Mark Saul

*A child’s insight*

“I know how to find out how many divisors a number has. You factor it into primes….” Alejandro was with a virtual group of four enthusiastic ten year olds, in the midst of exploring a problem. He gave the usual result, using his own somewhat makeshift words. But not too distant really from what I would have said: If $N$ factors as $p_1^{a_1}p_2^{a_2}p_3^{a_2} \dots$, then the number of divisors is $(a_1+1)(a_2+1)(a_3+1)…$. His description was less economical, but still accurate.

His virtual friend Xue said: “That’s great. Let’s look it up on Wikipedia.”

Then, “No. Let’s not look it up. Let’s pretend we don’t know it and see if we can prove it.” It is this insight into his own learning, not any mathematical breakthrough, that I remark on in the subtitle to this section.

Dear Reader: I swear to you, on Galois’ grave, that I am not making this up. Nor the rest of the vignette I will be recounting here.