So you want to be an A-level Mathematician?

Mathematics is a wonderful, terrible, elegant, messy, exciting and tedious subject.  Students considering an A-level (or two) in this subject may rightly be daunted by the work-load, and intimidated by the huge stack of new ideas they are expected to assimilate and regurgitate two years later, but should take care not to get stuck in the rut of ‘what I need to be able to do for the exam’.

Many courses come with a recommended reading list – a mesmerizing heap of relevant literature which, if not exactly essential to your course, will broaden your horizons and develop you further than the course textbooks alone would do.  When a student starts A-level Maths or Further Maths, their ‘suggested purchases’ list consists of a £50 calculator and… um.  Yeah, that ought to do.  Inspiring, no?

The list that follows is deliberately not a recommended reading list.  There are plenty of more qualified reviewers of mathematical literature out there (Cambridge University have something of a reputation for that sort of thing, so check out their recommended reading list if that’s what you’re after.)  My list is a collection of people / websites / channels that I suggest my students follow.  They lean more towards the interesting, recreational and non-examinable, but are all the better for that, since there will be plenty of the must-learn mathematics going on in the classroom.  My motivation for these recommendations is to insure against students losing their love of the subject.  Mathematics in year 12 or 13 (and even more so Further Mathematics) is a very demanding course, and while it is full of interesting and exciting ideas, it is still possible to be overwhelmed and intimidated by it all, and end up losing track of what made you want study it in the first place.  Call it the reverse Tom Sawyer effect, if you will (he turned a chore into a sought-after activity, and if we’re not careful we can turn the privilege of maths education into a chore).  So if you’re a student of maths, or a teacher looking to inspire your A-level & would-be A-level mathematicians, these are a few of the wonderful internet people who inspire me:

Numberphile: Primarily a YouTube channel which specializes in short explanations of interesting mathematics by a whole range of enthusiastic mathematicians including some of my personal favourites, James Grime and Matt Parker.  The mathematics can be quite a high level, but is always presented in an accessible fashion.  There is also such a wide range of subject matter, from number theory to compass construction, picking up on maths in the news, in popular games, or discussing new mathematical proofs in interesting and engaging ways.  You will also get a sense of the development of mathematics through history with anecdotes and insights into famous mathematicians.  Below is one of my favourites – Matt Parker demonstrates how computers perform calculations by building his own out of dominoes:

Vi Hart: Another YouTube channel to follow, Vi gives a wonderfully creative take on the mathematics that students will be encountering as they progress through their A-level course.  The video style alone is a work of art, with hours of scribbling condensed into a few minutes, with a concise and superbly scripted narration to go with it.  She combines music with interesting mathematics in a unique presentation that is a joy to watch.  Her video ‘Reel‘ is probably the most beautiful description of the way we feel about imaginary numbers you will ever encounter.  Here’s another favourite of mine, where she leads you unsuspecting from counting to adding to multiplying and suddenly you find yourself understanding logarithms:

XKCD: ‘A webcomic of romance, sarcasm, math and language’, these comics are worth a look for some light – if geeky – entertainment, but the really clever stuff comes in the What If section, where some really fun questions are answered from a scientific point of view.  The author used to work at NASA, so he really knows his stuff, and the way he presents solutions, along with his intuitive approach to tackling each problem, is superb.  Plus, who doesn’t want to know what would happen if a baseball travelled near the speed of light, or how many BB guns you would need to fire at a freight train to get it to stop?  Sprinkled with amusing illustrations, these are a great way to investigate daft problems.  One of my favourites has to be Train Loop: this question was in response to a Norwegian ad where a train performs a loop-the-loop.  The original ad is here:

Veritasium: This YouTube channel mainly investigates physical phenomena, and yields surprising insights by means of identifying misconceptions.  Essentially, the host will ask a bunch of people a question, identify the common misconceptions that we have about something, and then explain (or get an expert or two to explain, if it’s really tricky) what’s actually going on, such as this one about how we perceive temperature.  Some videos are designed to have more of an interactive feel, where an experiment is carried out, and the audience (us) are given the chance to predict the outcome.  One that will come in handy for getting to grips with the mass vs weight confusion is Misconceptions about falling objects.  But a personal favourite of mine hits a very common area of confusion – is there really no gravity on the International Space Station?

Vsauce: Another Youtube channel that’s not strictly mathematical, but this guy crams so many interesting phenomena and ideas into every video that there’s bound to be a few things to make you think.  The mathematics is less explicitly involved, but it’s definitely there.  One of my favourites is a really clever description (plus visual simulation) of what would happen if the Earth were flat:

Brilliant: After getting a little distracted by all the cool science bits, let’s get back to a bit of maths.  I’ve only recently discovered this website, but I’ve been really impressed.  It’s essentially a bank of multiple choice maths questions, ranging from pre-GCSE to post-A-level, separated into categories and difficulty levels.  Because of the format, it lends itself really well to smart phones, so download the app and fill those spare minutes brushing up on anything from geometry to calculus.  Also includes sections on some of the more mathsy bits of science.

WolframAlpha: An incredibly useful tool for all sorts of calculations.  It works rather like Google’s built-in calculator (which, by the way, can draw graphs for you and all sorts), but is connected to Wolfram’s powerful ‘knowledge engine’, allowing you to integrate functions directly (not just get numerical approximations), work out your binomial expansions, solve equations and much much more.

GeoGebra: This free program has revolutionized the way I teach certain concepts.  It is easy to learn and intuitive to use, and has sufficient built-in capacity to develop all sorts of interesting demonstrations.  Students can download it for free, use an online version or download an app for their tablet.  They can use it as a graphical calculator, drawing graphs (including implicit functions and parametric curves), performing differentiation and integration, investigating shapes, learning matrix transformations, understanding projectile motion, and all sorts of cool stuff.  I’ve made my own short introduction to the program, designed for students to quickly get to grips with the basics.  By searching the uploaded files of other educators on GeoGebraTube, you can build an intuitive understanding of all sorts of hard-to-grasp concepts, as well as have your eyes opened to all sorts of new and interesting mathematics.

Finally, a quick mention of my own website, TheChalkface.net.  It doesn’t quite belong with the links above, but I do upload my teaching and learning materials, and the A-level Maths section is particularly good (in my opinion) for revision resources.  It has links to AQA exam papers (downloadable in bulk instead of one by one), hundreds of worked exam solutions, my Not-Formula Book revision booklets and their further condensed Essentials revision cards, as well as a host of worksheets and activities designed to fit within the A-level syllabus.

The links I’ve listed above are just the beginning – loads of people are doing fantastic work out there inspiring students (and teachers!)  If there are any sites you think should have been on my list, please put them in the comments below – thank you!


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