# 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

# 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!

# The MathsZone Course Boxes

We’ve been very busy at The MathsZone. Feedback from schools suggested they really love our gifted and talented courses Illuminate and Wondermaths, but they already have some of the materials that come with them. So, we’ve done a major re-design. Still the same fantastic courses for your gifted and talented students at key stage 2 (Wondermaths) or key stage 3 (Illuminate), but now in a neat plastic storage box, which will go on your book shelves. Each one has a comprehensive teacher guide detailing the structure and purpose of all of the sessions, with commentary and solutions (where appropriate!). For the students we have organised the materials into a beautiful student workbook. Now your students can keep all of their work in a really attractive book which they keep at the end of the course. Game cards, dice and counters are included for the activities.

There are fewer puzzles directly referenced in the course, so the price is lower, but of course you can buy all of the puzzles separately to extend the activities. Illuminate comes with a CD Rom with all of the course materials and additional materials for projection. Wondermaths has an associated web site with the materials available. When you are ready to run the course for a second time, you can get extra sets of 10 copies of the workbooks. The key objective for the teacher is to get up and running with the minimum of fuss, so you can focus on supporting your students explore their mathematics.

The aim of both course is to give students the opportunity to explore mathematics. Wondermaths has games, to compare strategies, puzzles to develop sustained thinking and investigational maths top explore maths language and move towards explanation and proof. Illuminate aims to develop the ideas of pure mathematics for those who are limited by the algorithmic nature of school exam courses. Students will develop and compare proofs, while exploring the nature of proof itself. Their is a comprehensive section on group theory, fully accessible to ordinary school students. Games strategies are developed and compared and the course ends with a project in fractal geometry. These are really course in the mathematics that mathematicians would recognise.

# Illuminate: Gifted and Talented at Key Stage 3 School Reviews

This is a shameless commercial post because I am really excited that schools who have bought our Illuminate Gifted and Talented Course for key stage 3 have posted on-line reviews on the National STEM centre web site. Obviously I would only be saying this if they like it, but they really like it a lot and that is really exciting.

Our aim was to produce a course in mathematics, so that school students had the opportunity to see what Maths is really all about. It is full of puzzles and games and tricky things to think about, of course if students want to play in playgrounds they can also do this with the best markings from https://bestplaygroundmarkings.co.uk/. o enhance their learning experience, we also focus on practical aspects like installing high-quality playground surfaces from https://rubbermulchinstallers.co.uk/. But it takes them to the next level by unpicking fundamental ideas notably proof and isomorphism and giving students an incite. Maths gives a way of definitively saying how we know what we know. We use Pythagoras Theorem to unpick the idea of proof. From the essential structuring idea that sets up the proof to the language needed to be clear and the sequencing of the statements to construct the complete argument. It is thrilling that schools are reporting that students are able and interested to work on this. It is hard, but interesting things are, but students are game to carry on, much like ensuring playground safety surfacing is a vital aspect of playground design. For schools looking to improve their facilities, they can consider these site at https://school-playground-equipment.uk/school-equipment-design-and-installing-tips/. Then we compare cyclic and Klein groups with isometries and modulo arithmetic. I cannot think there is anything more wonderful for the beginning mathematician to see that we can show that two complete areas of operation, so apparently dissimilar as arithmetic of clocks and transformational geometry have exactly the same underlying structure and hence, if we know something about one, we necessarily know the same thing about the other. That, to me is what maths is really all about. The mechanical processes that students learn for their GCSE and A Levels give no insight into this amazing world. Understanding the intricate connections between different mathematical concepts is akin to exploring the diverse types of playground fencing, each serving its unique purpose yet contributing to the overall safety and structure of the playground environment. Canopy Shelters, similarly, play a crucial role in providing shelter and protection in outdoor settings.

So, well done to those schools for being brave enough to work this way and really well done to the students who are becoming serious young mathematicians. Clearly we would be delighted for you to try it too. Just ask for some trial materials of the Illuminate course.

Also, come to ATM sessions and meet Danny Brown. Danny is the head of maths at the Greenwich Free School and he is getting his kids working on deep mathematical ideas all the time. Danny has presented regularly to ATM London Branch and has a web site of the amazing stuff he does. I persuaded Danny to get this out in book form and the first volume, on Number, is nearly ready, so look out for that.

