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. The work of Bruner, Piaget and Vygotsky most significantly underpin their thinking. Physicial manipulative and iconic presentations proliferate. Bruner’s enactive, iconic and symbolic modes of representation and his notion, built on Vygostsky of scaffolding are the basis of the approach. However, in Shanghai, the key point is to ensure all students succeed together. Charlie Stripp on the NCETM site tells us that differentiation, by giving easier work to students identified as less successful can be “very damaging”. So, to those of us old enough to held the flame, ‘mixed ability’ teaching may be set to return. Notably, in Shanghai everyone practices but with “procedural and conceptual variation” i.e. well structured exercises developing the mathematical ideas by focusing on the key variation. An idea that would be familiar to those involved in the best text books from an earlier era.

I was sent a complete set of eight volumes of WOW maths from an Indian Publisher; E3 EduSolutions. This is an extremely comprehensive text book series with course books, work books and teacher manuals. Originally published in 2013 and now in a second edition, this is intended to embed the Singapore Approach for Indian schools up to roughly the equivalent of an English GCSE. It is interesting to see how in England we are only now reaching this stage. An central feature not seen in England is the availability of proper board games and other games and indeed the physical manipulatives that are embodied in the enactive and iconic phases: Dienes blocks, Mulitlink cubes and Cuisenaire blocks; these are all embedded in the scheme. The principle iconic mechanism taken from the Singapore approach is the so called ‘bar’ method. This essentially amounts to using blocks formatted in one or two dimensions for comparisons. A fraction wall is a classic example of this approach, however, here it is used as a jotting – a thinking tool. This practice is used throughout the series, so would be become second nature to users of the books. Also, Polya‘s general approach to problem solving is given a page and a photo at the start of each book, so problem solving is foregrounded.

It would be impossible to do justice to the details of such a large scale work in a blog post. However, this is a comprehensive and valuable work and it is fascinating to see the embedding of such a range of approaches very familiar to older UK maths teachers, which have largely disappeared from our text books. Everything starts from a diagrammatic or recognisable object, which is translated into an iconic form, which carries the mathematical content and thus to the symbolic form. There is an enormous amount of practice, which develops each idea in a coherent way. The whole is interspersed with open projects and ‘Lab Activities’. These are recognisable as ‘investigations’ the former with less structure and guidance the latter directed to the teacher, with more. Side bars provide additional thinking and discussion points ‘in daily life’, ‘discuss’ remember’ etc. Also, to the UK reader, the details of Indian life and indeed counting system differences are fascinating. It is very clear that these are directed at an Indian audience and are not generic.

Text books have a profound influence on how maths is taught. The influence of Shanghai and Singapore on English maths education is now very significant. The WOW maths series is a very serious effort to engage with these influences and is well established in India and therefore of significant interest to educators in England.

HP Prime in the Palace of Westminster

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.

The activities were formally launched in a committee room just of the old Westminster Hall in the houses of parliament yesterday. Maths was well represented with an activity called ‘Fitness Tracking’ which uses HP Prime’s 2-variable statistics App to compare a target improvement rate with the unfolding actual improvement rate as someone works through a fitness programme. This is a general purpose piece of maths that works great for any fitness programme, so pupils can choose what it is they want to develop, decide on a measure for it and a timescale.

House of Commons/UK Space Agency

House of Commons/UK Space Agency

Take a benchmark measure to get started and decide on their target at the end. I had about 5 minutes, so with my trial subject (a year 6 pupil who just did fantastically … such confidence!) we saw the improvement in her reaction time on a web based timer on my phone. The leader board on the App said 137ms was attainable and my subject started at 640ms, so we had a benchmark and a target for which we gave 8 attempts. You see me entering the target into the HP Prime emulator in the background. She got off to a good start, improving by about 30ms per go, but then the pressure (and possibly the phone) got too much and the improvements disappeared!

However, I think the set up makes very clear how a linear model can be used to generate a target improvement rate (in red) to compare against a linear model tracking the actual improvement (in blue) and so the maths worked really nicely.



