Homework and Problem Solving
When doing homework, you are actively using the knowledge you learned to solve problems. Reading textbook and attending lectures are somewhat "passive." You get a much better understanding after doing some problems.
In some sense, material is learned by doing many, many problems, especially the harder problems. A fatal mistake many students make is, after arriving at the solution once, they assume they have mastered the material. This is not enough practice to commit the material to long-term memory, so when the material is presented on the exam, students are unable to recognize the solution because they had not practiced enough.
It is extremely important that you do enough problems after you have learned the material, i.e., after your reading, lecture and review. Getting the answers right is not your goal (we already know the answer). It is the path of how you get the answers that is important. Only after you used the theorems, formulas many times, can you have a solid mastery of them and now you can say that you really "got it''.
The most important secret to being a good problem solver is simply paying attention to the techniques, methods, or "tricks'' found from examples, proofs and some of the homework problems. You should study each method thoroughly, keep a list of them, and know when and where to apply them.
Normally, each chapter has several typical problems. A problem becomes "typical'' either because of its relation to a theorem, a formula, an application, or because of the solution method. You should be able to recognize the problems, make a thorough study of them, know their possible variations and keep a list of them.
More vs. less, forest vs. tree:
You should do some well-selected problems very carefully to get the depth, to know all the details. After that, you should do more problems but less carefully to get general ideas and to become "wide.'' You can even just read the problems and do them in your mind without writing the solutions down.
One vs. several:
It is very beneficial to try to use one method to solve as many problems as you can, or to try to use several methods to solve one problem.
Getting something out of each problem:
You should always try to get something out of each problem you solved. After you have done your homework problems, you should think about them again briefly to see what you have learned and what methods are worthy of keeping. This step is important for you to retain what you learned. Without this step, most of the effort you made doing the homework will simply be wasted.
Summarize the material for each chapter:
Once you have practiced and mastered the material in the chapter, you will have confidence that you have learned the chapter's material. Now is the time to summarize. For example, calculus problems can be classified as follows:
- Drills. Direct application of theorems and formulas;
- Typical examples;