- “When am I going to use this?”
- “Why do I need to learn to do this by hand when my calculator/Google can do this for me?”
- “Why are you explaining how we came up with the formula instead of just giving us the equation?”
Many mathematics professors are used to hearing these questions. There are a plethora of ways to answer these questions and many reasons your professors may require that you learn the theory behind what your calculator is doing. The most notable answer is the importance of training students to think creatively and logically and become master learners and problem-solvers.
The creativity and problem-solving techniques that we use in our mathematics courses are sometimes obscured when students instead focus on the processes and rules that are used to complete exercises. In part, this is due to time constraints and the fact that many mathematics courses are prerequisites for future courses, so they must provide skills and techniques to be used in later classes. However, this is also because problem-solving is time-consuming and challenging!Remember that we are not problem-solving if we already know what to do.
Conducting research is a natural way for us to really “see” problem-solving in action partly because we are not as limited by time constraints. When we start a research project, we embark on a journey into the great unknown. We are asked to create something that has never been asked before or answer a question that has never been answered. It is quite exciting but also daunting. Below we share some general principles and techniques when tackling problem-solving.
General Problem-Solving Techniques:
- Understand the problem
- What is the unknown? What are the conditions and assumptions of the problem at hand? What data do you need to obtain? What is the goal of this problem? Do you understand the definitions and models involved in the problem or in your approach? Do you understand all the words used in stating the problem? Can you restate the problem in your own words? Have you researched the background skills required for such a problem?
- Do Something!
- Problem-solving is not a passive activity. You have to start somewhere. Don’t let the big task paralyze you! You can’t expect to fix your plumbing problem if you insist on your hands staying clean the whole time. Get in there! Devise a plan of attack! Remember that it is more than okay to be wrong. You may not know something is wrong until you try it!
- Be persistent
- Did your first attempt not pan out? Learn from your mistakes and try something else! Keep track of techniques you have already tried and what you learned so that you do not repeat failed attempts.
- Be patient
- This is very important. Exercises from class assignments might have taken you minutes to solve. True problem-solving is different– a good problem could take hours, days, or years to solve. This is normal!
- Learn from mistakes and dead ends
- You will make mistakes, and you may inadvertently waste time going down a dead end. This is all part of problem-solving! Learn from your mistakes and move on!
- As with all activities, we improve by practicing. The more academic papers you read, the better you will be at deciphering them. Practicing problem-solving helps you gain experience that can help guide you in the future!
- Don’t stop just because the
problem has been solved.
- Can you pose a new problem suggested by the previous problem, result, or method? This is what research is all about! For example, some non-euclidean geometry grew out of the question “What happens if we drop the parallel postulate?”
- Don’t stop just because the
problem is obviously impossible.
- Can you modify the problem so that it is possible? In mathematical modeling, we use models that we hope will predict the future or explain natural phenomena. Perfect models do not exist, but that doesn’t mean they are not useful or that we should disregard them. Sometimes our task is to do our best to find approximate solutions for unsolvable problems!
In addition to the general problem-solving techniques listed above, it is nice to consider some more specific problem-solving techniques that you can utilize if you are stuck and do not know what to try next. Note that this is not an exhaustive list, but rather is some of our favorite techniques.
Specific Problem-Solving Techniques
- Simplify or try a simpler problem
- Can you change the problem slightly to make it easier? Sometimes just looking at smaller cases or subsets can give some valuable insights.
thinking/”Wouldn’t it be nice” approach
- Is there something you could add or change about the problem that would make it easier or give some insight into the nuances of the problem? Wouldn’t it be nice if you could relate the problem to a known problem or technique that you already know how to do?
- Look for extreme or special cases
- Can you solve the problem for particular situations? Can you narrow the scope down?
- Try variations of the problems
- Play around with the assumptions and boundaries of the problem.
- Work backwards
- Start where you want to end up and think about how you would get there. This is a particularly great technique for proof writing!
- Try some examples
- An example won’t prove something is true all the time, but sometimes just trying an example can give some good insight into the problem. Sometimes a good counter-example is all one needs to answer a research question.
- Guess and check
- This doesn’t always pan out, but this can be a way to quickly collect data and find dead ends.
- Generalize the problem
- Sometimes loosening the requirements or scope of the problem can unlock insights into the problem.
- Look for patterns
- As you play with the problem, is there a relationship that keeps showing up?
- Draw a picture and/or look for
- Can manipulatives or pictures help you visualize the problem? Can we represent the problem in another way which will help us solve it?
- Look for invariants or
- Sometimes you can find a quantity that is unchanged in the problem no matter what you do (an invariant), and sometimes you can find a quantity that changes only in one direction (a monovariant).
- Organize and collect data
- Are there multiple cases that need to be examined separately? Making a log/table/chart in order to keep track of what you know along with what did and did not work is also very useful.
- Take breaks
- Going back to the problem after working on something else or taking a break is not only helpful for your mental health but can also yield useful insights as well! Sometimes when you go to bed, take a shower, exercise, etc., your mind may start wandering and you can come up with fresh ideas.
We would like to acknowledge that these techniques are important particularly with respect to what is going on in the world today. For many of us to work together to eradicate racism, we must first understand the problem and how it is rooted in different aspects of society. We should educate ourselves by understanding data and listening to the stories of those affected. We must also be persistent and patient; great change takes time. We do not need to give up on the goals of affecting change. We need to listen to each other and learn and give each other room to speak. We need to do something and not be passive with the injustice that we see. Past MAA president, Dr. Francis Su, suggests these two simple actions to help make a difference:
- Vote: especially in local elections
- Advocate for Reform: focus on specific policies tailored to your community. This advocacy toolkit has great reform ideas
Written by Drs. Amanda Harsy and Brittany Stephenson – Lewis University Mathematics Professors
George Polya’s How to Solve it.
Emily McCullough and Tom Davis’ “So You’re Going to Lead a Math Circle”
Joshua Zucker and Tom Davis’ “An Interesting Way to Combine Numbers”
Joshua Zucker’s “An Introduction to Problem-solving”
Peter Tingley’s “Listen, Share, Play”