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I agree mostly with your comment (check out my sibling one). >but it turns out to be impossible to demonstrate understanding, much less engage real-world probl

I agree mostly with your comment (check out my sibling one). >but it turns out to be impossible to demonstrate understanding, much less engage real-world problems, without knowing a lot of handy facts. That's why the most idealistic, anti-rote-learning teachers still want their kids to learn that sin^2(x) + cos^2(x) = 1. They can't demonstrate the real-world relevance of mathematics or help kids understand abstract concepts unless the kids have some basic problem-solving competence that doesn't involve spending hours searching through the book for fundamental facts. Might I suggest that they take the top-down approach instead of the bottom-up one? If you have a clearly defined real world problem (no matter how simplistic), it's easy to see what information is missing and needed, and so it's then clear why a certain proof or relation is required. Then it's more reasonable to delve into the theoretical and abstract, when it is required by the real world. Then you can make the real world problems harder to delve deeper into the theoreticals. You're more grounded in this way. When you take the opposite approach and teach sin^2(x) + cos^2(x) = 1 starting from lines, angles, triangle relations, even your best students say "so fucking what?". In fact, I'm fairly certain that the teachers I had who could keep the whole class' attention and who could teach even the weakest students consistently displayed the top-down approach. Edit: As an example, I first encountered trigonometry when I was building a rubber-powered trebuchet that I made for a middle school physics class. You could move a crank to lower or heighten the part the rubber was attached to, effectively changing the angle and thus the shooting distance. My dad was showing me how to calculate the angle of the elastic to the ground by knowing the length of the two sides. I instantly "got" why we needed trigonometry and could visualise how changes in the lengths of sides would affect the angles. It seems intuitive now but it wasn't at the time. Of course, I wouldn't solve problems until much later but I feel like a lot of my mates never made that same leap.

