Category Archives: Maths Technology

Public Mathematics in a Pandemic

Not so long ago I had a very minor Twitter spat with a former student of mine in which I advanced the view that knowing the Fibonnaci sequence was more useful than instant recall of 11×12. I was shot down, with the suggestion that I didn’t care about working class kids. Well, I guess, I would like to claim that I care about all kids, regardless. And, it seems that being able to read and understand mathematical information has suddenly become a whole lot more important than previously. Oddly enough, memorising specific number combinations has not proved to be of any great importance in this. Nor, to be fair has the Fibonnaci sequence cropped up. But, of course making sense of patterns and relationships absolutely has. We now live in a world in which it is expected that we can look at a graph with a logarithmic scale and read it sensibly. Also, we are being told (as I write) that the rate of increase in case numbers is decreasing. From this we need to know if the case numbers are going up or down and thus, what is in fact decreasing.

In school mathematics, the myth of application is ever present. Mathematics solves problems in the real world we are told. Except that we start from a curriculum containing specified mathematics, that must be taught, generally in a given order. So, of course, the real world is bent and twisted to fit the mathematics that needs to be learned. In our teaching of distance and time, how often does a car travel at a uniform speed, showing a straight line on a distance time graph starting at the origin? (Instantaneous acceleration is a useful phenomenon in maths lessons, but nowhere else). The pandemic has not thrown up many linear models, or even quadratic models. Learning the maths presented in a distorted and damaged context is worse than unhelpful. Either the learner knows enough about how the real world works to recognise that the maths lesson version is simply not real and therefore the maths is not a model of it, or worse they build their knowledge of the world through this damaged notion that cars really do travel with contant speed, starting from rest. So, when stuff actually matters, as it really does in this pandemic,  learners of school mathematics do not have the tools (they were not allowed to play with functions, instead they had to learn linear, then quadratic and never quite reached exponential) and would therefore expect reality to be required to fit. So, the growth rate graphs with the exponential scales, that then look much more linear, show the death rates in the USA only ever so slightly higher than those in Germany. This can only be because that is the way it was.

If we are to claim that maths has some use value and relavence to the word outside the maths classroom (as this year it really has like never before), then we need to engage our students in sense making. Looking at live data and playing with the maths to see how and where it might fit. With graphing software it is no harder to draw an exponential than a linear function and when it is driven by data that the learner has some investment in, then they will want to know. Building relationships between a felt and experienced world and mathematical objects useful for representing it, is complicated and muti faceted, but possible if we allow the setting to lead the maths rather than the other way round. I still have some ultrasound distance sensors which we would use connected to real time graph drawing software. Students would walk on a straight line towards and away from the sensor to create different distance time (and later velocity/time) graphs. In this way the relationship is created and really felt. The motion to create a sine curve as a distance/time graph is a very lovely thing. Equally our Pizza project had students find models for the cooling of a pizza from short term data (it looks very linear over the course of a 10 minute experiment, but what does that say about what will happen in the medium to longer term?) The implications of a model can only be engaged with in a context which behaves as the real world actually does.

I picked the Fibonnaci sequence as an example of interesting variation. 11×12 is not interesting, nor is it useful (except, I did find this example). To make sense of a world that behaves as the one our students actual live in does, they need to see what they learn as a examples from a range of useful tools that can, when critically applied, give us ways of seeing what happens in that world in a more manageable way. Linear and quadratic are examples of functions, of which there are many, many more. Times tables are number relationships of some power and Fibonannaci, Square, Triangle numbers also. So, when we see variation and relationship we take a flexible, open and always critical view of it.

The Mastery of Multiple Representation

I was invited speak at the Maths Mastery conference in London last month. My essential brief was to engage with using technology to support mathematics learners. I have increasingly wanted to take a wide view of things so I interpreted technology very liberally. Having trained teachers in using graphing calculators for both TI and HP for many years, I am well versed in the language of multiple representations. The possibility is there to see a function represented graphically, symbolically and as a table of values and to swap quickly between them and to see how each representation gave you different insights into the nature of the function.

For example, a linear function has a particular algebraic form, it has a straight line graph and a table of values with a common difference. We exploited all of these in our Pizza project, showing that the natural tendency when watching change over time (the declining temperature of a cooling pizza) is to look for a linear change. The difference is roughly equal over the minute intervals we used, for the 10 minute length of the experiment. This is forcefully confirmed visually when a real time graph being drawn is very nearly a straight line. So, we feel empowered to hypthesise a linear function which symbolically can be used to calculate extrapolations. It is these that undermine our initial thoughts (put time = 24 hours into the function and we quickly see there is something wrong). Then we can go back to the graph and change the axes to see the nature of the slight curve and look again at the nature of the differences from equal differences, which themselves have a pattern.

Continue reading The Mastery of Multiple Representation

Technology and Maths Exploration

20151008_110555 - Copy (450x800)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. Continue reading Technology and Maths Exploration

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!) Continue reading HP Prime Instant Set Up and Update

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). Continue reading Talking Maths in the Esoteric Domain: HP Prime Wireless

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. Continue reading Wireless Prime has arrived!

Pinch to Zoom on the HP Prime!

If you have an HP Prime (if you haven’t then naturally, you must get one!), plug it into your computer via USB, launch the connectivity kit and prepare to be amazed by the smoothest upgrade you’ve ever seen! On launching, the connectivity kit asks if you want to upgrade the kit and the calculator. You certainly want to do both. When the connectivity kit is finished, launch it again and turn the calculator on. You will be promPted to continue, but aside from that it will get on AND update the calculator’s firmware. Total about 3 minutes, with the the calculator part taking a minute or so. Continue reading Pinch to Zoom on the HP Prime!

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 (chris@themathszone.co.uk).

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

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.

Classroom Maths Dialogue with a Handheld

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.

Calculators ListThere 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’.). Class viewThese 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. MessagesThis 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.