In STEM courses, students solve problems in problem sets, as a way to learn through practice. Answers are often provided for some problems (e.g., the odd numbered problems in a textbook) and not for others. When a student is working on a problem that they have an answer for, they have a choice about how long to work on it before looking at the answer. If the goal is to learn as efficiently as possible for long-term understanding, what is the optimal amount of time to spend on a problem before looking at the answer?
Another way to think about this question: Imagine that a student has access to a problem bank with an unlimited supply of questions and answers. So at any point they can look at a solution and move on, without ever running out of questions. If they have a fixed amount of time to practice, will they achieve a better understanding of the subject if they set a shorter time limit and cover more questions, or set a longer limit and spend more time struggling with the more difficult problems? Assume that the problems are moderately difficult (not just "plug numbers into a formula" problems), but are not open-ended research problems or tricky challenge puzzles. I.e., They are written for the purpose of teaching fundamental results or techniques.
The answer to this question may depend on how you value the benefit of students struggling with a problem and discovering their own insights, compared to the benefit of learning the "correct" procedure for solving a problem, and going on to practice that procedure on other problems. An argument for the latter point of view is that by spending too much time on each problem, students risk not having enough time to discover and practice the range of problem types they need to learn the class material. An argument for the former point of view is that struggling with problems is more important for understanding in the long run. I don't know which argument is better supported by evidence.
- When to give up on a hard math problem? (Math Stack Exchange). I think there may be a distinction between problems where the difficulty is inherent to the problem, which is more what this MSE question is about, and problems that are hard due to lack of experience or knowledge.
- Keith Devlin's concept of mathematical thinking. Devlin is opposed to math instruction that emphasizes procedures for solving problems. I think he would favor a longer duration for each problem, since his goal for students is to learn how to think mathematically, not learn "correct" procedures.
- There are many similar Quora questions -- e.g., How long should I stay working on a math problem I am stuck on before asking?, How long should one attempt to solve a math problem before looking at a solution?, and How long should I keep trying to solve a maths problem before asking for help? I'm hoping for a more evidence-based answer.