Course Purpose
This course starts out by exposing students to puzzles and problems needing different kinds of reasoning. Once these have set the stage, we establish the motivation for problem solving, and how to participate in this course to get the most out of it. What follows is introduction and practice of various general problem solving strategies. The course ends with reflection and discussing teaching approaches for problem solving. These teaching approaches have been directly applied to the design of the course itself, informing its structure and activities. This means emphasizing depth, discovery and practice, and avoiding trying to cover a broad range of techniques shallowly. This is then reflected upon at the end of the course.
Course Learning Outcomes
CLO 1: Define transferable hands-on problem-solving skills in mathematical contexts.
CLO 2: Use research methodology for developing successful approaches to unseen problems.
CLO 3: Evaluate different proof techniques and various mathematical tools effectively.
CLO 4: Construct problems and solution arguments using rigorous mathematical language.
Course Content
| Module | Description |
|---|---|
| 1. Logic Puzzles | An introduction to deductive reasoning and foundational "hands-on" thinking skills using grid logic and truth puzzles. |
| 2. More Puzzles | Exploration of lateral thinking and pattern recognition through collaborative, puzzle-based learning in STEM. |
| 3. Problems | Transitioning from closed puzzles to open-ended mathematical exploration and identifying the nature of a "problem." |
| 4. Understanding Problems | Focusing on Polya’s framework: restating the problem, identifying unknowns, and defining rigorous constraints. |
| 5. Attacking Problems | Practical application of research methodology, including data gathering and testing small cases to find a foothold. |
| 6. More Strategies | Deep dive into core mathematical tools such as the Extreme Principle and the concept of Invariance. |
| 7. Strategies Continued | Advanced tool evaluation including Symmetry, Parity, and the application of modular arithmetic to unseen problems. |
| 8. Hard or Impossible? | Recognizing unsolvability and constructing rigorous mathematical arguments and proofs of impossibility. |
| 9. Learning Curves | Metacognitive reflection on the discovery process, analyzing cognitive hurdles and the research journey. |
| 10. Teaching Practice | Discussing the pedagogy of problem-solving and applying course structure to pose and facilitate new problems. |