Math Circle 5.11.2014


(May 13, 2014) “It’s time talk about the criminals,” I announced to an exuberant group of kids who were not quite ready to settle into math circle.  The word criminals got their attention.  We tackled Smullyan’s Inspector Craig mystery puzzle #74.  I expected a huge struggle for 2 reasons:

  • more clues to keep track of, and not all of the students can read, and
  • exposure to our first conditional statement (“if… then… “).

The students did struggle for a minute with the conditional statement, but then they solved it, just like that, much to my surprise.  As I learn, again and again and again, I better not get too attached to my expectations.  I had expected to spend about 15 minutes on this problem.  Fortunately, I had some other activities planned.


“I think I made a big mistake last week when we were using blocks to make the ants march in arrays,” I explained.  “I gave each of you 12 blocks, and we found that some numbers of rows didn’t work, like 5.  I realize what the problem was.  We were starting with the wrong number of blocks!  I think that if we start with 15, we can get the ants marching 1 by 1, 2 by 2, 3 by 3, and all of the numbers.”

As we sang, the students got right to work arranging blocks and testing to find what numbers are factors of 15.  (I didn’t yet use the word factor.)  Soon they realized that 15 is not a multiple (another word I did not use yet) of all of the numbers from one through ten.  Then came the conjectures.  Many of the students suspected that 20 would work.  N thought a higher number – 100 to be precise – would be needed.  They tested some more.  No matter the number, they couldn’t find a number that worked.  Most were happy to leave the question unanswered for now, but to retain the conjecture that Some Number must be a multiple of these Ant Numbers.  I mentioned that mathematicians do sometimes take breaks from problems at times like this.  F, however, decided that it would be more interesting and productive to redefine the question.  She kept happily working away, observing the properties of a variety of numbers.


We then visited another iteration of the Princess and Tiger logic puzzle from last week.  I wasn’t going to do it, as I’ve seen students struggle with it in the past, but this group is handling the logic puzzles so collaboratively and successfully that I couldn’t resist:

Statement 1, posted on a door, states that “Either this room contains a tiger or the other room contains a princess.”  Statement 2, posted on the other door, states that “A princess is in the other room.”  Either both statements are true or both are false.  Which door should you open?  

Based upon prior experience with this question, I expected the kids to have trouble negating the “or” statement.  But what happened is that the kids had trouble understanding the “or” statement.  My various explanations (via leading questions) weren’t getting through.  Most kids gave up and returned to the block table where G discovered – on his own – cubic numbers.* Only F remained with the logic problem, determine to solve it.  And solve it she did, albeit with a different solution and different definition of the word “or” from mine.  I told her that I wasn’t thinking of the word “or” in the same way that she was.  She asked me what I was thinking.  I couldn’t explain it in a way she understood.  I told her I’d need some time to figure out how to communicate my thoughts.**


I’d been thinking a lot lately about (1) how the kids in my prior group (age 5 and young 6’s) had struggled so much with Function Machines because of inexperience with numbers and (2) how this group, a smidge older and with a bit of overlap in students, just can’t get enough of these wooden unit blocks.  Aha!  I’d combine those two observations into the world’s greatest function machine – one that’s a little more concrete because it uses handfuls of blocks instead of numerals for the “in” and “out” items.  I set a whiteboard horizontally on a table top.  Then the kids dictated the requisite parts:

  • the drawing of the machine
  • the sound it would make
  • where things would go in
  • where things would come out
  • what to put in

One student set a handful of 3 blocks on the “in” place.  I set 5 on the “out” place.  Within the 60 seconds or so that we were discussing what the “rule” might be, the action shifted dramatically and I realized that I had actually facilitated Air Hockey.  ‘Nuff said, right?


I turned their attention immediately back to the big round table, where I asked them to sit while I retold the problem of The Wolf, the Goat, and the Cabbage.  I asked the kids to help explain it to G, who had been absent last week.  We didn’t make any more progress toward the solution, but I saw something pretty amazing.  One student, A, had solved this puzzle with his dad over the course of the week.  And he didn’t tell!  The self-control*** that this must have required still boggles my mind.  I mean, these kids are 6 and 7!  That solution would be hard for me, a little older than 6 or 7, to hold tight.  By the end, I did let him give the rest of the group a hint of his choosing, so he told them that the goat should go first, and that it’s okay for the peasant to carry something back across the river to the original side.  Then time was up.  To be continued next week…


*The Teacher in me thought “Good grief, I’ve been trying to set them up with activities to discover triangular, square, and cubic numbers on their own, but what they really needed was to be away from me.”  The Math Circle Leader/Secretary/Facilitator in me thought “Hooray!  Three cheers for the most powerful type of discovery: independent.”

**Communication is, as evidenced here, an integral part of mathematics.  Who cares about our ideas if we can’t share them?  To quote Zvonkin, “Mathematics is a human activity.”  To paraphrase his explanation of this statement, We must be able to explain our reasoning and also to understand the explanations of other people.  (p144)

***I think that perhaps it wasn’t just self-control here, but that kindness was involved.  A knew for himself what a joy it was to solve the puzzle, so he wanted his colleagues to experience this same joy.  Were our math circle by design a competitive, rather than cooperative, endeavor, would this have happened?  Just wondering; not going for a halo effect where I get some of the credit for A’s kindness.  Still, though, it makes you wonder, right?  Might the collegial environment of our Math Circle increase the likelihood that we could all bask in the glow of his kindness?

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