Problem Solving:
Things that might help
This is what you do, mostly, in Mathematics! Yes, it's nice to see interesting equations or nice geometric constructions, but the activity which brings you most often to math is solving problems.under construction

Puzzles (see my Math Puzzle of the Week) are great ways to sharpen problem solving skills because of the built-in rewards for correct solutions and often the problem depends on some kind of cool trick to solve.

You can't teach all the tricks which help to solve problems - all you can teach is some general principles. See also my page of helpful tips. Here are steps you should take to solve math problems.

Make sure you understand the problem. You must understand all the terms used. Some terms have more than one possible meaning depending on context. The entire problem, as you understand it, has to be consistent - it must make sense. One way to check this is to try a simple solution even if you have no hope it is correct - just checking this solution will help clarify the problem and illustrate one or more difficulties in solving it.
Try to limit the kind of math you use to something appropriate to the problem. If the problem looks like algebra, use algebra to solve it. If the problem has a geometric figure, use geometry. You can be even more specific. HOWEVER, once in a while going outside the obvious branch of math can be the only route to a solution. Again, the only way to learn how to judge such exceptions is lots of practice with puzzles.

Be really clear that if you GET a solution, you will be able to check that it IS a solution. This is not trivial. You need to have a clear procedure step-by-step to test the solution. Also, by clarifying this part, you often get helpful clues about how the solution has to work.


Be patient. Often the problem looks impossible, or way outside your experience. So don't expect an immediate solution. Take your time. Do little bits of the work and be satisfied you are making progress. Mull it over. Write it down. Go do something else for a while. Your brain is smarter than you are (!) and often will bring up key ideas that strenuous worry cannot uncover.


DON'T JUST STARE! Be active. Underline. Draw pictures. Circle words and numbers. Doodle. Mumble or talk to yourself. Hum. Scratch your head, or fidgit or stand on one foot. The more parts of your brain you activate, the better it works.


Make educated guesses. Estimate the size of the answer. For instance, if a problem is asking for the price of something you suspect costs about $5 in the real world, this may be a good estimate. See if $5 meets, at least approximately, the conditions of the problem. Then work on the problem. If the answer you get isn't something close to $5, either you made a mistake, or you did not understand the problem correctly. You may need to adjust your guess.


PLAN your solution process. Knowing the math is crucial, but knowing how to apply it is also. Write down the steps if you get confused. Check off the steps as you complete them.


Learn something new from every success. You should even learn from your failures. Underline key computations or equations. Put a star by the breakthrough.


Think about alternative ways you might have solved the problem. Often there is more than one successful method. Sometimes a less obvious method is faster than the straightforward way. These interconnections within mathematics illustrate its richness.


Use technology: calculators or geometry programs or modeling software are examples. You still need to know the math but you can save yourself a lot of work.

FINALLY, if you need it, GET HELP! Don't just give up on a problem. The only way to grow in math power is to learn the solutions. If you do, you will be the winner, not the problems.

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