This page describes some proven ways in which you can be more
efficient and effective when actually doing mathematics with paper and
pencil. It focuses on
**how** to do what you do. (Almost all the of other pages in this
course focus on **what** to do.)

Students often think that the only purpose of figuring things out is to get the answer with the minimum amount of fuss and effort, and to get it over with. Actually there are several objectives:

- First and foremost, you want to get the
**right**answer. Mistakes are easy to make, and they occur with dismaying frequency. In many cases they are not immediately obvious and they invalidate everything that follows. As a result people waste huge amounts of time. - You want to be able to go back while you are figuring and see what you did, and what went wrong, if you did make a mistake.
- You want to recognize any mistakes as soon as possible so that you don't waste your time on meaningless calculations.
- If your work is going to be read (and perhaps evaluated) by someone else you want that person to understand what you did.
- In a math class (like most certainly in this one) everything you
do builds on what you did before and so you want to be able to go back
to what you did some time ago--even if you already turned in the
answers and got credit for them--and understand what you did back
then. In the same vain, you want to
**understand**and**remember**what you did because you are bound to need it in a future problem. - You want to learn what there is to learn from the particular piece of figuring that you are doing.
- Subject to accomplishing the above objectives you want to spend as little time as possible on any particular problem.

Luckily, the last objective is perfectly consistent with the others if you think in terms of the whole set of problems and exercises that you do in the course of the semester. If you guard carefully against errors and ensure that you can correct them easily soon after they occur you save time. If you learn what there is to learn in each problem then you have to do fewer problems and exercises overall, and you are able to do future exercises more quickly and with less frustration.

Of course, how to figure things out is a highly personal process, and what works for you may not work for somebody else, and vice versa. However, I wrote this page because over and over I see students approach problems in a way that does not work at all, for them, or anybody else. So here are some suggestions:

- Before you start a problem think about your expectations. If a contradiction to those expectations arises as you work the problem pause and figure out what happened.
- Students usually seem to be in a rush when trying to solve a
problem. That's understandable, there is much to do and little time.
But actually it is much more efficient, and
**faster**, to go about solving a problem carefully and deliberately, taking small steps,**writing down each step,**and making sure each step is correct before going on. Doing so reduces the number of errors, makes you more alert to errors when they do occur, and reduces the time you spend on identifying and correcting errors. - When algebraic expressions are equal write an equality sign between them. Write the expressions in the sequence in which they occur. Don't just scribble them unconnected all over the page wherever there is some as yet unused space.
- Use engineering type graph paper instead of blank or lined paper. This is particularly useful when drawing graphs. Don't use "scrap" paper with unrelated information on one side. It will only confuse you and your reader.
- Use a soft pencil and an eraser. Don't use a pen, since scribbeling out errors and perhaps writing over them makes your writing incomprehensible. Don't use a hard pencil that writes only faintly, it's hard to read.
- When writing a sequence of equations or steps line them up so the logical connection is clearly apparent.
- When you change an expression or equation don't modify what you wrote. Don't erase or cross out things (unless they were wrong). Instead write the entire new expression or equation. (Sometimes it makes sense to cross out terms that cancel. In that case do so, but make sure it remains clear what actually did cancel.)
- Take a note of what you did in each step, don't just do it, and then later wonder why you could take that step.
- Continue to ask yourself whether what you have currently makes sense.
- Think about the physical meaning of what you are doing. This is one of the major reasons to use variables rather than numbers. You can't add a distance to a weight, for example, and so if your figuring calls for that then you know that something has gone wrong. You can recognize this in the expression but not in the expression .
- Keep your writing neat and organized.
- Label your axes and note what your variables mean.
- Remember that
**upper and lower case letters are different in mathematics**. - Use meaningful names for your variables (e.g., for height, for distance, for weight, for mass, etc.).
**Always check your answers.**If you solved an equation substitute your answers in the original equation. Compute the same answer in two different ways. If your answer is a formula see that it gives the right particular value in a case where you know the answer. Check identities (equations that are true for all values of the variables) by substituting particular values. Draw a picture and see that your algebra is consistent with the picture. Ask if your answers meet your expectations. Do they make sense?- When you are done take a moment to reflect on what you learned. Chances are you will need to use this new knowledge in a subsequent problem, and thinking about it now will make it easier to recognize when you need it in the future.
- Keep your notes, including all your worked exercises, organized for future reference.
- A major mathematical problem solving technique is to simplify a difficult problem, solve the simple problem, and apply what you learned in the process to the difficult problem. In the context of a class like this, if you are stuck on a problem, chances are that you already solved a simpler problem that's relevant to your current problem. So if you are stuck, look back over problems you solved recently to see if something you did there applies to what you are doing now.
- Don't spin your wheels. If you still can't solve a problem go back over your notes, read in your documentation (these web pages or your textbook), talk with friends, tutors, or your instructor, or set aside the problem and solve some others before returning to the obstacle.

You'll find examples for many of these techniques throughout these web pages and the solutions of the homework problems.