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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

#61

Post by gibspaulding »

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 remember discovering this for myself when tutoring math in college. It's amazing how difficult it is to explain a lot of "basic" math without using "advanced" math. Trying to solve limits without L'Hospital's rule is excruciating.

Re: A new way to make quadratic equations easy

#62

Post by Anon84 »

pflats wrote: post #45
sykick wrote: post #41

Earlier quoted context omitted.

I’ve taught elementary algebra for many years at a community college. The method described in the article is truly, in my opinion, very nice. I will be using this method from now on in elementary algebra. Students in elementary algebra don’t know what parabolas are. They haven’t been taught to complete the square and don’t know how to work with radicals. They have just been taught what square root is and basic binomi…

> To me it is obvious that the method in the article is far superior than teaching completing the square. I disagree. I would need some convincing that "two numbers that multiply to C and sum to B must have an average of B/2, so they must be B/2 + z and B/2 - z, so (B/2 + z)(B/2 - z) = C" is by any means obviously superior to completing the square. Neither is immediately intuitive; both will require prompting and tea…

It's phrased in a funny way, but this: ""two numbers that multiply to C and sum to B must have an average of B/2, so they must be B/2 + z and B/2 - z" is pretty obvious.

If x+y=B then the average(x+y) = (x+y)/2 = B/2

B/2 is then the number in between x and y so you can represent x and y as B/2 + z and B/2 - z (where z is just half the distance between x and y, or |x-y|/2)

Re: A new way to make quadratic equations easy

#63

Post by abnry »

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.

Re: A new way to make quadratic equations easy

#64

Post by ajarmst »

I think there's an unacknowledged premise here that is fundamentally misguided: that the point of teaching math is to minimize the complexity of the formulas and recipes the students have to memorize, or that skill in finding the roots of a quadratic is so important that we must optimize the experience of learning it. I work with mathematics teachers and engineers and I know of no one who has willingly factored a quadratic by hand since we put our slide rules away.

The goal, fundamentally, is not the skill of factoring a quadratic. The goal is understanding of the relationships of numbers, operations, and domains that allow algebra to be a powerful set of tools for solving huge classes of problems. I _never_ teach the quadratic equation. I teach completing the square, because it's an illustration of a useful way of algebraically manipulating a relationship into a form (square of a linear binomial) that they recognize and can easily factor. I usually do a quick proof of the quadratic equation using completing the square, but generally as an illustration that if you properly understand the associated algebra, you don't need to memorize formulas and algorithms. The use of the field axioms to manipulate polynomials, and the goal of manipulating them to a tractable form, is what they need---not the quadratic equation. This 'new' technique (which appears to be a simple riff on the standard technique of looking for a pair of numbers whose sum is B and whose product is C that is taught as a matter of course in every middle school on the planet) is missing the actual goal of the lesson.

In particular, the reason I even teach completing the square is because it's a precursor of me bringing up the roots of $x^2 + 1$, which takes us in to an introduction to complex numbers, the fundamental theorem of algebra, and the discovery that the same old algebra over the Reals they learned in high school can now be used to do things like solve differential equations. Later, we introduce matrices and I can return to our old friends the algebraic field axioms to solve whole systems of equations. If they're lucky, they also get modular arithmetic and can be shown Galois fields and start using their understanding of algebra for problems in logic and set theory. All of that is easily within the reach of students in their first or second year of college, and I think if we bothered to try, we'd discover it's easily in the reach of secondary school students---or would be if we'd stop pretending that teaching them that "mathematics" is making change and pushing numbers from a word problem into the blanks of some generic formula they've memorized.

The task isn't teaching them to factor quadratics. The task is teaching them algebra.

Re: A new way to make quadratic equations easy

#65

Post by sykick »

6gvONxR4sf7o wrote: post #58
sykick wrote: post #55

Earlier quoted context omitted.