This was for principally primary school pupils and the general public (plus MPs and space people!) However, I think we were all ready to talk about (linear) regression by finding the improvement rate in the fit equation, and then of course we would critique the linearity of the model;, … But the time ran out and that was saved for another day! However, in a lower secondary class, this is exactly what you would be able to do. There is a presentation, a teacher pack and a student sheet to support this in schools. Let me know how you get on.



HP Prime Instant Set Up and Update

I had the opportunity to visit the very beautiful Swiss city of Zurich last week to run a session for Swiss senior high school maths teachers. It is always very interesting to see the differences in the way that mathematics teaching and indeed schools are in different countries. Firstly, the school is for ages 16-19 and has roughly 3000 students in a city centre environment. Striking interior architecture, excellent catering facilities, a range of fascinating teaching spaces and a very well equipped presentation room were all very impressive (and that’s not to mention the three floors of underground heated car parking!)

In Switzerland, they make things. Engineering is highly valued and of extremely high quality. So, a relatively small proportion of these students will go to University and a relatively large proportion into apprenticeships. So, they are interested to see technology that enhances students’ ability to engage with the subject. The discussion was a comparison between HP Prime and TI-nspire. This is a good thing. No-one needs convincing that graphing technology is useful. People are not saying “I’ve just bought some iPads, what can I do with them”.

So, where does HP Prime win? Well, firstly the free teacher emulator and connectivity software kit makes the overall purchase price of Prime rather lower. Secondly the data streaming kit was seen to be very impressive. Instant collection of data, followed by analysis. No set up, just plug in the sensor, press start and watch. I did the experiment where you attach an accelerometer to the end of a steel rule, hold the other end on the desk and ‘ping’ the end with the sensor. It produces a beautiful exponential with trig function, where you zoom in and see the sin curves perfectly smooth. You can easily measure the wavelength to compare with other ruler types and model the decay rate with a two click export to a stats app. However, the biggest win is the simplicity of the wireless system. We were in a very long room indeed and all of the devices connected and stayed connected throughout the session. The total procedure was to plug the USB aerial into my laptop. Plug a dongle into each handheld. Launch the connectivity software and click the monitor button. Select the network on each handheld with three taps of the screen. That is enough to see what everyone is doing, find something interesting, expand that student (teacher/delegate)’s work and ask them to explain to the class. We collected data to compare height and shoe size and almost immediately were having a discussion about about the strength of the correlation (they said low) and it’s measure r=0.91 (which didn’t sound too low!)

The two issues we need to work on are to be sure the software will run on Macs and this should be soon now. Secondly, students develop their work as a project report. So, how can they integrate their algebra and graphs into a text report with commentary? On a desktop, running the emulator together with a word processor will work well enough. However, using the handheld or even working on a tablet, the integration may not be so smooth. I will have a think about these issues and post soon.

The final issue that is really working well is exam mode. With the wireless connectivity this is very impressive. On Prime the exam mode is so highly configurable that exam boards in many countries are now convinced that it is indeed suitable for use in public exams. But the set up takes seconds. The Swiss teachers seemed genuinely impressed by the control this gave the teacher even in a classroom setting. In France, the requirement is that the calculator has it’s memory wiped, which is sad if you have personalised it. But the backup mode is now so simple there is no issue. In the connectivity kit you right click on the calculator name. Choose backup. This creates a zip file of everything on your machine which you can suitably name and stores it in a specially created backups folder. Wipe the machine for exam use and then afterwards, plug it in again and right click again. Choose restore and it will find your backup files, choose yours, click OK. That’s it.

In a school setting with busy teachers, things just have to work. It was good to be in a teaching environment and that is exactly what happened. HP Prime vs. TI-nspire? Well they are different, despite having overall similar functionality, but it seems that Prime does genuinely have that plug and play simplicity that school use demands.

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).

