Recently, an article with a breathless headline (since revised) was published on the ABC website about how a year 12 student in regional Victoria, Mubasshir Murshed, published a proof of the above result in an academic journal. I think this a good achievement and to be commended, but the way it is framed is…problematic.  The original headline, “Teenager’s parabola equation blows away maths world” could have done with some input from mathematicians (I understand how news works, the journalist probably had little choice in the matter). The quote from the editor of the Australian Mathematics Education Journal also seems to me to lack perspective. Publishing in an education journal is not the same as publishing in a mathematics journal, which requires something genuinely previously undiscovered.

I’m all for furthering the idea that everyone can discover interesting mathematics themselves, even if it was already known. That’s part of the fun in mathematics, figuring out how to understand existing work in a new way, or even the joy of realising you discovered a piece of ‘real’ mathematics that someone else had thought of. I definitely had fun deconstructing the proof and rewriting it (below!) in a way I felt more comfortable with. But heralding someone who does this as a “17-year-old mathematics genius” is, I feel, counterproductive to the promotion of mathematics more broadly. It can definitely prompt people to put themselves in the “I’m not a genius, I can’t do mathematics” box, when this is just not true. My colleague David Butler engages with people of all kinds of backgrounds with public mathematical play, and they often are doing little pieces of mathematical discovery without even realising it. Praising up doing mathematics as a work of “genius” alienates people.

It’s easy to praise such an achievement as Murshed’s, but the article missed an opportunity to highlight the way in which discovery in mathematics is cheap: you don’t need any special equipment! With access to the internet, and pen and paper there is little limit to what kinds of theoretical mathematics you can do. The idea that problems such as the result in the title are even amenable to proof is genuinely important. That abstract-looking mathematics can have a real-world impact (parabolic reflector design!) is important to know for non-mathematical policy-makers at the all levels, but also for people genuinely if there is to be an appreciation of mathematics in today’s society. There is also the message that mathematics is not ever truly finished, and that there can be more than one way to approach a theorem and its proof (see below).

With the grumpiness out the way (and I’m happy to discuss the pros and cons), I want to give a variation on the theorem and proof, assuming less than Murshed’s version. I am happy to take as a black box one of the equations he derives, because that is done by elementary geometry.

Theorem: A plane curve given by the graph of a differentiable function and with a unique focus is a parabola.

Continue reading “The only curve with a unique focus is the parabola” →