In math, confidence doesn't count

Originally appeared November 18, 2006, in the Salt Lake Tribune.
Appeared various places after that.

About a month ago, the Brookings Institution's Brown Center on Education Policy released a report that, among many other things, produced two sets of statistics from K-12 students in various countries around the world.

The first set recorded achievement on certain standardized math exams. The second attempted to measure the confidence and enjoyment of students by tabulating the percentage who agreed with statements like "I usually do well in mathematics" and "I enjoy mathematics."

The results were surprising. Countries where students displayed confidence in their mathematical abilities invariably recorded lower exam performances, and vice-versa. The United States fell somewhere in the middle on both achievement and confidence, but even here the results raised eyebrows.

U.S. students scored higher on confidence and enjoyment, for instance, than their counterparts in Singapore, but even the least confident Singaporean students outscored the most confident American ones on achievement.

But is any of this really surprising? I've devoted about half of my life studying to become, and now practicing as, a research mathematician. In those years, I've used a lot of adjectives to describe math, but "fun" was one that only rarely surfaced.

Most of the time, my days are spent in utter frustration, stuck in impossible confusion, trying desperately to put together a grand puzzle of which I understand only a few pieces (and those rather dimly). I am reminded constantly of my mathematical limitations and my failings.

An imperfect analogy might be a giant Sudoku puzzle, 81 by 81 say, with absolutely no assurance that the puzzle is arranged to have any solution at all. Of course, it's ultimately not all torment. A tiny fraction of the time I'm visited by a flash of insight when a portion of the puzzle snaps into clear view. But quickly that portion suggests new questions -- new frustrations -- and the process begins anew.

I suspect that the previous paragraph applies with little modification much more broadly. Is writing a significant poem fun? Training to run a four-minute mile? Climbing Everest? I doubt it.

I suspect these are mostly terribly frustrating, exhausting experiences, which even in the best engender deep feelings of inadequacy and at times a complete lack of confidence. But there are brief moments of indescribable exhilaration too -- reaching the summit, crossing the finish line -- which make it all immensely worthwhile and satisfying.

In fact, as I frequently tell my students, I've observed that the truly meaningful experiences in life are often the ones that are the most frustrating. The principle also admits a kind of converse: If something is too easy, then most of the time it is not significant. It's a simple proposition, but one that took me a long time to absorb properly.

Which brings me back to the Brookings study. Do students in Singapore really develop an earlier understanding that the process of meaningful learning is invariably not all that much fun? It's not likely, since that sort of understanding requires quite a bit of experience and maturity.

But I bet that their teachers possess this understanding in some form and are not afraid to embrace it. In the end, that's what a student really needs.

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