Math Circle at Talking Stick

The green tape on the rug hinted at a quadrilateral shape, but was actually composed of 4 line segments that did not meet at corners.  Is it safe to say that the lines would definitely meet and form corners were they to be extended?  If so, can you call this shape a square or a rectangle?  If so, how can you determine which of those it is? 

 

These questions arose as we began this week’s Math Circle.  Every week, our group asks more good questions – such an important skill to a mathematician.  Regarding the tape on the floor, we finally agreed to assume 2 things:  (1) that the lines met at corners and (2) that we could measure (with rulers or our fingers) to determine which shape it is, but that we didn’t want to.  O lobbied for calling it a “whatchamacallit,” but the collective opinion of the group leaned more toward calling it a “squaretangle,” and so it was dubbed.

 

I dropped a pile of pencils into the squaretangle for a round of the Takeaway Game.  The students quickly ascertained which one was gone, so I took away three pencils the next time.   I expected the children to say that three pencils were gone, but most of the kids described in detail one pencil that was missing.  P said that three were gone, but was unable to describe any of them.  Since each child had initially focused on just one attribute (number or appearance), I was curious to see if they would focus on two attributes now that they were aware of the possibility.  I put the pencils in a pile again and immediately M and A starting counting them before closing their eyes.  I asked why they were counting, and V’s eye lit up as he said “so you can subtract afterwards and know how many are missing!”  With the emergence of this new strategy, everyone started counting, but no one got the same number, so we all counted together.  One person counted in Italian, one in Spanish, and yet another in Roman Numerals.  Then they closed their eyes for the last time, I removed four pencils, and everyone happily announced with open eyes, “seven are left; you took four!” 

 

And speaking of the number seven, since Halloween is coming up, I mentioned a spooky thought about the number seven:   I’ve heard that to some people long ago, the number seven was considered a bad-luck number because it does not correspond to an obvious number of things in the human body.  The children brainstormed various body parts and their associated numbers; then we returned to the squaretangle.

 

I dropped a pile of wooden geometric shapes into the squaretangle.    While the children’s eyes were closed, I removed the sphere.  When they opened their eyes, N declared that “the circle” was missing.  All agreed.  I showed them the sphere and asked, “What is different about this from a circle?”  “Oh, a circle is flat.  This is a ball.”  I told the children that mathematicians had a special name for balls, and O called out “sphere!”  We had fun identifying the other shapes, especially the hexagonal prism.  That one was fun in three ways:   another name for a curse is a hex; “prism” sounds like “prison” (we wondered whether we could lock someone in there?); and “hexagonal prism” is just an awful lot of fun to say aloud.  People described the octahedron as a diamond and the cylinder as a can or a tube.  O said that we should put some beans in it.  We played the Takeaway Game with the solids once or twice, then moved the shapes and ourselves to the table, where a box of apples was waiting.

 

We used the apples for Maria Droujkova’s “Apple Math” activity, in which children predict how many knife slices are needed to cut an apple into a particular solid shape.  N first asked to cut it into a sphere, and the children delightfully called out “zero!”  They agreed that while the apple was not an exact sphere, that we could approximate and call it one.  Each child took a turn suggesting a shape, and we used a show of hands to make conjectures on the number of slices it would take.  Consensus regarding conjectures on number of cuts was infrequent.  My 12-year-old helper (and photographer and editor) Rachel kept track of predictions and results on the board.  We arrived at a prickly juncture when one child said we could cut off the stem first, and then cut the top off.  Others said we didn’t need to use 2 separate cuts for this result.  It turned out that we needed to clarify the question to specify that we wanted the minimum number of cuts. 

 

For every shape, we made conjectures before checking them with actual cuts.  We came up with definitions of “top” and “bottom” as they regard to apples.    We arrived at another prickly point when M conjectured that the bottom of the apple did not need to be sliced off horizontally to make the base of a cube since the apple lay “flat” on the tabletop.  However, we had already done so.  As we were eating the cast-offs from our cube, M examined the bottom piece and refined her conjecture:  the bottom was not flat, it was “flattish.”  Definitions are, of course, important in mathematics, and we discussed the definition of a conjecture. 

