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-t... [2]: http://j2kun.svbtle.com/things-mathematicians-know-proofs-ar... [3]: http://j2kun.svbtle.com/things-mathematicians-know-more-than... [4]: 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).

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.

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.

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.

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.

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.

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.

>it is really inefficient to derive things from scratch every time.

Well yeah, it would be really inefficient to derive the formula and then plug numbers into it, but you don't have to do that, you can just teach people to complete the square instead. It's pretty much as fast as using the formula (since it's the same operations), but it builds on your existing equation manipulation skills and so you understand every single step. And if you ever need to actually know the equation for whatever reason, you can derive it easily.

Isn't it the case that 99% of the population just needs to know it exists, and, in the very, very, very rare case they ever need it again, they would simply look it up?

I'd rather spend our children's time letting them know that these formulas exist, making sure they can perform them as needed, and then have them move onto something new. No need to drill / memorize trivial (as opposed to fundamental and foundation) stuff that is 2 seconds a way with google.

The worst kind of math education was that endless dribble associated with memorizing useless patterns that could easily be looked up in a book. At one point I had (painfully, oh my god, so painfully) memorized about 20 different patterns so I could chunk up integrals into products of u(x)v(x)only to spit them back on exam day, and never again look back on any kind of calculus. That was not a pleasant day week in high school. Would have been much better spent learning Geometry, Prob/Stats, Discrete Math, Linear Algebra, or any other host of mathematically oriented topics.

Kids forget. If you just introduce a concept, spend a day or even a week on it, then move on to a different concept, most kids (if not almost all) will forget it by the beginning of the next school year.

Personally, it think the time for manipulation of symbols has passed. Math should be only about concepts and hardware or software should do he manipulation. But then you run into systemic issues like "how do I test all students". The factory-styled learning environment needs to die.

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?

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.

>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?

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.

Just because the solutions are not guaranteed to have a closed form in radicals doesn't mean that my statement is false. In the field of real numbers any polynomial with a degree higher than two is reducible.

http://en.wikipedia.org/wiki/Irreducible_polynomial#Real_and...

I remember reading (with horror) the methods for factoring cubics and quartics and thinking, "Well, I guess that's what they did before they had Newton's method and computers." Ewwwww.

I think I've actually used the quadratic formula maybe twice outside of an academic math class in which the quiz question was designed to invoke the quadratic formula.

I'm a computer scientist, one of the more mathematical occupations.

I've never been in a life-or-death situation where I thought, "aha, thank God I memorized that formula in high school!". Also, I don't actually remember it anymore, other than "negative b plus or minus the square root of b squared minus four-a-c, all over two-a".

Well huh. I guess I do remember it, but apparently at a verbal-auditory level rather than visual-symbolic. As usual for me.

Still: I've never had occasion to use that recitation I just made.

1. I'm implemented the quadratic formula in software more than twice over the years, I think. In, you know, real programs I was paid money to write.

2. Have you ever been in a life-or-death situation where anything you learned in class in high school was useful to you? "Well, I would have died, but then I remembered that Julius Caesar's last three words were not actually 'Et tu, Brute!' according to Shakespeare..."

It's useful in "shoot a ball out of a cannon at 50 m/s at a 27' angle, how far does it go?" problems.

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.

"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.

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.

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?

As I said, the problem is that when a kid is having trouble, it's very difficult to figure out where he's going wrong if he isn't showing his work. To take a simple example, let's try factoring a quadratic. The kid doesn't show any work and just writes down "x = 1 and -5." He's wrong. Well, how did he get there? Did he make a careless mistake when factoring it? Did he try the quadratic formula but mess up a term? Is he just guessing? I don't know because he didn't show his work.

Meanwhile, if he shows that he's factoring the polynomial and writes a 5 where he should have written a 3, I can immediately tell that he knows what's going on but made a careless mistake. Alternatively, if he writes down a bunch of gibberish, it means that he doesn't know what's going on and needs someone to go over the concepts again.

It's like a compiler. Do you want a compiler that compiles really, really fast but just throws opaque error exceptions, or do you want a compiler that is slower but gives you detailed warnings and error messages? I'd rather take the latter. Maybe once I'm perfectly sure that my code works, I'll do it with the former.

