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I'm a former math teacher, now a programmer. I think he leaves out a few considerations: 1) You need to be able to do basic calculations before you can do adv

I'm a former math teacher, now a programmer. I think he leaves out a few considerations: 1) You need to be able to do basic calculations before you can do advanced proofs. I taught a lot of high school seniors, and I had a ton of students who were smart enough to handle abstract concepts, but couldn't follow along when I showed them cool proofs because they got caught up on the basic calculations (because they hadn't learned them well in middle/high school). 2) Good high school teachers DO do a lot of pattern recognition/abstract reasoning. That's the entire idea behind a discovery lesson and constructivist teaching - having students learn formulas by discovering patterns and reasoning about them. 3) Again, as he points out, American high schools do do proofs in Geometry. He thinks they're really pedantic, but there are good reasons why 2-column proofs are so tedious. For one, students seeing proofs for the first time freak out, so giving them structure helps. For another, if the students write out every single step, it's easier to identify who really knows his/her stuff and who's BSing.

vkjv 12y ago
1. This. And many of the smarter students also gravitate towards these things. 2. It's easier for many students to grasp things when they are not abstract. 3. Yes, it still rattles my brain that Algebra teachers force students to memorize the quadratic formula. It's ridiculous. The method of completing the square is straight forward, more applicable in other situations, and can even derive that verbose formula. It only fosters the, "memorize every possible form of the question that could be on the exam" type of learning.
- Grue3 12y ago
  The formula for quadratic equation solutions is not that hard to memorise. It is useful to remember since it makes it possible to get the solutions instantly, whether numerically or algebraically. It also exposes discriminant of the equation, another useful concept (to instantly determine the number real solutions). Of course, the derivation of these formulas should also be taught, but it is really inefficient to derive things from scratch every time.
  - julius 12y ago
    It was unpleasent for me to memorise it. Could you elaborate on why it should be a goal to make students calculate solutions to a lot of quadratic equations? It seems to me that training to derive a lot of stuff would enable them to solve more kinds of problems, would it not?
    - Grue3 12y ago
      Quadratic equations are common. Very common. In physics, geometry, differential equations and so on. It is also the next step beyond linear equations. It is nice to be able to solve these quickly and to be able to tell their properties just by looking at the coefficients. The sum of roots, the product of roots. The axis of symmetry of the parabola. Better yet, any polynomial of higher degree can be theoretically factored into a product of linear and quadratic polynomials, so it basically always comes down to linears and quadratics.
      - eli_gottlieb 12y ago
        >Better yet, any polynomial of higher degree can be theoretically factored into a product of linear and quadratic polynomials, so it basically always comes down to linears and quadratics. Could someone have told us that in high school?
        deleted 12y ago
        [deleted]
        eximius 12y ago
        Yes, but this is actually false, so it's probably for the best. Anything of degree 5 or higher is not guaranteed to have solutions solved by radicals (that is, a solution that can be expressed as some rational number to some exponent). For example, x^5 - x + 1 = 0 cannot factored into linear and quadratic polynomials in this way.* The proof for the insolvability is actually quite elegant. Even so, factoring a polynomial from degree 3 or 4 into quadratics or linear terms is hard. The most general way I can think of is using the rational root theorem and plugging a few values in. * - You can factor using ultraradicals (yes, it's a thing), but that is far above highschoolers or undergrads, even.
- DigitalJack 12y ago
  I remember struggling greatly in algebra because it was like a boatload of recipes to remember. There was never much discussion on what these actions meant or why you do them. There is this awful commercial in the states for an online tutoring project where the student asks "how do I find the area if a triangle?" The response is "well, Cindy, the formula for the area of a triangle is 1/2 b*h, so you take half the base and multiply by the height and that's how you find the area of a triangle." Non of that is false, but all the poor girl in the commercial learned was yet another reasonless recipe.
  - XorNot 12y ago
    I have to disagree here. Recipe's are a very helpful fallback where you might be struggling with understanding the origins of material in a rigorous manner. You can inspect a triangle all you like, but at the end of the day it's much easier to simply remember the formula. This, I am finding, is the only way I'm managing to actually understand complex analysis - take the formulas for the results, and remember how to apply them. It's revealing to me that what looks complex gets very simple in that manner (and also that I still get tripped up by elements of basic integration). If I couldn't do this, then I'd be lost - and in a test simplification remembering how to work through the definition isn't possible (and isn't required thank god) because that's a path which leads to me spending 6 hours figuring out and trying to picture something in a way which makes sense.
dfan 12y ago
Totally agreed with #1. I think it's easy to forget how deeply you need to get calculation and symbol manipulation into your fingers so you don't get stuck later on.
- sliverstorm 12y ago
  It would be like trying to learn "software engineering" with a weak understanding of syntax and variable manipulation. You can do it, but you are building on top of a house of cards.
cafard 12y ago
