University of Colorado, Boulder

Doing mathematics homework involves a set of specific skills.

The first skill is **patient devotion**. Since homework is the single best way to cement your understanding of the material in the course, I expect you to do it diligently and do it all. For my part, I will work hard to pick worthwhile problems instead of overwhelming you with volume.

The second most important skill is to **challenge yourself**. The point is *not* just to get it done (the grade is only a motivator): the point is to spend time challenging your brain in specific ways. Generally, the harder you work, the more you learn. This means you have to spot and avoid the ways your brain tries to be lazy. For example, it's easier to do a homework problem side-by-side with a textbook example. But by closing the text book, you challenge yourself more. The key is to make it as hard as you can while still being able to make progress. So sometimes you peek at that textbook example, try to understand the reason behind the method, and then close it again. The brain is a muscle. Taking the elevator gets you to the top floor faster and easier than the stairs: in math, you have to learn to force yourself to take the stairs. Because the elevator won't always be there.

Below are some notes and strategies.

- Foremost,
**do two copies**of your homework. The first is where you work it out, and the second is where you write it up. Part of the skill you are developing is mathematical writing, so take the writeup seriously. In your writeup, include english sentences indicating which theorems or definitions you are using, and explain your reasoning. Disorganised homework angers your grader, and angry graders are harsh graders. Incomprehensible solutions will be graded as if incorrect. Never write down guesses β understand everything you write. If you are having trouble explaining your work, and find yourself just blindly mimicking textbook examples without knowing why, get extra help. - Before starting your homework,
**review**the section of the textbook we've been covering, make sure your trusty Definitions and Theorems are laid out clearly, and take a glance back at the examples we've done. - When reading a problem, make a mental (or written)
**list of the terms, ideas and strategies**that might be relevant. Write down clearly what you need to know and what you already have. Think of ways you've bridged similar gaps in the past, even if the problem is phrased in a novel way. Most of the tools used to bridge such a gap are Theorems we've covered recently. - If you are having trouble with one problem,
**find similar but easier problems**in the problem section of the appropriate chapter, and do a few. Choose ones with the answer in the back, and once you feel confident, go back to the one that was giving you trouble. Often you will find you have new understanding. Close the book now as you try it again; it is best to avoid working side-by-side with an example. -
**Don't erase**when you get stuck; you may want it later. And as soon as you successfully solve a problem, neatly write up your final copy right away, before you lose track of it. If you don't, you'll go back later and suddenly realise what itβs like to be a grader faced with a messy solution. -
**Doing homework with friends**can be helpful, since explaining solutions is good practice. When explaining, try to lead by the Socratic method. However, if you find that you receive more explanations than you give, work by yourself before meeting. And of course, always write up the solution by yourself and in your own words. - It's okay to look to the internet for help, or to
**ask**a tutor. But don't ask for answers to exactly the same homework problem. Come up with specific questions that will help you learn, such as "how can I tell whether an equation is a plane?" instead of "how do I solve #34?" - If you are having trouble solving problems, perhaps you should work on
**problem solving strategies**. Ask a tutor, or in office hours, for specific help with this. - Working on your homework sets at a
**Drop-In Tutoring Centre**can be a good way of getting extra help. Take advantage of this and other resources at your school. -
**STAPLE YOUR HOMEWORK**. You can buy a lifetime supply of staples at β where else? β Staples for under a dollar. Yes, the lifetime friendship of your grader is that cheap. Please also write your name clearly at the top, and keep the problems in the correct order (indicating any that you skipped).