 

When we cut the apple into a cylinder and each child received a disc of apple to eat, some noticed that the seeds made a star shape.  I asked what geometric shape you get when you connect the points of a star, and no one was able to visualize it.  I put it on the board, connected the points, and talked about both shape and building name “pentagon.”

 

At this point, V asked, “Is this one of the games that the children at the zoo played to help with their concentration?”  I was surprised to realize that I had forgotten about our story.   Our continued narrative has added continuity, interest, and context to our Math Circle.  I answered in the affirmative, and also told the children that the magical animals in the zoo required food that had been cut into these geometric solid shapes.  We then returned to the ongoing dying unicorn situation.

 

We recalled that we ended last week with failure when the slowest pair of zookeeper children crossed the bridge first to try to save the unicorn.  I reminded them that last week we made two plans: to use M’s suggestion that we see what happens when the fastest go first, and J’s suggestion to use our imaginations (instead of our bodies) to figure it out.  P was not at all interested in returning to this problem, and I wasn’t sure whether A and N had lost interest, so I changed the wording of the question to insert some hope.  I told the children that I had recently learned that another zoo of magical creatures had experienced this exact same problem and had solved it mathematically.  “Now we know it can be done.”  Our question is no longer, “Can it be done?” but “How can it be done?”  Fortunately, our group remembered how to combine rates, and was able to do the arithmetic mentally.  N quietly said the correct sum when several people added incorrectly.  

 

Unfortunately, the unicorn still died when our characters Ginny and Ron crossed the bridge first (in our minds) and Ginny returned to give the flashlight to the slower walkers.  I reminded the Math Circle children that thiscan be solved.  Someone proposed that we use our power to go back in time and give the unicorn another chance.  We decided to make another mathematical attempt to save the unicorn next week.  Then Math Circle ended with each child excitedly taking a whole apple home as a present.

“Something is in the air today,” said Talking Stick co-director Angie.  The kids came in brimming with energy, and most came early.   As we waited for the last child to arrive (still early), four of the kids were at the table writing newspaper articles.  Soon I asked them to put their papers on the windowsill.  They complied a bit reluctantly, and I pulled out a small musical instrument in the shape of a triangle.  I asked, “Who knows the name of this instrument?”  “A wind chime!” guessed J.   No one knew for sure so I gave the hint that its name is a shape name.  “Triangle” called the group in unison.  I instructed, “When I strike this and you think the sound has ended, it will not have and you’ll be wrong.  Listen harder.  Then put your head down when it’s really done.”  I struck it, heads went partially down, back up, and then down again.  M asked whether eyes should be open or closed, and I said “whatever you think – you could even try both ways.”  O said “You mean like this?” and closed one eye.  N said “I can’t do that,” so I suggested covering an eye with a hand like a pirate’s eye patch.  We focused our attention with three triangle chimes before I asked them to recall what was happening in our zoo story last week.  

“They were feeding the animals,” replied V.  What else?  “We were racing cars,” said someone, to which M added, “and we’re going to do it with people today!”  I had the kids line up behind the green tape line and repeat several of last week’s exercises.  “We’re being spectators,” said P and O as they returned to the table, newspapers in hand.  J sat on the windowsill as all kids tried to predict what speed cars glued together could do.  Most had forgotten the conclusion from last week and simply added the rates, but once we did dramatized it with cars the kids understood that the slower car determines the rate when the cars are glued together.  Then we talked about attached people.  A number of the kids did not transfer this concept of “attached” rates from cars to people and once again added the rates.  So M stood in between P and V and walked at her maximum but slow rate, while P and V held hands with her and walked at their faster maximum rate. M asked why she had to be the slow one and I said “because you are strong.”  The need for strength became apparent when P and V pulled forward and M pulled back and they all finished at the slower rate.  Then the whole group was able to generalize the concept to “things can do less than their maximums but not more than their maximums” and “slow things hold back fast things.”  Everyone agreed. Transferring and generalizing math concepts are important skills, so I was curious whether they’d be able to now apply the same concepts to our zoo family.