There was something on HN years ago about the problems students encounter when they are missing a bit of the preparatory work for the next steps in math. Wish I could find it.
j2kun 12y ago
I have taught enough lectures to high school students (presenting "advanced proofs") and talked to enough geometry teachers who abandoned the two-column geometry proof to know that 1) is not true and 3) is not worth it. The problem is that people build up proofs like they're something to freak out about, or that the proofs that are presented are inherently mechanical because that's what students are taught. You can't expect someone learning to write proofs to be perfect any more than you can expect a first-time drawer to color within the lines. It's okay and should be embraced as an opportunity to reflect and improve. A proof is not complete just because you "got to the answer." It's complete when it's simple, elegant, and easy to explain to others. I can and have explained beautiful proofs without the need for mechanical proficiency to ten year olds and mathphobes alike. Here are a few examples: [1]: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-theory/ http://jeremykun.com/2011/06/26/teaching-mathematics-graph-t... [2]: http://j2kun.svbtle.com/things-mathematicians-know-proofs-are-beautiful http://j2kun.svbtle.com/things-mathematicians-know-proofs-ar... [3]: http://j2kun.svbtle.com/things-mathematicians-know-more-than-one-infinity http://j2kun.svbtle.com/things-mathematicians-know-more-than... [4]: http://jeremykun.com/2011/06/26/tiling-a-chessboard/ http://jeremykun.com/2011/06/26/tiling-a-chessboard/ The world is full of these cool problems and proofs. I could literally teach an entire course and do nothing but puzzles involving chessboards. That many teachers ignore these great topics is a problem, but it's certainly for a good reason (the myriad of other problems with high school education).
chanced 12y ago
"You have to show your work Chance" was the sentence that drove me to despise school. As a 6th grade visual spatial student in a "gifted" algebra class, I could see the answer as if I were reading english but struggled to show my work. I read/write slow and I have an incredibly hard time memorizing anything so I rebelled. Even after I got my act together, got my GED, and went to college I suffered through the various levels of Calculus because it was the same tune all over again. Classes like Linear Algebra were a lot harder for me to "see" but it was still faster & easier for me to take the time to visualize it.
- omegaham 12y ago
  My girlfriend's brother has this same problem - he definitely knows the material, but he gets really frustrated because it's so tedious to write out something when you can just write the answer down and go onto the next problem. When I went over it with him, I showed him several spots where he made careless mistakes - he added where he should have subtracted, he multiplied where he should have divided, he screwed up a decimal point, whatever. I told him, "It's easy to spot your errors now because these are easy problems. But when you get to harder math, it's going to be much harder to find out what you did wrong, and a teacher isn't going know whether you made a careless mistake or just don't know it at all. By showing your work, you show the teacher that you actually know it." Nowadays he understands the reason why he needs to show his work but still hates it. I'm hoping that he'll be like this only while the math is easy.
  - NoMoreNicksLeft 12y ago
    By showing the work, he's busy proving himself instead of learning. This demonstrates that, intentionally or not, public schools have become primarily credentialing institutions and not teaching institutions. I offer a thought exercise... If you could be given a magical amulet that let you teach students better, more quickly, and more permanently than ever before but at the expense of never being able to test them to see exactly what it was that they learned, or you could be given a magical apparatus that let you test them perfectly so that you knew exactly what it was that they learned and did not learn but gave no insights or help into how to teach them those things that they failed at, which would you choose? Which would your local school administrator choose? Which would your children's teachers choose? Which would the legislator writing education policy choose? Everything else is post hoc rationalization. Having decided what it is that we want public education to be, we need to have some sort of justification for it even if it doesn't make sense. Do you know what people who don't show work do when they move on to more difficult problems? They start scribbling it out on paper, without any prompting. The more difficult problems are interesting enough that they want to get them right, and when they notice that basic mistakes are interfering they strive to avoid those. Or, in some cases, they just don't bother. When you solve the Poincaire Conjecture (spelling? didn't want to cheat and look it up) no one gives a crap whether or not you "showed your work" because most of the other mathematicians can also "just see" the boring details, and are interested primarily in the truly insightful portion of the proof. I suspect that we're actually selecting for accountants and not math geniuses when we harp on "showing your work". How many Perelmans did we discourage and how many math stooges were praised last year in public schools?
nbouscal 12y ago
Point 1 is only true for a proper subset of mathematics. This subset tends to be the only mathematics that are taught in high school (or even in undergrad unless you're a math major), which I think is a huge part of the problem. I can't recall the last time I did a proof of something in abstract algebra, category theory, or algebraic topology that actually involved a calculation of any kind, so clearly a facility with basic calculations is unnecessary for those proofs. Instead what is required is a facility with understanding rigorous definitions and abstractions, which is extremely valuable and important, and of which the average high school mathematics education provides essentially none.

