I ran across an interesting post on Medium written by Junaid Mubeen called The time I ‘nearly’ solved the Twin Prime Conjecture. The story is about how, in the summer before starting at Oxford, Junaid thought he had solved the Twin Primes Conjecture, even though he didn’t really come particularly close.

The point, however, wasn’t really his failure to solve the problem. It was on the difference between thinking like a mathematician – what he was doing when he tried to solve this conjecture – and what students are taught in math class before going to college (as we call it here in the states).

For me, this was the money quote:

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.

In high school, I had trouble with math. Well, be honest, I had trouble with everything. But math was especially bad. My test scores and middle school work were good enough to get into an advanced first year algebra class, but I failed it completely and had to repeat it. Meanwhile, I loved English. I read everything, and also enjoyed writing. I had great English teachers for both my Freshman and Sophomore years of high-school.

Eventually, the disaster that was high school gave way to college. Before embarking on that adventure, however, I had a math tutor. I don’t remember her name, but she was amazing. I particularly remember her getting me to derive the quadratic formula. I was delighted! I had never realized that mathematics was something that could be worked out from first principals rather than being memorized and recited.

Before that experience, math was all memorizing facts and using them to solve problems designed to demonstrate we had properly memorized the facts. This was different. This was, I think, more like what Junaid meant by “thinking like a mathematician”.

I enrolled at Illinois Wesleyan University as an English Major. One more great math teacher (this time I remember him: Ron Sandstrom) and I was hooked. I switched my major to math. (Dr. Sandstrom also taught one of two or three programming courses at IWU around 1980. One of them included a project that led directly to my leaving the school to earn a living writing computer programs. That’s a good story for another post!)

But back to today. I completely agree with Junaid. I’m pretty sure My kids have never had the experience I had deriving the quadratic formula. I want them to have that. He linked to these two web sites: NRICH and Mathalicious as examples of places kids could get an idea of what math was really like.

I haven’t looked at Mathalicious yet, but I went to NRICH, clicked on “Upper Primary”, and was immediately confronted with a problem I remember enjoying solving when I was younger. Here is the first problem I saw:

 1 a b c d e
x          3
------------
 a b c d e 1

(Sorry for the crude formatting. I should learn a better way to present that.)

I’ve always liked problems of this general form, and I’d love for my daughters to learn how to solve them. The general process is along the lines of:

1. \(3 e \equiv 1 \; (\bmod 10)\), hence \(e = 7\)
2. \(3 (d+2) \equiv 7 \; (\bmod 10)\), hence \(d = 5\)

and so on.

However, the problem presented here is a special case:

\[\begin{align} 1abcde \cdot 10 & = 1abcde0 \\ 1abcde \cdot 10 - 100000 & = abcde0 \\ 1abcde \cdot 10 - 99999 & = abcde1 \end{align}\]

but the original problem stated that \[1abcde \cdot 3 = abcde1\]

So, let \[abcde = x\] then, \[\begin{align} 10x - 999999 & = 3x \\ 10x & = 3x + 999999 \\ 7x & = 999999 \\ x & = \frac{999999}{7} \end{align}\]

Now a calculator can tell us that \(999999 \div 7=142857\). Note that the rightmost two digits agree with my initial logic that \(e=7\) and \(d=5\).

Because this was a special case, It was possible to use simple algrebra to solve the problem. Usually, my puzzle magazines had problems that were not so simple.

The NRICH site had two puzzles that fit the above pattern: that is, they could be solved using simple algebra. One did not need to be concerned with the more general (and abstract) approach of determining what each individual digit can be (excellent introduction to modular congruence though that might be).

The site gives (in this case, two) possible paths toward a solution that students are supposed to look at only after trying to come up with an approach on there own. Both of these used my digit-by-digit approach. But I can’t help the feeling¹ that it was not no coincidence that the two problems given were amenable to an algebraic solution.

Ok, what am I trying to do here? Did I simply want to show off that there was another way of solving this specific problem that might not be readily apparent to a high school student as it was to a 55 year-old engineer who uses math in his work every day? Or did I want to get people thinking about what it really means to think mathematically?

Ok,

1. I’m a sucker for ancient memes.
2. I have no idea why the alt-text appears twice below the image. Something to be debugged in my blog software?

Cheers.