Whether you're deciding how to study or deciding how to solve a problem, have a goal in mind and a plan to achieve it. When studying, it's not a good idea to simply lay out all your course materials and decide to leaf through them one at a time. Think about particular concept that you don't quite understand. What do you know? What questions remain unanswered?

In a problem, it's rarely a good idea to simply start scribbling equations to "get to" the answer. Think about what you are trying to solve for. What are the steps to finding the answer? If even this is difficult, sometime you can figure out the steps by thinking backwards from the answer (before you start working backwards). Consider the following example. What would I need to know in order to write an equation for a plane? How do I determine the normal vector to the plane? How do I find two vectors in the plane whose cross product I can take?

As you work out the problem, keep bringing yourself back to the meaning of your calculations. The longer you spend pushing through a string of symbols without thinking about meaning and purpose, the greater the chance you're heading into a dead end. A similar warning is applicable to reading of your math textbook, or any textbook, and to listening in class. If you find yourself reading without purpose, without some question in mind or some point you're trying to figure out or something you're trying to connect to your prior understanding, it's likely you'll remember little. If you find yourself writing furiously in class but not thinking furiously, you'll again not take away very much. The point of class is not to produce more text which you can look at later; you've already got a thick text. The point of class is to construct meaning behind the symbols; brief note-making, rather than furious note-taking, is a useful way of reminding yourself of what you understood in class. In short, knowledge has to be constructed by you; osmosis works for a semi-permeable membrane but not your brain.

Finally, if you're having trouble gauging your understanding of the concepts and deciding what to study, take a look at the Concept Check section at the end of each chapter. This section has good questions that will provide direction to your reading and studying.

Plan on understanding concepts while doing problems, and stick with this plan. It can be tempting to just finish a problem set with a little direction from friends or from glancing over a similar example in the book. Beware that this pushes off really understanding a concept until the exam. It can also lull you into the belief that you understand a concept when you can only follow the lead of an example. In both cases, the short-cut affects your understanding of future material.

When you're studying, specially for the exam, remember that looking at a problem and then looking at the solution and saying to yourself, "I could do this," is not really studying. You should put away your course materials, and really think about what's involved in solving a problem. Often the struggle to figure it out will give you a very good idea about concepts you need to better understand the next time you look at your book.

Although the amount of time each person requires to understand the material will vary, on average you should expect to spend three hours outside of class for every hour of lecture. This includes reading before class, working on the homework assignment, and reviewing occasionally.

The following tips can be generally helpful when you're solving problems. The first tip is to try and break down a problem into pieces that are manageable. Sometimes when you first look at a problem, it can seem confusing or overwhelming. Often it helps to break it down into steps, each of which you know how to traverse.

The second tip is to take several perspectives on the same problem: think about it in pictures, in equations, and in words. Often, each perspective will offer a piece of insight in solving the problem. You should practice translating between these different views to ensure mastery of all of them.

Finally, the last tip is just a reminder. Often, a problem in "the real world" or in the Math 21a world will not suggest the method of solving it. This does not mean that the problem is vague or poorly worded or a trick question. It requires that you've synthesized the various concepts covered in the course and can apply them more than simply as recipes.