vkjv 12y ago

1. This. And many of the smarter students also gravitate towards these things. 2. It's easier for many students to grasp things when they are not abstract. 3. Yes, it still rattles my brain that Algebra teachers force students to memorize the quadratic formula. It's ridiculous. The method of completing the square is straight forward, more applicable in other situations, and can even derive that verbose formula. It only fosters the, "memorize every possible form of the question that could be on the exam" type of learning.

Grue3 12y ago
The formula for quadratic equation solutions is not that hard to memorise. It is useful to remember since it makes it possible to get the solutions instantly, whether numerically or algebraically. It also exposes discriminant of the equation, another useful concept (to instantly determine the number real solutions). Of course, the derivation of these formulas should also be taught, but it is really inefficient to derive things from scratch every time.
- julius 12y ago
  It was unpleasent for me to memorise it. Could you elaborate on why it should be a goal to make students calculate solutions to a lot of quadratic equations? It seems to me that training to derive a lot of stuff would enable them to solve more kinds of problems, would it not?
  - Grue3 12y ago
    Quadratic equations are common. Very common. In physics, geometry, differential equations and so on. It is also the next step beyond linear equations. It is nice to be able to solve these quickly and to be able to tell their properties just by looking at the coefficients. The sum of roots, the product of roots. The axis of symmetry of the parabola. Better yet, any polynomial of higher degree can be theoretically factored into a product of linear and quadratic polynomials, so it basically always comes down to linears and quadratics.
    - eli_gottlieb 12y ago
      >Better yet, any polynomial of higher degree can be theoretically factored into a product of linear and quadratic polynomials, so it basically always comes down to linears and quadratics. Could someone have told us that in high school?
      - deleted 12y ago
        [deleted]
      - eximius 12y ago
        Yes, but this is actually false, so it's probably for the best. Anything of degree 5 or higher is not guaranteed to have solutions solved by radicals (that is, a solution that can be expressed as some rational number to some exponent). For example, x^5 - x + 1 = 0 cannot factored into linear and quadratic polynomials in this way.* The proof for the insolvability is actually quite elegant. Even so, factoring a polynomial from degree 3 or 4 into quadratics or linear terms is hard. The most general way I can think of is using the rational root theorem and plugging a few values in. * - You can factor using ultraradicals (yes, it's a thing), but that is far above highschoolers or undergrads, even.
DigitalJack 12y ago
I remember struggling greatly in algebra because it was like a boatload of recipes to remember. There was never much discussion on what these actions meant or why you do them. There is this awful commercial in the states for an online tutoring project where the student asks "how do I find the area if a triangle?" The response is "well, Cindy, the formula for the area of a triangle is 1/2 b*h, so you take half the base and multiply by the height and that's how you find the area of a triangle." Non of that is false, but all the poor girl in the commercial learned was yet another reasonless recipe.
- XorNot 12y ago
  I have to disagree here. Recipe's are a very helpful fallback where you might be struggling with understanding the origins of material in a rigorous manner. You can inspect a triangle all you like, but at the end of the day it's much easier to simply remember the formula. This, I am finding, is the only way I'm managing to actually understand complex analysis - take the formulas for the results, and remember how to apply them. It's revealing to me that what looks complex gets very simple in that manner (and also that I still get tripped up by elements of basic integration). If I couldn't do this, then I'd be lost - and in a test simplification remembering how to work through the definition isn't possible (and isn't required thank god) because that's a path which leads to me spending 6 hours figuring out and trying to picture something in a way which makes sense.

dfan 12y ago

Totally agreed with #1. I think it's easy to forget how deeply you need to get calculation and symbol manipulation into your fingers so you don't get stuck later on.

sliverstorm 12y ago
It would be like trying to learn "software engineering" with a weak understanding of syntax and variable manipulation. You can do it, but you are building on top of a house of cards.

cafard 12y ago

There was something on HN years ago about the problems students encounter when they are missing a bit of the preparatory work for the next steps in math. Wish I could find it.

