(October 7, 2014) Since I had forgotten my Gardner books last week, this week I began class with reading aloud his versions of the problems. “His wording is confusing,” commented one of the kids. (To me, his wording is remarkably clear; I suspect that this is more a vocabulary issue.)
First we revisited “A Switching Puzzle.” The kids basically re-solved it because they couldn’t remember their prior solution. They wanted to be sure that all of Gardner’s requirements were met.
Then we revisted “Folding Money Fun.” This time I had prepared a 2’ x 3’ paper dollar bill. Bickering over who got to draw in George Washington’s hair foreshadowed other bickering to come today. (Is there such thing as TOO much enthusiasm?) I asked that everyone handle a different section of the bill, and as I read Gardner’s directions, the kids folded. They did it several times. By this time, everyone was sure of how to trick their friends, and of the explanation of the result.
In retrospect, the bill should have been at least 4-5 feet long so that all the kids could hold it all the time. In this problem, orientation matters – the kids have to all be on the same side for the folding to maximize clarity.) You may want to ask your kids to demonstrate this trick, if they haven’t already.
Now we revisited “The Folded Sheet,” in hopes that collaboratively the kids could figure it out with the help of an extra-large number map. I read aloud Gardner’s explanation of how to do it as the kids folded. This time, the bickering intensified as one very excited student took charge of the map folding. Another got frustrated and walked away. Yet another played the peacemaker and brought that student back. I asked everyone to work together, but asked too late. The students did work together, but grudgingly. One student suggested that everyone fold their own map. (If you read last week’s report, you’ll know that this was the approach then. I brought the single map b/c I thought last week’s approach wasn’t collaborative enough!) I didn’t have extra paper with me, though, so we tried to one large map approach for a bit, equanimity finally restored. The kids got no further than I did toward the solution. [Pedagogical Note To Self: intervene sooner when things get too intense. My goal is to become invisible, but kids do need help sometimes.]
These two experiences open up a great opportunity for a discussion about collaborative versus individual approaches to mathematical problem solving – first on the agenda for next week!
Now we were more than ready for something new….
THE MAZE OF THE MINOTAUR
We’ve taken advantage of the beautiful fall weather and our lush natural setting1 by having each class outdoors so far. (Q for another time: for these kids, how does plopping down in the middle of nature affect thinking, collaboration, and frustration tolerance?7) We have 6 picnic tables, which enable me to engage in strewing. (Strewing is a common unschoolers’ practice of leaving interesting things lying around without comment or instruction.) Each week I put things out on 4 tables, watch where students’ eyes gravitate as they arrive, and then determine the order/choice of the activities with a physical component.
In keeping with this week’s theme of one large object to work with, I had prepared2 a large laminated copy of Gardner’s “Maze of the Minotaur,” and had left it strewn about with some smaller copies of seemingly the same thing. Kids moved over to this activity without my needing to direct them.
Most of the students were well-versed in Percy Jackson’s adventures3, and were asking repeatedly “Is this the labyrinth of Theseus….?”
“What do you think?” I replied. Debate ensued. Then students started tracing the large path with their fingers. Some picked up the small ones. As people worked, I told them the story of my experience preparing the maze:
“Last night, I read Gardner’s description of the maze. He said, ‘No one has ever drawn a maze that looks easier to work, but actually is so difficult.’4 I set to work attempting his small diagram in the book. Lo and behold, I solved it in about 2 minutes! In that moment, I got that smarty-pants feeling that “Wow, this is something I’m pretty good at.” I also felt certain that if I could do it so quickly, then our math circle group could also do it without much frustration. So I then set to work drawing a larger version for multiple kids to work simultaneously. After drawing it, I decided to double check against copying mistakes by solving the large copied version. But now I couldn’t solve it. I tried and tried, and if just didn’t work. Had a made a mistake in my copying? I looked for the pattern that I thought I had identified in my earlier success, but that seemed to be gone. Then I spent half an hour double checking my work using line counting strategies until I was sure my copy was accurate. Why couldn’t I solve it anymore? Was my first success a fluke? Maybe Gardner was right after all about difficulty. Would this be too hard for the kids? Maybe. I surmised that if the lines of the maze path were different colors, then the maze would be easier. I then traced over some lines in other colors, and here it is, along with my doubts.”
What I didn’t tell the students is that in my struggle to re-solve the maze, the really interesting mathematical questions bubbled up in my mind:
- Is there actually a pattern in the solution?
