It seems clear that despite grade 4 being the critical judgement point this year, it will not be for long. Top Universities and the professions are sure to be looking for grade 5 as soon as they catch on. See this UCAS briefing for Universities as early evidence.

The outcome of this effort is a plan to prepare input sessions for year 11 students in the Spring term to support them in exam preparation strategies. Broadly, there are two critical question types which are of a different character to previous exams; (i) context based questions with problem solving, (ii) ‘show’ (or of course ‘prove’) questions in the sense of mathematical proof. The latter are the re-emergence of a traditional question type and need technical mathematical skills. In this session, we only dealt with the former.

I will share some of our observations that lead to action points for students:

• EAL is an issue which is increasingly ignored as students show high levels of facility in spoken English. Please read British Council Report on EAL Teaching to remind yourself that technical English takes time to acquire. So, action 1: read through the question slowly, twice. Underline the key words that give information you will use and that tell you what to do.
• The issue is to unpick the question, recognising it solely as an exam question. Resist the temptation to see this as ‘problem solving’ or ‘real world’ in any way. The best example being the 5 metal strips in a rectangular framework with one diagonal included. The diagram is clearly not five metal strips, it is a geometric diagram even to the extent of including the right angle marks. The maths teacher is well versed in ignoring the relationship between ‘context’ and diagram, so it seems trivial. To learners believing this really is somehow about the ‘real world’ it can be deeply destabilising. So, action 2: highlight the set of words in the question that say exactly what is shown in the diagram. 
• We noticed that students who were successful, frequently had worked something out as their first step, before really knowing what they were going to do with it. This seemed to secure their involvement in the question. In the metal strips example, the obvious thing to work out is the length of the diagonal using Pythagoras’ theorem since the diagram (but NOT the context) shouted Pythagoras at you. So, action 3: work something out to get yourself started.
• Then, successful students seemed to organise the information such that they knew things to work out. The best example of this was the two stage trans-Penine journey where the average speed, distance and time for the two phases and the average speed for the whole journey needed to be found . Writing the formula for each stage, then filling in what was known and working out the rest, then for the whole journey, was effective (but done in a very different organisational way by different successful students). So, action 4: write down formulae or organise the information for the things you need to work out. Fill in the things you know, work out the others.
• The question often then asked you to do something with the outcome of the calculations and successful students had identified this clearly. The metal strips asked about the weight in the last line. It would be easy to have assumed it was a Pythagoras question and left it that. So, action 5: go back and check through the question to make sure you have answered the actual question that was asked. 
• We also saw evidence where students had done all of the things and still got nothing, because at an early stage they failed to work something out and then gave up. In multi-step questions continuing a well worked out strategy with one wrong value will get most of the marks, but these were missed because it was assumed it had gone wrong. So, action 6: if one calculation has gone wrong or you don’t know what to do, carry on with what you have or make up a sensible value and carry on.

This is a first stab at setting out principles. It is very important to note that students who followed these principles did it in different ways, so trying to set out anything hard and fast or too prescriptive will be damaging. Also, students who were successful with one question of this type, were sometimes not so with others. We will aim to gather a working group to continue this work as preparation for the student sessions.

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.

In school mathematics, the myth of application is ever present. Mathematics solves problems in the real world we are told. Except that we start from a curriculum containing specified mathematics, that must be taught, generally in a given order. So, of course, the real world is bent and twisted to fit the mathematics that needs to be learned. In our teaching of distance and time, how often does a car travel at a uniform speed, showing a straight line on a distance time graph starting at the origin? (Instantaneous acceleration is a useful phenomenon in maths lessons, but nowhere else). The pandemic has not thrown up many linear models, or even quadratic models. Learning the maths presented in a distorted and damaged context is worse than unhelpful. Either the learner knows enough about how the real world works to recognise that the maths lesson version is simply not real and therefore the maths is not a model of it, or worse they build their knowledge of the world through this damaged notion that cars really do travel with contant speed, starting from rest. So, when stuff actually matters, as it really does in this pandemic,  learners of school mathematics do not have the tools (they were not allowed to play with functions, instead they had to learn linear, then quadratic and never quite reached exponential) and would therefore expect reality to be required to fit. So, the growth rate graphs with the exponential scales, that then look much more linear, show the death rates in the USA only ever so slightly higher than those in Germany. This can only be because that is the way it was.

If we are to claim that maths has some use value and relavence to the word outside the maths classroom (as this year it really has like never before), then we need to engage our students in sense making. Looking at live data and playing with the maths to see how and where it might fit. With graphing software it is no harder to draw an exponential than a linear function and when it is driven by data that the learner has some investment in, then they will want to know. Building relationships between a felt and experienced world and mathematical objects useful for representing it, is complicated and muti faceted, but possible if we allow the setting to lead the maths rather than the other way round. I still have some ultrasound distance sensors which we would use connected to real time graph drawing software. Students would walk on a straight line towards and away from the sensor to create different distance time (and later velocity/time) graphs. In this way the relationship is created and really felt. The motion to create a sine curve as a distance/time graph is a very lovely thing. Equally our Pizza project had students find models for the cooling of a pizza from short term data (it looks very linear over the course of a 10 minute experiment, but what does that say about what will happen in the medium to longer term?) The implications of a model can only be engaged with in a context which behaves as the real world actually does.

I picked the Fibonnaci sequence as an example of interesting variation. 11×12 is not interesting, nor is it useful (except, I did find this example). To make sense of a world that behaves as the one our students actual live in does, they need to see what they learn as a examples from a range of useful tools that can, when critically applied, give us ways of seeing what happens in that world in a more manageable way. Linear and quadratic are examples of functions, of which there are many, many more. Times tables are number relationships of some power and Fibonannaci, Square, Triangle numbers also. So, when we see variation and relationship we take a flexible, open and always critical view of it.