“Something bad happened at the zoo,” I told the kids.  “Someone freed the unicorn, not knowing that its magic required 4 people holding hands around it to be able to eat.  Now it was on the other side of the river, starving, with only 17 minutes left to live.”  Immediately eyes lit up as those who attended our demonstration class remembered this story:  “There is a unicorn dying at the end of a bridge.  He has 17 minutes left to live, unless 4 people can join hands around him and recite a magical spell.  There are 4 people on the other side of the bridge, but it is very dark, they have one flashlight, and the bridge can only hold 2 people at a time.  Ginny can cross in 1 minute, Ron in 2, Fred in 5, and Percy in 10.  Can they save the unicorn?” 

“Those numbers add up to 18, so it can’t be done,” announced V.  “Maybe it can,” someone suggested, since the people can cross two at a time.  I pressed the kids on how we could know for sure, and they came up with a plan to act it out as we did the rates problem.  It took a few minutes to come to something close to agreement on who would play each part.  A few kids wanted to return to their newspaper writing at this point, but became spectators instead.  The rest of the kids suggested that the two slowest walkers cross together first.  The kids were able to accurately predict and demonstrate the rate of crossing (math success!) but were sorely disappointed that the unicorn would die with this method.  The spectators started rolling around on the floor now, and J announced, “And the Nimbus 2000* whisked in and carried the unicorn to safety!” 

With a few kids rolling around on the floor, another flying on an imaginary broomstick, and the rest still wanting to further test possible solutions to the unicorn story, I quickly drew a broom on the board and said “Actually, the Nimbus 2000 is also a function machine.”  I told them how it worked and what sound it made and immediately everyone was sitting around the table, totally engaged. 

The 60 seconds of disorder in out classroom reveals something about how people enjoy math:  different people enjoy it in different ways.  We all have what the Kaplan’s refer to as “the architectural instinct” – a natural desire to pursue revelations about structure.    But some of us seek structure numerically (i.e. the stepstool problem of last week), others geometrically (i.e. map coloring), others logically (i.e. finding loopholes in a scenario to allow more possibilities), and so on.  In our group, different styles are emerging, and it will be fun to see if and how they change over time. 

 

And speaking of loopholes, let’s get back to our Nimbus 2000 function machine.  At first, the machine did a function that the kids said did “minus one.”  The kids suggested adding feet to the machine and increasing the sound, and the result was a machine that the kids said “figured out what half of the number is and then minuses that from the number.”  (In my mind I was dividing by 2, but there are multiple approaches to every math problem, which is one reason that math is an art.)  A few of the kids were calling out the functions before I was able to get multiple inputs and outputs on the board, so I decided to try to really stump some of them when they asked me to add ears to the machine.  In went 10, out came 6. After a conjecture about “minus 4,” in went 100, out came 51.  After another conjecture was disproved, in went 4 and out came 3, then in went 20 and out came 11.  P posited that the rule was changing for each input, but I assured them that a function always does the same thing no matter what.  The room was quiet; almost everyone had a smile yet a furrowed brow.  I gave the hint that the machine with ears did the same thing as the machine with feet but with one extra step, and quickly several students called out the function.  One said, “I don’t get it,” so I demonstrated and encouraged that excellent strategy of counting on fingers (and toes if needed).  Once it appeared that everyone understood the function, M said, “But you changed the rule.  You can’t change the rule.”  I explained that when a technological advance comes along (such as feet or ears on a function machine or a camera on a cell phone) a machine can perform a new task, or even a task involving multiple steps.   

At that point our time was up so I passed out snacks and told them they were done.  No one left.  They drew, wrote, ate, and continued to discuss function machines.  O talked about a “function box” he has seen at school, and we figured out how that differed from the function machine.  One parent asked what progress we had made on the unicorn problem, and I told him (so that the kids could hear this recap) that we now are convinced that a slower person’s rate determines a faster person’s rate when walking together, and we know that the unicorn dies when the 2 slowest people go first.   One child said that next time we should try the fastest people first.  J suggested doing it with “imaginary people.”  I asked the group if this problem would be easier to solve in their heads instead of by acting out.  The divergent replies showed that once again, there is more than one way to enjoy math.   Finally, I ended by collecting supplies and promising to return to feeding the animals at the zoo next week.

