Preparing the Ground for Mathematical Creativity
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Let me share my own experience with—and advice for—mathematical creativity. Mathematical creativity is usually required in order to develop a proof, to figure out how to mathematically model something of a type that has never been mathematically modeled before, or to figure out what kins of statistical method will work in an analysis unlike previous analyses.
The first piece of advice based on my own experience—backed up by scholarly work such as that in the paper “The Characteristics of Mathematical Creativity” is that you really want to get your subsconscious in gear working on the math problem. In the workings of our brains, consciousness is a tiny window on a large mansion. Simply because the throughput in consciousness is so small, most of the activity of the brain is unconscious. So getting the rest of your brain to work on a problem is a key to mathematical creativity. (Call it the subconscious, the unconscious or the nonconscious—for this discussion it is all the same.)
But how do you get your subconscious working on a math problem? That answer: it takes a lot of hard work on a math problem before your subconscious takes the hint that it should work away at the problem. A lot of that conscious hard work may feel like banging your head against a wall, but it credibly communicates the urgency of the problem to your subconscious. This is analogous to the memory principle that it is those things which you make an effort to remember (and in large part, only those) that your brain assumes are worth storing in long-term memory—a principle I discuss in “The Most Effective Memory Methods are Difficult—and That's Why They Work.”
The second piece of advice is that it helps to develop your ability to think productively about a math problem without a piece of paper in front of you. This makes it easier for both your conscious and unconscious mind to think about the problem when you are at your most creative. Also, simply trying to keep track of a math problem without a piece of paper in front of you requires continually going back to the basics and trying to remember what the overall objective and intermediate objectives were. That is, not having a piece of paper in front of you forces you to focus on the forest, rather than on the trees.
One of the times I am most creative is when I am on a walk. That is true for at least two reasons. One is that most people are smarter during exercise. The other is that exercise and interesting surroundings can calm the mind.
In order to think about a math problem while on a walk, it is helpful to do preprocessing of the problem to strip off the fundamentally easy but distracting parts of the problem from the hard kernel of the math problem. Then thinking about a math problem on a walk requires persistence and patience. I lose my thread of thought many times as I think about a problem in this way, but every time I pick up of the thread after losing it strengthens my mathematical ability a little bit.
The third piece of advice is to strengthen one’s mental muscles for thinking about a problem in different ways. For me that often means practicing thinking geometrically. See my post “Why Thinking Geometrically and Graphically is Such a Powerful Way to Do Math.”
The third piece of advice is that after getting some picture in your head of the kernel of how to do a math problem, you then need to sit down with a piece of paper or computer screen in front of you and begin to work out the details. An alternative is to get someone to try to explain the solution idea to. However you begin working through the details, you will often discover new things in that process of working out the details.
Overall, in my experience, the path for mathematical creativity is conscious—>subconscious—>conscious. (Sometimes there are many more alternations between conscious and subconscious than that.)
The fourth and final piece of advice is to revisit math problems from time to time, even long after you originally found a solution that seemed reasonably satisfying at the time. Once you have gotten your subconscious working on a problem, it often doesn’t stop, even after you have accomplished whatever your immediate goal was in working on the problem. And the passage of months and years often leads you to bump into ideas that can help with a deeper solution to the problem. The bottom line is that, if you are like me, you will often have a deeper insight into an old problem if you think about it again months or years later.
Mathematical creativity is a lot of fun. And it is more learnable than most people think. I have a favorite book to recommend for those who want to build their mathematical creativity: George Polya’s How to Solve It:
Don’t buy into the idea that mathematical ability is mostly determined by what you were born with. Instead, read “There's One Key Difference Between Kids Who Excel at Math and Those Who Don't” and “How to Turn Every Child into a 'Math Person'.”
I have links to many other posts on math in my post “Gabriela D'Souza on Failure in Learning Math”
On calming the mind, which can enhance creativity, see “Judson Brewer, Elizabeth Bernstein and Mitchell Kaplan on Finding Inner Calm” and the links at the bottom of that post.