# Public Mathematics in a Pandemic

Not so long ago I had a very minor Twitter spat with a former student of mine in which I advanced the view that knowing the Fibonnaci sequence was more useful than instant recall of 11×12. I was shot down, with the suggestion that I didn’t care about working class kids. Well, I guess, I would like to claim that I care about all kids, regardless. And, it seems that being able to read and understand mathematical information has suddenly become a whole lot more important than previously. Oddly enough, memorising specific number combinations has not proved to be of any great importance in this. Nor, to be fair has the Fibonnaci sequence cropped up. But, of course making sense of patterns and relationships absolutely has. We now live in a world in which it is expected that we can look at a graph with a logarithmic scale and read it sensibly. Also, we are being told (as I write) that the rate of increase in case numbers is decreasing. From this we need to know if the case numbers are going up or down and thus, what is in fact decreasing.

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.

# 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.

# 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

# Technology and Maths Exploration

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

# 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!

# 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.

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!

# IT supporting kids learning in maths: what is the problem?

I completed my PGCE in 1983 (oh my!) and went to work in a comprehensive school in Corby new town in the East Midlands. (Then it was the largest town in England without a railway station, somewhat depressed by the closure of the largest steelworks in Europe). The walls of my classroom had a large bench running all the way round. On this bench were set out about 8 RM 480Z work stations. For anyone who doesn’t remember, these were competitors to the BBC Micro. When I taught transformational geometry, I could pause in the lesson and get my students to gather round the computers and engage with an activity I set up for them where they would create a shape and transform it using LOGO. They would make hypotheses and test them, seeing the result immediately, visually, dynamically.

I have recently observed a number of lessons on transformational geometry in London comprehensives. Despite every classroom being fully equipped with a networked computer and an interactive whiteboard and in every case, the teacher having been trained within the last year on using GeoGebra to teach transformational geometry, not one single diagram moved at all in any of the lessons. Students were shown object and image and asked what transformation connected them. An agreement was reached (often with much disagreement and uncertainty) and that would be that. There was no way that anyone could validate the agreement or see the transformation enacted. This is the traditional teaching method of ‘proof by teacher says’ or its slightly more inclusive counterpart ‘proof by agreement’. Now, just in case anyone who was there in the room with me can recognise themselves, I should share that everything else about all of those lessons was really good, sometimes quite outstanding. It is simply that giving kids experience of the mathematics, rather than showing them how it works, seems to be such a long way from conventional school practice, that even with everything else in place, teachers find it hard to achieve. Yet in 1983, it was just what you did and we had reliable technological tools ready in the classroom to support it.

I have had lengthy discussions about technology in the classroom with colleagues in teacher education and most recently I have heard about the various classroom manager systems that are being developed by the hardware companies and the IWB people. The essential premise is that you connect to handheld devices that the students have. The screens of their devices are available in thumbnail format on the teacher machine and hence the classroom screen (and able to be enlarged to show the whole class the work of an individual). The software has polling and analysis, so questions and messages can be sent and answers received and engaged with. With this level of technology available, it will again be possible to do what I was happily doing in 1983, interrupting an ordinary lesson in an ordinary classroom to engage with an idea dynamically using technology and seeing what the students are doing (I wandered round and looked at the screens and if I saw something interesting, I got the others to come over and see). At the moment, teachers feel they have to book the computer room to achieve this effect and we all know how unlikely/impossible that is.

But it is a compelling thought. Now, the teacher can manage the dialogue, setting a task, students can engage with the software and discuss the issues. When ideas emerge these can be shared with the whole class. A real dialogic engagement. So, what’s stopping us? Wheel the trolley of laptops in and they will connect seemlessly to the network with no fuss and then it’s OK? Of, course it doesn’t/won’t. Not least because controlling dynamic software from a track pad is a nightmare, but have you ever made a half class set of Laptops connect to a network? So, bring in the set of iPads the school just massively invested in. Agh. No manager software and as yet only a very cut down version of GeoGebra.

The Holy Grail is that everyone turns their smart phones on and launches the iOS or Android app they need, and we get some generic tablets for those that don’t have smart phones and these all connect. Even then we would need better software (unless you invest £30 a head for TI-nspire on iOS which is really good). I hope to get delivery of a trial set of HP Prime wireless graphing calculators very soon. Naturally, they do everything that that I have said. The massive difference is that they have an auto detecting dongle (the same as the ones that make wireless keyboards work). No installation, no logging in, if the device is in the room, the screen appears on the teacher machine. People say: ‘what’s the point of graphing calculators these days?’ I say: it is a piece of bespoke hardware with an optimised interface for the range of maths functions you need, with really well developed and well thought out maths software. Moreover, compared to iPads they are really cheap. They are small, easy to carry and importantly easy to charge. You just have to be able to grab the box on your way into lesson and hand them out the same as you would hand out rulers and compasses and they just work when you turn them on. Only then can we get back to 1983 and have technology seemlessly integrated into ordinary lessons in ordinary classrooms. Only now we’ve got rather classier software to play with.

I would like to work with anyone who is using any comparable kit that can achieve the same effect. I would be delighted to set up a research project where we can examine the actual classroom use of these technologies. I would be keen to hear from schools who think that this sort of kit will solve the problem of static teaching and would think they could use such technology all the time (not just special occasions). I would happily support such work with loan equipment and support materials. Contact me (chris@themathszone.co.uk).

Apart from the dodgy hairdos and the rusty cars, 1983 had things going for it!

# 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. You can learn more about modern finance if your visit Skrumble. You can also check out this helpful resource at foxinterviewer.com to navigate the tax.

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.