In Problem Solving II, we are using games to explore mathematics.
The goal is for students to think about "solving" a math problem as the first step in a longer process.
Suppose you are playing a game on a chess board with a queen, a king, a knight, a bishop, and a rook. These pieces move like normal chess pieces, but this is no normal chess game! In this game, players take turns moving one of the chess pieces to a square that is above and/or to the left of its current position. Pieces are allowed to share squares. The first player to run out of valid moves loses.
We opened class with this game last week. Students had ten minutes to prepare before facing off as a class against me. They scrambled around the room, filling in charts, checking them with each other, and writing important information on the board. They took turns choosing moves for their team, and they won the game--a great start for the day!
Afterward, we dove into a discussion of the mathematics of this game. We carefully articulated the strategy that the students had used to win, and sketched a proof that this strategy works in general. Then students spent the rest of class working on individual writing projects.
This structure is not uncommon in this class. Each day, we cycle through activities designed to engage the students in collaborative problem solving, full-group discussion, and solo writing. Through the group work, students learn to view each other as mathematical resources, and to practice communicating their thoughts to someone at their own level. In our discussions, they get to hear from other groups who used different strategies. Then they have time to think about how to take the ideas from their own solution and from the group discussion and write them down formally.
The goal is for students to think about "solving" a math problem as the first step in a longer process. The final product is a piece of rigorous mathematical writing that someone at their level can read and understand. The writing exercises in this class are not easy, but students rise to the challenge. Our in-class writing time buzzes with activity. Students compare solutions, ask each other to check proofs, or call me over to ask for advice and feedback. The results have been impressive. They are getting better at explaining their ideas to each other verbally, and they are making great strides in their writing.
Our focus on mathematical communication allows us to explore various beautiful areas of mathematics. The game theory unit has given students the chance to practice structuring rigorous induction arguments. Material from math logic has given them the opportunity to think about what it means for a statement to be "obvious," and to write highly-structured, axiom-based arguments. They are also learning to think critically about their own thought processes--in one recent assignment they wrote both a rigorous proof, and an informal explanation of the problem-solving process that led them to the proof. We are currently transitioning into a unit on geometry and topology, where students will be working to write convincing arguments based on pictures.
It has been a real pleasure working with this class so far. As we progress toward the end of the block, I look forward to watching them continue to grow both as mathematicians and as communicators of mathematics.
-- Susan Durst