financial education. This will be built in to the mathematics and the citizenship curricula. See this article from the Daily Telegraph. Notice the picture of school life that they choose to illustrate the article with. This is how students learn; in rows at individual desks, looking seriously bored! The trouble is that a good proportion of the materials available for ‘financial education’ in schools is perfect for this scenario. Standard worksheet based discussion and practice activities are the norm and work in the same way that makes PSHE such a disappointing subject, taught by non-specialists, with no exam, it is hard to see the purpose when you are in school.

What makes these important things come alive is getting students into the setting. They have to care about the issue in order to engage with the ideas. I have seen fantastic drugs education sessions where former users and dealers have come in and talked to teenagers about their experience and where they are now. It is edgy, but it is real and they certainly listen!

Finance is tricky. Kids in school rarely have any real need to save with interest and if they have a bank account, their main worry is losing their cash. Certainly, they cannot borrow beyond their means or need to budget in a life changing way. Some, certainly, have life experiences that may necessitate any or all of these, but they are a small albeit important minority.

We have been working in financial education for over a decade. As a development of the Number Partners project which I was director of for many years, we designed a series of large format board games designed to set up scenarios in which players have to make important financial decisions: how to invest a small amount of capital, to generate profits to reinvest. Managing money between cash and different bank accounts, to enable purchasing but retaining security. Budgetting for a holiday and managing exchange rates. Making the life transition from school to work, while meeting your life goals.

The power of a board game is that the social setting frames the decision making. You are playing with real people who you have to engage with, framed by the settings of the games. The games were trialled in very ordinary schools, in classrooms with groups of students and have been widely used in different settings since. The effect is impressive. Students talk to each other about their financial decision making, developing strategies to succeed in that setting. Naturally, winning strategies involve good financial decision making.

We set up a web site to showcase the games. So see what you think. All of the games have teacher guides with extra materials and school use ideas. Please get back to us with your questions and thoughts. But, when you plan to deliver financial education this September, get your students into a setting in which they care. Only then will they be able to make decisions in a way that matters to them.

Playing Maths Games Makes You Do Better at School!

OK, so I came to this by being responsible for public maths events for maths year 2000. We had 22 shopping centre events, at the end of January 2001, where we set up staffed table stands with maths activities. It was humbling to see ordinary shoppers give up on Sainsburys and spend the day doing maths puzzles and games. So, the I end up being a part time shop keeper selling maths games and puzzles. It is just great to keep being reminded that people love doing this stuff. Continue reading Playing Maths Games Makes You Do Better at School!

Finding Good Maths Resources

The internet for teachers, blessing or curse? In the past, you would have a set of text books or work cards as your basic resource. The department would have bought a small library of additional books and materials from people like the ATM. If you needed a good idea, you would never have to look beyond the maths office or the maths cupboard (do you still have those?) Every department would have a pile of good physical manipulatives like centicubes and logic blocks, cuisenaire rods and probability kits. A set of large compasses and ruler for board work and a good collection of games and puzzles for activity days. There would be copies of those wonderful books by Brian Bolt (which are still available) for practical problem solving and a set of Points of Departure books for maths investigations. Always excellent, always to hand. Continue reading Finding Good Maths Resources