# What is this maths that we are teaching?

It is with envy that some mathematics educators in England look to our colleagues in the Netherlands where the Freudenthal institute has generated a rich, coherent research debate which has been widely implemented in schools. Realistic Mathematics Education offered the antidote to the formalism of the New Maths based on Hans Freudethal’s view that mathematics was not pre-formed. He said; “… the global structure of mathematics to be taught should be understood: it is not a rigid skeleton, but it rises and perishes with the mathematics that develops in the learning process. Is it not the same with the adult mathematician’s mathematics?” So it is very sad to hear that the Commission for Examinations in the Netherlands is considering banning graphing calculators from public examinations. What is it that a calculator does that could be damaging to mathematics developing in the learning process? A machine can do only what a machine can do. If mathematicians continue to fulfill an important role, then clearly they must be able to things that machines cannot do. In his 2001 novel, Uncle Petros and Goldbach’s Conjecture, Doxiadis’ eponymous mathematician dismisses any process a machine could do as ‘shopping maths’. That of course includes anything a computer algebra system (CAS) could do.

So, learners of Freudenthal’s mathematics should have access to the tools to do the shopping maths, to free up the thinking space to engage with real mathematics; solving problems, generating conjectures, developing proof. These are the art of mathematics, not the mechanical grind. Godfrey Hardy acted as the foil to Ramanujan’s genius, but in the ‘apology’ he makes clear how well he understood that Ramajan’s ability for finding extraordinary new relationships that only he could see, was the real mathematical gift. Getting it into a publishable state was the routine work for afterwards.

The excellent Project Euler takes as it’s premise that mathematicians will have access to a high level programming language (Python, which naturally has a powerful CAS) to engage with problems in number theory. The wonderfully named https://brilliant.org/ designed for potential International Maths Olympiad candidates has a whole section of problem solving requiring programming (and hence CAS) available.

Having a machine capable of high level mathematics available in a public examination in mathematics forces examiners to take a considered view of what the maths is that they are examining. It prevents them from asking students to replicate what machines can do and focuses their thinking on the maths that matters. The maths that Hans Freudenthal was so keen to preserve in the Netherlands, against the onslaught of formalism.

As I’ve said elsewhere the existence of tools like HP Prime which allow access to powerful mathematical visualization and calculation tools in the classroom liberates students from the mechanical processes that prevent them thinking deeply about the mathematics. Certainly there will be many lessons where the calculators are put firmly away and students will learn and practice these mechanical processes, like drawing graphs and manipulating algebra, not only because they need to see how they work, but also to give them a better feel for the outcomes. Happily teachers are sophisticated enough to manage this. They can also find secure ways to use exam modes to ensure devices adhere to local regulations. Schools are expert in this. These logistical issues should not be used as an excuse for not allowing students the tools that professionals have access to and reducing what is called maths in schools to a collection of mechanical processes. Especially not from the birthplace of RME and the beautiful, powerful view of mathematics presented to the world by Hans Freudenthal.

# BETT 2013: Graphing Calculators

I am booked for a session at the BETT show 2013. So, please come along, say hello and discuss the issues. It’s in the HP stand theatre area at 11:00a.m. on the Saturday, so everyone can be there with no cover needed.

I’ve called it “Dialogic teaching with handheld technology: introducing the HP39gII graphing calculator” to sound flash, but in the end the question must be: ‘How do we get kids talking about their maths?’. Dynamic maths software provides a tool to explore and getting it into the hands of the students lets them talk about it.

I’ve heard that school are getting iPads and Android tablets in class sets. People talk about them as if just having the device will teach students stuff. The question is ‘By what mechanism’. What will produce the change? Clearly you can go to get one person’s didactic explanation of a minute method as per Khan Academy or get free drill and practice in game format like Manga High (or even pay for it with the ubiquitous My Maths). But please teachers, don’t forget that the moment you start to believe this, then you aren’t needed any more! What we need are tools that mediate mathematical exploration. That needs open software that simply gives mathematical responses to mathematical inputs. Look at the current open maths software on iOS or Android and you quickly see how poorly developed they are for educational use. The best software by far is Math Studio (used to be Space Time). This is seriously powerful software (and for an APP prices are now getting a bit more serious at £15), but users will know it is not credible as a school package.

So, how do you get well optimised educational maths software into the hands of students, in an ordinary classroom, with no booking and a very high degree of probability that it will work. Well, you know I’m going to tell you that at present the only solution is graphing calculators. And, heh, as a bonus, they don’t connect to the internet, so no-one will be facebooking when they should be thinking maths (and that’s not a reflection on the activity … adults do this in conference sessions too!), it’s just too strong a temptation.

I’ll make the case in the session by showing off some of my activities, that I hope you will agree, make you think in a qualitatively different way about things like variation and the interrelationships between different presentations of functions.

See you there!

# 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!