When I was young, everyone considered me to be "good at math." What do we mean when we use that phrase? It implies a certain innate ability, one that most people can't achieve. I had classmates who were told that they succeeded only by "working hard" at math, while for me the ability was supposedly inborn.
My daughter Ruth is also good at math. At this point she has taken more of it than I ever did, she helps others with their math and physics homework at a tutoring center, and she is studying to be an engineer. She never had the stellar test scores that I did in high school, though, in part because it takes her a little longer to work through the problems on a timed test. Does this mean she isn't really "good" at the subject? I think not: On a recent construction project, I deferred to her better-informed calculations.
As a culture we subscribe to the myth of genius: the idea of a lone individual, graced with some amazing innate ability that forever sets them apart from others. The "differentness" of the great genius is in fact one of the most common arguments used in attempts to rationally prove the existence of God: The fact that such a person could do such incredible things must mean that humans are indeed "only a little lower than the angels" in our creative powers (Heb. 2:7). On the other hand, the same differentness also traditionally predisposes the great creative genius to loneliness, substance use, and even madness. My master's thesis was on this topic, measuring rates of manic symptoms among students with creative and noncreative occupational trajectories.
Despite the continuing cultural appeal of the lonely genius, however, the best available research suggests that strong ability in any area is developed through repeated practice. I have argued that creativity is not so far outside the norm of skilled performance, and that skilled performance is a result of the Intuitive Mind. Any type of skill results from a set of Intuitive-level abilities that a person has, at least as much as from conscious effort. But the way in which the Intuitive Mind acquires those abilities is through practice. The "ten thousand hour rule" is the idea that one can develop exceptional ability in any area simply by spending 10,000 hours engaged in it. Malcolm Gladwell popularized this idea, and gave it the catchy 10,000-hour name, in his 2011 book Outliers: The Story of Success. The number of hours is not, of course, a single cut-off. The more time one spends practicing a given activity, the stronger one's abilities in that area compared to others who have not had that amount of practice.
For young people in particular, even small differential amounts of practice might be enough to set one apart from one's reference group. In the case of my mathematical "ability," teachers noticed early on that I could do problems faster or more accurately than my age-mates. But I was no "Good Will Hunting" discovering new theorems: all of my math performance was fairly rote. The reason, most likely, is that I was a socially awkward child who entertained himself with reading and writing. I liked to write out the multiplication tables in particular -- something appealed to me about the symmetry of the table, with its reflection of results across a central diagonal (6*7 is the same as 7*6), or the way that multiples of 9 consist of 2 digits that together add up to 9. It's an odd hobby for a child, I confess, but I found it enjoyable, so I did more of it than my classmates. No wonder I got a little faster and more accurate at that type of problem. When I got to algebra, a whole new world opened up: I enjoyed sliding numbers around from one side of the equation to the other, simplifying terms, and discovering how things that looked different were in fact the same. I saw in those problems the essence of algebra, the reason why the Arab scholars who developed this method named it al jabr, or "transposition." My peers might not have seen it that way. Once again enjoyment led to practice, practice led to mastery, and mastery led to recognition for a particular type of skill. But practice is cumulative; eventually I set math aside (except for one subset, statistics, that is still a love of mine). Ruth, who has continued to practice, can crank through calculus problems that I no longer remember how to approach.
Although Ruth's math skills eventually came to exceed my own, I never had the sense that she enjoyed numbers in the same way. (My efforts to show her the glories of the multiplication table fell on deaf ears). Her own most incredible gifts come in the area of music. She was concertmaster of her high school orchestra, successfully auditioned for Colorado's all-state symphonic orchestra, and lately was invited to tour with a college group that includes faculty and professional musicians. Again, the 10,000-hour rule applies: Ruth was just 3 years old when she saw Elmo speaking with a young violinist on Sesame Street, and asked if she could learn to play. My wife put her off, saying "let's talk when you're four." But on her fourth birthday Ruth remembered that conversation, and asked if she could please learn to play the violin now? Her mother relented, and a life-long love of music began. By the time Ruth entered school, then, she had already been playing violin for a year; her elementary school had a Suzuki program that many students started in first grade, but by then Ruth had nearly 2 years of practice ahead of them. Like my math story, Ruth's tale of course begins with the student's interest. The reason she and I both practiced is that the activity made us happy in some way. Enjoyment led to practice, which led to mastery, and then to recognition for the skill.
K. Anders Ericsson was the psychology researcher who conducted many of the foundational studies that Gladwell relied on when he formulated his 10,000-hour rule. In his book Peak: Secrets from the New Science of Expertise, Ericsson argued that not just any practice will do. What helps people to develop from strong performers to exceptional ones is an intentional effort to work on the areas that are weak. For instance, this year Ruth's college instructor showed her some bow techniques to help get more volume out of her violin. She has been practicing those specific methods in a deliberate way, trying to improve the quality of sound she is able to produce. There's an experimental feel to this, trying one thing and then another, and noting the results. The one area of math where I have continued to develop my skills, statistics, is like that as well: I read about a new technique for missing data imputation or visual display of information, and look for opportunities to try it out. What keeps the activity interesting is not just comfortably performing at one's usual high level, but instead always working on the edge of one's expertise. Ericsson suggests that this is the way to improve at any skill, not just math or music.
What, then, about the true outliers? At the top of this page is an image of Mozart, who is perhaps the archetypal case that people cite as a "natural genius." Isn't it true that Mozart composed his first symphony by the age of 7, and performed as a teenager for the kings and queens of Europe? Ericsson provides a close reading of the historical facts that suggests otherwise. Mozart was indeed a talented musician, but so was his father, who started preparing Wolfgang Amadeus from a very young age to follow in his footsteps. Like Ruth, the teenage Mozart had already experienced a decade of practice before those famous performances; furthermore, part of his fame was simply the novelty of his youth. If an experienced performer had done the same, perhaps it would have been considered a workmanlike effort, but coming from a teenager it was a revelation. And Mozart's early compositions have also been called into question -- he definitely did work on them, but they also benefitted from great improvement and refinement by his father the experienced composer. Part of Mozart's luck was not genetic endowment, but having been born into the right environment to nurture his skills early on. Even in Mozart's case, the equation seems to hold: enjoyment -> practice -> mastery -> recognition.
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