4 ms·
Let's get the criticisms of the article out of the way: --It has terrible formatting and typos. --It puffs up something more important than it is. --It is mo
by abnry 7y ago
Let's get the criticisms of the article out of the way:
--It has terrible formatting and typos.
--It puffs up something more important than it is.
--It is more about pedagogy than a mathematical idea.
--It suggests the idea is original, when it almost certainly is not.
--It doesn't link to the original (and better source).
Okay, here is the good things about the approach:
--It is good to shift your thinking about mathematical derivations and proofs and think about them as code that runs on people's brains. You input a derivation into someone's brain and they return a boolean value (this is true, it makes sense, etc.). Pedagogy is trying to optimize the code for less powerful architectures. Just like when you are optimizing code tiny little details of instruction orders matter, the same with mathematical derivations.
--Fundamentally, algebraic manipulations are uncomfortable and nonintuitive for students. They feel like tricks. Going forwards from (x+a)^2=x^2+2ax+a^2 makes sense but going backwards as in the case of completing the square is hard. It's not the same case for x^2=a vs sqrt(x)=a. This is kind of a similar case to math students feeling confused by adding and subtracting the same quantity when doing calculus limits. For any trained mathematician, this is obvious, but it really feels like a trick at first. The nice thing about this approach is that it avoids this issue and gives you a good reason WHY the -b/2 term shows up. Additionally, it avoids the problem of substituting, which tends to bog students down (try teaching the chain rule someday).
Students should still understand completing the square but I don't think this is a bad way to introduce them to the quadratic formula. It highlights the symmetric of the roots (at least for real values), which makes sense if you plot a quadratic.
- infocom6502 7y agoI agree. This is nothing novel and a fairly standard way of solving the equation.
- jfengel 7y agoI have never understood "completing the square". I mean, I get why and how it works, but it's completely unintuitive to me, or why you'd do it that way. I think it is a historical hangover. It makes more sense viewed geometrically, and the technique is attributed to the same al-Khwarizmi for whom algorithms are named. But it's less intuitive as part of algebra, and we focus a lot more on algebra today than in the medieval times, when geometry ruled.
- edflsafoiewq 7y agoAlgebraically, my intuition is that the x^2 + bx part of x^2 + bx + c = 0 "looks" pretty close to the expansion of (x+b)^2. If you relabel b so this becomes x^2 + 2b, it's even clearer. So you can try to fiddle with the constant term to make the LHS exactly that square, and you get completing the square. This is similar to how you solve a first order linear DE, y' + f(x) y = g(x). The idea is that the LHS "looks" like the derivative of a product, (hy)' = h y' + h' y, so you fiddle with an integration factor to make the LHS exactly that derivative.
- abnry 7y agoIt's not intuitive for someone new to mathematics, but as someone with two math degrees, it is just the strategy of "do stuff to something until something about it looks similar to something you've seen before" that mathematicians use ALL the time. The additional part of this strategy is to dream optimistically about how you can make the thing you are dealing with "nice" after doing stuff to it. In the case of completing the square, you are hoping you can just straight up take a square root. That would be easy. Turns out if you try enough stuff, you can.
- bigred100 7y agoOnce you get to quadratic forms you can use it again.