dxbydt 14y ago
Finding some connection between trigonometry and angles of rubber-powered trebuchets is incidental and quite frankly, this approach isn't going to take you very far. One simply can't find real world mappings to atleast 99% of what is properly called math. Even if you'd like to strictly operate within the confines of high school geometry , it is a study of points, lines, planes, angle relations, norms & distances motivated by axioms & theorems that follow by symbol pushing. For example, it so happens that if you push the right symbols, you'll eventually work out that the area of a triangle is related to its sides ( http://en.wikipedia.org/wiki/Herons_formula http://en.wikipedia.org/wiki/Herons_formula ). But this is a consequence of your definition of area and triangle and side and so forth. The relationship falls out of those definitions. Its not about constructing a real wooden triangle out of real sticks & then measuring its area and then the lengths of its sticks and saying - hey, I told you so! Those things ie. the real wood and the real sticks,...reality - is just a sideshow. Real world problems are at best a distraction. If you were to focus only on these sort of real-world mappings, you'd end up as an accountant - because that's were 1 would map to $1, 2to $2 and add would mean 1 + 2 = $3 and so forth. That's what corresponds to reality and the real world. That isn't math. Math is where you say 1 + 2 = 0 because the abelian group G3 with three elements {0,1,2} can satisfy the addition rule and have three distinct elements only if 1 + 2 was 0. For if 1 + 2 was 1, then that would mean 2 was equal to zero by cancelation laws. Similarly if 1+2 was 2 then 1 would equal 0. Since you do have 3 elements, it stands to reason that 1+2 must be 0, because it can't be anything else in that system. That's math.
- Dn_Ab 14y ago
  You sound like someone from the Konishi polity as described in Diaspora. I don't agree with you. The brain uses an associative learning algorithm and learning difficulty is proportional to how easily a concept can be integrated with what is already known. And it's an iterative process, so new knowledge can be built on old knowledge that was once itself abstract. So you can start with concrete problems and generalize from there. No one starts out learning about unitary matrices, hilbert spaces, and natural transformations. By the time they get there they can be taught in terms of more concrete things like rotations in 3 dimensions, euclidean spaces and parametric polymorphism (if they are programmers). That is why math is hard when you first start. It's a highly compressed subject where every word is more an index than it is a full concept. So you have to maintain a cascade or hierarchical graph of intuitions.
  - dxbydt 14y ago
    >you can start with concrete problems and generalize from there You could. But that isn't the only way. You should really read up on the way math is( was ?) taught in gottingen & eastern europe & romania & russia. Speaking of which, I learnt most of the machinery behind matrices in India in high school, with no reference to where it came from or what its applications were. I could compute the determinant of square matrices, compute adjoints, inverses, multiply two matrices....all of this without knowing wtf a matrix was or what its real life use might be. This is true of most high school students in India, simply because of the way the curriculum is structured. Its only when I took at a graduate level computer graphics course in my late 20s did I see that linear transforms were actually matrices, so you could do rotations and translations of vectors by matrix multiplication. So real life applications don't have to precede abstract concepts - you can make a hell a lot of progress by approaching stuff the other way around.
    - jessriedel 14y ago
      > I could compute the determinant of square matrices, I learned to do this too---and then promptly forgot---several times over until someone finally clued me in to the fact that the determinant is the scale factor of the linear transformation associated with the matrix. Then everything clicked. It's not just that I couldn't remember how to define/calculate the determinant, I also couldn't apply it usefully in my (proof-based) linear algebra class because I didn't have any intuition for it or any way to connect it to other concepts.
dkarl 14y ago
As an example, I first encountered trigonometry when I was building a rubber-powered trebuchet that I made for a middle school physics class.... I have to ask, do you realize how much of an outlier you are and how irrelevant your experience is to teaching normal kids mathematics? I mean... you immediately saw value in mathematically defining the relationship between the angle and the length. Teachers don't worry about kids like you (and me, and most of HN probably) because we'll be fine no matter how the class is taught. Wondering about how to get kids like you or me interested in math is like wondering how to get a cat interested in mice, or a Jersey Shore cast member interested in taking his shirt off. Teachers don't waste a second's thought on kids who will flourish regardless of the classroom environment. They worry about the marginal kids who might succeed if taught well but will flounder if taught poorly. A kid who really says "so fucking what?" to trigonometry is going to say "so fucking what?" to your trebuchet example, too. In the unlikely event he gives a damn about the length of the elastic rubber, he can just crank the attachment up and down and watch how the length changes, so who needs math? The "best students" have sat through enough math classes that they know the right default assumption is that whatever is taught is probably useful. They get that engineering and science (including social science) depend on math, and math builds on previous math, and the high school curriculum doesn't have room for stuff that never comes up again later (okay, with the exception of geometry class, but even that is mostly useful.) Heck, even the "pretty good" students understand that. Those are the kids who get excited about real-world applications of trig: the kids who already get it and never ask "so fucking what?" Plus there's another group of kids who will never seriously ask "so fucking what," the ones who care about grades and college applications. They already know how they're going to use math: they're going to use it to get good grades in math class. When students in trigonometry class really do say "so fucking what?" and "When are we ever going to use this?" they're always right. Because the kids who say that are the kids who really aren't going to use trig ever again in their lives, except maybe to scrape out a single math credit in college by taking a "College Algebra" course that's easier than the trig course they're taking in high school. The kids who say "so fucking what?" are the kids who are going to be business people, car mechanics, policemen, English teachers, coaches, corporate trainers, office workers, journalists, politicians, plumbers, soldiers, and grocery store managers. You know, the 98% of the planet that doesn't use trigonometry. There really isn't an answer you can give them. You can convince them that trig is essential for making video games, curing cancer, and any number of things they care deeply about, but what they're really thinking is, "I'm not going to be making video games. I'm going to be designing the ad campaign for the video game," or, "I'm not going to be curing cancer; I'm going to be mopping the floors at the hospital." Or, "I'm going to work for my dad for a few years at the dealership and then run for city council, and I'm going to pass a $5 million bond to hire an urban planning agency that probably employs some people who know trig." I don't know how you motivate those kids to learn trigonometry, but you can't do it by convincing them they will apply trig in the real world, unless you are a very gifted liar ;-)

dxbydt 14y ago

Finding some connection between trigonometry and angles of rubber-powered trebuchets is incidental and quite frankly, this approach isn't going to take you very far. One simply can't find real world mappings to atleast 99% of what is properly called math. Even if you'd like to strictly operate within the confines of high school geometry , it is a study of points, lines, planes, angle relations, norms & distances motivated by axioms & theorems that follow by symbol pushing. For example, it so happens that if you push the right symbols, you'll eventually work out that the area of a triangle is related to its sides ( http://en.wikipedia.org/wiki/Herons_formula http://en.wikipedia.org/wiki/Herons_formula ). But this is a consequence of your definition of area and triangle and side and so forth. The relationship falls out of those definitions. Its not about constructing a real wooden triangle out of real sticks & then measuring its area and then the lengths of its sticks and saying - hey, I told you so! Those things ie. the real wood and the real sticks,...reality - is just a sideshow. Real world problems are at best a distraction. If you were to focus only on these sort of real-world mappings, you'd end up as an accountant - because that's were 1 would map to $1, 2to $2 and add would mean 1 + 2 = $3 and so forth. That's what corresponds to reality and the real world. That isn't math. Math is where you say 1 + 2 = 0 because the abelian group G3 with three elements {0,1,2} can satisfy the addition rule and have three distinct elements only if 1 + 2 was 0. For if 1 + 2 was 1, then that would mean 2 was equal to zero by cancelation laws. Similarly if 1+2 was 2 then 1 would equal 0. Since you do have 3 elements, it stands to reason that 1+2 must be 0, because it can't be anything else in that system. That's math.

