## Postgame Analysis: the Towers of Hanoi

I recently gave my juniors the classic Towers of Hanoi puzzle to play with in small groups.  It went something like this:

You have three plates, and plate #1 has a stack of 5 pancakes, in order from the largest one on the bottom to the smallest on top.  The puzzle is to get the stack onto plate #2 using as few moves as possible.

Two rules: (i) you can only move the top pancake on a stack, and (ii) at no time can any larger pancake be on top of a smaller pancake.

They spent a couple minutes getting familiar with the mechanics of it, and then settled into working together, shifting pancakes and keeping a count of their moves.

It didn’t take them long to get into a pretty strategic approach: get the smallest two pancakes onto plate 3, then move the third pancake to plate 2, now get the smallest two pancakes onto plate 2 also, move the fourth pancake to plate 3, etc.

A couple of related observations struck me as I watched them do this.  First, every group got preoccupied for a while about whether their first move should be to plate 2 or plate 3.  And second, even after they’d begun working systematically, they still made each move manually, rather than abstracting the process of moving a substack.  For example, even after they got to here…

…and knew that it had taken them seven moves to move the substack of the smallest three pancakes from one plate to another, they still proceeded to manually make the next seven moves to get to:

Eventually, each of the groups managed to complete the puzzle using 31 moves, but when I asked each group whether that was the best they could do, they were uncertain.  Now, on the one hand, I think it’s pretty likely that if they had made the leap of abstraction to treating moving a whole substack as a “thing”, they would have been able to dispense with their doubts pretty easily.

On the other hand, the fact that they hadn’t made this leap prompted some pretty nice exploration.  In trying to decide what was optimal, several groups started looking at the 1-, 2-, and 3- pancake cases and saw the “two times the previous plus one” pattern.   Naturally, this did give them more confidence in their answer of 31 for the 5-pancake case, even if it wasn’t a full proof.   Realizing that they hadn’t really explained why the 2*previous+1 pattern would hold, one group started keeping track of how many times each pancake is moved in the course of solving the problem.  They found a remarkable, delightful result (I’ll not give it away here), but still didn’t quite manage to justify the pattern.

On reflection, I think the reason that the students didn’t naturally make the leap of abstraction is that there were too few pancakes—few enough that it wasn’t actually inconvenient enough.  There’s a sort of “sweet spot” to many problems, a perfect size where the problem is small enough to be accessible, to feel like it’s possible and therefore worth trying, but at the same time large enough that it soon becomes impractical or inconvenient to do it concretely.   In the end, I showed my class the abstraction in this problem, but I didn’t feel great about having to do that.  I think if I’d hit the sweet spot—the range of ideal inconvenience—they very well might have been prompted to make that abstraction themselves, in response to their own authentic desire to simplify and speed up their thinking.

1. Jenny Burton

Anand,
Can you explain what you think would happen if they had realized it without your saying? Do you think that they would have been done and satisfied with the problem earlier, or do you think there was more to get from it from just the abstraction?

My real question is, if you have one student who gets the abstraction and he tells ito you, what strategies do you use to make sure that he doesn’t (or does?) give it away to the rest of the class? Do you bring it to the classes attention and then just ignore its brilliance and make the class judge it themselves? Do you confirm/deny/act like you don’t believe the kid while (s)he is talking to you?
I would love to hear you thoughts,
-Jenny

• Posted December 8, 2011 at 3:56 pm | Permalink | Reply

@Jenny – I’m afraid I’m going to have to answer with the ever unsatisfying “it depends”. Usually, when a student comes up with a key insight like this, s/he tends to direct it at the other students in the small group (my kids sit in tables of 3 or 4), rather than directing it to me.

I suppose if a student saw the abstraction very early on, I might try to sort of “quarantine” it within that small group, by going over and discussing the idea with them, probing it with questions, etc. Other times, if the other students seemed to have gotten in plenty of good thinking, I’d probably just sit back, put on my best poker face, and see how the conversation unfolded. (And I do, as you’re saying, like to see how the students judge an idea/solution on their own.) Typically, the student who came up with the abstraction will end up at the board, explaining her idea to several intrigued classmates, while a few others are ignoring the exchange and continuing to work on their own.

I’d say those are my two most common responses… along with, when a student does actually talk directly to me, redirecting him back either to the small group or the whole class.

• Jenny Burton
Posted December 8, 2011 at 9:57 pm | Permalink

Anand,
I’m actually not a teacher yet (working on it!), but if I was I would tell my class at the beginning of the year, and be consistently reminding them that I would not tell them if they get a problem wrong, and that its their job to figure it out and to question each other and not just believe what others tell them. That way when situations like that occur that would feel the need to do a very solid proof to convince me, (but really themselves) that there answers is correct. And then they would want to really make sure that the class believed it so they didn’t have to worry about missing a concept. I like this because then getting the class’s approval wasn’t to boast their ego but to actually have a actually philosophical mathematic debate where the genuine goal is to come up with a correct mathematical formula no matter who comes up with it first.
You probably think that this idea is crazy, I doubt I will ever get hired if I told a class that I wouldn’t tell them when they are wrong, everyone tells me thats not the place of a teacher.
What are your thoughts, and do you think I should try it when I get my first job (if ever?)

2. Posted January 4, 2012 at 7:23 pm | Permalink | Reply

I’ve done this many times with kids, but never thought myself (or had kids ask) about how many moves each pancake makes. Nice!

3. blaw0013

Jenny, I would hire you in a minute.

If you are serious, please let me recommend you have three possible additional layers ready for your response to people’s (interviewer’s) questions about your approach? Are you willing to let children leave your class thinking they know something that is incorrect? How might you manage student frustration with this change in the [math classroom] “game”? And finally, some form of “its your job to tell students how to do this the right way” (maybe more likely from a parent or student), not an interviewer.

4. Posted July 9, 2013 at 5:44 pm | Permalink | Reply

fantastic issues altogether, you simply won a new reader.