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A new way to make quadratic equations easy

technologyreview.com

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Re: A new way to make quadratic equations easy

#51

Post by jerf »

pflats wrote: post #47
jerf wrote: post #44

Earlier quoted context omitted.

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 n…

> 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 i…

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.)

Re: A new way to make quadratic equations easy

#53

Post by fyp »

LatteLazy wrote: post #28
fyp wrote: post #21

People are criticizing this because it is still the same quadratic formula. But of course it is! Math is consistent. But representation matters. A good chunk of mathematics is just about rewriting the same mathematical fact in a different way. For example the equation of a line could be written with coefficients or in slope/intercept form or in polar coordinates or in homogeneous coordinates or etc etc. Here the clai…

Full disclosure: I critized the article in a comment below... Respectfully, that's not the reason people are critiquing the article. I fully agree mathematics is what works and many methods use identical underpinning logic, just expressed in different ways. I'm fine with that. But that doesn't mean all methods are equally good. This method is no quicker or easier or less error prown than the quadratic formula it "rep…

I think it's useful to think of this as refactoring code to make it more readable (and therefore more teachable).

If you're a memorizer, your code might as well be obfuscated code:

    def quadratic_formula(a, b, c):
        return [
            (-b - math.sqrt(b ** 2 - 4 * a * c)) / (2 * a),
            (-b + math.sqrt(b ** 2 - 4 * a * c)) / (2 * a)
        ]
If you understand the importance of the expression under the square root (i.e. the sign of the discriminant) you can rename one subexpression:

    def quadratic_formula(a, b, c):
        discriminant = b ** 2 - 4 * a * c
        return [
            (-b - math.sqrt(discriminant)) / (2 * a),
            (-b + math.sqrt(discriminant)) / (2 * a)
        ]
Of course you can refactor it further with pointless stuff like denom = 2a, but that doesn't add much semantic value. So the above is more or less the vocabulary we have about quadratic equations today.

Loh's contribution is a specific way of refactoring the code by first dividing by a:

    def quadratic_formula(a, b, c):
        b = b / a
        c = c / a
        discriminant = b ** 2 - 4 * c
        return [
            -b / 2 - math.sqrt(discriminant) / 2,
            -b / 2 + math.sqrt(discriminant) / 2
        ]
Which then unlocks the ability to talk about the subexpressions in relation to the roots (a la Vieta's formula):

    def quadratic_formula(a, b, c):
        b = b / a
        c = c / a
        sumOfRoots = -b
        productOfRoots = c
        averageRoot = sumOfRoots / 2

        # Want roots [averageRoot - delta, averageRoot + delta]
        # such that:
        #   productOfRoots == (averageRoot - delta) * (averageRoot + delta)
        #                  == averageRoot ** 2 - delta ** 2
        delta = math.sqrt(averageRoot ** 2 - productOfRoots)

        return [
            averageRoot - delta,
            averageRoot + delta
        ]
Your code is no longer using single-letter variable names!

Re: A new way to make quadratic equations easy

#54

Post by wrycoder »

czrnb wrote: post #38

There is an error in the (pixelated) example that is given int the MIT tech review article (z² = 3 instead of -3). I think the fact that the author of the news missed this, that he probably took a screenshot of the formulas rewritten in Word, and that he was compelled to write such a long article on such a simple topic speaks for his level on the topic. The fact that he only lists the formal article as a reference in…

And the third pixelated equation has an error - the final term on the left side should be C, not Cx^2.

I think the point of the arXiv article is this is a more straightforward /proof/ of the quadratic formula. That article has some interesting historical commentary that shows exactly where the author thinks his contribution fits - he is not naively coming up with something "new".

As far as computation of the roots goes, it is a slightly streamlined approach:

First, put the quadratic into canonical form by dividing by A, if necessary.

Then, take B/2 into a new variable, call it F. Get F^2.

The roots are then -F +- sqrt(F^2 - C)

Re: A new way to make quadratic equations easy

#55

Post by sykick »

6gvONxR4sf7o wrote: post #50
sykick wrote: post #33

Earlier quoted context omitted.

I teach mathematics at a community college. This includes teaching a lot of elementary algebra courses. These are pre-college level math courses. In elementary algebra we introduce solving quadratic equations by factoring. In the next course intermediate algebra we teach the quadratic formula. Students in elementary algebra are not equipped to understand change of coordinates. This is too hard of a concept at that st…

How aren't they equipped for it? Do you mean change of coordinates in its full generality? I'm talking about simple substitution of x' = x + k, connected with the intuition sketched below. They know how to perform simple substitution by then, and they can surely follow this logic: x 0---1---2---3--> x' 0---1---2---3---4--> x' = x + 1 ==> x = x' - 1 y = f(x) ==> y = f(x' - 1) If you can talk about Napa being an hour n…

Well for one thing the simple substitution you mention is a hard concept. They certainly will have a hard time with f(x-1) and introducing a new variable is a mental block at this stage. At my college we don’t introduce functions until the next course. I suggest your viewpoint is clouded by the fact that you know this stuff so well that you no longer remember what the pain points are for students learning it for the first time.

