One of the most basic questions in mathematics is: how do you solve problems in general? This is traditionally why students tremble in terror at “story problems”— instead of being asked to mimic well-known algorithms, as in the majority of their school exercises, suddenly they are in a situation where they are not presented the clear path to the answer. Yet problem solving is one of the most critical skills you can learn in mathematics classes, and many of us, especially those in science and engineering fields, spend a lifetime continuing to sharpen our skills in this area. Even in non-math-based professions, people often encounter dilemmas where the solution is not obvious. So I think it’s worth taking a look at ways to improve our problem solving abilities in general. And surprisingly, modern neuroscience can provide us some strange methods to try when simple linear reasoning fails us.
Probably the most famous book on this topic is “How To Solve It” by the late Stanford math professor Georg Polya. Polya lays out a general 4-step process for approaching any problem, in a book full of useful examples from basic areas of algebra and geometry. First, you need to understand it: what are the current information, the unknowns, the goals, and the restrictions that apply? Second, find a way to connect the data and the unknowns, in order to plan your approach. If this is not obvious, look for related problems, or a smaller subset of the problem that you can solve. Third, carry out your plan, taking care to show that each step is correct. And finally, examine the solution: is there a way to independently check the result, or use it for other problems?
While Polya’s method is very useful, something about it seems a bit too simple. After all, if it is easy to understand a problem, plan the solution, and carry it out, why are there so many unsolved problems out there? Why hasn’t someone definitively solved each millennial problem, like the P=NP question we discussed in podcast 13, and taken the million dollar prize? I think one key is that a lot of problems require a flash of intuition, or a conceptual leap that is very difficult to arrive at by linear reasoning. And that’s where the neuroscience comes in. Recently I’ve been reading an intriguing book by Andy Hunt called “Pragmatic Thinking and Learning”, which offers a number of strategies for stimulating your mind to solve problems in different ways.
As you’ve probably heard somewhere, many modern scientists believe our brains exhibit two main modes of thought. Commonly these are called “left brain” and “right brain”, but Hunt points out that the strict connection with the brain hemispheres isn’t quite right, so he suggests the terms “L-mode” and “R-mode”, with the L standing for “linear”, and R standing for “rich”. You can think of the two modes as being the two CPUs of a multiprocessing computer system, potentially working in parallel at all times. Your L-mode brain excels at analytic, linear thinking, and is the primary user of methods like Polya’s. Your R-mode brain is what you typically exercise in artistic or creative endeavors. R-mode, while trickier to interact with due to its nonverbal nature, can also provide intuition, synthesis, and holistic thinking— it probably won’t come up with a mathematical proof, but can lead to do discover a conceptual leap you need to get past a roadblock in one. But how can we effectively interact with our R-mode, or stimulate its activity, in order to leverage its power? Hunt suggests a variety of basic techniques for getting a dormant R-mode active and more involved.
One simple method is to try to use different senses than usual, in a way that engages your artistic side. While thinking about a problem with your L-mode, do some minor creative action with your hands that exercises your R-mode, such as making shapes with a paper clip, doodling, or putting together Legos. In one amusing example, Hunt describes a case where a team designing a complex computer program decided to get up and “role-play” each of the functional units, and soon had a variety of new insights about the system.
Another method Hunt suggests comes from the domain of computer science, but is likely applicable to many other fields: “Pair Programming”. The idea here is that one programmer is actually typing a computer program on the screen, inherently an L-mode activity, while the other is sitting next to him, observing, and making suggestions. Because the second programmer doesn’t have to worry about the L-mode task of entering the precise sequence of commands, he is free to use his R-mode to take a holistic look, and come up with intuitive suggestions about the overall method.
A third method that can be surprisingly effective is known as “image streaming”. After thinking about a problem for a while, try to close your eyes and visualize images related to it for ten minutes or so. For each image you can think of, first try to imagine it visually, then describe out loud how it appears to all five of your senses. This one sounds a bit silly at first— and I would suggest you don’t try it in an open cubicle with your co-workers watching— but can be a very powerful way to engage your R-mode.
A fourth suggestion is called the “morning pages” technique: when you wake up every morning, immediately write at least three pages on whatever topic comes to mind. Don’t censor what you write, or try to revise and make it perfect, just let the information flow. Because it’s the first thing in the morning, you’re getting an unguarded brain dump, while your R-mode dreams and unconscious thoughts are still fresh in your mind. If you were working on a hard problem the day before, your R-mode may naturally have provided new insights during the night that you now want to capture. As Hunt summarizes, “You haven’t yet raised all the defenses and adapted to the limited world of reality”.
These ideas are just a small subset of known techniques for leveraging your lesser-used R-mode— if you want to maximize your ability to use your whole mind for problem solving, I would highly recommend that you check out his book, linked in the show notes. I’ll be interested to hear from any of you who successfully use some of Hunt’s odder-sounding techniques to solve difficult problems. On the other hand, if you think everything I’ve said today sounds crazy, that’s probably just your L-mode brain over-exercising its linear, logical influence.
And this has been your math mutation for today.