4 Attitudes
“Many people who have not studied mathematics confuse it with arithmetic and consider it a dry and fruitless science. In reality, however, it is a science which requires a great amount of imagination.”
– Sofia Kovalevskaya (MacTutor 2025)
Effective problem solvers masterfully balance curiosity with discipline. They begin by probing, playing, and hypothesizing, then move on to justifying, checking, and reflecting on their findings. True progress originates from cycling through the four problem-solving phases, maintaining a keen awareness of one’s actions, and a readiness to adapt when new evidence arises (Pólya 1945). When progress stalls, adept problem solvers pivot swiftly and consistently take time to “look back” at their process.
4.1 Selecting and pursuing an approach
Begin by matching structure to strategy. Let the problem’s cues choose the method: a recurrence invites induction; symmetry points toward invariants; a geometric flavour asks for a diagram and auxiliary elements; unruly numbers suggest estimation or approximation.
Then commit to a strategy, but keep your course of action time-bound. Choose one plan and pursue it deliberately for a few minutes. Set a concrete checkpoint in advance so the decision to switch is not emotional or driven by frustration (“If I cannot determine the number of cases in twenty minutes, I will try another approach.”).
Finally, you are on the right track if relations clarify, expressions simplify, or the number of cases shrinks. If instead the algebra gets tedious, cases proliferate, or your reasoning seems circular: stop, reframe the problem, and select a different strategy.
Try answering these questions (see Schoenfeld 1985):
- What exactly are you doing?
- Can you describe what you are aiming to do precisely?
- Why are you doing it? How does it fit into the solution?
- How does what your are doing help you solve the problem at hand?
- What will you do with the outcome when you obtain it?