We all want our students to ‘think critically’ and ‘become better problem solvers’. But what does that mean, and how do we scaffold and provide opportunities developing problem-solving strategies? Obviously, this question is deeper and wider than a single blog post, but I wanted to put my first card on the table. First and foremost, this must all begin with ‘legitimate’ problems. To me, this means ones where a student must reach beyond plug-and-play; there must be elements of identifying which information is useful, choosing approaches that attack weak points of the problem, finding their own path rather than following ones already mapped out. Two cases that I think exemplify ‘real’ problems are Petals around the rose and (surprise surprise) my own PatternMaster.
The rest of this introductory post is dedicated to a distillation of George Polya’s advice to young mathematicians
(FYI, this is adapted from text I wrote for the Lab Manual for the Intro Bio course I once directed)
The general principles below are derived with modifications from George Polya’s text (cited below) via Sam Ward, who introduced them to me. However, strategies given to you advance you as a thinker vastly less than ones you develop or discover for yourself. So use these pages when you’re truly out of your own ideas. Problem solving is an immensely valuable skill, but one best learned using your own creativity, insights, and hard-won victories. Supplementing these experiences with other inputs can be valuable—if you go to the trouble of analyzing how you missed the insight yourself, and consider how to do better next time.
See “How to Solve It: A New Aspect of Mathematical Method“, G. Polya, 1985, Princeton University Press.
I. Assessing the Problem
A. What is unknown; what is the goal?
B. What are the data given or essential facts known?
—> Make sure you include relevant information not explicitly included in the problem statement.
C. Does the current state of the prob- lem provide for a solution?
—> Intermediate steps may be required.
D. Can you separate the problem into parts?
—> Make what progress you can, and new insights may occur.
E. Can you draw a figure to describe the problem?
—> Organizing your thoughts and representing the problem in a different format may reveal approaches or solutions.
II. Planning an Approach
A. Find connections between the data and the unknown.
B. Have you encountered problems like this before?
—> Apply this thinking both to the overall problem as well as to its parts.
C. Can you re-state or re-format the problem to resemble one you know?
D. an you solve a part of the problem?
E. Can you work backwards from the end or an intermediate point?
—> Thinking from the “other end” may show you the way.
III. Remaining Effective
A. If you are really stuck, take a break, or sleep on it and let your mind work in the background
B. Having a second idea is much hard- er than a first—the original keeps drawing you back. Try thinking outside that box, starting from a different point, achieving a differ- ent perspective, or stepping away and letting your brain work on the problem in its own way.
IV. Carry Out Your Plan
A. Check each step as you go along; apply “Examine your solution” techniques.
A. Does it make sense? Does the scale come out, is the sign as you anticipated, are the units right?
B. Can you check the result?
—> Can you look at it in a different way (say for an algebra problem–if you solved for X, what happens if you leave X in and put in Y for another value–when you solve for Y, do you get the value you removed?)
—>Do changes in data/calculation give the expected changes in outcome?