Math Homework

There are several meta-problems associated with solving math homework, depending on your point of view.

From the point of view of the instructor, there are two main goals:

1. For students to learn how to use the methods they are being taught.

2. For students to learn when to use the methods they are being taught.

To help with the latter goal there is an unwritten rule: If students are given a problem in their homework and they cannot use the methods they have been taught, make assumptions until they can. See "Spherical cow" on wikipedia for an example of this. This rule is useful because to use a method we almost always need to make simplifications or other assumptions which allow the method to be used.

We can also add other goals depending on the instructor. For me, I also add trying to share my passion for the field and an understanding of how these models can help in work or daily life.

From the point of view of the student, there are two primary meta-problems.

The first meta-problem has to do with how they approach their homework. Some people simply work each problem, in sequence, until they are done. Others tackle the problems that are easiest first, and build their way to the ones that are hardest for them. Still others know that the assignment should not be too difficult, and will re-evaluate their approach after "enough" struggle. Fundamentally, this meta problem is about balancing personality (how do I remain motivated enough to complete my work) as well as the effort and their goal for how well to do on the assignment in question.

The second meta-problem has to do with assessing where things have gone wrong. Imagine two students are working together on a problem and get different answers, or the instructor has shared the right answer and a student does not match. In both of these cases one student has effectively solved a different problem than they meant to. The meta-problem is to identify which problem they did solve, and how it is different than the intended problem. This trick is in fact how wrong answers to multiple choice questions (called distractors) are built. "What if we miss the step to divide by two? What answer do we get?" This set of meta-problems can be a very positive way to think of a mistake as it is now less of a mistake and more of a puzzle to figure out which problem you did solve.