By Steven Klee, Contributing Editor, Seattle University
When two grandmasters face off in a chess tournament, they are faced with a complicated bit of game theory. If you were in one of their positions, you would prepare for the match by studying your opponent’s games in great depth. You would study board positions they had created, looking for weaknesses in their defenses and blunders their previous opponents (or they themselves) had made. It would be safe for them to assume that you could have a strategy in mind to counter any of their strategies that had previously been successful.
Of course, your opponent would naturally study your body of work in the same way. Therefore, by the time you sat down at the board, there would be a natural expectation that you know that your opponent knows that you know as much as you possibly could about them, and likewise they have the same expectation of you.
As a consequence, the natural strategy for determining who is the better player is to try to avoid these positions in the first place. Don’t allow the board to get to a point where you have been defeated in the past. Don’t allow the board to get to a point where you have been successful in the past because your opponent might know how to turn that position to their advantage. Get away from what has been seen before and create a new position that truly tests the skill of each player. There’s a term for this – chess players call it going “off book.”
To chess enthusiasts, this moment is exciting. It’s the moment in the game when the board reflects a position that has never been recorded in a tournament. It is an opportunity for observers to experience chess history and witness the creation of new knowledge or strategy. Every move is new and the anticipation of what might come next is thrilling.
Why is this relevant to math education?
Employers want students to be prepared to tackle a variety of real world problems. These problems may be vague or imprecise. They may be posed without any sense of what the answer should entail, or perhaps without understanding what the problem truly is. We don’t need to look much further than the annual Mathematical Contest in Modeling to see a wealth of interesting real-world problems that can inspire a wide range of potential solutions.
Of course, the same principle applies to pure mathematics. Answers to research problems of any sort are exciting because they represent a creation of knowledge. There is a thrill that comes from writing down a formula or idea that no one has written before; from creating mathematics that has not been seen. This happens as a result of getting away from the books.
How can we create a similar experience for our students?
An obvious venue for this is through research experiences for undergraduates, ranging from formal summer programs to community/industry partnerships or projects that are part of a class. However, research experiences are not the only way to get students to think outside the textbook. For me, the most exciting day of any class is the one where a student asks me a question I can’t answer. I love these types of questions, and I try to be as transparent as I can about my thought process. I’ll say, “That’s a really good question! I don’t know the answer right now, and to me that’s really exciting because it means you’re thinking really deeply about the material we’re studying.” Sometimes, I need a few minutes or a night to think about the problem and figure out how to answer it. Sometimes I don’t know the answer because it is completely new.
For example, one time a calculus student observed the periodic cycle of derivatives of trig functions (the derivative of sin(x) is cos(x), whose derivative is –sin(x), whose derivative is –cos(x), whose derivative is sin(x)) and asked “Does that have anything to do with imaginary numbers?” On the way back to my office after class, I realized that his observation was related to Euler’s formula e^(it) = cos(t)+isin(t), which then inspired an exciting homework problem when we reached the chapter on derivatives of exponential functions.
In another instance, I devoted two days at the end of the quarter of a graph theory class to unsolved problems in graph theory. My philosophy was that attempting to cram new material into the last two days of the course, and subsequently testing the students on this new material on the final exam (which was to be given two days after the last day of class) was an unfair assessment of their learning. Perhaps we would be better served pedagogically by exploring applications of what they had learned in a quarter of graph theory.
I came to class with an open problem and asked students to spend 5 minutes in a group brainstorming potential approaches. The students shared their ideas with the class and then split into groups with peers who were thinking about the problem in similar ways. We spent the rest of class working on the problem. My role was to bounce from group to group, hear their ideas, and provide input as best as I could. My most common response was “I don’t know, but that seems interesting.” When I could, I would point to interesting special cases, share my intuition, or point to terminology or references that could be helpful. Some students wrote code. Others drew pictures. Others generated data. Others focused on special families of graphs. But everyone worked productively on something. Everyone generated new ideas. Everyone created new mathematics, or at least, mathematics that was new to us.
Working on unsolved problems allowed the students to showcase the variety of graph-theoretic ideas they had learned over the course of the quarter. It served as a good exercise in preparing for the final exam because they had to reflect broadly on the course material and think about applying those ideas to new problems in a relatively short amount of time. Beyond this, the activity showed the students how much they had learned and let them see that they were capable of applying their knowledge to unsolved problems. Research didn’t require a PhD or an internship in a fancy lab – it just required a blackboard and the willingness to say “what if…?”.
My hypothesis is that the students’ excitement to explore new ideas and ask interesting questions partially stemmed from my openness to hearing their ideas and my willingness to say “I don’t know.” This is not to say that we as professors should not have a deep understanding of mathematics. We absolutely must. We exhibit the breadth of our knowledge through our teaching in the way we present the material, or the different ways we can explain a concept to a student. At the same time, we can be honest when students make keen observations or ask thoughtful questions. We should be excited when our students push us to think deeply about our subject and celebrate their insights.
Your classroom approach is very inspiring and a breath of fresh air. It brings the students to think about the topics they are thought in an environment where the students are not afraid to ask questions to gain an understanding as deep as they can. The other benefit is one remembers something if one actually understands it well.
Thank you, Rene. I appreciate the support.