(April 8, 2014) We started out playing a few logic games – “Picking Fruit” and “Wearing Hats.” The students had fun using process of elimination to deduce what couldn’t be seen. The students announced one conjecture, then would immediately change their minds, then revert, etc. (Once again, insight is fleeting.) One student didn’t want to commit to any conjecture. The student sitting next to her said something like “Join us. Say blue is the answer.” He put his arm around her in solidarity. This blew me away because the same student who said this was the one who last week changed his own answer to match his friends’. If you read last week’s report, you’ll recall that I assumed he was expressing mathematical self-doubt upon hearing another possibility. Boy, was I wrong. Flashback to my ninth grade biology teacher in his green disco suit repeatedly warning, “You know what happens when you assume. You make an…” Here I am trying to train students to notice and question their own assumptions when I reveal my own humanity by not spotting *my* assumptions. The student probably changed his answer last week in a show of group-building. A future diplomat.

At the end of the activity, I emphatically stated that “sometimes you can use what you can see to figure out what is hidden away.” (A hint to The Very Clever Prince.) I asked the kids what kind of thinking they were using to solve these problems. “Thinking it through,” said one person.

“Thinking it through – that sounds funny,” replied another. *Thinking it through* was a catch phrase for the rest of the session, although I also let them know that this is also called logic.

We then moved outside for some number line jumping. We’ve been playing with opposites the past few weeks, so I decided to introduce more mathematical language:

- “Instead of saying ‘move 3 lines forward’, I will say ‘move positive 3 lines.’”
- “Instead of saying ‘do the opposite of move 3 lines forward,’ I will say ‘move negative 3 lines.’

Sixty seconds into the activity, a few kids started running around the outside of the building. I gathered them back and reminded them that attending math circle is optional, but if you want to be here for it, you need to participate. “But we want to do something else,” said L diplomatically. I’m assuming that his statement was a polite way to say, “I know you think your activity is the bee’s knees, Rodi, but we find it boring.” He was right; it was boring.

“Let’s play *Simon Says*,” I suggested. (First I had suggested the game *Mother May I*, but none of the kids knew that one.) We then joyfully did negative/positive number line jumping within the context of the game until it was clear that the kids had a basic understanding of negative numbers – a basic understanding *in that moment*, for, as we’ve seen, insight can be fleeting.

Someone then said excitedly, “Today’s the last day, right? Today’s the day you’re going to tell us how the very clever prince saved his life!”

“Remember,” I said, “I promised you a hint, not the answer. The games we’ve been playing have been your hint. We’ve just been playing games about opposites, and using what you can see to identify what you can’t see.” I gave them a moment to ask questions or form new conjectures, but nothing new arose. I said that they will figure it out at some point, and that they no longer needed me. I promised that if they ask their parents to email me, I’ll email the solution to parents.^{1}

The students also asked for a hint to the Boy in the Elevator^{2}, so we moved back inside and acted out the story. Zvonkin’s students solved this puzzle by actually riding in an elevator and seeing for themselves what was going on. But we’re in a trailer classroom here, so no elevator. My hint for the students, then, was to suggest to their parents a ride on a real elevator and talk about the problem there. Acting it out was just not an adequate hint.

Finally, we returned to our weekly clock-jumping activity. This week I had drawn clocks on a whiteboard. The students helped me draw in lines to make the patterns we’d already jumped. Today, the students suggested jumping by 5s, then 6s, then by 12s. They first made predictions, then jumped. We spent a lot of time looking at the patterns that emerged – both in the whiteboard tracing of the paths (see photos), and the chalkboard table of numerical results. The students were amazed that when they jumped multiple revolutions by 5s, they landed on every number. They checked and rechecked, found no mistakes, and then accepted that for some strange reason, 5 behaved differently from the other numbers. It is prime relative to the clock. We didn’t have time to further explore this revelation, though.

We didn’t have time for a lot of things in this four-week circle, so here are a few suggestions of fun activities to follow up with at home:

- Function machines – just play
- The Very Clever Prince logic puzzle – try to solve at home as a family (email me if needed)
- The Boy in the Elevator logic puzzle – act it out in an elevator, then email me if needed
- Simon Says on a number line with negatives (have kids write in the numbers)
- Clock jumping arithmetic – try it with bigger jumps – 7, 8, etc. Record the patterns in diagrams and tables. Star factoring might be a natural extension.
- Number conservation tasks – the questions of how many coins are in each row, and how to prove that the numbers are equal, remain unresolved in our circle. Email me for more details if desired.
- Keep on talking about opposites!

Thanks to all of you!

Rodi

^{1} Those of you who’ve been reading these reports all along know about my current favorite math book: Zvonkin’s Math from Three to Seven. Today I saw that this logic puzzle is in that book! Zvonkin calls it a “folklore problem,” and refers to it as “life versus death” in the index. In the foreword to the book, Paul Zeitz describes a math circle as “not so much a classroom as a gathering of young initiates with elder tribespeople, who pass down folklore.” (p. viii)

^{2} Zvonkin, pp. 47-48

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