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PROVING STUFF: Mathematical Thinking with Five-Year-Olds, 3.0

Math Circle 4.1.2014


(April 1, 2014) Things started out smoothly enough.  The clock was taped to the floor, and the kids were ready to jump on it.  Will you land on every number if you count by four?  They did this without me, and added a new approach:  everyone started on a different number, and half the kids moved counter-clockwise.  (Worth noting, though, is that M forsaw difficulties in this approach and strongly urged everyone to move in the same direction.  I suggested trying it both ways.)

Before commencing the jumping, I asked for predictions. Everyone called out their conjectures.  However, after one particular student whose recent conjectures have been pretty accurate made his prediction, another student changed his to match.  It made me wonder:  When/why/how does that mathematical self-doubt/defer to others start?  I’ve observed every student having mathematical aha moments.  So why the doubt?  This one student doesn’t always have more mathematical insight than the others.  He is, though, very outgoing, and able to verbally express himself well.  That observation naturally begs the questions (1) do those skills trump mathematical insight in some situations?, and (2) what role does confidence play in mathematics?

After some vigorous jumping, the kids returned to our circle of chairs.  Some reported having jumped on three numbers, others on all, and another was not sure.  We all agreed that M’s concerns had been valid, and tried jumping again in an orderly manner.  After that, everyone but B agreed that when you count by four on a twelve-clock, you land on three numbers, no matter what you start on.  (Parents, are you starting to notice a pattern here?)

B insisted that he had landed on every number while accurately counting by four.  I asked him to demonstrate.  It turns out that he was using his hands to help balance as he made the huge four-jumps, and had also counted the numbers that his hands touched.  (Another great example of the mathematical task of refining your question.)  Seems that I needed to specify that we’re looking for how many times we “land on our feet on a number.” I stressed that mathematicians need to refine questions all the time.  And now finally everyone agreed on an answer.

I could tell that the kids were looking for patterns in our cumulative list on the board.  The table listed

  • What number did you count by?
  • Did you jump on every number when you counted by it?
  • How many numbers did you land on?

I loved that I didn’t suggest looking for patterns; the kids just did it – more evidence that humans naturally seek structure, predictability, regularity, pattern, etc.  (Is it our desire to be safe?)  The students even talked a bit about what the table meant, but so far, no generalizations occurred.


We moved to the floor, where I had stacked dimes and nickels.  Which pile has more coins?

“This one,” everyone said together, pointing to the taller nickel pile.

How can we be sure? 

“Count them,” came the reply, so we did.  I placed the nickels in one row and the dimes in another, and the counting began.  Most students agreed that counting yielded 10 in each row.  “That can’t be right,” said someone, remembering how the nickel pile was taller.  Everyone counted, and recounted, and recounted.  One student didn’t get the same result as the rest. I could have gone into counting strategies here, but toy airplanes came out of someone’s pockets, leading to distraction.

The questions of how many coins are in each row, and how to prove that the numbers are equal, remain unresolved in our circle.  I’m not positive that all members of our circle fully grasp the concept of invariance of number, but we’ve made progress in that everyone uses counting as a proof strategy.

I refocused the somewhat scattered group with some very unexpected questions.

PROVING THINGS                 

“What do you use to see?”  I asked.

“Eyes,” said everyone, pointing to their own eyes, and looking at me like I really ought to know this.

“I don’t,” I said contrarily.  “I hear with my eyes and see with my ears.”  The kids giggled and argued against me.  I demanded that they prove that I don’t.  I asked if any of them knew what “prove” means.  Only one did – he said that when he thinks that his sister stole something from him, his mom demands that he prove it (with evidence).  This was a great definitional example, which required only another sentence or so for everyone to understand what “prove” means.  So they set about proving that I see with my eyes and hear with my ears.  It took several very enjoyable minutes.  Then the student who had earlier changed his conjecture to match another’s spoke directly into my ear.  (This was the same student who had prior experience with proof.)

