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I just found this comment, left by Vlorbik:
math is hard to understand.
but it's easier to understand than anything else.
I love that, though I'm not sure I know what V means by it. For me, this puts into words something I've been struggling with. Most non-math types, I'd put money on it, assume math is easy for people who are good at it. I was stunned when I read Carolyn's On having a math brain post, in which she said that everyone 'hits the wall' in math at some point. On having a math brain is one of my send-outs now, one of the essays & commentaries I keep on hand to give to people. I'll probably be sending it around our school district shortly. The reason it's important for non-math types to read On having a math brain, or something like it, is that (IMO) a huge number of Life Decisions are getting made on the basis of wrong data, namely that if math is hard for you, then you're not good at it and you should find a different career (or a different math track if you happen to be in 3rd grade). I keep mentioning Ed's experience as a freshman at Princeton. He had taken all the hardest math courses at his high school, and it sounds like they really were demanding and serious. When he got to Princeton freshman year, the guidance counselor looked over his grades & coursework, and told him he belonged in the advanced calculus course for engineering majors. So that's what he took. It just about killed him. He didn't understand what was going on, the course moved too fast, all the other students were too smart, etc. He went to the professor's office constantly, and the professor, a good teacher, spent a great deal of time explaining concepts directly, one-on-one. Ed ended up with a C. He concluded from this experience that he didn't have a math brain. Today, having hung out with math-types here at ktm for awhile, I'm horrified by that story. A C in advanced calculus for engineering students at Princeton. And this was back before grade inflation. A C in advanced calculus for engineering students at Princeton told him he was no good at math! Later on he drew the same conclusion about economics. He took economics, which he was keenly interested in, found it difficult (not as difficult as advanced calculus for engineering majors), and concluded he wasn't good at economics. This has nothing to do with work ethic. Ed is an insanely hard worker; he did four hours of homework every single day of his high school career. (That is something I can't even imagine.) It has to do with wrong assumptions. Ed assumed that a person who was 'good at math' found math easy. Ditto for economics. A person 'cut out' to be an economist is a person for whom economics courses come naturally. This wasn't as extreme as I'm making it sound. Ed had gone to school with a kid who sounds like a bona fide genius; I think this boy had graduated high school and earned his doctorate in physics by age 20 or something. (I'll check.) That kid was Ed's standard for: do I have a math brain?
late bloomers in math
* I think it was Andy Joy (or Anne Dwyer?) who left a comment saying that she'd loved math as a child, and liked doing multiplication worksheets. I think Wickelgren, in saying most children aren't motivated to study math, is talking about children not being motivated to push themselves through a math textbook at top speed. He was describing an alternative school where children were free to choose whatever they wanted to study. Apparently just about none of the children chose to study math intensively (or even non-intensively, he may have been saying).
-- CatherineJohnson - 16 Oct 2005 Back to: Main Page.
I just found this comment, left by Vlorbik:
math is hard to understand.
but it's easier to understand than anything else.
I love that, though I'm not sure I know what V means by it. For me, this puts into words something I've been struggling with. Most non-math types, I'd put money on it, assume math is easy for people who are good at it. I was stunned when I read Carolyn's On having a math brain post, in which she said that everyone 'hits the wall' in math at some point. On having a math brain is one of my send-outs now, one of the essays & commentaries I keep on hand to give to people. I'll probably be sending it around our school district shortly. The reason it's important for non-math types to read On having a math brain, or something like it, is that (IMO) a huge number of Life Decisions are getting made on the basis of wrong data, namely that if math is hard for you, then you're not good at it and you should find a different career (or a different math track if you happen to be in 3rd grade). I keep mentioning Ed's experience as a freshman at Princeton. He had taken all the hardest math courses at his high school, and it sounds like they really were demanding and serious. When he got to Princeton freshman year, the guidance counselor looked over his grades & coursework, and told him he belonged in the advanced calculus course for engineering majors. So that's what he took. It just about killed him. He didn't understand what was going on, the course moved too fast, all the other students were too smart, etc. He went to the professor's office constantly, and the professor, a good teacher, spent a great deal of time explaining concepts directly, one-on-one. Ed ended up with a C. He concluded from this experience that he didn't have a math brain. Today, having hung out with math-types here at ktm for awhile, I'm horrified by that story. A C in advanced calculus for engineering students at Princeton. And this was back before grade inflation. A C in advanced calculus for engineering students at Princeton told him he was no good at math! Later on he drew the same conclusion about economics. He took economics, which he was keenly interested in, found it difficult (not as difficult as advanced calculus for engineering majors), and concluded he wasn't good at economics. This has nothing to do with work ethic. Ed is an insanely hard worker; he did four hours of homework every single day of his high school career. (That is something I can't even imagine.) It has to do with wrong assumptions. Ed assumed that a person who was 'good at math' found math easy. Ditto for economics. A person 'cut out' to be an economist is a person for whom economics courses come naturally. This wasn't as extreme as I'm making it sound. Ed had gone to school with a kid who sounds like a bona fide genius; I think this boy had graduated high school and earned his doctorate in physics by age 20 or something. (I'll check.) That kid was Ed's standard for: do I have a math brain?
when math is hard for kids
I was on track to make the exact same mistake on Christopher's behalf until the moment I read Wayne Wickelgren on children and math. Obviously, math wasn't easy for Christopher. It didn't come naturally. He was a 3. When I read Wickelgren, who said that most children aren't particularly motivated to study math* no matter what their level of innate talent for the subject, the scales fell from my eyes. On that day I stopped using 'how easy is math for Christopher to learn?' as my standard of judgment. These days I judge by international standards. If an average child in Singapore studies and masters algebra in 8th grade, then Christopher should study and master algebra in 8th grade, too. Simple.back to Vlorbik
Vlorbik's koan (if that's the word) captures Ed's experience, for me. I've been trying to think why it is I would say 'math is hard' but 'history is easy.' History isn't easy; history is impossible. (So is writing.) But taking a history course -- learning history -- isn't hard; at least it wasn't hard for Ed. Math seems to be hard even at the learning stage, while the soft sciences & humanities become hard at the 'producing' stage, the point at which you yourself are going to try to write history or non-fiction books, or whatever it is you've chosen. That brings me back to V. For me it's true: math is hard to understand, but once I do understand it (leaving aside the fact that you never 'finish' understanding math) it's easier to understand than most other things. Reality is opaque. When I try to think about why things happen the way they happen, what the 'big picture' is, why constructivism has been winning for 100 years....it's hopeless. Why and how a fraction can mean and be four different things--that I have a shot at. Why and how we have school districts teaching lattice multiplication; forget it. I'm never going to figure that one out.late bloomers in math
* I think it was Andy Joy (or Anne Dwyer?) who left a comment saying that she'd loved math as a child, and liked doing multiplication worksheets. I think Wickelgren, in saying most children aren't motivated to study math, is talking about children not being motivated to push themselves through a math textbook at top speed. He was describing an alternative school where children were free to choose whatever they wanted to study. Apparently just about none of the children chose to study math intensively (or even non-intensively, he may have been saying).
-- CatherineJohnson - 16 Oct 2005 Back to: Main Page.