Yesterday, I was covering a colleague’s math class at the last minute, and he had made photocopies of a chapter 1 to 7 review. I looked at the review sheets, and the grade 10 students in front of me, and decided that it was unlikely that the review sheets were going to be useful. I handed them out, and then started putting puzzles up on the board.
The first puzzle I put up was the Seven Bridges of Königsberg problem. Within a couple of minutes, every student was trying to figure out the path across the 7 bridges that doesn’t cross any of the bridges more than once. Before the students got completely frustrated with this problem (since it is deceptively simple to state, but "difficult" to solve), I put up a couple more problems, including a gem from Dr. Gordon Hamilton. I added the frog hopping problem to the board, and taught two students the game of Nim.
Each problem had some students who were working on it intensely. Every student found some problem which was interesting to them, and almost all students were working in small groups on the problems and puzzles. Eventually, a small group of students gave up on all of the puzzles and worked on the review sheets while the rest of the students continued to work on the puzzles until the end of class.
Some students asked for a hint on the bridge problem, and I led them (through questioning) to Euler’s formulation of graph theory. From this, we discussed that there could be at most one starting spot, and one ending spot, and that only a starting and ending spot could have an odd number of paths leading in and out of it. I then put up the 5 rooms puzzle, which one of the girls said within seconds was unsolveable by applying Euler’s analysis to the graph.
A group of boys worked on the frog problem, and went from struggling to even find a single solution to the 3 frog problem to being able to generalize a solution for n-frogs on either side (and a formula for determining the number of moves for each frog puzzle).
The next day, I spoke to my colleague, and asked him if he was okay that I had not done the worksheet with the students. As expected, he was fine with it. I asked him what the students said. He said that students said that they enjoyed the day before, but one student had said, "We didn’t even do any math yesterday."
I’m not sure I agree with that student, and I’m slightly distressed that he didn’t see the problem solving activities we did as being part of math. What do you think? Are problems like these important in mathematics? If so, why aren’t more of them in our curriculum?
Doug Whitehouse says:
Great post. Today, instead of doing an exam review with my students, I posed three problems to them: The Kongisberg, the Towers of Hanoi, and another. My first class did them without much fanfare. My 2nd class though got into the work and I actually had 3 students bring problems to the board for their fellow students to solve. Class was exciting as students tried to solve each other’s problems.
May 9, 2012 — 8:12 pm
Ted Lewis says:
I am a little bit dyslexic with numbers, and I had a bad experience with arithmetic in elementary school. It is perhaps strange that I eventually became a mathematician, but one of the reasons was that some very good high school math teachers regularly presented the class with puzzles and problems like the ones you mention. Working on the puzzles taught me that there was more to math than just acquiring computational skills.
May 11, 2012 — 2:56 pm
Chris says:
It must have been a day for puzzles yesterday as I did the four 4’s puzzle with a couple of my classes
May 11, 2012 — 5:43 pm