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.

*The venue*

This spring, in reaction to the COVID crisis, I was part of a team developing an online ‘webinar’. The team was from the Julia Robinson Mathematics Festival (JRMF) program. At the time, I was its Executive Director. In normal times, we run non-competitive after-school mathematics events (“Festivals”) in which students are offered interesting games, puzzles and problems, assisted by a facilitator. Since face-to-face work with students has lately not been possible, we have sought to continue the work virtually.

The program has met with success. The JRMF team works on the presentation of a problem each week, polishing it for a group of about 200 students who `tune in’ to the event internationally. The students are split into groups of fewer than ten, and put in breakout rooms to discuss the problem. An adult facilitator guides the discussion, not to achieve a particular goal, but as a moderator, letting the students’ insights emerge naturally. Facilitators meet for half an hour after webinar, to pool their experiences and offer ideas for refining the program.

Problems are “Low threshold, high ceiling”. That is, very young students can work on them, have fun, and achieve insights that will eventually take them farther. More advanced students can use them to engage in thorny issues or deep mathematical concepts.

For examples of such problems—and an open invitation to participate in these webinars—see www.jrmf.org. We post the problem on Mondays and discuss them on Thursdays so participants have time to explore the question. On Thursday, we ask students two questions: (1) How long have you worked on the problem? (2) What is your age? They are assigned to breakout rooms depending on their answers to the questions.

Attendance has grown steadily. We find that students who come to the webinar tend to return. Thus we have created a virtual community, all around the globe, of students who enjoy mathematics.

*The Zoom Room*

Last week, I was assigned a room with four energetic and highly motivated young students, ages about 10. The facilitators were familiar with these four. When we first started the program, we found them difficult to work with. They had often gone far into the problem: the amount of time they spent on it could not tell us that. They often had bits of mathematical background that other students lacked. And their youthful and overflowing exuberance made it hard to integrate them into a group. They were always a challenge to any facilitator.

So we decided to create a special breakout room for them, the “Zoom Room” where they could race ahead. The success of this effort varied with the mood of the children. At best, they urged each other forward and vied with each other for insight. At worst, they would try to show off to each other what they already knew, without contributing to either the group effort or their own knowledge.

This past week the group clicked. I was delighted to find that the four boys (they were all boys) worked beautifully together as a team, and got further than any one or two of them could have in the short time available. I led them with but a light touch of the reins.

They did not solve the given problem. They didn’t even work on it. They created their own, and the last thing I wanted to do was confine them to what I thought they should be learning.

Here was the problem we had set (briefly): Given a large square with integer sides, how can you tile it with smaller squares. also with integer sides? The problem was presented in a more structured way, to offer `on ramps’ to the mathematics. An interesting problem, combining elements of combinatorics and geometry. And, as is typical of JRMF problems, it can be worked on many different levels. I was eager to find out where the discussion would go with my four young students. It took a turn that I could not have predicted—or prepared for.

They looked at the first problem and immediately answered that for $1 \times 1$ squares, you can tile any $N \times N$ square. The important point here is that they saw this as a special case: it was a sophisticated insight for children that young. They then went on to consider the question of tilings with $2 \times 2$ squares. I asked if you could tile a $7 \times 7$ square with $2 \times 2$ squares. They again saw that they couldn’t, and articulated the reason: 2 does not divide 7.

So I asked, “If $a$ does not divide $B$, then clearly an $a \times a$ square cannot tile a $B \times B$ square. But is that enough?” My point was new to them. It was the difference between a necessary condition and a sufficient condition. Very generally, I find that the core difficulty in learning mathematics—for anyone, at any level—is the logical structure behind the assertions or computations. Even these very experienced students had to take a minute to understand what I was saying.

In fact the condition is sufficient as well as necessary. They seemed to understand this particular example, but I am not so sure that they will understand the distinction between a necessary and a sufficient condition in another context. No matter. They are ten years old.

To guide the discussion a bit, and to get what I could out of their intense engagement in it, I asked how many ways they can tile a $7 \times 7$ square with identical squares. Dan (I am not using the students’ real names here) immediately said, “Only with $1 \times 1$, because 7 is prime”.

“No,” countered Alejandro. “You can tile it with one big fat $7 \times 7$ square. Does that count?”

“Well,” said Titus, “A prime number has only two divisors: one and itself. So we can use the same idea to count these tilings, if we count $7 \times 7$ as a tiling.”

Titus may have wanted simply to show what he already knew. But this seemingly innocent and perhaps boastful remark turned out to be a fertile one. Dan generalized immediately: “For an $8 \times 8$ square, there are four tilings.” (He meant tilings with identical squares, and everyone knew it.) “That’s because 8 has four divisors: 1, 2, 4, and 8.”

And this is where we came in. Alejandro took up Xue’s challenge, and his ten-year old explanation was wonderfully simple. “Say there are two primes, $p$ and $q$. Say the number is $p^2q^3$ You just make a picture.” And he drew this on the shared screen:

In another group, Alejandro’s explanation would have been a mystery. But these four looked at it and understood.

“You need a 1 to count the 1,” said Dan, “and also the singles: $q, \ q^2, \ q^3$.”

“Right,” said Xue. “So if $p$ is squared, you have three columns, not two. That’s why we add one to the number on top.” He meant the exponent.