After learning to solve simple quadratic equations and then the quadratic formula we typically introduce variable substitutions to solve things like x^4 + 5x^2 - 6 = 0.

Re: A new way to make quadratic equations easy

#56

Post by pdonis »

dooglius wrote: post #24

I disagree that this proof is better pedagogically; it assumes the quadratic case of the Fundamental Theorem of Algebra (the correspondence between factorization and solutions), which at this stage would have to be taken on faith by students, whereas completing the square is fully justified.

> it assumes the quadratic case of the Fundamental Theorem of Algebra

No, it doesn't; it proves that case. The key step in the proof is the fact that, over the complex numbers, every number has a square root: from that it follows that the factorization used in the proof must always exist.

Re: A new way to make quadratic equations easy

#57

Post by sethammons »

LatteLazy wrote: post #28
fyp wrote: post #21

People are criticizing this because it is still the same quadratic formula. But of course it is! Math is consistent. But representation matters. A good chunk of mathematics is just about rewriting the same mathematical fact in a different way. For example the equation of a line could be written with coefficients or in slope/intercept form or in polar coordinates or in homogeneous coordinates or etc etc. Here the clai…

Full disclosure: I critized the article in a comment below... Respectfully, that's not the reason people are critiquing the article. I fully agree mathematics is what works and many methods use identical underpinning logic, just expressed in different ways. I'm fine with that. But that doesn't mean all methods are equally good. This method is no quicker or easier or less error prown than the quadratic formula it "rep…

> this is not new

While the mathematician claims to not find historical evidence of this, it is suuuuper similar to something we went over in high school in Calculus. It was related to finding the vertex of a parabola and noting the roots will be equally distant to both sides of the mid point. At the time, it was used as a "see? Neat. It all works out" type lesson.

This was around 2001.

Re: A new way to make quadratic equations easy

#58

Post by 6gvONxR4sf7o »

sykick wrote: post #55
6gvONxR4sf7o wrote: post #50

Earlier quoted context omitted.

How aren't they equipped for it? Do you mean change of coordinates in its full generality? I'm talking about simple substitution of x' = x + k, connected with the intuition sketched below. They know how to perform simple substitution by then, and they can surely follow this logic: x 0---1---2---3--> x' 0---1---2---3---4--> x' = x + 1 ==> x = x' - 1 y = f(x) ==> y = f(x' - 1) If you can talk about Napa being an hour n…

Well for one thing the simple substitution you mention is a hard concept. They certainly will have a hard time with f(x-1) and introducing a new variable is a mental block at this stage. At my college we don’t introduce functions until the next course. I suggest your viewpoint is clouded by the fact that you know this stuff so well that you no longer remember what the pain points are for students learning it for the…

I freely admit that my viewpoint is clouded by fluency, but I didn't come up with this today. This is how I've done it since I was still in school and I remember being annoyed by the opaque "just memorize this formula" approach from early on, where the concepts just seemed much clearer. I admit to not being the typical math student. But I think my approach was my competitive advantage, not something that makes my experience inapplicable.

To be clear, you're saying that if you tell a student

    y = x^2 + b*x + c
and you tell them

    x = z + 2
they aren't yet equipped to learn to combine those into

    y = (z+2)^2 + b*(z+2) + c.
Is that right? Why do you teach quadratic equations at this stage? I'd consider those to be much more advanced than simple "replace x with (z+2) everywhere you see it" plus some "alice's house to bob's house to carol's house" problems along a single axis for the concepts.

It seems like you're saying they're taught quadratic equations before they're equipped to poke around with them, which seems to be setting up for the black box/memorize-the-formula version of math.

Re: A new way to make quadratic equations easy

#60

Post by gibspaulding »

LatteLazy wrote: post #23

I must be missing something here... The "standard" quadratic formula at the top of the article is just as quick and painless to solve the equation he uses as an example. And its easier for many other versions (basically any time B/A or C/A are not integers). Plus, this isn't new: I was taught to do exactly this IF it simplified the whole equation. That was in 2000 in London in a pretty standard secondary school. (Als…

The focus is not on getting students to be able to solve quadratics (if it were, then the quadratic formula is a great choice!), but rather to understand how solving a quadratic works. That is, how to derive the quadratic formula for themselves. Generally this is done by completing the square. This is another method which some students may find more intuitive.
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