Kate used a sociological analysis known as Social Activity Method (SAM) devised by Paul Dowling of the Institute of Education, London. He suggests that a practice (in this case mathematics education) has discourse in one of four domains of action. If the content (e.g. solving an equation, constructing a proof) would be recognised as mathematical and the symbols and technical vocabulary recognised as mathematical (e.g. evaluate 3x+1=10), then this is ‘esoteric domain’ discourse. This is contrasted with ‘public domain’ discourse where the content and the symbols/vocabulary would not be recognised as mathematical (but nonetheless a discourse in maths education). Importantly, the task of the mathematics educator is to induct learners into the esoteric domain of mathematics.

Kate found that students only rarely maintain any discourse in the esoteric domain. The teacher would mostly restrict their discourse to metaphor (in solving an equation: “get rid of the x’s”) or make appeals to common sense knowledge (“What is a square? It’s like one of those ceiling tiles”). Perhaps unsurprisingly, lower achieving students had very little discourse in the esoteric domain, while higher achieving students had at least some. However, this was in response to the restricted discourse of their teachers, not necessarily to what they could achieve.

So, I billed this blog as being about HP Prime Wireless. Well, later in the day, I had my first opportunity to use the system in a classroom setting. The class was a group of teachers and maths educators and I gave them an activity to explore conics starting with the form x^2 + y^2 = 9 in the advanced graphing App. I could observe the class’s conics by monitoring their screens in the Connectivity Kit monitor. When I saw an interesting example (a larger circle) I would double click on the screen and show it to the class. I asked; “How did you make the circle bigger”, to which their were two response to the two times this happened; “I changed the constant” and “I zoomed out”. This immediately sets up a rich discussion about the relationship between graph and function and the scaling of the graph. I then said; “Has anyone found a non-closed curve?”, which led to a new burst of activity. When I saw one I could ask; “How did you do that?”. Here, the teacher discourse is generally just teacherly prompting. However, the student discourse is predominantly in the esoteric domain of mathematics. The HP Prime only gives access to esoteric domain mathematics (the graphs and functions in symbolic form) positioning students to make esoteric domain responses.

The second activity was a new way of doing a classic. I sent out a poll asking for shoe size and handspan data. My class entered the data on their handhelds and pressed send. Within seconds a whole class worth of data was available for analysis. In the poll results screen in the connectivity kit the points are plotted and a line drawn, showing an overview of a possible relationship. However, selecting the HP Prime emulator and sending the data to it, generates a new APP on the emulator with full two variable statistics facilities. So, we can see a relationship. We can see the correlation coefficient to see it is a  weak relationship. Then we validate the relationship by seeing if my hands and feet fitted the model. I was a poor fit, so we could discuss why my hands/feet relationship was different from the group (they were all women, which suggested a new hypothesis to test). Issues of experimental design were discussed. Within a few minutes of setting up an experiment we were having a well framed and well informed discussion, entirely within the esoteric domain of statistics.

This was unexpected. Kate’s research suggests it is very difficult for teachers to sustain discourse in the esoteric domain they aim to induct their students into. Harder still for students to work in that domain. Yet by putting students into a setting where they work with technology that only communicates in this domain and by keeping the discourse framed by the technology based activity, the vast majority of discourse is generated in the esoteric domain. See my previous post for a description of the software and how to set up the polls and the monitor. Suffice it to say it is not difficult to set up. Inevitably there are a few teething troubles (notably my 13″ LapTop screen is just not big enough to see the screen of enough connected Primes). Also, it is amusing to see how classroom management techniques are still needed. Calling the class to order and announcing the arrival of a message with instructions is still necessary (even with teachers!). But, teachers using a HP Prime wireless kit could use it in almost any lesson, so they will quickly become fluent in the changed classroom environment.

Please get in contact to share your thoughts or if you would like to see the system in operation. ( Click here to link to the HP Prime pages at to see pricing etc.

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.

The auto-update feature on the software is a joy. The emulator and the connectivity kit both prompt you on launch to update. It is two confirmation clicks and the software is updated. In both cases these are important updates, so you must say ‘yes’! If you plug in (via USB) a calculator while the connectivity kit is running, then that will check to update the firmware. Again if you have any which are not brand new, say yes. It takes about 30 seconds. The wireless kit and the calculators both come with a CD which gives direct access to the software if you haven’t installed it before. There is a small aerial stand which is a USB connection to your PC. This is installed without anything else being needed. The wi-fi dongle is very small and neat and fits under the slide cover. Once this is installed and the software is running, the calculators are listed and their screen visible in the connectivity software on the teacher PC and hence projected for the whole class..