Rodi

*Harry Potter’s broom

(Photography and editing credits this week go to Rachel Steinig)

 

On the rug, the children sat around a green tape square which enclosed some toy cars, office supplies, a plastic dog, a book, and other sundry items.  I explained that in our continuing zoo story, the owner had four children - Ginny, Ron, Fred, and Percy - who needed to be brave to feed some of the animals.  These four played games to enhance their bravery, and this set-up on the rug was one of them.  I asked the kids to look at the assortment for 10 seconds and to then close their eyes while I removed something; then they opened their eyes and tried to discern what was missing.  The game went quickly until there was a bit of debate over whether the book was “green with 3 men on it” or “green with 4 women on it.”  The book was Women in Mathematics by Lynn M. Osen.   I called on the children who were not calling out answers and found that as usual, just because certain people were not volunteering answers didn’t mean that they weren’t successfully engaged.  We played “The Take-Away Game” a few more times before I asked them what this might have to do with math.  We recalled the feelings of frustration encountered last week when we attempted to determine a chromatic number.  Our group realized quickly that this game helps with noticing, which helps increase attention, which helps increase frustration tolerance, which hopefully increases bravery (and gets that tree pixie and shark and mermaid fed).  We talked about how math requires the bravery of Neville Longbottom of Harry Potter fame: the courage to make attempts without certainty of success.

And speaking of feeding these animals, this zoo had some pretty tall ones.  The carnivorous vacuum, the unicorn, and the dragon each had a feeding slot that required a height of 7 feet .  I reported that the four children were 3, 4, 5, and 6 feet tall respectively, and that they had stepstools of heights 1, 2, and 3 feet.    From a lively discussion about which person could use which stool emerged the math concepts of maximum and minimum.  When asked which stool 6-foot-tall Percy could use, the participants said “any of them.”  “So is there only one answer to that question?” “No!”

 I next asked whether math questions always have just one answer, and discovered that not everyone in the group realized exactly what “math” is.  The group decided that this stepstool question is identifiable as math because it deals with numbers.  We will surely expand this discussion in the future.  This would have been a nice place to discuss what numbers are, but alas, we only have 55 minutes each week.  The idea that math questions don’t always have a single answer felt like a turning point in our circle, which has been characterized by frequent musings about what “the answer” could be.

All the talk about feeding the animals got pretty exciting, so we refocused our attention with an interlude of bobble-head doll on the floor, then continued the zoo story.  I reported that the 4 children also liked to play with toy cars.  Their favorite games involved driving the cars at different but constant rates.  “Rate” was a new word for everyone in the room, so we talked about it first, and then got out the cars.  We took turns driving the cars at different rates.  While 2 people drove, the rest of counted (in various ways).  We practiced covering our set distance in 5 seconds and in 10 seconds.  We talked briefly about how we were measuring rate in way similar to, but not exactly the same as, “the signs on the highway.”  Our final exercise was to imagine that the blue car (“the fast one” that could drive in 5 seconds) was glued to the gold car (“the slow one” that could drive the same distance in 10 seconds).  I asked what was the minimum amount of time they could cover the distance together.  Some said 5, some 10, and some 15.  Then we acted it out:  2 children held the cars together while the rest counted.  The drivers drove at the maximum speed they could, considering the circumstances.  When the cars crossed the finish line at the count of 10, the kids proclaimed, “They did it in 10!  Now let’s do it with people!”   We ended our circle with a promise that next week, we will. 

I’d like to thank Maria Droujkova and Dmitry Sagalovskiy for direction in planning this session.

We had a few visitors today, so we started with a collective recap of our voting scenario from last week.  Kids immediately started suggesting still more voting/counting  methods.  M, who has asked if we could get back to the Dying Unicorn story (we will!), asked what we will do if we get “the answer” really soon.  I reminded them that the question is whether there is a system for voting that is fair.  O suggested multiplying votes in different positions (first, second, etc.) by different numbers.  I briefly told them about Borda Counts, which count votes in such a way.  P suggested some sort of competition among the candidates to determine who should win.  Some kids were getting distractingly physical with some of the math manipulatives on the table, so we engaged in an attention- focusing activity:  the Bobble-Head doll.