Dn_Ab 14y ago
You sound like someone from the Konishi polity as described in Diaspora. I don't agree with you. The brain uses an associative learning algorithm and learning difficulty is proportional to how easily a concept can be integrated with what is already known. And it's an iterative process, so new knowledge can be built on old knowledge that was once itself abstract. So you can start with concrete problems and generalize from there. No one starts out learning about unitary matrices, hilbert spaces, and natural transformations. By the time they get there they can be taught in terms of more concrete things like rotations in 3 dimensions, euclidean spaces and parametric polymorphism (if they are programmers). That is why math is hard when you first start. It's a highly compressed subject where every word is more an index than it is a full concept. So you have to maintain a cascade or hierarchical graph of intuitions.
- dxbydt 14y ago
  >you can start with concrete problems and generalize from there You could. But that isn't the only way. You should really read up on the way math is( was ?) taught in gottingen & eastern europe & romania & russia. Speaking of which, I learnt most of the machinery behind matrices in India in high school, with no reference to where it came from or what its applications were. I could compute the determinant of square matrices, compute adjoints, inverses, multiply two matrices....all of this without knowing wtf a matrix was or what its real life use might be. This is true of most high school students in India, simply because of the way the curriculum is structured. Its only when I took at a graduate level computer graphics course in my late 20s did I see that linear transforms were actually matrices, so you could do rotations and translations of vectors by matrix multiplication. So real life applications don't have to precede abstract concepts - you can make a hell a lot of progress by approaching stuff the other way around.
  - jessriedel 14y ago
    > I could compute the determinant of square matrices, I learned to do this too---and then promptly forgot---several times over until someone finally clued me in to the fact that the determinant is the scale factor of the linear transformation associated with the matrix. Then everything clicked. It's not just that I couldn't remember how to define/calculate the determinant, I also couldn't apply it usefully in my (proof-based) linear algebra class because I didn't have any intuition for it or any way to connect it to other concepts.

dkarl 14y ago

As an example, I first encountered trigonometry when I was building a rubber-powered trebuchet that I made for a middle school physics class.... I have to ask, do you realize how much of an outlier you are and how irrelevant your experience is to teaching normal kids mathematics? I mean... you immediately saw value in mathematically defining the relationship between the angle and the length. Teachers don't worry about kids like you (and me, and most of HN probably) because we'll be fine no matter how the class is taught. Wondering about how to get kids like you or me interested in math is like wondering how to get a cat interested in mice, or a Jersey Shore cast member interested in taking his shirt off. Teachers don't waste a second's thought on kids who will flourish regardless of the classroom environment. They worry about the marginal kids who might succeed if taught well but will flounder if taught poorly. A kid who really says "so fucking what?" to trigonometry is going to say "so fucking what?" to your trebuchet example, too. In the unlikely event he gives a damn about the length of the elastic rubber, he can just crank the attachment up and down and watch how the length changes, so who needs math? The "best students" have sat through enough math classes that they know the right default assumption is that whatever is taught is probably useful. They get that engineering and science (including social science) depend on math, and math builds on previous math, and the high school curriculum doesn't have room for stuff that never comes up again later (okay, with the exception of geometry class, but even that is mostly useful.) Heck, even the "pretty good" students understand that. Those are the kids who get excited about real-world applications of trig: the kids who already get it and never ask "so fucking what?" Plus there's another group of kids who will never seriously ask "so fucking what," the ones who care about grades and college applications. They already know how they're going to use math: they're going to use it to get good grades in math class. When students in trigonometry class really do say "so fucking what?" and "When are we ever going to use this?" they're always right. Because the kids who say that are the kids who really aren't going to use trig ever again in their lives, except maybe to scrape out a single math credit in college by taking a "College Algebra" course that's easier than the trig course they're taking in high school. The kids who say "so fucking what?" are the kids who are going to be business people, car mechanics, policemen, English teachers, coaches, corporate trainers, office workers, journalists, politicians, plumbers, soldiers, and grocery store managers. You know, the 98% of the planet that doesn't use trigonometry. There really isn't an answer you can give them. You can convince them that trig is essential for making video games, curing cancer, and any number of things they care deeply about, but what they're really thinking is, "I'm not going to be making video games. I'm going to be designing the ad campaign for the video game," or, "I'm not going to be curing cancer; I'm going to be mopping the floors at the hospital." Or, "I'm going to work for my dad for a few years at the dealership and then run for city council, and I'm going to pass a $5 million bond to hire an urban planning agency that probably employs some people who know trig." I don't know how you motivate those kids to learn trigonometry, but you can't do it by convincing them they will apply trig in the real world, unless you are a very gifted liar ;-)