...so...I realize that in lengthy thread like this one it's sometimes easy to read posts by several different people and mentally blur them together.
...That is what happens when I hastily try to write a post in the morning.
And finally I'm making the observation that the methods that are apparently being used to teach math very often don't result in this sort of understanding. Rather, they result in people who don't understand and therefore apparently need to go through all sorts of silly and unnecessary complicated stuff to do very simple things...because all they know to do is follow a generic recipe.
And I think that's not so good.
This is about the US math education system, right? Since YMMV, I can't really comment on it as I was not taught there. What I can say after having read Lockhart's paper, however, is that at some point you need to start teaching mathematical formalism. By all means, you can write proofs elegantly or inelegantly, all that matters is that the assumptions and arguments are true. However, it must be written in a way that prevents confusion. Natural languages have this problem due to semantics. That is why we have formalism. This doesn't
exclude the presence of natural languages. Sure enough, you can fill the gaps with whatever amount of fluff you want as long as the actual proof remains formal. A purely formal proof just happen to be concise, to the point and easily understood by one's peers and can stand the test of time regardless of changes in the natural languages.
When doing calculations, if you're the one who's only ever going to read it then by all means leave out obvious steps. I do that all the time. If you're showing it off to someone else, however, some of those steps may need to be written out to avoid confusion.
I mean if I'm solving the integral of cos(x)
2 at university for an assignment, this is an answer I can give:
∫cos(x)
2dx = ½∫[2cos(x)
2 - 1 + 1]dx = ½∫[cos(2x)+1]dx = (1/4)∫2cos(2x)dx+x/2 = x/2 + sin(2x)/4 + C = ½(x+sin(x)cos(x)) + C
I do not need to write out all the identities, they're implicitly understood at this level of education. At best I might point out that e.g. 2cos(x)
2 - 1 happens to be cos(2x). Sometimes I might even get away with leaving out the third step.
If I was writing a proof on the other hand, then the standards would obviously be higher especially if it's for proving a thesis or theorem.
Now I don't know much about the US education system, but I'm pretty sure High School is a secondary education level. At that point, the students are at least 14-15 years old, they've developed the basic mathematical knowledge. If they're not supposed to learn mathematical formalism here, then boy will they have a surprise waiting for them in tertiary education levels.