Hacker News Reader

Comment by jerf | Hacker News Reader

I believe the objection is that deriving the formula itself from completing the square is the challenge, not the direct use of completing the square as you show in your example. (Although I wonder where you learned that, because I was certainly never taught that so directly.) The problem with deriving the quadratic equation that way is that for most people, that is a lot of symbols to keep track of, and you have to not only do an unintuitive "completing the square" step but you have to unintuitively do it fully generically.

For a professional mathematician this is barely a warmup, of course, but for the average person in middle school or early high school, this is distinctly nontrivial work.

(I'm not taking a side here, just trying to describe what the issue is, since you asked. I'm ambivalent. Neither of my kids are quite this far yet, so I'm not quite here yet.)

View on HN

> The problem with deriving the quadratic equation that way is that for most people, that is a lot of symbols to keep track of, and you have to not only do an unintuitive "completing the square" step but you have to unintuitively do it fully generically.

"Unintuitive" depends entirely on your introduction to the topic. If you're already completing the square, using it to solve quadractic equations you cannot factor is not unintuitive.

As far as doing it fully generically, well, how else do you get a generic formula? When teaching this, we would do some completing the square to solve quadractics, and then tell our students:

"You know, this is annoying to have to complete the square EVERY TIME. What if we just decided to solve it the really hard way once, with A, B, and C in the equation instead of the numbers, and see what we get?"

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.)

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.

Reply on news.ycombinator.com