Category Archives: School Issues

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.

See here: http://www.stemdirectories.org.uk/scheme/wondermaths-gifted-and-talented-maths/#comments

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. 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. 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.

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.

This sad situation was brought to my attention through the English translation of a response by Erik Korthof to an advert for the new HP Prime graphing calculator. He suggests that the absence of graphing calculators in the past allowed the construction of ‘proper exams’. The task of mathematics education should not be to make the lives of examiners easy. Clearly, asking a student to complete a mechanical task that would be simply done by a machine is very simple. To construct a question knowing that the student has access to such a machine is hard. Specifically so, because the question must demand genuine mathematical thinking and that puts great demands on examiners. In the UK, the most progressive mathematics education project (MEI) for A Level students (age 18) have just had their first cohort complete an examination module with a CAS calculator available. The result is thoughtful, highly mathematical questions of exactly the type University maths courses are excited to see. The link will take you to their answer to Erik Korthof’s question: “Is secondary education served with a Computer Algebra System?”. Clearly they answer a resounding yes and MEI are major players in the future of maths education in England.

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.

HP Prime The New Future

HP_prime_front_pictureHP Prime will be launched ready for September and the new school year. Have a look at the teaser YouTube HP released to show you what it looks like. Last week I had one in my hands at a launch workshop in Prague led by GT Springer the lead designer. GT has been central to most of the major innovations in graphing calculator design and he has put all of that experience into a genuinely wonderful new device. Read the interview GT gave to the US tech blog Cemetech. First impressions matter to schools who want to show the smart new kit they are buying and to students who want something really flash in an era where new tech does indeed look good. It is interesting that after a stunned response at the NCTM conference in Colorado there has been a lot of buzz around tech sites like Slashgear and Ubergizmo. Well that’s good, because if the tech savvy think it’s worth talking about then bright young teachers and their equally bright students will take a look.

Being an old fogey myself, all I can say is that it looks very smart indeed,with a brushed aluminium front and a smooth bright screen. The colour is bright and very sharp with extremely clear detail and you just have to keep reminding your self that it is a touch screen and that you can drag and move objects and navigate drop down menus. The touch is smooth and very accurate. Younger folk than me will do this instinctively, I’m sure that they will be wondering how it could be done any other way. It is very well made and feels sleek and smooth all round. It is about 300g which feel sufficiently heavy to be solid but easy to hold and it balances really nicely in tho hands with your thumbs over the Home screen and the CAS button. You really feel you are holding a classy piece of kit. So, part one of the battle is won, savvy young people will want one and schools will be proud to show off that they bought them. So, what does it do?

The biggest headline is: wireless connectivity. Files can be transferred via the connectivity software. However, if you plug a small USB dongle (which you purchase seperately) into the top of the PRIME, it will immediately be recognised on the computer, notably the teacher’s computer in class. Files and settings can then be transferred wirelessly. (Only from PRIME to PC not from PRIME to PRIME). More than that, the PRIME screen can be shown on the teacher’s screen. Then their will be class polling functions allowing the teacher to set a question from her computer and students to offer responses from their PRIMES with the results shown in table and chart form. Just like the polling systems many schools are getting which only do this. That will be just the start of what can be done. The critical point is that this a plug-and-play system. No set up, which is a critical factor for classroom use.

The software itself initially looks like an up-rated version of the HP39gII, which it is, so you will find all of the Apps in the HP39gII working exactly the same. So, anyone who has used a HP39gII will get started immediately. However, there are three new Apps which make a big difference. There is a mathematical spreadsheet, a dynamic geometry system and the advanced grapher. Together these represent a major advance in providing an space to explore mathematical ideas. These tie together with the big pause for breath moment. The CAS button.There is no CAS/non-CAS option. A mathematical machine must speak algebra and this one does. There are two home screens; a CAS screen which deals with exact objects and the traditional home screen which deals with approximate objects. The Apps can use the last object from each of these screens and the choice is always there; CAS screen or Home screen. This recognition of the fundamental pure/applied, exact/approximate distinctions is central to an underlying philosophy which has the potential to transform the way we think about exploring mathematics. For me, this is the thing that will determine future research into maths education technology. The spreadsheet, the dynamic geometry and the advanced grapher can all take CAS and non-CAS statements and allow users to explore the results. Just to get a feeling for what this means, have a look at GT’s handouts from the NCTM conference.

Now the sad thing is that exam boards are scared of CAS and we look forward to a future where CAS systems will transform maths exams by getting beyond procedural questions and towards mathematical problem solving. Well done to MEI for getting an A-level module approved allowing CAS and look to Germany and Australia for examples where CAS is embraced. But in the UK CAS is not allowed. Well, no, CAS is not allowed in public maths exams for which any calculator IS allowed. So, it is quite clear that this machine has a CAS system, so could you use it in an exam? To be sure the answer will be yes, the machine includes a comprehensive exam mode. A menu system allows a vast range of features to be turned on or off, CAS is one of the, but suppose a particular exam disallowed solver apps, they can be turned off too. The system is password protected and the user will simply be greeted with a little round exclamation mark if they try to access or disallowed function or suppressed apps will simply be missing from the menu. For school use, the teachers sets the settings they want e.g. turn off the CAS, creates a password and then beams this setting to all of the connected PRIMES, wirelessly. A series of bright LEDS light up in the same sequence while exam mode is engaged. It is immediately clear to the exam secretary that the machine has only those facilities allowed in exams. In discussion with teachers, it became clear that this feature sets up the possibility to allow younger learners to get started with the machine in a simplified mode and actually presented exciting pedagogic possibilities too.