“But what if it’s like $p^2q^3r^4$?” asked Alejandro… and answered his own question. “Oh. It’s the same thing. You can just list the twelve divisors we have already down the side, and list $r, \ r^2, \ r^3, \ r^4$ on the top.” As facilitator, I squirmed a bit at the error. But in this virtual environment, no one saw it. And knowing these kids, I remained silent.

“No,” said Titus quickly. “You need five columns: $1, \ r, \ r^2, \ r^3, \ r^4$.”

“That’s right,” said Alejandro. My silence had paid off: the point was made better and faster than I could have. The interaction at once exploited the benefits of kids working together and increased the bond between them. Boastfulness and ego were quickly put on the back burner.

I didn’t want to rest there. They could recite the formula. They could prove it. I wanted to make sure they could use it. So I asked them a question that they were unlikely to have seen before: What two-digit number has the most divisors?

Their thought was swift, and collective. They quickly saw that they had to look at prime divisors and balance the number of divisors with the exponent in the formula. All this without writing anything down.

Titus led off: “It probably should have lots of 2’s and 3’s. Because we don’t want the number to get too big.”

Xue: “Well, it can’t have more than six 2’s, because $2^7$ is already 128. And $2^6$ is 64 and has seven divisors.” He had intuited the formula for the case of a single prime. I did not need to call his attention to this special case.

Titus again: “What if we put in a 3? Three times 32 is 96. It has. . . ” He thought a minute. “ It has $6 \times 2 \dots$ twelve divisors.” I didn’t have to ask him to explain.

Indeed, I didn’t have time. Alejandro jumped right in: “It depends on the exponents. The primes don’t matter. They just can’t be too big.”

Xue: “Can we have a 5 as a prime factor? Well, we can’t have two 5s. We can, but that will give us 25, 50, 75, and they don’t have enough divisors.” He was imagining what applying the formula would do, and his intuition told him (correctly) that these numbers would have fewer divisors than the 12 that they already saw for the number 96.

Dan: “And if we have one five, the rest of the number is 20 or less. We would need 6 or 7 divisors for that kind of number. Can we do it? ”

Silence.

Then Dan again: “Seven divisors can’t work. It’s prime. Six divisors? It’s $2 \times 3$, so we need $pq^2$. That’s $2 \times 3^2$ or $2^2 \times 3$. Eighteen or twelve. Five times these give 90 or 60. Each of these also has 12 divisors.”

Alejandro: “I don’t think we can beat 12. We just have to look at 2’s and 3’s. No. We can’t get 13 or 14 divisors. We would need too high a power.” (I didn’t stop him—everyone seemed to understand.) “Can we get 15 divisors? We’d need $2^2 \times 3^4$. That’s too big. Or $2^4 \times 3^2$. What is that? $16 \times 9$. No, still too big.”

Titus: “So only 12 divisors.”

I asked, “Which two-digit numbers have 12 divisors?” The list came tumbling out of them, and they all contributed.

*Generalization*

Unbidden, the group asked the next question: “What three-digit number has the most divisors?” They started working on this, and the ideas flowed. Ramsey Makan, my techical assistant, himself quite young, had been listening. The number 720 came up, and someone remarked that this was $6!$.

Ramsey asked them, “How many divisors does $6!$ have?” They worked it out. Then of course started thinking about factorials in general.

Titus was out of the discussion for a few minutes, then came back. “I wrote a Python program to list the divisors of $n!$.” They all wanted to see, so Titus ran it, for $n = 1$ through $6$.

“Can it do $10!?$” someone asked. Titus ran it for $n=10$. The screen went blank.

“The numbers are pretty big,” he finally said. “So it’s going slow.”

And indeed it was. The program was using brute force. I wanted to keep the momentum of the group up, so I said: “Can you figure it out yourselves? Maybe you can beat the machine.”

And they did. When the number finally popped up on the screen, it matched their result.

With time running out, I wanted to leave them with something to work on. So I said: “Suppose you know the number of divisors of 12!. Suppose some wizard told you how many there were. Would there be more divisors of 13!? Or fewer?”

The group responded easily: “More.” And then Dan said, “Twice as many. Because 13 is prime.” This was met by a chorus of “Oh, yeah.”

“But it wouldn’t work for 14! if you knew 13!,” said Xue. Then, a moment later, “What would work?”

They started thinking. Titus said: “Four times as many…”

Titus’s idea was not quite right. But the time was up. The breakout room was closing. I said goodbye and the webinar came to a close.

*Conclusion*

Teaching online can be tough. You lack certain means of communication: gestures, looks, posture. And if a student is silent, it’s hard to tell if he or she is engaged. My experience with these four students may not generalize easily. But it does give us a picture of what can happen when students encounter each other virtually.

And it gives us another picture. Sometimes it is argued that we must not do anything special for students who need more mathematics. It inflates their egos, makes them think that somehow they are ‘superior’. Well, it can, if done badly. And it can damage a child if we value the gift and not its bearer. The experience I chronicle here shows how important it is that such students meet other such students, that they come to see themselves as no different from a whole group of peers. My experience has shown is that once they are in such a group, they grind off each other’s rough edges. Being a ‘nerd’ or a ‘brain’ is a costly defense, like a suit of armor. And the defense, in such situations, is quickly and eagerly shed.

*Acknowledgments*

I would like to thank Ben Blum-Smith and Yvonne Lai for their invaluable input into the writing of this piece.