But I found a great illustration of the issue, and why it at least continues to produce that "cognitive dissonance" inside me about what I should be valuing in math class and in education in general that compelled me so much that I forced myself to write and publish this, even though I have other things I really need to get done.
So I'm grading this Honors Algebra 2 test. This is one of the high difficulty level problems. Traditional way of solving it (the way I taught it as well, and how the textbook examples in the books I use illustrate it) is to square both sides twice. With the algorithm on the test generator (which I should have worked out the problem to check the numbers, but was bad and didn't :/ ), this one actually turned into a particularly nasty, but still factorable quadratic at the end. In fact, most of these problems are doctored to have rational solutions. (Cue a debate about factoring and real-life situations not having nice solutions like this where the problems have been rigged in order to be factorable, but that's a story for another day.)
Anyways, here's a student with the problem worked out correctly by the standard way:
As I'm going through these, trying to get them done because quarter grades are due the beginning of this week, I come across this solution. First thing I notice is that the answer is wrong. OK, it's wrong, so I'm just going to quickly glance at it to figure out how I would rate it. I have to pause for a second, because I can't figure out exactly what's going on. Luckily, now I'm curious, and have to look at it more closely. And that's when it hits me. I'll let you look at it first to see if you can see it. She did her work on the right side of the page.
Here's a better view:
I re-work it this way myself, just to figure out what's going on. Then, I finally figure out that she took the square root at the end instead of squaring it. So in one way, it's wrong, and that's kind of a big conceptual gap. (Regardless, she didn't even bother to set the other factor equal to zero since she saw it was not going to result in a good solution.)
BUT THE IDEA IS F-ING BRILLIANT! How did she see the quadratic form that was already there? And, the numbers are so much easier to factor or put into the quadratic formula.
For whatever reason, I had never thought of approaching the problem this way. I immediately had to try it with a bunch of other problems. (Does it always work? Does it work if both radicals have two terms in it? etc...) All the other ones I tried, I could work it. It requires a really keen sense of structure and arranging and re-arranging the terms, but it works!
So even though there's a wrong answer, here's a student who has taken mathematical thinking and "Look for and make use of structure" to heart, has totally schooled her teacher, and has restored his faith in mathematics teaching when a week ago it was at a really low point.
But the answer's wrong...
So do I penalize her for not getting the right answer? (Maybe, also depends on what the rest of the equations she solved look like, too, I guess.). Do I tell her this is brilliant and embodies what "doing math" actually means? (Yes).
And do people see why I continually doubt and question everything about teaching mathematics, and teaching in general? If all I wanted was right answers, would this student have had the guts to try this? I do emphasize thinking and problem-solving over procedures and memorization, although the part I struggle with is if grading doesn't mesh really well with this.
I guess it's a matter of what do I want to worship? Is it right answers and grades? Or deep, creative, good mathematical thinking? I think I know what my heart says, but my head keeps getting in the way.
