[thoth]

>I felt like it was nothing but a bag of tricks... had to substitute a "2" with "1/2 + 3/2".

Many techniques in math are "tricks" like this. Think of solving a quadratic by completing the square, or integrating by substitution or integrating by partial fractions, etc. You could arrive at these techniques on your own, but that is a lot of trial-and-error, deep understanding of theory, and applying it, which all takes a huge amount of time. Meanwhile, previous mathematicians figured this out and we get to benefit from their work. ;)

Maybe your instructor didn't present it well - plopping out the answer without a good enough explanation of the technique, why it works, etc.

[ethanbond]

The implication seems to be that students can't do the trial-and-error, understand then apply the theory. I agree with parent comment that this is awfully defeatist.

But even if it is futile to teach true understanding, why are we knowingly teaching computation in its place? What if we left the computation to the computers from the get-go? Could we then have enough time to teach true understanding?

I can't tell you how much time I spent trying to memorize my multiplication table – a 12x12 grid of numbers that for most students became arbitrary 3-number sequences. "3, 3, 9" is different from "3 sets of 3 is equal to 9." Most students learn the former in place of the latter.

[thoth]

>The implication seems to be that students can't do the trial-and-error, understand then apply the theory. I agree with parent comment that this is awfully defeatist.

All teaching is a balance between various factors, and perhaps the instructor didn't get right in this specific instance.

Ideally the instructor would teach the theory, allow some trial-and-error as students grapple with new information, and then step in with prodding towards how to do it - but fundamentally, there is a limit on how the instructor can let the class wander without needing to move on to cover the rest of the material (this wasn't a special topics class covering how to solve this one specific problem).

Hence, go through the information but provide the technique involved after suitable time passes. That's just the nature of teaching/learning when the students don't have infinite time to essentially re-invent the material they are trying to learn.

[marcosdumay]

Yes, trial and error isn't viable at all.

But Google is a better teacher of that kind of stuff than any professor can claim to be. If this is the kind of knowledge you pass on during education, it's basically worthless.