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 exp…

In elementary algebra students can do this for the most part:

Evaluate x^2 + 2x + 4 for x = 3.

Quite a few will struggle with:

Evaluate x^2 + 2x + 4 for x = –3

Almost all will struggle with:

Evaluate -x^2 - 2x + 4 for x = -3

I don't think they'd handle replacing x with z+2.

In elementary algebra they hate fractions. Many struggle with 4 – (–5). I like the approach in the article because of how it relates to factoring and difference of squares. This reinforces those concepts and shouldn't be too great of a leap at this stage. Also, there is a nice geometry behind the (b/2 + z)(b/2 - z) idea. The approach is nice precisely because it isn't a memorization approach. It's an approach that says, "Hey, let's analyze what factoring trinomials is all about and what the relationship between b, c, r, and s are.". It says, we know that r and s have to have the property that r+s is b and rs is c. We know this because of our analysis of multiplying binomials and our experience with factoring trinomials. We are using patterns and pattern recognition and from this we are constructing solutions to an equation that we can't solve by isolating the x like we did with linear equations. This to me, is true mathematics and it's nice to show students this exploration. To show how mathematicians think and approach problems. It's not a black box. It builds upon previous ideas and uses them to solve problems we weren't able to before.

Note that I'll be using this method in elementary algebra from now on but will not be using it to prove the quadratic formula. Indeed I will not even tell them what the quadratic formula is. I will save that for the next class.

Re: A new way to make quadratic equations easy

#67

Post by eyeundersand »

In Nepal, you are taught around the sixth grade that you can solve ax^2 + bx + c = 0 by splitting b into m and n such that mn = ac and m + n = b ( or m -n = b, if a and c have opposing signs). This "discovery" the author claims and some other corollaries were quite commonly known.

I find it absurd and, frankly, laughable that this is being heralded as something new!

Re: A new way to make quadratic equations easy

#68

Post by carapace »

sykick wrote: post #41
robinhouston wrote: post #6

This is bizarre. It’s no less a mathematical trick than completing the square, and it doesn’t seem to be any simpler to use. The example given is to find the roots of x² - 2x + 4 = 0. Completing the square gives (x - 1)² + 3 = 0, from which you can immediately see that the roots are 1 ± √3 i. If anything this seems easier than the method of the article. Am I missing something? Added : The argument seems to be that yo…

I’ve taught elementary algebra for many years at a community college. The method described in the article is truly, in my opinion, very nice. I will be using this method from now on in elementary algebra. Students in elementary algebra don’t know what parabolas are. They haven’t been taught to complete the square and don’t know how to work with radicals. They have just been taught what square root is and basic binomi…

Apologies in advance for the harshness of my tone, but I nearly spit up my coffee...

> Students in elementary algebra don’t know what parabolas are.

The mind boggles. Have you never shown them a sliced cone? ( https://en.wikipedia.org/wiki/Conic_section )

I mean, they know what parabolas are: they live on a planet, with gravity. Every ball, every jump: parabolic motion, yeah?

"Michael Jordan exerts his muscular power to enter a low-altitude earth orbit..." ~some Nike commercial in the 90's.

They know what parabolas are, you just have to connect the dots.

Re: A new way to make quadratic equations easy

#69

Post by carapace »

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…

*error prone

Re: A new way to make quadratic equations easy

#70

Post by [deleted] »

sykick wrote: post #65
6gvONxR4sf7o wrote: post #58

Earlier quoted context omitted.

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 exp…

In elementary algebra students can do this for the most part: Evaluate x^2 + 2x + 4 for x = 3. Quite a few will struggle with: Evaluate x^2 + 2x + 4 for x = –3 Almost all will struggle with: Evaluate -x^2 - 2x + 4 for x = -3 I don't think they'd handle replacing x with z+2. In elementary algebra they hate fractions. Many struggle with 4 – (–5). I like the approach in the article because of how it relates to factoring…

[deleted]
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