[jerf]

I think you're speaking from the perspective of someone very casually comfortable with symbol manipulation. This does not describe the average middle school student. When I said "unintuitive", I was speaking from the perspective of an average middle school student, for whom this is all either at the edge of the ability, or, often, a bit past it, and for a non-trivial number of them, way past it.

In high school, I was a tutor for a non-accelerated, non-honors class that was about at this level in high school. After years of being in the accelerated course, it was a bit of an eye-opening experience. There's a lot of people who are just passed through this stuff with a C-, and I'm not even sure that's wrong, because there's a lot of people who just aren't ever going to get to the point where they can fluidly derive any of these equations. What you, and probably a great deal of the HN commetariat experience as "average" is actually way above average.

(And the students I was tutoring for, in the parlance of the day, would still mostly be considered "privileged". I would still not be calibrated for the mathematical skill of the truly disadvantaged.)

[pflats]

I teach mathematics and write math curriculum professionally, and have literally taught the lesson I describe above for the better part of a decade. I say "we" in that post because in more than half of those classes, I did not teach it alone, but in a co-taught inclusion class for special education and general education students together. None of the classes where I used this lesson was an "honors"/"accelerated"/"pre-AP"/etc. class.

Every single one of those students was able to derive the quadratic equation by completing the square. It was not easy for some of them, but every single one did it.

Your parenthetical also implies to me that you think that the "truly disadvantaged" have less "mathematical skill". I would encourage you to reflect on that.