My first authentic mathematical experience
It was the summer of 2003. I was in optimistic mood, having received my A Level results that confirmed my place at Oxford. I would be studying maths with the greatest minds in the country (or something to that effect; I can’t quite recall the precise wording in the prospectus). My only trepidation was that I was chosen by mistake — what if I was that guy who the tutors let in through error or misplaced sympathy?
Still, I was on my way to the dreaming spires and, with six weeks still to go, I had time on my hands. From the not-so-dreamy spires of Coventry, I dabbled in some extracurricular maths. As I rummaged through the popular maths literature (yes, such a genre does exist), I came across an intriguing unsolved problem, involving so-called twin primes.
The Twin Prime Conjecture
Two numbers are said to be twin primes if they are both prime (no surprises there) and they are spaced two apart. So, for example, the pairings
5 and 7,
17 and 19,
29 and 31
are each twin primes, but the pairings
7 and 9,
37 and 41
are not (9 is not a prime, the difference between 41 and 37 is not two).
Here comes the problem:
How many twin primes are there?
Mathematicians have hypothesised that there are infinitely many occurrences of twin primes; the so-called Twin Prime Conjecture. It is the boldest of claims because we could never count them all manually, so it demands a sophisticated argument that can be applied beyond our familiar realms of the finite. To this day, no proof has been found, although mathematicians have long known that primes themselves are infinite in supply.
I was astounded by the sheer simplicity with which the Twin Prime problem is stated. That it has confounded the best mathematical minds only added to the intrigue. With six weeks to burn, I got to work.
My ‘proof’
I scribbled down what little I knew of primes. As I recounted Euclid’s elegant proof that infinitely primes exist, I was struck by a new thought. What if his argument could be adapted to the case of twin primes? Thus was born my very own attempted proof of the Twin Primes Conjecture, cribbed from the work of one of the most celebrated mathematical minds.
Following Euclid’s proof, I convinced myself that twin primes could be generated by multiplying the first N primes, for any N. So, for example, multiplying the first two primes gives 2x3=6, from which we can get a prime on either side (5 and 7, the twins). Proceeding further, we can generate the twin primes 29 and 31 from the prime product 2x3x5. And if we continue this way, since there are infinitely many primes (thanks Euclid!), we can also generate infinitely many twin primes. QED, as the mathematicians say.
Euclid was my inspiration and, for a short time, I believed I was about to rise to his legendary status. Gone were the doubts I had over my readiness for university maths. A proof of the Twin Primes Conjecture was just the tonic I needed to persuade my tutors and my peers (and myself) of my mathematical competencies.
My mind was bouncing with excitement. What were my next steps? I could email my tutors-to-be to inform them of my breakthrough, or perhaps go straight to a journal.
I carried my excitement through the night, hoping sleep would clarify the best course of action. I could not escape a niggling doubt. My proof seemed too…simple. Surely this proof would have been developed by now? Then again, I could not see the obvious flaw.
I did not sleep comfortably.
The truth unravels
The next morning, it suddenly dawned of me. With as much simplicity as my proof itself, a counterexample presented itself: 209 is not prime (it is 11x19) and yet, according to my argument, the product 2x3x5x7 should give rise to the twin primes 209 and 211. A day earlier, I ran a quick check and had mistakenly persuaded myself that both were prime (in retrospect, confirmation bias was at play).
This image might have saved me some pain back in 2003
As I rifled through my notes (okay, just a few lines), I realised the error in my proof. Euclid’s proof generates a new prime by assuming that all other primes are known. My numbers, however, are generated from just a small handful of primes at a time — which means they are not guaranteed to be prime (as the example of 209 confirms).
Damn.
I swiftly erased the draft email I had prepared for my tutors-to-be. Good job too, as it may have been the death knell for my maths career, confirming to the tutors that my admission was ill-judged.
I buried the experience deep in the recesses of my memory — nobody needed to know about Junaid’s failed experiment as a mathematician. And yet I realise now, all these years later, that this brief episode was my first authentic mathematical experience. Although my Twin Primes ‘proof’ was hopelessly shallow in its approach, and my response to it naïve, it helped to develop my identity as a mathematician.
I would discover at university that a mathematician’s job is to play and contend with the unknown, and to build on the ideas of others as they navigate uncharted territories.
It is perfectly natural for mathematicians to develop flawed proofs as they inch towards new truths. They do not wear failure with pride, but recognise it as an inevitable part of the problem-solving process.
The absence of authentic maths in the curriculum
I should not have had to wait 18 years, and a chance encounter over a long summer, for my first shot at authentic mathematics. The school curriculum delivered zero opportunities for me to engage with maths as a mathematician rather than as a passive consumer of knowledge.
At school, primes were reduced to a perfunctory role in helping us to calculate highest common factors and such ilk. We never once questioned how large the bucket of primes is, nor did we probe the erratic behaviour of primes that leads to such interesting ideas as the Twin Primes Conjecture. Everything we were taught about primes was true and absolute. There was no space to even consider the unknown or to develop hypotheses around what might be true.
Euclid — perhaps in despair at the state of the maths curriculum to come
A common retort is that this level of problem-solving belongs to expert learners; that novices should focus instead on acquiring knowledge. That’s a cop-out. It is absurd, of course, to suggest school students are prepared to tackle mathematical proofs with all its rigour (my eighteen-year old self could barely handle it). But maths should always be experienced in a manner that engenders the learner’s sense of identity as a problem-solver.
Authentic maths relies on having students develop and confront their intuitions as they straddle the lines between truth and uncertainty.
Sites such as Nrich and Mathalicious, with their vast repositories of deep, open-ended problems, show that this level of thinking can be nurtured at every level of schooling.
If it doesn’t look, sound or feel like authentic mathematics, then there’s a reasonable chance students are being short-changed in their maths education.
I am a research mathematician turned educator working at the nexus of mathematics, education and innovation.
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