Next week Tim Peake is going to the international space station. This is an important moment in UK space. The UK space agency has organised a whole raft of school activities, events, resources and so on to coincide with his mission. One of these is the Space to Earth Challenge. The essential idea here is that pupils will find wacky and amusing ways to travel the distance that Tim will travel in returning to Earth and track their progress in doing this. I was commissioned by the project and HP to write a set of maths activities to develop the mathematical thinking involved in the parallel science and PE activities that had been developed. Continue reading HP Prime in the Palace of Westminster

# Category Archives: Gifted and Talented

# EAL in Maths? Problem Solved!

Where we were working in South East London, a number of students would arrive in England for the first time in the middle of secondary school. They would have very little English language and would try to get into local secondary schools. The schools would turn them away because they assumed that these students would end up with poor grades and compromise their exam statistics. So, a unit was set up to support these students make the transition to school. I got together with Gwyn Jones to produce a course designed to teach the mathematics content of GCSE with the minimum of language, but developing the key technical vocabulary of maths and of school while they learnt. The materials were supported by online interactives to see the maths dynamically and practice the ideas in an open format. There was a very low language pre-test, so that the student could show what they already knew, a tracker sheet to choose the maths they now needed to work on, a large collection of activity sheets to develop the maths and a post test with the same language demands of a normal maths test to show the schools how good they were.

In the very first group of students to use the first version of materials there was a student who had just arrived from East Africa. He had been rejected by every school in the borough. He took the pre-test and got 100%. He worked on the advanced materials and did the same on the post test. He took his work as a portfolio back to the schools and immediately found a place. Within 18 months he had an A* in GCSE maths.

We are proud to announce that we have now redesigned and updated this course and made it available to schools. Called Access to Mathematics it comes as one of our course boxes (like our well known gifted and talented courses; Wondermaths and Illuminate). There is a comprehensive teacher guide with notes on running the course. Ten copies of the comprehensive student book (120 pages) and access to the online interactives, test, answers, etc. in the Access to Mathematics web site. Priced at £195 this gives access to mathematics for all of your students for whom English is an Additional Language from those who have just arrived with no English to those who appear to have conversational English, but cannot access or succeed at maths in lessons.

Everything is described diagrammatically, putting the maths into a visual structure. Two colours are used to emphasise the structure and the maths is practised through this structure, gradually peeling it away to leave the formal symbolic maths. The course worked well supervised by non-specialist teachers as it is designed largely for self-teaching. However, with access to a specialist teacher, the materials could be used for a whole range of learners where reading and language demands of any sort are an issue.

Once you have the box, further copies of the students books are available in packs of 10 priced at £45. So, you can use them as a standard class text if you want. The overall content is covers about 90% of a higher level GCSE.

We are very proud of this publication. We have so often seen excellent mathematicians languishing in low achieving sets simply because they are still learning English and find accessing conventional books difficult. Now, they can quietly and quickly show everyone how much they know and can do, while learning the essential school language that they need.

# 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.

See here: http://www.stemdirectories.

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 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.

# Gifted and Talented at Maths

About four years ago, I was asked if I could run a course for year 6 students from inner city London primary schools, who had been identified as ‘gifted and talented‘ in maths. Now, I’m troubled by this idea in general. What measures schools might use to identify gifted and talented is very hard to tell. My idea of a good mathematician is almost never the person who correctly answers all the arithmetic questions. However, the truth of the matter was that they were ordinary kids in ordinary schools. Continue reading Gifted and Talented at Maths