> As I said, the problem is that when a kid is having trouble, it's very difficult to figure out where he's going wrong

The correct (though inefficient) approach is to keep trying until you see that he starts understanding. However, this is impossible when there are 25 other students in the classroom. Each might require a different manner of teaching to "get it", or learn at different speeds. And so if you're trying to crank out graduates on an assembly line this just won't cut it.

So instead of figuring out a solution where each student can get the education he deserves as a human being, we instead seek to change the student so that he can be programmed with the education that is possible in an assembly line system. This also explains the dearth of highly competent, highly respected teachers... you don't staff your factory with gifted artisans who could carve the pieces. You want someone who will push the button and have the product stamped out in 0.75 seconds.

If you calibrate everything perfectly, some number of students will get a highly optimal (for them) education where everything was timed perfectly, using the easiest-to-understand lessons. For everyone else, for the slow and learning disabled, for the quick and talented... it will be an awful experience. And, whether you call it luck or circumstance, neither of those groups will be educated well enough to be able to express their criticism easily.

> It's like a compiler. Do you want a compiler that compiles really, really fast but just throws opaque error exceptions,

But a compiler isn't a person, and a person isn't a compiler. I don't want to treat people as if they were machines... I especially don't want to treat children like they are machines, it's almost certainly even more damaging the earlier that happens to them.

I'm a programmer too, I do this for a living. I know all too well how easy it is to think of human circumstances and other people as if they were machines to be debugged, and it feels awful. Imagine what the 7 year old kid feels like in school when he's a bug to be solved on the teacher's trouble ticket system. Especially when he's probably marked "low priority, fix when time allows".

You're no longer talking about a system where learning is considered the primary goal. It may not even be a goal at all.

> If you calibrate everything perfectly, some number of students will get a highly optimal (for them) education where everything was timed perfectly, using the easiest-to-understand lessons. For everyone else, for the slow and learning disabled, for the quick and talented... it will be an awful experience. And, whether you call it luck or circumstance, neither of those groups will be educated well enough to be able to express their criticism easily.

This is a really good point - the public education system isn't an artisanal workshop; it's a large, industrialized factory where "raw materials" are turned into "product." Every grade is another step in the factory process. And while I guess it might be optimal given the very limited resources that we devote to education, it's heartless and doesn't work very well from an absolute standpoint.

Personally, I didn't get a lot of my education from school. Sure, I was there ten hours a day, but I mostly learned from my father and the homework that I did. I would get assignments, and my father was the one who really taught me whenever I ran into problems. I would then go back to school and pass tests.

Unfortunately, my situation was atypical and very lucky; I was blessed with a loving father who was fascinated with a large variety of topics and loved teaching. Most kids don't get a resource like that and get stuck with school as being the only avenue for learning. How can you reach them? I think the only answer is more money, which will go toward more teachers. Cut down the class size to ten kids per class, and you'll get a much more individualized curriculum. As long as you have 25 kids in the classroom, you're going to end up with the factory approach.

Lots of kids see the answer directly for the simple problems, and so don't see the point of showing their work.

The point of showing the work is to learn the mechanics of solving the problem. If you do not learn the mechanics of solving for simple problems, you will not be able to apply the mechanics to more complex problems where one can no longer intuitively see the answer.

Yep, I understand the motive behind the requirement. There's a lot of good that can come out of it; students are forced to learn the "steps" or "mechanics," teachers can see where things went south and give partial credit or offer assistance, and finally it is a great way to circumvent cheating.

I totally get it and it makes total sense except for when it doesn't. The problem with our education system is that it's fundamentally flawed. It is designed to work best with the typical student being taught from a skill set chosen for optimal widget-making. People learn/think differently and yet we cater more and more to the mantra of pump-and-dump where children with the ability to retain the most frivolous information wins.

It took until my junior year as a comp-sci student for me to figure out what I personally needed to learn the material. I absolutely had to understand the big picture before I could ever attempt to solve the problems. If I didn't and I relied entirely on memorization then I was destined to fail.

In order for me to understand the big picture, I had to ask questions, sometimes a lot of them. Abstract questions would annoy some professors and certainly other students. Engineering classes, like most college classes, are full of people brought up in a system where the slide-show-after-slide-show of formulas, facts, or bullet points was all they needed. My questions were irrelevant and an interruption to their note taking.

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.