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The typical math education relies a lot on rote learning This has been known and lamented for a century, and it long ago transitioned from being a radical idea

by dkarl 14y ago

The typical math education relies a lot on rote learning This has been known and lamented for a century, and it long ago transitioned from being a radical idea to being conventional wisdom. It has been orthodoxy for decades (although recently there has been some reaction against the orthodoxy by a few experimental charter schools.) So, if current math education involves a lot of rote learning, it's not for lack of awareness or lack of attempts to fix the problem. In rebellion against facts and rote learning, teachers have elevated understanding above problem-solving as the highest goal of mathematics education, 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. And yes, they do want kids to get to the point where they can "forget" that sin^2(x) + cos^2(x) = 1 and regenerate that knowledge from visualizing the unit circle and applying the Pythagorean Theorem. That ideal has also been orthodoxy for decades. It just turns out that in practice, deeper understanding emerges from practice, and practice requires competence. Rote learning is a way of bootstrapping competence so competence can be turned into understanding. Believe me, progressive, "kinder and gentler" teachers were the norm in all the schools I've gone to, and they all tried every trick they knew to help us skip straight from ignorance to deep understanding with as little dry, repetitive work as possible, but they could not get around the need to learn many facts and techniques by heart. By the way, the route through calculus you describe reflects that you are basically the kind of kid that the schools don't worry about. I was the same kind of student. It might seem strange to us, but to most students, the history of a subject and the problems that originally motivated it are the very definition of dry and disconnected. Nobody gives a damn about the awesome intellectual journeys of some dead nerds except kids like you and me who will learn the subject regardless of how it's taught. Differentiation is introduced first precisely because it makes the subject less dry to most students, because it allows them to immediately apply what they learn to simple concrete problems involving rates.

makmanalp 14y ago
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.

by dkarl 14y ago

makmanalp 14y ago

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.