When the dongle is first fitted to the calculator, it is necessary to find the network. This simply involves touching the info panel in the top right hand corner of the Prime screen and touching the network icon that is there.This directs you to a screen with a drop down list which shows that a network is available (HP_Classroom_Network).

On launch the connectivity kit will not show the screen monitors. So, just click the buttons in the top right hand corner to show the monitor and the messages. CC_Monitoring

Here you can see screen shots from 10 Primes plus the emulator. The messages pane is open above.


Although it is very tempting to maximise the screen area for the monitors, You should avoid this, so there is space in the middle to set up an instant poll. (The grey area; if you cover this all up then you will not be able to see the poll window).


When we see a student has done something interesting, we can just double click their screen and a freely resizable window opens, to show to the class and get them to explain.

Click the add (+) button and add a poll. This can have many questions and a multitude of pre-set formats. This is a range of variations on; choosing from list, one or two number input or free text. You can set up a whole class quiz, but better to gather data as you go. For example do the classic hands and feet survey. set up a poll asking for a ‘point’ i.e. two variable data. You click on ‘add’ choose ‘poll’, type a name for the poll and then the poll interface comes up.


Go to the entry for your poll in the content list on the right hand side, right click and choose send. (Before you do this, make sure that none of the screens in the monitor have been clicked as this selects specific machines to send the poll too. you should see a button in the messages pane saying Send to Class (All) and then you know they will all get it!)Capture_5

Now, every machine has the instructions and can go on to enter the data. (Clicking send when they are done).

Now double click on the results entry that will have been created automatically in the content list. Capture_6It defaults to a suitable view according to the type of data making the speed and ease is very impressive.

With emulator running you can choose to send the data either to all of the students or just to the teacher emulator, for further analysis. (Click the machines’ screens you want it sent to). Capture_7



When you do this, a new App of a suitable type (1 or 2 variables stats) will be created with the name of your poll. Capture_8




So, now we can find the regression line and model the relationship and get the summary stats.

This opens up a whole world of dialogic classroom opportunity. Classics like the class survey become incredibly easy to get the data and get ready for analysis and comparison. However, we can now ask all students to use the function app to enter an equation parallel to y=x. We generate 30 responses immediately. Pick out different examples. Ask students what is the same and what is different. We can poll students to choose a whole number between 0 and 10. Look at the distribution of their responses in a box and whisker plot. What do we think of as ‘between’? Now we get deeper conversation, with everyone involved, with software that allows thoughtful mathematical responses.

This cannot be done with iPads or Android tablets. Even if it could, the software is still not a patch on what Prime can do mathematically. For the maths (and science and computing) teacher this remains the way to go. Please get in touch if you would like me to show you the kit in operation or try it out with your students.




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 (

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

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.


Update for HP Prime

Calling all HP Prime owners. Make sure your ROM is up-to-date. The latest update was released early in December. A quick check is to see if you have the full calender function which is new in this release. (Tap in the top write corner to show the setting, then tap on the date. A full calender will come up if you have the new version, if not, go update!). The procedure is unbelievably simple.

You need the connectivity kit installed on a PC and your HP Prime ready to connect with a USB cable. (The connectivity kit software is on the CD or available to download on my web site. Click here). Run the software. It will probably prompt you to update it when launched. Do this and follow the prompts to complete the installation. When complete, launch the software again and plug the calculator in with your USB cable. You will see the your calculator listed. Right click on the name HP Prime and choose update firmware. Follow the prompts. That’s it. Your Prime should be ready. Double check by marveling in the new Calender facility. Check here for a video showing the checks.

The HP community is in full swing now, check out these videos for HP Prime games by the wonderful MicNic Tetris and ‘Angry Birds‘ and MasterMind from Tony Galton.

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:

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.