The exams battle is a big one and many schools still think you cannot use any graphing calculator in a maths exam, so we will need to talk with exam boards and the JCQ to make sure the message is clear enough: you can use this machine in a maths exam and without disabling it as an amazing teaching tool.

I’ve always been a fan of calculators as a learning tool. I’ve said elsewhere that tablets are exciting, but you don’t work and think like that, you need different technological tools for different functions and the resilience of the calculator as a form factor is remarkable  I think for this reason. It’s a highly portable, personal thinking space. I am really excited about PRIME because it has all of the maths you could possibly want with an intuitive touch driven interface, wireless connectivity to support proper classroom dialogue in a package that everyone will want to own.

Please get in touch with me if you see the video and want to be part of early development to get really exciting maths back into our classrooms. I would be delighted to talk to you about the support I can offer.

The New National Curriulum

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.

Graphing Calculator Workshops

I’ve had a really nice time doing a round of workshops for teachers and for PGCE and GTP students on handheld technology. I’ve always thought that ICT provides opportunities for teachers to invent interesting activities that give students deep insights into how maths works. It is interesting that today, pupil’s in primary schools are no longer to be allowed to be examined in their ability to solve problems with numbers harder than those they could handle by written or mental methods. Calculators are banned. Bizarely, the minister responsible justified this move in terms of the need to be able to handle numbers because maths “influences all spheres of our daily lives”. This maths is routinely done by engineers and scientists who would never stoop to using a calculator or indeed a computer to support their number work. The failure to get the sums right in the recent Virgin trains debacle was presumably caused by over use of calculators, except that the culprits will have been educated in an era when they did have get enough number work. An era that clearly never was.

I start with the neat teacherly trick of playing ‘guess the function‘ here the participants see a calculators giving values of f(x) for their values of x, letting them choose to get a feeling for the variation. I only show them a graph, when they have already formed a reasonable view, then watching as they focus on the details. The first thing is to realise that experienced teachers and well qualified trainees struggle to see a quadratic just form a small table of values. No doubt because the drill and practice pedagogy the present government is so enamored with means many will have only ever encountered a quadratic already knowing that was what the five points they were given to plot would show. But it is good to get a feeling for things and they see this. So, playing the game on the handheld with their partners strengthens the insights and makes them more flexible.

It turns out that lots of schools are buying sets of iPads, demonstration that there is plenty of money around. But the maths software available for iPads isn’t a patch on any graphing calculator and the storage, security and battery issues for anything you have to recharge means they will be no more reliable than laptops. In one group of 25 trainee teachers after about 4 weeks in schools only one had seen any handheld machines possible to use in ordinary classrooms. That was a school where every student carried a laptop with them at all times. It only came out later that in all this time they had not been used even one single time in maths lessons. A set of 15 HP39gIIs stored in a bag in the maths office with a few spare batteries and you just pick them up on your way in to class. I make the case that it is the teacher who prevents the use of technology. That is a bit harsh. Mostly it’s the technology. So use something which is no more expensive than a couple of textbooks and is almost certain to work.

Then we get back to seeing the resources we have as sites to conjour up really clever ways in to mathematical ideas. That’s what makes our job fun. Look at bag of dice, counters, centicubes and we should always be saying, OK what could I do with those that encapsulates a mathematical idea. A graphing calculator is just the same, it’s something we can use to give students deeper insights. It is in fact a calculator, and the scientists and engineers of the future should certainly learn to use it to support their number work, their algebra, whatever, so they can focus their brain power on being brilliant with the science and the engineering. But also it’s a pedagogic device. A clever piece of kit for clever teachers to do what is most creative about our jobs. Something that supports kids doing clever thinking.

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

Why Learn Maths

Just before Easter I ran a session for the A Level mathematics groups in the Harris Academy group in South East London. I can tell you it was pretty daunting in the small hall, but with something like 120 sixth form students, who had chosen me over another talk, about options, I think. However, can I publicly thank (a) the teachers at Harris Crystal Palace who invited me and most especially (b) the students who attended for reminding me what fun it is to talk about maths to young people. I’ll be applying for a teaching job again, next … Continue reading Why Learn Maths

Brunel AND Nelson in King’s

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

Maths Trails

The ATM/MA London branch meeting today was a wander round Parliament Square, up Whitehall and Round Trafalgar Square. Four groups of maths teachers made the trek and were intrigued to see this most famous bit of London in a different light. The trail is one of a number that I prepared during my time working for maths Year 2000 and it’s great to see it used again. To support the session, I set up a new web sight with the great URL of www.mathstrails.org.uk . You’ll find PDF and Word versions of all of my trails plus links and details of a load of other trails and trail related materials. Please visit and most especially, please contribute, you maths trail fans with your own ideas, materials and stories.

In the end, it’s just great to get out and about and look at things in a different way. So, take the opportunity and get your students out too!