j2kun 12y ago

I have taught enough lectures to high school students (presenting "advanced proofs") and talked to enough geometry teachers who abandoned the two-column geometry proof to know that 1) is not true and 3) is not worth it. The problem is that people build up proofs like they're something to freak out about, or that the proofs that are presented are inherently mechanical because that's what students are taught. You can't expect someone learning to write proofs to be perfect any more than you can expect a first-time drawer to color within the lines. It's okay and should be embraced as an opportunity to reflect and improve. A proof is not complete just because you "got to the answer." It's complete when it's simple, elegant, and easy to explain to others. I can and have explained beautiful proofs without the need for mechanical proficiency to ten year olds and mathphobes alike. Here are a few examples: [1]: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-theory/ http://jeremykun.com/2011/06/26/teaching-mathematics-graph-t... [2]: http://j2kun.svbtle.com/things-mathematicians-know-proofs-are-beautiful http://j2kun.svbtle.com/things-mathematicians-know-proofs-ar... [3]: http://j2kun.svbtle.com/things-mathematicians-know-more-than-one-infinity http://j2kun.svbtle.com/things-mathematicians-know-more-than... [4]: http://jeremykun.com/2011/06/26/tiling-a-chessboard/ http://jeremykun.com/2011/06/26/tiling-a-chessboard/ The world is full of these cool problems and proofs. I could literally teach an entire course and do nothing but puzzles involving chessboards. That many teachers ignore these great topics is a problem, but it's certainly for a good reason (the myriad of other problems with high school education).

chanced 12y ago

"You have to show your work Chance" was the sentence that drove me to despise school. As a 6th grade visual spatial student in a "gifted" algebra class, I could see the answer as if I were reading english but struggled to show my work. I read/write slow and I have an incredibly hard time memorizing anything so I rebelled. Even after I got my act together, got my GED, and went to college I suffered through the various levels of Calculus because it was the same tune all over again. Classes like Linear Algebra were a lot harder for me to "see" but it was still faster & easier for me to take the time to visualize it.

omegaham 12y ago
My girlfriend's brother has this same problem - he definitely knows the material, but he gets really frustrated because it's so tedious to write out something when you can just write the answer down and go onto the next problem. When I went over it with him, I showed him several spots where he made careless mistakes - he added where he should have subtracted, he multiplied where he should have divided, he screwed up a decimal point, whatever. I told him, "It's easy to spot your errors now because these are easy problems. But when you get to harder math, it's going to be much harder to find out what you did wrong, and a teacher isn't going know whether you made a careless mistake or just don't know it at all. By showing your work, you show the teacher that you actually know it." Nowadays he understands the reason why he needs to show his work but still hates it. I'm hoping that he'll be like this only while the math is easy.
- NoMoreNicksLeft 12y ago
  By showing the work, he's busy proving himself instead of learning. This demonstrates that, intentionally or not, public schools have become primarily credentialing institutions and not teaching institutions. I offer a thought exercise... If you could be given a magical amulet that let you teach students better, more quickly, and more permanently than ever before but at the expense of never being able to test them to see exactly what it was that they learned, or you could be given a magical apparatus that let you test them perfectly so that you knew exactly what it was that they learned and did not learn but gave no insights or help into how to teach them those things that they failed at, which would you choose? Which would your local school administrator choose? Which would your children's teachers choose? Which would the legislator writing education policy choose? Everything else is post hoc rationalization. Having decided what it is that we want public education to be, we need to have some sort of justification for it even if it doesn't make sense. Do you know what people who don't show work do when they move on to more difficult problems? They start scribbling it out on paper, without any prompting. The more difficult problems are interesting enough that they want to get them right, and when they notice that basic mistakes are interfering they strive to avoid those. Or, in some cases, they just don't bother. When you solve the Poincaire Conjecture (spelling? didn't want to cheat and look it up) no one gives a crap whether or not you "showed your work" because most of the other mathematicians can also "just see" the boring details, and are interested primarily in the truly insightful portion of the proof. I suspect that we're actually selecting for accountants and not math geniuses when we harp on "showing your work". How many Perelmans did we discourage and how many math stooges were praised last year in public schools?

nbouscal 12y ago

Point 1 is only true for a proper subset of mathematics. This subset tends to be the only mathematics that are taught in high school (or even in undergrad unless you're a math major), which I think is a huge part of the problem. I can't recall the last time I did a proof of something in abstract algebra, category theory, or algebraic topology that actually involved a calculation of any kind, so clearly a facility with basic calculations is unnecessary for those proofs. Instead what is required is a facility with understanding rigorous definitions and abstractions, which is extremely valuable and important, and of which the average high school mathematics education provides essentially none.