Examples of three different tasks and the groups who produced them.
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 …
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.
It is quite astonishing to see pupils in years 4 and 5 (aged 8 to 10) able to solve these puzzles almost instantaneously. Their teachers certainly can’t, my PGCE students with top maths degrees can’t. So how is it done?
I talked with Bob Sun, inventor of the 24®Game in Easton, Pennsylvania and he gave me a copy of a book by journalist Daniel Coyle called The Talent Code. Coyle examines a series of instances in which exceptional performance is found in different fields and looks at the elements that came together to produce it. A great coach is always included, so teachers, you know you are important! However, the coming together of real desire and serious hard work with lots and lots of practice are the principle elements. In the end, the final few percent are achieved through an intangible element that can be called ‘talent’. But, for sure these kids can beat there teachers because they have worked hard at it.
Now, playing the 24®game is not like memorising your times tables. It involves flexibility of mind. You generate a whole raft of relationships which make up parts of the 24, like looking for 8 and 3 or 6 and 4, or 23 and 1 made up of pairs or triples of the numbers available. So, you are juggling lots of combinations. The outcome is young people who see numbers and are aren’t interested in seeing if they can remember the answer, but recognise the need to fiddle with what they’ve got to unlock routes to the answer. You can’t get more like true mathematical thinking than that in a 9 year old!
So, the Count On Us Challenge provides the desire. Compete for your school, win the prize, get to walk across the top of Tower Bridge. It doesn’t matter, it was a great day out for everyone, but everyone involved was ready to compete because they cared and they’d worked hard at it. Net result, 100s of young people with much better and more flexible number skills than their teachers. That’s good!
All of the 24®Game sets are only available from The Maths Zone. There are class kits, tournament packs, the competition standard one and two digit sets, primer sets for early practice and tougher sets for advanced challenges.
There will be a Count on Us Challenge next year. So, start practising now. The kids from Kender School are very good. Very good indeed. They will take some beating! (And please don’t think it is a school with any special advantage, not at all, it is a very straightforward urban primary in SE London. They work hard at it and their kids practise and my are they sharp with their number skills. Well done to them.
We have produced a guide to help you run a number challenge tournament in your school or your area. You can use the 24®Game cards as they do in the Count On Us Challenge. If you want a more equally weighted tournament, we also have SuperTMatik, which is a card game from Portugal where they have a National (and World!) championship, but the problems are seeded so you can have pupil’s competing at different levels in the same game. Finally there is Target Maths, where the numbers are combined to make a different target each time. Try this one (the target number is in the middle).
So, an in-school tournament to provide the desire, then plenty of opportunities to practice, practice and young people get really good. And then in secondary school what happens? They forget, because they stop practising. What does Andy Murray do before during and after every tournament? He practices hard. That’s why he is so good (and he may just have a bit of that extra few percent too!). It was humbling to see how good the kids from Kender (and indeed all of the other schools) are. Teachers can get all of their kids to that level with enough desire and a lot of practice. Good luck for next year!
© Suntex International Inc, All rights reserved. 24®, 24 Challenge® are trademarks of Suntex
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 (firstname.lastname@example.org).
Apart from the dodgy hairdos and the rusty cars, 1983 had things going for it!
From September 2014 it will be compulsory for schools to teach financial education. This will be built in to the mathematics and the citizenship curricula. See this article from the Daily Telegraph. Notice the picture of school life that they choose to illustrate the article with. This is how students learn; in rows at individual desks, looking seriously bored! The trouble is that a good proportion of the materials available for ‘financial education’ in schools is perfect for this scenario. Standard worksheet based discussion and practice activities are the norm and work in the same way that makes PSHE such a disappointing subject, taught by non-specialists, with no exam, it is hard to see the purpose when you are in school.
What makes these important things come alive is getting students into the setting. They have to care about the issue in order to engage with the ideas. I have seen fantastic drugs education sessions where former users and dealers have come in and talked to teenagers about their experience and where they are now. It is edgy, but it is real and they certainly listen!