“What are you doing that for?” I asked in a gentle way.  He said nothing, but I knew that he was proving me wrong.  He just didn’t have the words yet to explain his reasoning.  Then another student instructed me in some eye exercises:

“Look directly at my hands,” he ordered.  He placed his hands in front of my eyes.  “What do you see? … Now look at my hands.”  (He moved them to next to my ears.)  “What do you see?”  After I replied, he said triumphantly, “Then that means you see with your eyes, not your ears.”  Everyone nodded in agreement.  They had just done their first proof.  Not just that, but it was collaborative.  Each thing a student did or said in this process resulted in someone else having an idea.  BTW, I got this idea from Zvonkin1, who writes that the proof for the eyes lies in closing them.  I told the kids that there is more than one way to prove something, and that their way was entirely convincing to me.

We tried another of Zvonkin’s proofs for young children:  “Prove that clouds are nearer to the earth than the sun.”  This didn’t work out so well because not everyone in the circle accepted the statement as true.  One student, A, claimed that the sun is closer.  I asked how she knew that.  Her reason, which I can’t remember, was excellent.  I realized that to do this proof we’d have to first do some scientific observation – beyond the scope of this circle – so we moved back to the coins.

“How can we prove that the number of nickels is the same as the number of pennies?”  We talked about it for a moment, but no one had any ideas.  Then B said, “Hey Rodi!  I thought that this was supposed to be a MATH Circle.  I don’t understand why all we do is sit around and talk about things and try to prove them.”  Now, if I were writing an article on the kinds of discussions you want to have with kids in a math circle, that question would be right at the top.  This is what mathematicians do!2  How wonderful that the class (as opposed to me) came up with this observation. The kids were so surprised to learn that math is not just about manipulating numbers.

We spent a few minutes revisiting the logic puzzles we’re working on (The Very Clever Prince, and The Boy in the Elevator).  I asked A to help explain last week’s progress to E, who had been absent.  We didn’t outwardly move closer to solutions.  But who knows, maybe in this discussion someone phrased something differently that will get wheels turning for next time.  One part of me is regretting my promise to give them a hint if they haven’t solved the puzzles by next week, our final session.  The students will feel glorious, I expect, when they eventually work it out for themselves.  Another part of me, though, is glad of the promise.  I think these kids need a success at this point.  My challenge is to come up with a hint that is not “leading.”  I wouldn’t want to steal any of the joy of realizing the solution.

Why have some of the kids gotten a bit squirmy this week?  A few conjectures:

  • It’s April 1, a beautiful day after a long snowy winter.  Has spring fever has set in?
  • I’ve been working their brains extremely hard.  Too hard?
  • L was absent.  Was the group dynamic markedly different today?
  • The last 2 sessions were so magical.  Have I become overconfident?
  • We’ve been asking the same questions for three weeks.  Are we in a rut?

Anyway, I took the squirmy group outside.


First we made some piles of rocks.  As I did last week, I asked, “What’s the opposite of a pile?”  The dissension of last week (negation only, versus the vector approach in which you move the same distance in the opposite direction) remained.  Then things got even hairier.  The kids each made 2 piles.  “What’s the opposite of two piles?”  I got four answers from five kids:

(1)     No piles

(2)    Two holes (or “destroy two piles”)

(3)    One pile

Interestingly, people’s opinions had changed from last week.  The big proponent of the vector approach was now in the negation camp.  The vector camp now included two additional students, including the one who deferred to another’s position earlier.  He had no doubt of his answer now that we had moved into the realm of physical, versus verbal, mathematics.  (He was clearly not deferring/following, since the student who was so eloquently verbal had now switched his opinion in the opposite direction.)  This shifting makes me realize how fleeting insights can be.

“Is the opposite of building a certain number of piles to build a different number of piles?” I asked.  The student who proposed this said yes, and then several students, who had other answers a minute ago, agreed.  I realized that we weren’t going to solve this thing with talk only.  We needed a change of approach.


I had drawn a chalk number line on the ground ranging from -8 to +8.  (I did not draw positive/negative signs, only the absolute value of the number.)  I asked everyone to stand on zero, then gave I gave some commands:

“Move forward three lines.”  (Consensus.)

“Move forward two lines.” (Consensus.)

“Do the opposite of move forward three lines.”  (Everyone moved back three lines.)  We continued along this vein (do something and do its opposite) for a few minutes, with the kids moving in unison like a chorus line.  Then I asked them to all return to zero.

