### Graphing apps on smartphones

I’ve been using GeoGebra for over ten years now and it has had a huge impact on how I think about mathematics and teaching mathematics. I’ve also been massively impressed with Desmos though I don’t know it inside-out in the way I do with GeoGebra. My use of these graphing tools, and Autograph before it, was always based on using them on a computer and this means I have a preference for the computer software over the phone apps: I find them much more usable on a large screen with a proper keyboard and mouse. This works fine for me but, as I’ve commented on here before, I think the greatest impact of technology in the mathematics classroom is as a result of students using it. Consequently I’ve long been an advocate of the use of graphing apps on phones by students, especially in A level classrooms. However, even though I’d promoted the use of phones I had always thought that this was an inferior device for students to be using the software on and that the ideal would be to have them using the software on laptops if it were possible.

What has changed for me this year is that I no longer consider phones an inferior device for using the software for students. Today’s generation of students have grown up with small touch-screen devices and find it completely natural to operate software in such an environment. For many students their phone is the device they are happiest using and so the opportunity to situate such fantastic bits of software as GeoGebra or Desmos in this environment is one that we should be capitalising on. The quality of modern phones is such that the apps work beautifully on them. In particular I think the GeoGebra 3D app on Android phones renders the image better, and is more responsive, than the equivalent computer-based version.

This year we recorded some more PD videos for the Further Mathematics Support Programme – short 5 minute videos that address a particular bit of pedagogy. In one of these we looked at using graphing apps in the Further Maths classroom. In this video you’ll see students using them and hear the teacher comment on how surprised he was that the students didn’t find the small screens an impediment to them using the app effectively. This video, along with questions for reflection, can be seen at furthermaths.org.uk/pd-videos-further-pure#video5

At MEI we’ve developed a series of classroom tasks that are designed to be compatible with the phone apps for both GeoGebra and Desmos. These can be found at mei.org.uk/geogebra-tasks and mei.org.uk/desmos-tasks.

### Representing functions as mappings between number lines

At the BCME conference at Warwick in April I attended a talk by Martin Flashman, a visiting professor from the US, about the use of mappings between number lines for graphical representations of functions. This is in contrast to the usual convention of plotting them on a pair of cartesian x-y axes.

This was an idea I’d seen before but I hadn’t really thought about it in detail. It was a revelation to see how a change of representation could give whole new insights into mathematical ideas I thought I’d understood. In particular I think the representation of composite functions and their inverses is much clearer in this way. I was also amazed at how this could be related to the cartesian system: for example a “linear” function is the set of points that lie on a line; if this is transferred to a mapping between number lines this becomes a set of lines that pass through a single point. What is more, these two representations are the duals of each other!

Martin went on to give a lot more detail to how this system could be extended to represent ideas such as calculus and functions of complex variables represented as mappings between two planes in 3D space. There is a lot more information, including presentations from his BCME conference sessions, at: users.humboldt.edu/flashman/

Overall this has had a wider impact on me than just thinking about representation of functions: it has made me consider how the way an idea is represented impacts on my understanding of it. For example I think setting out long multiplication in a grid helps me focus on the two-dimensional nature of a product and I’m now searching for more such examples…