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 …
It is always good to get the chance to actually teach real kids in a real school. I have always said that if you are going to teach students about linear functions, it would be a crime to do it without any technology. So, up comes the topic and in I go with the technology. We have organised a class set of HP Prime handhelds for the school and my job was to get students started using them, so they would be sufficiently familiar at the start of the linear functions topic.
My argument is that if we have a device that speaks mathematics, then we can ask open mathematical questions and then students can explore. So we started with the obvious ‘tell me some interesting calculations that make …’ (e.g. 27 or 3115 or 5.2). Clearly the purpose is to get used to the calculator but when students are allowed to use any function, they really start exploring.
Next, to find the correct app and enter functions, I simply told them the format (something x plus or minus something) to enter in the symb screen on the function app, then asked them to create pairs of functions that matched some pictures I gave them (different slopes, parallel or not).
The exciting thing is to note how quickly students feel comfortable with the technology. The teacher is nervous and not keen to become familiar on their own, but kids feel no fear. The trick for the teacher is managing the classroom in this setting. Make sure the students write down in their note books the outcomes they find (otherwise they will be lost when they need to feedback and so we can see that they are progressing). They are fun to touch, so whenever we had some spoken feedback time it was essential that all students placed the HP Prime face down on the table. Changing mode from paired investigation to class plenary needs some clear management.
It was clear that the students liked the machines and that is nice, but mostly they felt comfortable to explore and try stuff out without fear of being ‘wrong’. The next lesson(s) got students to explore the details of relationship between the functions and their graphs and emphasised the technical mathematical vocabulary used to describe them. I’ll tell you more about this soon.
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. (email@example.com). Click here to link to the HP Prime pages at hp.com to see pricing etc.
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).
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!)
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. It defaults to a suitable view according to the type of data making the speed and ease is very impressive.
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.
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.