Finance is tricky. Kids in school rarely have any real need to save with interest and if they have a bank account, their main worry is losing their cash. Certainly, they cannot borrow beyond their means or need to budget in a life changing way. Some, certainly, have life experiences that may necessitate any or all of these, but they are a small albeit important minority.
We have been working in financial education for over a decade. As a development of the Number Partners project which I was director of for many years, we designed a series of large format board games designed to set up scenarios in which players have to make important financial decisions: how to invest a small amount of capital, to generate profits to reinvest. Managing money between cash and different bank accounts, to enable purchasing but retaining security. Budgetting for a holiday and managing exchange rates. Making the life transition from school to work, while meeting your life goals.
The power of a board game is that the social setting frames the decision making. You are playing with real people who you have to engage with, framed by the settings of the games. The games were trialled in very ordinary schools, in classrooms with groups of students and have been widely used in different settings since. The effect is impressive. Students talk to each other about their financial decision making, developing strategies to succeed in that setting. Naturally, winning strategies involve good financial decision making.
We set up a web site to showcase the games. So see what you think. All of the games have teacher guides with extra materials and school use ideas. Please get back to us with your questions and thoughts. But, when you plan to deliver financial education this September, get your students into a setting in which they care. Only then will they be able to make decisions in a way that matters to them.
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.
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.
Now that HP prime is launched and I have a few of them to play with (oh and to run training sessions with …) the implications can be tested of having handheld maths technology that connects easily to the teacher computer in a classroom. Ultimately the connection will be wireless but the dongles will not be available until early next year. So, I bought some 5m micro USB cables and could test it in teaching situations. So far only teacher groups, but I’m getting the hang of it.
There is free emulator software and the connectivity kit is freely available too. You can try this out yourself by opening a number of instances of the emulator and the connectivity kit on your PC. Just go to http://www.hpgraphingcalc.org/hp-prime-links-and-resources.html for the files. The most impressive thing is how utterly seemless it is. Install the connectivity kit and launch the software. Down the side there are three tabs: ‘Calculators’, ‘Content’ and ‘Class’. Just plug in an HP Prime calculator (or launch an instance of the emulator software and you will see it listed. Here, I’ve plugged in one real machine and am running my emulator. Clicking next to the device brings down a view of all of the content on the machine. Clicking the content tab allows us to author new content and then save it to the machine. You can create tests and polls. (Right click on the thing you want to create and ‘new’ comes up. Click that. When you are done, click the small save icon in the top left of the screen. The dialogues are pretty self-explanatory so I’ll leave you with that. When you are done, go back to the new item you created and right click to ‘send to class’ and it will be on the machine. Imagine being in a classroom in which simply by launching the software and machines being in the room, you can share content with them. No connections no log ins, just get started. But to me it bursts into life when you click the ‘class’ tab. Now you can see the screens of all of the connected machines. (Remember, when wireless is there, connected just means ‘in the same room’.). These refresh every couple of seconds (although right click and refresh speeds this up if needs be. Also, right click and ‘project’ creates a resizeable image to show the whole class. So, now I can ask my class to work on a problem and watch what they do. When something interesting happens I can (if I want) bring up that screen to show the whole class. But, there is more, … I can open a messages window. From there I can send a private message to a single machine or a message to the whole class. This appears on their screen and they can even respond and we can discuss. Personal in-class responses to individual students work monitoring the outcome of their thinking from the maths appearing on their screen. Now, I always say get students to work one between two, so we’ll have 15 screens up in a full classroom. I tried this with 6 Greenwich PGCE students and it is a bit frantic trying to respond individually. However, really, I should be more teacherly and give the occasional prompt looking our for the ‘Aha’ moments. The possibility to get to the nub of student understanding is tantalising. What do you think? I am dying to get the wireless dongles and try this in a routine classroom with ordinary kids. Then we’ll see what they can do. Exciting times.
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. 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.
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.