It’s fine to not solve a problem.
“I’ve got it!” said X, J, M, N, A, G, and D. Almost everyone came up to the board (excitedly and without invitation) to demonstrate a solution to the famous Konigsberg bridge problem. “Uh, wait… I just had it,” said X, J, M, N, A, G, and D. Each returned to her/his seat to think some more. Some came up more than once. We practiced presented our conjectures as weather reporters would: facing the audience while writing on the board.
We don’t raise hands, take turns, or ask for permission to come up to the board in Math Circle. We let our enthusiasm (or lack of it) dictate our actions so that we can access our natural human desire to seek structure. We play weatherperson to make sure that our colleagues understand our thoughts, therefore increasing collaboration.
“We need a strategy to remember,” I suggested. The kids suggested paper and pencils, and started drawing maps. We briefly discussed how helpful writing things down can be, as it has been rare in our Math Circles that the participants have thought to put their thoughts on paper. A few came back up with solutions that did not work.
The leader does not suggest strategies, and ideally does not even voice a need for one. Sometimes we do, though, to get past an impasse. As humans, we are much more likely to remember and repeat strategies that we thought of ourselves.
“Why don’t you connect the bridges?” suggested X. I had drawn a map on the board of the town when I first introduced the problem. I asked X for clarification, she clarified her idea, and N added the clarification “like a maze.” I amended the map on the board and gave the group more thinking time.
Math Circle participants repeatedly clarify their ideas. This promotes increased explanatory ability, but also, as we clarify, we refine our own ideas. Extended silence is welcomed to increase the creative thought that leading questions can sometimes stifle. This silence could be overwhelming for an individual working alone or for a group working competitively, but our open collaborative approach allows ideas to flow. All ideas are welcome; the “unproductive” ideas often lead to breakthroughs.
As the kids were thinking with both their minds and their pencils, I gave a detailed history of this problem. Euler’s treatment of the Konigsburg Bridge problem led to the creation of a new field of mathematics. As I quoted a rhetorical question from a historical account (“Why would Euler concern himself with a problem so unrelated to the field of mathematics?”*), N piped in, “Because it’s fun!” D wanted to know exactly how old Euler was when he wrote his first letter to the people of Konigsberg. No one was able to calculate his exact age by subtracting years, but to everyone else’s relief, G knew how to approximate by rounding off. I ignored the subtraction lapse to preserve the momentum of the prevailing question. Trust that they will ask how to do this when they need it.
Stories make the math real for people. A good fictional or historical narrative can make the math come alive and instills the desire to solve the problem. This problem was so captivating that no one touched the Polydrons, which were out on the table the whole time.
I held up a Polydron construction and mentioned that Euler enjoyed tracing the edges of solids …. “We already talked about that!” interrupted several people, before I had the chance to finish the statement. They seemed concerned that I might try to get them to count vertices, etc., again. But I finished by saying, “… to see if there was a path that covered every edge only once.” This intrigued them, as it is so similar to the Bridge problem. We debated whether such path counting is pure or applied mathematics, and if it matters what paths you are counting (whether bridges or geometric solids).
I have a conjecture that one reason math classes do not often promote the joy in discovering and using mathematics is that the classes present neither pure nor applied mathematics. Think about it.
Then N showed his paper with a drawing on it that looked much like a Eulerian graph of the Konigsberg Bridge problem. He explained it to the group and they collaboratively dictated it to me for placement on the board. “There are no distractions” with this approach, noted Anna about the graph in comparison to the map. Everyone was re-energized with this streamlined approach. They found this shift from a graphic to symbolic representation to be quite useful.
Every person in the group contributed to this breakthrough. It came from the collective desire to solve the problem and from everyone seeing each other’s failed attempts.
While they worked, I continued the historical narrative, including the addition of one bridge and the later subtraction of another. A few kids picked up compasses and had some fun with circles. J came up with a solution that would work from 1875 until 1944, when the city had 8 bridges. M commented that “this would be so much easier if we knew the name of each bridge.” (We knew the names, but not which was which.) We were out of time, and still left without a complete solution.
It’s fine to not solve a problem. It gives people something to think about for fun, builds frustration tolerance, and also demonstrates how real math developments happen: over years.
As I cleaned up the room and parents were leaving, J was still staring at the board, deep in thought.
(Thanks to Mike Manganello for the “weatherperson” approach to sharing ideas.)
*This excellent historical account is from the Mathematical Association of America’s website. Mathematical approaches for our age group can be found elsewhere.