One study of 65 subjects suggests that creativity prefers to take a slower, more meandering path than intelligence. “The brain appears to be an efficient superhighway that gets you from Point A to Point B” when it comes to intelligence, Dr. Jung explained. “But in the regions of the brain related to creativity, there appears to be lots of little side roads with interesting detours, and meandering little byways.”This sounds like the diagram I use with students, to explain how to do proofs or other creative math work.The task is to go from some question (

**?**) to a resolution (

**!**) through structured inquiry. Which is a fancy way to say trial and error. Read the diagram from left to right.

Each of the straight segments is an attempt to solve the problem. The way I've drawn it, the first five attempts don't work, but give some hint about the real solution. It might not be so straightforward, of course. Real problems might look more like this:

Here, multiple lines of inquiry finally meet up in an Ah-Ha! moment to provide the solution. Andrew Wiles gives a nice exposition of this process in The Proof, the Nova show on how he proved Fermat's Last Theorem. His analogy about creative exploration is of exploring a darkened room. You can see a minute of it on YouTube here. The whole show is highly recommended. It's powerful.

Note that in order to do the creative exercise, one has to be able do the deductive part of the work. This entails knowing how to write and think in the target discipline, and knowing the tools available. Without this background, it's impossible to make even reasonable guesses, or to check your work for logic errors. Checking solutions is generally much easier than finding them to begin with.

**There are other points of view expressed in the article.**For example:

According to Kenneth Heilman, a neurologist at the University of Florida and the author of “Creativity and the Brain” (2005), creativity not only involves coming up with something new, but also with shutting down the brain’s habitual response, or letting go of conventional solutions.This resonates as well. Trying the same thing over and over and failing isn't productive, and is part of the reason math proofs are hard for students to learn. Initially they have a limited number of ways of looking at a problem, so are constrained in what they can attempt. Maybe the lesson there is to train them early to ask the question "what's another way to look at this?" That may be the heart of creativity.

**How could we assess this in practice?**In my last post I mentioned that the visual and performing arts faculty are interested not only in the final product of a creative work, but also the evolution of thinking that went into making it. This may include a portfolio of intermediate work, for example, something like the lines on the schematic above. "How many ways can you look at this?" could be an interesting question to routinely ask. There are established theories for many areas, and these can be used as lenses. In the end, creativity may be more about the process of creation than the final, visible result. This attitude could have an impact on the way we teach technical subjects too: given a problem to address, list all the approaches you can think of before trying to solve it. This meta-inquiry leads one to seek out new sources of inspiration. One time when I was stuck on a math problem I went to an art show and found what I was looking for. This sort of determined curiosity may not be valued as much as it should be because it may be under the radar, so to speak. In my experience, we tend to expect curiosity and creativity to just bloom like spring flowers, but paying more attention to the details could help a great deal.

**As evidence that the "many views" approach is useful**, I submit the Polymath Project by Tim Gowers. You can read a great account of it in the Science News article "Mathematics by Collaboration." Here's the setup:

Late last January, University of Cambridge mathematician Tim Gowers decided to run a little experiment. Was it possible, he wondered, for a large number of mathematicians to collaborate openly on the Internet, pooling their ideas around a single problem? If it were possible, would it be easier, more efficient, more fun? Could the mathematicians together solve a problem they might not be able to solve individually?The answer is yes! This hints at another possible way to look at creativity--if multiple views are important, then collaboration can be a good thing. I've experimented with this by allowing students to work together on exams. It depends on the class, but it can work very well. They tend to sort themselves into groups by ability level, which is interesting. It also might lead us to question the whole idea of assessing skills. What if a person has the skill of being a really good collaborator, taking others' ideas and stitching them together in novel ways? In the usual way courses are run, this could easily be called cheating. And since assessments of intellectual ability are focused on students as individuals, such traits wouldn't be assessed, and probably wouldn't be valued. Interesting questions. Why don't we focus more on group problem-solving or creation, develop techniques for organizing and documenting it, and find ways to assess it? Maybe this is already being done somewhere?

At a mid-Atlantic state university that is about to welcome a cohort of students from China, one professor cautioned the rest of the faculty to be "patient" with the foreign students, because their ideas of "plagiarism" are culturally different from those prevailing at most western universities.

ReplyDeleteIn China, she said, that activity is viewed as "sharing knowledge."

While this concept seems alien to an assessment system that considers individual achievement as the primary (if not only) basis for academic evaluation, further investigation might (or might not) point to "sharing knowledge" as an avenue towards a more systematic kind of collaborative creative problem-solving.