I don't think people who are good at math have some radically different way of thinking that ordinary people are incapable of. They're just using the same kind of common sense logic people use every day, for example when solving puzzles. However, what makes them good is that they are very determined to understand *why* something is true, instead of just accepting it on the authority of some textbook or some teacher.
If a bad math student is solving x+2=9, he'll say, "the equals sign is a bridge, and when the 2 walks across the bridge it has to change its sign." wtf?? Some people actually teach it this way. If a good math student sees this equation, he'll think, "okay, x+2 is literally the same number as 9, so if I subtract 2 from x+2 then I'll literally get the same number as if I subtract 2 from 9, so therefore x is 7." Or maybe he'll just notice that 7 satisfies the equation, therefore the answer is seven. (But wait---is that the only number that satisfies the equation?? Yes, if x is any bigger than 7, then x+2 is too big, and if x is any smaller than 7, then x+2 is too small.)
Someone who's good at math will often ask: how could I have figured this out? How do I know it's really true? Could it be some mistake that has never been caught?
Also, people who are good at math are always checking things with specific examples. Suppose you forget whether or not sqrt(a+b)=sqrt(a) + sqrt(b). (This is a common mistake.) If a good student is unsure if this is true or not, he'll just check it with some numbers. Is the square root of 4+4 equal to 2+2? No. Yet somehow students make this mistake again and again on exams. (I was a TA, I saw it a lot.)
Everyone has this kind of common sense ability. But most people aren't willing to spend the very large amounts of time that it takes to understand math well. It takes a lot of time. I can understand why they don't do it, because it's hard work and they have other things to do. But if you want to learn math, chances are very good you can do a good job of it if you just invest a lot of time.
Start hanging out at www.artofproblemsolving.com. That place will make you good at math.
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I don't think people who are good at math have some radically different way of thinking that ordinary people are incapable of. They're just using the same kind of common sense logic people use every day, for example when solving puzzles. However, what makes them good is that they are very determined to understand *why* something is true, instead of just accepting it on the authority of some textbook or some teacher. If a bad math student is solving x+2=9, he'll say, "the equals sign is a bridge, and when the 2 walks across the bridge it has to change its sign." wtf?? Some people actually teach it this way. If a good math student sees this equation, he'll think, "okay, x+2 is literally the same number as 9, so if I subtract 2 from x+2 then I'll literally get the same number as if I subtract 2 from 9, so therefore x is 7." Or maybe he'll just notice that 7 satisfies the equation, therefore the answer is seven. (But wait---is that the only number that satisfies the equation?? Yes, if x is any bigger than 7, then x+2 is too big, and if x is any smaller than 7, then x+2 is too small.) Someone who's good at math will often ask: how could I have figured this out? How do I know it's really true? Could it be some mistake that has never been caught? Also, people who are good at math are always checking things with specific examples. Suppose you forget whether or not sqrt(a+b)=sqrt(a) + sqrt(b). (This is a common mistake.) If a good student is unsure if this is true or not, he'll just check it with some numbers. Is the square root of 4+4 equal to 2+2? No. Yet somehow students make this mistake again and again on exams. (I was a TA, I saw it a lot.) Everyone has this kind of common sense ability. But most people aren't willing to spend the very large amounts of time that it takes to understand math well. It takes a lot of time. I can understand why they don't do it, because it's hard work and they have other things to do. But if you want to learn math, chances are very good you can do a good job of it if you just invest a lot of time. Start hanging out at www.artofproblemsolving.com. That place will make you good at math.