In this blog post, Professor Colin Foster describes his recent project, producing over 160 short video explanations of mathematics, for school and university age students. You can watch the videos at twominutemath.org.

Introduction

As they gain experience, school mathematics teachers and university mathematics lecturers get very used to explaining the same things over and over again. Each year, a new set of eager (or not so eager) students appears, but the mathematics we teach hardly changes. As time goes on, teachers and lecturers develop ways of explaining various concepts, informed by the difficulties their students present with. The knowledge we develop doing this might be categorised as an aspect of pedagogical content knowledge.

While the content of mathematics taught at school and beginning university may change less than the content of other subjects, one thing that does change is the technology available to learn it. Mathematics teaching at both school and university level has come a long way since the days of ‘chalk and talk’. Teachers generally now have easy access to free technological tools, such as GeoGebra and Desmos, which allow live graphing, geometric construction, visualisation in 3D and handling large data sets and running simulations.

The power of animation

In this context, I’ve become increasingly aware of the power of an extension package called TikZ that runs inside LaTeX (a document preparation system and markup language) to generate vector graphic animations. University mathematics lecturers will be very familiar with LaTeX, as the standard way of formatting mathematical text, and TikZ as a convenient way of drawing diagrams and graphs, whereas school mathematics teachers are likely to be more familiar with Word and PowerPoint.

But although I have used LaTeX and TikZ for many years, I only recently realised how easy it is to use TikZ to generate animations. It is quite accessible, even if you don’t have previous experience of these languages. (See the notes at the bottom for technical details if you want it.) Because the generated images are vectors, the animation file sizes are tiny – often just kilobytes, or a few megabytes, so they are very quick to download and watch in a web browser, on a computer or even on a phone.

The role of AI in coding

A big part of what now makes this so easy is the ability of AI to help with writing and debugging code. Gone are the days of spending half an hour hunting down a missing semicolon! My most common prompt to AI these days seems to be, “Debug my code”, which I find it now does with impressive efficiency.

Over the summer, I have produced more than 160 short video explanation of mathematics. I’ve called the website Two-Minute Math, although some of the videos are a bit longer than that – occasionally up to 5 minutes. Not all the videos use animations – I have tried to avoid being ‘a hammer in search of a nail’ – but many do. And I could not have produced anywhere near so many videos over such a timescale without AI help with the coding.

The style of the videos

The videos are minimalist in style, and I’ve tried to be informed by the design principles we used in developing the LUMEN Curriculum, including the Cognitive Theory of Multimedia Learning.

I have been very inspired by the classic mathematics videos from the Open University produced under John Mason, which I first watched 40 years ago, and by Dave Hewitt’s writing about the pedagogical power of ‘canonical images’.

Below are three examples, one on algebra, one on geometry, and one that tries to explain what correlation is in statistics. I’d be delighted if you would pick one and take a look, and see what you think.

Ways to use the videos

I could envisage the videos being used in a few different ways:

1. For Students: Students could watch them to help with concepts they have found difficult – either from their current learning or from some time previously. I have suggested on the site a possible workflow for this based around self-explanation. It is easy to engage passively with videos, whereas if students are pausing, rewatching parts, making notes and asking themselves questions, it is possible that more active engagement is happening that will lead to longer-term learning.

2. For Teachers: Teachers might watch them together and consider what they like and what they don’t. Perhaps a continuing professional development opportunity could consist of going away and making notes on what they would do differently (or perhaps even making an alternative video), and then coming back together and comparing and discussing. The discussion aspect might be more valuable than any of the individual videos produced.

3. For Trainee Teachers: Trainee teachers might get ideas for their own explanations from watching them. They might craft an explanation of their own, in their own style, based on some of the ideas used in these.

4. For Experienced Teachers: More experienced teachers might use the videos with their students without the sound, pausing where necessary to support their own explanation and to invite discussion. I think that’s how I would use them myself.

What should I animate next?

If you have comments – or suggestions for additional videos to make – please let me know!

About the author

Colin Foster is a Professor of Mathematics Education at Loughborough University. His research focuses on the learning and teaching of mathematics in ways that support students’ conceptual understanding.

Technical note: How the stop-motion pipeline works

The way the animations are produced is based on traditional stop-motion animation. The .tex file acts as the ‘digital flipbook’ generator: instead of drawing shapes or graphs by hand in software like PowerPoint, the TikZ code instructs the computer where every axis, dot, line and text label should sit across incremental steps.

You then run an R script that automates the entire video pipeline – compiling the document rapidly with LuaLaTeX (via tinytex), extracting the high-resolution PDF pages as frames using pdftools, and stitching them into a crisp 60 frames-per-second 1080p MP4, using the av package (powered by FFmpeg).

Finally, I add a bit of voiceover, and it’s good to go!

For more details see https://www.luatex.org/.