Is modern math education broken?

I was helping my daughter with her math homework the other day. It was multiplication of various 2-digit numbers (ie 80x52=?). The instruction were to select a 'method' and then solve using that 'method'. The 'method' was one of a half-dozen or so 'tricks', which the previous days homework had listed and demonstrated. I only remember two of the names ("friendly numbers" and "halving and doubling"), and I only remember how one of them works (using halving and doubling, 20x 12 = 2 x 1/2 x 20 x 12 = 2 x 10 x 12 = 2 x 120 =240).

My daughter only remembered "friendly numbers", and she had turned in the previous homework, thus had nowhere to go for the list of 'methods'. Many tears were shed as we struggled through these problems. One problem, after giving her a hint as to a way to make it easy to solve (changed the previously mentioned problem to 8x10x52), she argued that she couldn't do it because the didn't know the name of the method, and In frustration I told her to call it "the smart way" (I really hope I didn't just kill her straight A streak with that).

Having shared that story, I am going to try and not rant for a minute...

I remember a math teacher by the name of Scott Ziegler. Our first couple days of class he had us start building a notebook. On the first few pages we listed properties, by name and description.
(example:
 Commutative property of addition:    a+b = b+a
Distributive property: (a+b) x c = (a x c) + (b x c)


We then went on to build theorems in the book, which would allow for faster solving of problems, those theorem had to be demonstrated to work, with a valid proof.

Then, when we solved homework and test problems, we were expected to show our work, each step, with the property, postulate or theorem we applied to each step documented to the left of the line on which the step was performed.
 example:
80x52 =
= (8x10)x52       --> Substitution
=8x(10x52)       --> Associative property of multiplication
=8x520            --> substitution
=4160              --> solve



If we didn't explain a step with a property, or if we used a theorem for which we didn't have a valid proof, we'd lose points.

Based on what I have seen of my daughter's homework, and based on posters I have seen in at least one math classroom of one middle school, this mathematical rigor I experienced has been replaced with memorization of tricks. The math classroom I walked through a few weeks ago had a poster, which more or less listed the additive and multiplicative properties, but they were presented with no rigor, no precision. Instead they were presented as 'tricks'.

This concerns me. I have known people who relied on the 'trick' method. They were able to solve problems very quickly, so long as they were problems which fit a 'trick' that they new. But they were completely lost if the problem did not fit one of their problem solving recipes. they were largely incapable of solving novel problems.

I am concerned that this method of teaching (common core? That was written at the bottom of the homework pages) is doing our children a great disservice. They are being trained to be nothing more than code monkeys - able to quickly churn out solutions to problems which have already been solved, but unable to truly innovate, unable to really advance science, technology, or human knowledge in any meaningful way.

I realize I may be looking at this through my particular filter. So I am trying to keep an open mind as I reach out to those working in education. Can you please help me understand this? Is there something I am not understanding about the current teaching method? Or is education seriously broken?