The Bobble-Head doll (who is “a distant relative of the man who owns the zoo”) sat in the middle of the table.  I tapped his head, which is on a spring, and told the kids that they had to watch until it stopped moving, then put their own heads down.  The doll never stopped; with every fidget (and possibly truck passing outside) the bobbling/vibrations increased.  At this point, attentions were sharpened, and we decided to put him away and try him on the floor next time.  I told the kids that the doll is somehow related to math, and with that, we were ready to return to our story.

The kids were curious to hear what the man decided to do, and I said that he did a second election where people indicated their likes and dislikes (“approval voting,” as invented last week), and I showed the results on the boards.  The kids immediately called out “The monkey did win!  Hooray!”  I told them that the anteater’s supporters were pretty mad about this, and had extracted a promise from the man to explore other voting methods before choosing a new animal the next time one dies, hence keeping the door open for more play with voting.   

I told them that since the possum’s pen had been split into two (cemetery and anteater) that a new map was needed, but since the man was still struggling for money, he needed to save money on printing costs.  He wanted to distribute a colored map to visitors that had each animal’s region a different color from its adjacent animal’s, but that used the minimum number of colors. We looked at a world map that used many colors.  I passed out maps of the zoo and colored paper  squares .  I noticed aloud how N got right to work placing the squares onto the regions of the map, and my comment to her got the rest trying it out.  I said,”You can work with a partner if you want,” but no one did.  This was the first time the members of our circle worked independently on something.

J, who prefers cooperative work, immediately left her seat with the comment, “I don’t want to do this.”  I said, “That’s fine.”  She spent the next 10 minutes approaching the other members and their work, and in a gentle and cooperative manner, pointing out their mistakes.  Fortunately, no one minded. N was “tiling,” as opposed to “map-coloring”, hers, so I told her that what she was doing was also something mathematicians work on (“tessellations”).  I let her know that the goal of tiling is that no pieces overlap or have gaps.  I wondered whether she’d continue with that approach, but when I walked back to her side of the table, she had switched to map-coloring by category (yellow for the animals, red for the man’s personal regions, etc), prompting me to tell the group about a map of Rome I have at home that is colored by categories: the city regions are all one color, the suburbs another, the countryside yet another.  As I looked at each person’s completely colored map, I asked, “Now can you do it in fewer colors?”

Soon P announced that it could not be done in fewer than 4 colors.  The rest were still industriously working, so I gave him a map with more regions (including the neighbors properties too, increasing the number of regions from 15 to 29).  He very quickly colored that in with 4 colors also.  A and V had colored theirs in with 5 and were staring at them.  M and O were still rearranging.  At this point a few people looked frustrated.  I asked a few questions that didn’t always help, so we talked about how mathematicians handle frustration.  (P asked if any mathematician had every gotten so frustrated as to cut off his own arm?  I don’t know, but to anyone who’s reading this and knows the answer: please let me know!)    We also talked about the origins of the map-coloring problem, the later-disproved multiple attempts at proofs over 150 years, and how long it often takes to prove something in mathematics.  How helpful it is to hear about unsuccessful attempts pushing mathematics forward. 

We decided that anyone who was frustrated could come to the other side of the room and play function machines.  Most of the group did so for the last 5 minutes, and 3 kids bounced back and forth between the activities.  The kids really enjoyed trying very big numbers in the function machine.  As we dismissed, some of the kids wanted to take their maps and coloring squares home to play with some more, and I gave out a United States map to color for all those who were interested.  I told them to email me if they wanted something even more challenging because “I have two more very interesting maps that we are not going to do in the group.”  I told them that we’ll talk more about their discoveries next week, after anyone who wants to gets to play more with them at home.  “But of course,”  I said, “there’s no homework in math circles.”

You never know which idea in a math circle will take root in someone’s imagination.  Hours later that same day, J was playing with something like an old-fashioned whirligig, and said to me, “This has something to do with math, doesn’t it?”  “Yes; how could you tell?”  “It just seems like it would.”

We began this week’s math circle sitting in an ogre - not a circle, not an oval, but an ogre, as suggested by the kids.  This used the math skill of mentally shifting from the concrete to the abstract. 

