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 …
Here are links to my collection of maths trails:
The Lewisham Trail (the-lewisham-town-trail-v4)
The Stratford Trail (the-stratford-town-trail-with-clip-art)
The Bank of England Trail (bank-of-england-museum-trail-for-11-to-14)
South Kensington and the Science Museum (sk-trail-booklet)
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.
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.
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.
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.
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.
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. (firstname.lastname@example.org). 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!
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