- What’s the difference between a maze and a labyrinth?
- Is the solution manageable?
- Are there useful strategies for drawing the maze?
I used this list of questions to stimulate discussion while the kids worked. I didn’t really need to, though. Some of the same questions bubbled up from the group before I asked them.2
After a while, finally someone successfully solved the maze, so I had confirmation that my copy was accurate. At this point, any bickering left over from earlier had dissipated. The kids were very calm. Was this calmness the meditative effects of walking our fingers in a labyrinth?7 (I did posit this aloud.)
Unfortunately, I hadn’t anticipated that only 2 kids could work the maze at a time. Fortunately, the small labyrinth printouts5 gave everyone something to do. The kids took turns with the big one, but I’ll bring it back next week so everyone can have success.
SOME QUICK GARDNER INFO
I read aloud some snippets from the new Scientific American article about Gardner. The kids seemed to really enjoy hearing about his playfulness. It was nice, too, for them to see a picture of him, and to realize that he’s not a guy from ancient history.
SCRAMBLED BOX TOPS
I ended class with a Garner mathematical puzzle that involved no physical objects – no pen, paper, chessboard, poster, giant dollar bill, no nothin’:
“Imagine that you have three boxes: one containing two black marbles, one containing two white marbles, and the third, one black marble and one white marble. The boxes were labeled for their contents – BB, WW and BW – but someone has switched the labels so that every box is now incorrectly labeled. You are allowed to take one marble at a time out of any box, without looking inside, and by this process of sampling you are to determine the contents of all three boxes. What is the smallest number of drawings needed to do this?”6
The kids made some progress toward the solution, but then we were out of time, so we’ll revisit this problem another time. When we revisit it, I plan to facilitate it differently so that everyone is making conjectures; today 2 kids dominated most of the discussion. Interestingly, those 2 kids had totally different approaches to the problem, and both approaches fed each other. (One was sketching out a visual representation of the problem, the other was mentally working many steps ahead the way I imagine chess players plan out their moves.) Now we need to find out what the rest of the group is thinking!
I’ve had a lot of time to reflect on this circle since I started writing this report, and I’ve gotten some positive emails about the circle from some of you parents. (Please, feel free to send constructive criticism too – that always helps!) Here are the pedagogical points I am still pondering:
- Where exactly is the balance between having too much confidence as a leader/teacher, and simply feeling ready? (confidence as in kids’ abilities and levels of interest, and as in our own ability to facilitate inquiry)
- Is the way I’m presenting the topic of this circle (math via Martin Gardner) providing sufficient depth of mathematical thinking? While collaborative work is happening, many of these questions are on the short side, and I feel like I’m struggling to find a good balance between depth and breadth. All of the questions have a lot of mathematical depth, but some delve into topics that would need the kids to have more mathematical experience than they do. For instance, The Mutilate Chessboard problem described in week 1 would be easier to explain with some knowledge of parity. The questions we’re using are more than sound; it’s my approach that I’m questioning. BTW I’d like to thank Sandy Lemberg for suggesting the 2 books that I’m using so far. I am finding his prediction that PPTT4 is less mathematical and MBLMP6 is more difficult to be 100% on the money.
1 Talking Stick Math Circle and Talking Stick Learning Center both meet on the grounds of Awbury Arboretum in Philadelphia, for those of you following these reports from a distance.
2 This class is keeping me humble, always a good thing. And it continues to excite me, year after year, decade after decade of working with kids, how empowering it is for kids to hear tales of their teachers’ struggles.)
3 Rick Riordan’s series of books bringing the Greek gods into contemporary times
4 Perplexing Puzzles and Tantalizing Teasers, p20
5 I printed out a generic labyrinth path from Wikipedia
6 My Best Mathematical and Logic Puzzles, p3
7 Katie Gomm at the University of Utah published a paper that begins to address the questions of the effects of a natural setting and/or walking in a labyrinth: http://gradworks.umi.com/15/56/1556050.html. BTW the students in our group who attended Talking Stick more than a year or so ago are familiar with labyrinths, as there is one on the grounds of our prior location: St. Thomas Church in Whitemarsh. I have not visited that one, but have walked one on the campus of Bryn Mawr College – that one closely resembles the small example I handed out. While poking around online I found a worldwide labyrinth locator if you’d like to find one in another location.