Problem solving
A general problem solving strategy
How do you solve a problem you haven't been shown how to do? It's relatively easy to look up answers to plenty of problems, but the creative process of problem solving is much more valuable, generally speaking.
Ultimately, the best way to learn how to solve problems is to consistently and mindfully practice solving problems. The following is a general outline of a problem solving process based on George Pólya's classic How to solve it.
Understand the problem. Easier said than done—this step is often half the battle. Identify what you know and what you don't know. Make sure you understand the terminology and notation in the problem. Rephrase the problem in your own words and organize the information in whatever way makes the most sense to you (in a list, table, diagram, etc.).
Make a plan. This step can require some ingenuity but gets easier with practice and experience. Identify any patterns or relationships you might be able to exploit to break the problem down into simpler parts. In particular, look for relationships between what you know and what you want to know.
Carry out the plan. Carefully execute the plan you came up with in the previous step. Avoid paralysis by analysis and try something! You'll either solve the problem or you'll learn something new, so there's really nothing to lose.
Reflect. Look back at what you've tried. What worked? Can you explain why it worked? What didn't work? Why didn't it work? Can you fix it? Being mindful of your thought process and the strategies you've employed can help you devise plans for future problems.
Try again! If necessary, reconsider your understanding of the problem, rethink your strategy, and try again. The problem solving process is iterative—you may not be able to solve it yet, but you'll get there if you're persistent.
Working on the problem sets
Here are some tips to help you make the most of your efforts on the problem sets. This list is far from exhaustive and I urge any student who is struggling with the course to contact me as well as Academic Services.
Start early. The rest of these tips require at least some preparation. It's also important that you keep up with the material as the class progresses. Don't start the night before! These assignments can be difficult and often require some creative thought (which usually can't be rushed).
Ask questions. You are expected to have questions and seek assistance when necessary. Bring any confusions to class, the Discussion Forum, office hours, the Math Lab, or a classmate. If you need additional help, contact Academic Services—there are plenty of resources available to assist you!
Take breaks. It may seem counter-intuitive, but you can actually be more productive by taking regular breaks. Take some time to reflect and recover. To avoid burn-out, try working in 20-25 minute intervals with 5–10 minute breaks in between.
Get stuck! Getting stuck on problems is a normal part of studying mathematics. It happens to even the greatest mathematicians, so don't take it personally. Give yourself time to get stuck on a problem every now and then—try to accept the state of being stuck.
Read the problem carefully. This is especially important for questions that ask for an interpretation, explanation, or that happen to contain a lot of information. After you've come up with you answer, reread the question. Does your response fully answer the question? It can be helpful to do this with a friend.
Use proper grammar, notation, and terminology. You're not necessarily graded on your grammar, but poor grammar and misused notation can result in your solution being hard to follow. Read over your answers to make sure they say exactly what you want them to say.
Double-check your calculations and show all of your work. Have a friend read your solutions. If you have to explain something out loud to them, you should consider adding that to your assignment to make sure your solution is as easy to follow as possible. Note, this means that you should expect to write several drafts of an assignment!