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.
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?