Creativity as the Pinnacle of Learning

In a recent post on the topic of memory, I noted that this skill was at the base of the revised Bloom's Taxonomy (or taxidermy?). This morning I woke up thinking about the other end: creativity. I've mused about the role of creativity before, and how to teach and it (here and here).

It's easy to make the unwarranted leap from creativity to aesthetics, because we associate art justifiably with a creative process. But I prefer to think of creativity as the production of new knowledge in any context. Let me give a pedantic example:

All men are mortal.
Socrates is a man.
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Socrates is mortal.

The conclusion follows deductively from the two statements above it, so it is not the production of new knowledge. This is the hammer that fell on Bertrand Russell in his quest for ultimate true by means of logic. Logical, rigorous deductive thinking is an essential skill, but it's not creative. In contrast, Aristotle's encoding of logic into language was creative, but I have a more interesting example.

The other day I saw an interesting problem posted on the math subreddit. The diagram below shows a laser beam coming from the right and striking a perfect mirror (the thick black line at the bottom) at angle b, bouncing off, and then striking another mirror placed at angle a with respect to the first one:

The gray line is imaginary here--it extends the line of the bottom mirror to illustrate angle b, but the light hits the very end of the mirror itself. The light will continue to bounce off the mirrors in some way. Where does the beam end up going? I will put the solution at the end, in case you want to think about it first.

The point is that I suspected there should be an elegant way to think about the problem, where the solution--all solutions--would be obvious. So I cast about, looking for it. This is rather like trying to find the light switch in a dark room, as Andrew Wiles put it (see my other posts on creativity for the link). I gave up before I found it. I found an inelegant solution, which was correct, but wasn't creative enough to be called elegant. It was sort of a plodding, "add up the cumulative effect" solution, where you sort of crush a problem with the weight of logical facts until it leaks out its secrets like a garlic clove exudes oil.

My lack of blazing imagination does, however, illustrate that the creative process itself deserves its own "taxonomy." In other words, there are qualitative differences to creative enterprise. Let's take a look.

Creativity as producing new information can start with sheer randomness. Flipping a coin and writing down the results is creative. This sounds too trivial to be counted, but it's not. In fact, it's the singular most important spark of novelty in history. I have two examples. First, physicists have wondered how galaxies formed. If the big bang started from a single point, for example, why wasn't everything thereafter perfectly uniform? Where did all the novelty come from? One proposal is that tiny differences in the primordial universe were seeded by quantum events, which we know to be deeply random. So the largest structures in the universe may have started with infinitesimal randomness. Cool, right?

The second example is the evolution of life, which explores via an ecology a vast space of possible designs for living things. This exploration proceeds by random mutation of genes, and other ways in which genetic material may get mixed around (like parasitism), or a bacteria's lascivious lifestyle with regard to DNA. This is not the deeply puzzling randomness of quantum mechanics, but the sort that emerges from complex systems that is sometimes called chaos.

Randomness is a great entry point into creative thinking. The casting about for novelty is a skill in itself. It requires courage to be wrong, a good idea of how to recognize your intellectual quarry when you've found it, and determination--because it takes a long time for randomness to hit the right target. Louis Pasteur's "Chance favors the prepared mind." has two parts: chance, and preparation. The latter is a formed in the laborious mastering of some discipline or subject.

The whole idea of serendiptipy is based on these two elements, and our culture has benefited handsomely from it: rubber, penicillin, radioactivity, and many more are on the list. Wikipedia has many examples here.

In the last post, I showed an example of a game designed for high schoolers that is aimed at creative thinking in a mathematical context. The essential skills are being able to understand the problem and do basic math (easy), and cast about for creative solutions (fun, I hope). These solutions will start with guessing.

Guessing is a step up from randomness. Humans aren't very good at true randomness--we have to depend on the world around us for that, like moldy bread crumbs falling accidentally into a Petri dish. I suspect that good guessing is an art unto itself, and that it can be taught and practiced. There's an MIT course on the art of making educated guesses with regard to estimation (how many gas stations are in the US, do you think?). Here's the course description from the Open Courseware site (it's free!).

This course teaches the art of guessing results and solving problems without doing a proof or an exact calculation. Techniques include extreme-cases reasoning, dimensional analysis, successive approximation, discretization, generalization, and pictorial analysis. Applications include mental calculation, solid geometry, musical intervals, logarithms, integration, infinite series, solitaire, and differential equations.

This is targeted at students with a good math foundation (everybody at MIT, I guess), but I find it exciting because it shows how to teach a whole course on guessing in the context of a discipline. There's no reason that this couldn't be done in other subjects just as well. Guess-and-check is a fundamental human skill that reinforced our knowledge of the world. Think about kids and the funny way they conjugate verbs at first because they are guessing based on simple rules (e.g "I eated my peas, daddy"). The guess is close enough to communicate, and as an additional reward, they glean information about new complexities of language, if someone is kind enough to point out the right way of saying it.

Problem-Solving might be the next step in the creative chain of being. This is a natural continuation of randomness and guessing, which results in the production of new knowledge in some applied context. This works in art as well as math, I think. It's the evolution from random doodles to purposeful artistic creation. Problem solving weds the analytical/deductive process, discipline-specific skills and knowledge with the trial-and error process that I've described in prior posts on creativity. This is the nuts and bolts of creative production.

Inspiration may or may not be teachable. If we help students to be good seekers of randomness, good guessers, and good problem-solvers, can we help them elevate themselves to inspired thought? I don't know, but I guess that we can provide a fertile environment for this, and foster it in individuals who might otherwise have not reached their potential. I don't really believe that we can take every math student and produce another Gauss or Euler, but we can ameliorate one of the great hidden human tragedies--the many, many inspired thinkers who never got the intellectual cultivation they needed to allow their talents to flower.

This is all first-draft thinking. An interested group of discipline experts could turn these rough ideas into something applicable to a curriculum or institution. To include ways to assess creativity at each step along the way. Disciplines can learn from each other and share approaches, opening up the possibility of interdisciplinary learning. I often though that the math students could benefit from watching art students critically review each others' work.

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Here's the solution to the problem. I have redrawn it, but I saw it first here. The original problem was in terms of a tiny billiard ball, but I changed it to a laser beam. The key insight is that reflections preserve angles, so that instead of imagining the beam bouncing off at the same angle (incidence = reflection), imagine it passing through the mirror as if it were a pane of glass. Then add another pane of glass where the return bounce would have occurred, so that copies of the mirrors look like spokes on a wheel separated by angle a. This illustrates clearly that the beam will swiftly exit the mirrors and go on its way in most arrangements. The whole process is laid bare. I've illustrated it with a=45 degrees below. This is an inspired and elegant solution, unlike my workable but problem-solving brute force approach (not shown).