“Do NOT move forward three lines.”  E stayed rooted on zero.  Everyone else moved back three lines.  Then some of them saw rock-solid E and moved up to join her.  Some stayed on -3.  Some wavered.  “Is NOT moving forward three lines the same thing as doing the opposite of moving forward three lines?”  Everyone realized that no, they are not the same.  (Phew, finally!)  But now came the moment of truth:  would five-year-olds be able to transfer that concept to another scenario?5  (I can almost hear you laughing as you read that.)

We returned to the rock piles.  And, you guessed it, some did and some did not transfer the idea.  I tried yet another approach to opposites:  going up and down stairs.  This was just preaching to the choir.  For several kids, squirming became running around.  A parent helped to bring those kids back (thank you!).  I told them gently, “You don’t have to be in Math Circle if you don’t want to.  But if you do want to be here, you have to stay here and do the activity.”

“But Rodi,” said one earnestly, nearly in tears, “I LOVE math circle.”  Oh no.  My gentle reminder had come across as a suggestion that they didn’t like math circle.  Their feelings were hurt and I felt terrible.  I’ll need to come up with a better line to use in these situations.  (Suggestions?)


Fortunately, I had my puppets with me.  I had forgotten them last week, but then it didn’t matter because we were living a charmed mathematical life back then.  Now, I put a puppet on each hand and the puppets gave an analogous problem, imbued with drama.  Everyone was instantaneously drawn in.

Penelope (the pig puppet):  “Last week, I was walking down the street and found two dollars.”

Koko (the gorilla):  “You’re luckier than I was.  I was walking down the street and the exact opposite thing happened to me.”

Me (to the kids):  “What happened to Koko?”

M:  “He didn’t find two dollars.”

M (reconsidering, a moment later):  “He dropped two dollars.”

This, I felt, was a mathematical triumph.  Everyone nodded and agreed with M.  Then I got super-ambitious again and tried to generalize/transfer the concept to the rocks.  This met with mixed results.  I was pleased in general, though, because each student had experienced at least one way of comprehending that negation is not always sufficient to demonstrate an opposite:

  • Piles and holes
  • Number line jumping
  • Money example with puppets
  • Stair climbing

Not everyone “got” every example, but that’s okay.  We all have optimal approaches to problem solving.


Again several group members needed refocusing, so I quickly resorted to another of Zvonkin’s proofs:  “What part of your body do you use for thinking?”3

“Your head!”  (“Your head, Silly!” was implied by their tones.)

“I don’t.  I think with my stomach.”

“No you don’t!”

“I don’t believe you.  You’re going to have to prove to me that I think with my head and not my stomach.”  While students now knew immediately what I meant by proof, this one was harder to prove than seeing with the eyes.  N asked me where my food goes when I eat.  “My stomach.”

“So see,” he said, “you’re using your stomach for eating, not thinking.”

“Good point,” I agreed, “but I think my stomach has two jobs – eating and thinking.”  Now kids were silent, and we were out of time.  I asked them to think about this during the week.  Then they good-naturedly went back into the classroom to clean up with me.  As we pulled the tape numbers off the floor, I asked “Do you think we asked more questions or answered more questions today?”  Opinions varied.  I used this last quick minute to make the point that one of the most important thing a mathematician does is ask questions, and that asking questions is sometimes more important than answering them.

Thanks again to older sibling L for set-up help.


1Alexander Zvonkin, Math from Three to Seven, p76-77

2Zvonkin describes proof as “The central – in fact, formative – notion of mathematics which sets it apart from all other intellectual disciplines.” p 76

3I had been saving this proof (Zvonkin p 77) as what negotiators call my “hip-pocket concession.”  I was grateful to have anticipated the need for it, and to not have used it prematurely.

4 Thanks to Maria Droujkova for this activity.  It is posted in great detail on the Living Math Forum listserve (message #12536).  Her post contains great ways for older students to use their bodies to understand various operations with negative numbers.

5Virginia Tech did an interesting study on concept transference in older students in STEM courses.  I haven’t read it in detail yet, but it seems to be saying that activities that are hands on, collaborative, and reflective (i.e. involving metacognition) may improve concept transference.

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