Once situated, we played a version of the shell game aimed at increasing our attention spans and developing the Jeffersonian hand required for extended mathematical inquiry.  (Next time we will do several focusing activities at different points to help contain enthusiasm just a smidge.)  Then we moved to our task at hand:  to create a scenario that would involve land plots and voting so that we can explore the topic of our semester, “maps, voting, and counting.”

I told the story of a man who collected animals and got his near neighbors to vote on a new animal to include in his zoo.  Of course, the children greatly improved the story, hence attaining ownership of the problem.  The problem is how to fairly set up a voting system and fairly count the votes.  L wondered aloud whether this was a problem with a single answer that I knew the answer to and was coaching him to figure out. I answered that we don’t know whether there is a single answer.

We discussed how in some places and times, certain classes of people were not allowed to vote, an idea that shocked most of the participants.   P said “that’s like saying that you can’t vote if you’re wearing a purple jacket!”  L said “that’s not fair to people who live in hotels!”   I told them about Marquis de Condorcet and Madame Sophie de Condorcet, figures from the French Revolution.  He was a mathematician who worked on fairness criteria in voting and also was a feminist.  Sophie was a highly educated woman who stood up to Napoleon on behalf of a woman’s right to an education.  Only V had heard of Napoleon and told the group “he was a guy who thought he owned the world.”

Then the group came up with its animal voting plan for the above-mentioned man’s neighbors:  each person would (confidentially!) write on a slip of paper which animal she preferred, and then on the board a check would go under each animal’s name for each vote.  The plan was to count the checks and whichever animal had the most would win.  Notice that the group (not “the participants”) came up with this plan (not “plans”).  The group is already moving nicely into a practice of collective inquiry.  One child had came into the class saying “I am the best at math here because I know my times tables up to 12,” but that competitiveness that so often mars math classrooms for people was totally absent once we got to work.

 I reported to the group that the neighbors had used this exact system, and the result was this:   tamandua (anteater) 6, clouded leopard 5, and emporer tamarin (monkey) 4.  (Note:  these are real animals that can be found online and in the London Zoo.)  There was excitement that the tamandua won!  But then I broke the news that all of those who voted for the leopard or the tamarin were totally opposed to the tamandua because of the possibility of unpleasant odors.  Excitement immediately evaporated.  V suggested combining the votes of the defeated animals.  L reported that 9 are against the tamandua.  P called out, “then the leopard should win!”  Most cheered this suggestion until M wondered whether the tamandua’s supporters might actually prefer the tamarin over the leopard, and hence would generate 10 votes combined.  (Sighs.) 

 M and J pointed out that this occurred because “we only counted the likes, not the dislikes.”  Excitement filled the room as several alternate voting methods were suggested.  I listed these on the board and our group voted on which one should be used to be fair.   I mentioned that I was bothered that we ended up with 7 voters and 8 votes (“someone voted twice – who was it?”), but since most voted for the same option, this incongruity was dismissed. 

I told the group that they invented something on their own that is actually used in real life:  approval voting.   (We’ll talk a bit more about this next time.)  I asked whether they thought there was one system of voting and counting that is always fair.  No one was sure;  we will revisit this.  Circle ended as folks raced up to the chalk board to vote for their own personal preferences.

“There is a unicorn dying at the end of a bridge.  He has 17 minutes left to live, unless 4 people can join hands around him and recite a magical spell.  There are 4 people on the other side of the bridge, but it is very dark, they have one flashlight, and the bridge can only hold 2 people at a time.  Ginny can cross in 1 minute, Ron in 2, Fred in 5, and Percy in 10.  Can they save the unicorn?” 

So began our younger Math Circle today (after playing Function Machines for a few minutes to make sure everyone was comfortable adding numbers).  Immediately the participants began doing what mathematicians do:  clarifying the question.  (Can they throw the flashlight?  Can they carry each other? How much does the unicorn weigh? Can they get across without the flashlight?  Can one person go at a time?  Did they bring any tools?  Can the unicorn use magic?  And so on.)  They asked many questions beginning with those 2 magic words that spark mathematical discovery, “what if….?”  (What if Ginny made return trips with the flashlight?  What if Fred and Percy travelled together?)  Some were looking forloopholes in the rules (a skill that will come in handy for proofs) while others were adding up the numbersto see what is possible.  While stymied with every attempt, enthusiasm grew. 

