One of the great joys of mathematics is the incontrovertible feeling that you’ve understood something the right way, all the way down to the bottom; it’s a feeling I haven’t experienced in any other sphere of mental life. And when you know how to do something the right way, it’s hard— for some stubborn people, impossible— to make yourself explain it the wrong way.
How Not to Be Wrong: The Power of Mathematical Thinking
by Jordan Ellenberg. There aren't many books that educate and enlighten non-mathematicains on the importance of mathematical thinking. Ellenberg has given us a rare gem and an inspirational book. This is one the best books of 2014 - A must read for everyone!!
I’ll give a misimpression of mathematics as an enterprise in which a few solitary geniuses, marked at birth, blaze a path for the rest of humankind to trot along. It’s easy to tell the story that way. In some cases, like that of Srinivasa Ramanujan, it’s not so far off. Ramanujan was a prodigy from southern India who, from childhood, produced astonishingly original mathematical ideas, which he described as divine revelations from the goddess Namagiri. He worked for years completely in isolation from the main body of mathematics, with access to only a few books to acquaint him with the contemporary state of the subject. By 1913, when he finally made contact with the greater world of number theory, he had filled a series of notebooks with something like four thousand theorems, many of which are still the subject of active investigation today. (The goddess provided Ramanujan with theorem statements, but no proofs— those are for us, Ramanujan’s successors, to fill in.)
But Ramanujan is an outlier, whose story is so often told precisely because it’s so uncharacteristic. Hilbert started out a very good but not exceptional student, by no means the brightest young mathematician in Königsberg; that was Hermann Minkowski, two years younger. Minkowski went on to a distinguished mathematical career himself, but he was no Hilbert.
One of the most painful parts of teaching mathematics is seeing students damaged by the cult of the genius. The genius cult tells students it’s not worth doing mathematics unless you’re the best at mathematics, because those special few are the only ones whose contributions matter. We don’t treat any other subject that way! I’ve never heard a student say, “I like Hamlet, but I don’t really belong in AP English— that kid who sits in the front row knows all the plays, and he started reading Shakespeare when he was nine!” Athletes don’t quit their sport just because one of their teammates outshines them. And yet I see promising young mathematicians quit every year, even though they love mathematics, because someone in their range of vision was “ahead” of them.
We lose a lot of math majors this way. Thus, we lose a lot of future mathematicians; but that’s not the whole of the problem. I think we need more math majors who don’t become mathematicians. More math major doctors, more math major high school teachers, more math major CEOs, more math major senators. But we won’t get there until we dump the stereotype that math is only worthwhile for kid geniuses.
The cult of the genius also tends to undervalue hard work. When I was starting out, I thought “hardworking” was a kind of veiled insult— something to say about a student when you can’t honestly say they’re smart. But the ability to work hard— to keep one’s whole attention and energy focused on a problem, systematically turning it over and over and pushing at everything that looks like a crack, despite the lack of outward signs of progress— is not a skill everybody has. Psychologists nowadays call it “grit,” and it’s impossible to do math without it. It’s easy to lose sight of the importance of work, because mathematical inspiration, when it finally does come, can feel effortless and instant. I remember the first theorem I ever proved; I was in college, working on my senior thesis, and I was completely stuck. One night I was at an editorial meeting of the campus literary magazine, drinking red wine and participating fitfully in the discussion of a somewhat boring short story, when all at once something turned over in my mind and I understood how to get past the block. No details, but it didn’t matter; there was no doubt in my mind that the thing was done.
by Jordan Ellenberg. There aren't many books that educate and enlighten non-mathematicains on the importance of mathematical thinking. Ellenberg has given us a rare gem and an inspirational book. This is one the best books of 2014 - A must read for everyone!!I’ll give a misimpression of mathematics as an enterprise in which a few solitary geniuses, marked at birth, blaze a path for the rest of humankind to trot along. It’s easy to tell the story that way. In some cases, like that of Srinivasa Ramanujan, it’s not so far off. Ramanujan was a prodigy from southern India who, from childhood, produced astonishingly original mathematical ideas, which he described as divine revelations from the goddess Namagiri. He worked for years completely in isolation from the main body of mathematics, with access to only a few books to acquaint him with the contemporary state of the subject. By 1913, when he finally made contact with the greater world of number theory, he had filled a series of notebooks with something like four thousand theorems, many of which are still the subject of active investigation today. (The goddess provided Ramanujan with theorem statements, but no proofs— those are for us, Ramanujan’s successors, to fill in.)
But Ramanujan is an outlier, whose story is so often told precisely because it’s so uncharacteristic. Hilbert started out a very good but not exceptional student, by no means the brightest young mathematician in Königsberg; that was Hermann Minkowski, two years younger. Minkowski went on to a distinguished mathematical career himself, but he was no Hilbert.
One of the most painful parts of teaching mathematics is seeing students damaged by the cult of the genius. The genius cult tells students it’s not worth doing mathematics unless you’re the best at mathematics, because those special few are the only ones whose contributions matter. We don’t treat any other subject that way! I’ve never heard a student say, “I like Hamlet, but I don’t really belong in AP English— that kid who sits in the front row knows all the plays, and he started reading Shakespeare when he was nine!” Athletes don’t quit their sport just because one of their teammates outshines them. And yet I see promising young mathematicians quit every year, even though they love mathematics, because someone in their range of vision was “ahead” of them.
We lose a lot of math majors this way. Thus, we lose a lot of future mathematicians; but that’s not the whole of the problem. I think we need more math majors who don’t become mathematicians. More math major doctors, more math major high school teachers, more math major CEOs, more math major senators. But we won’t get there until we dump the stereotype that math is only worthwhile for kid geniuses.
The cult of the genius also tends to undervalue hard work. When I was starting out, I thought “hardworking” was a kind of veiled insult— something to say about a student when you can’t honestly say they’re smart. But the ability to work hard— to keep one’s whole attention and energy focused on a problem, systematically turning it over and over and pushing at everything that looks like a crack, despite the lack of outward signs of progress— is not a skill everybody has. Psychologists nowadays call it “grit,” and it’s impossible to do math without it. It’s easy to lose sight of the importance of work, because mathematical inspiration, when it finally does come, can feel effortless and instant. I remember the first theorem I ever proved; I was in college, working on my senior thesis, and I was completely stuck. One night I was at an editorial meeting of the campus literary magazine, drinking red wine and participating fitfully in the discussion of a somewhat boring short story, when all at once something turned over in my mind and I understood how to get past the block. No details, but it didn’t matter; there was no doubt in my mind that the thing was done.
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