  We struggled with the idea of combined rates (how long would it take Ginny and Ron to cross together?), so we shifted to a historical anecdote about the mathematician Gauss and some of his early childhood math-related experiences.  I was ready to return to Function Machines to work on the rate issue, but the ideas, excitement, and questions about the problem just kept coming.  With one minute remaining, 1 young mathematician jumped out of her seat and came up to the board to illustrate her solution idea.  We left the question open to play with more at home, with the instruction to email me with questions and ideas.  A hint to the parents in the room:  question your assumptions.

I asked the older group, “Can you predict the number of pieces you get when you connect the dots on the edge of a circle?”  I drew a circle with one dot on the edge and asked, “How many pieces?”  We added a few more dots and connecting lines to realize that the question needed clarifying to specify the maximum number of pieces and what our explicit goal was.  (In a Math Circle, initial questions are intentionally vague to foster skillful question-asking.)  One person soon noticed that the number of pieces was always even.  We added a fifth dot and line and another member of our group piped in the idea that “they are doubling.”  I asked what is doubling to elicit accuracy of description, and the pattern was named:  “The number of pieces doubles each time you add a dot.”   I asked for a prediction for 6 dots, and they all agreed that there would be 32 pieces.  I asked how we can be sure, and they said to draw and count.  So we did.  We counted 31 pieces.  The group decided to redraw and recount several times, with the process becoming more systematic each time.  Their claim of “the drawing is wrong” became “the rule is wrong.”

At this point I asked how many examples we need to prove a rule (“a lot!”).  I mentioned empiricism(conclusions that arise initially from observation, as espoused by Aristotle) versus rationalism (theory first, then experiment, as espoused by Desartes).  We discussed which they thought was better, and how both can be used in real life.  We then talked about the Ptolemaic astronomers who struggled with making predictions based upon their observations and in light of more and more evidence of a sun-centered universe.  We took a quick detour into the question of whether the astronomer Ptolemy might have been related to Cleopatra, then returned to our original question.

The group decided to count the pieces for 7 dots, and discussed which counting strategy is better: adding to the current diagram, or starting anew with each revision.  We counted 57 pieces from 7 dots. We stared.  We thought.  We presented and then rejected conjectures about patterns.  Frustration mounted.  (One goal of a Math Circle is to be able to tolerate higher and higher levels of mathematical frustration and hence greater future rewards of discovery.)   Time was almost up when one participant said, “Look!  3 dots, 3 lines, that’s times 1.  5 dots, 10 lines, that’s times 2.  7 dots, 21 lines, that’s times 3. Every other one is times another number!”  When asked what we could do with that idea, they all agreed that we should draw a circle with 9 dots and see if we got the predicted 36 lines.  Despite only having a few minutes left, the consensus was adamant that we start from scratch with our diagram to ensure accurate counting.  We applied our most systematic drawing and counting strategies and discovered that the prediction held:  36 lines!  Delight filled the room.

We ended with hope that with time and effort, we could probably test this pattern on larger numbers of dots, and also continue to look for a pattern in the number of pieces.  We talked about how mathematicians play with problems like this for years, and take breaks, and come back again.  I took a picture of the last diagram (attached), and look forward to hearing ideas from anyone who plays with this some more. 

Below are some references if you are curious about any of the topics we explored today:

·        List of notable astronomers with brief bios.  This list includes Ptolemy, and also 2 women of more recent times: http://www.starteachastronomy.com/astronomers.html

·        Geocentric view of the solar system:  http://en.wikipedia.org/wiki/Geocentric   

·        Ptolemy, including discussion of his ancestry: http://en.wikipedia.org/wiki/Ptolemy

·        Dark bridge problem:  http://www.coolmath4kids.com/math_puzzles/b2-darkbridge.html  (but please, play with this one first before looking here, and if parents can’t resist peeking at this, please give your kids the chance to work on this themselves over time!  Remember, our goal is for them to think like mathematicians, so revisiting this problem after the passage of time is valuable.  This link is reluctantly included for parents only.)