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A new way to make quadratic equations easy (2019)

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Re: A new way to make quadratic equations easy (2019)

#61

Post by tirpen »

crispyambulance wrote: post #54
iandanforth wrote: post #36

The author should really remove the words 'simple', 'easy', and 'intuitive' from this article. They clearly know nothing of why most people detest math. If you're already familiar with the quadratic equation and you care about math and even enjoy it you might find this new equation interesting, useful, or intuitive, but I guarantee it is still a garbled mass of un-intuitive numbers and symbols to "[m]any former algeb…

The key difference between this method and the traditional "completing the squares" technique is that there's no guessing involved. Students get frustrated with trying out different numbers to get the right sum and product. It's easy for them to make a mistake or think they made a mistake. This method is more mechanical and also looks easier than just chugging on the quadratic formula. Is it dramatically better than…

There is no guessing in the "completing the square" method either.

Re: A new way to make quadratic equations easy (2019)

#62

Post by fifilura »

jimhefferon wrote: post #13
finite_jest wrote: post #7

I don't think this is new at all. This looks like just a standard proof of the quadratic formula. I believe most people probably have seen a variation of this in high school. Depending on your preferences, this might be a slightly better or worse presentation than what you have seen before. The fact that they couldn't (or haven't) publish it in a journal also supports this. [1]. (The ArXiv pre-print [2] is dated Dece…

I think there is a different standard way in Europe, compared with the US. Dunno about other places.

I am from Europe (Sweden) an I learned it the way described in the article, so when I read it I had to read it twice to try to figure out what I was missing.

But I had a very inspiring (or stubborn ymmv) maths teacher. He insisted that we learned sine cosine by drawing our own unit circle and measure from with a ruler. Calculators were forbidden.

Re: A new way to make quadratic equations easy (2019)

#63

Post by gizmo686 »

svat wrote: post #49
finite_jest wrote: post #17

Earlier quoted context omitted.

Surprisingly, the article seems to be written by a smart mathematician, not a crackpot or a social scientist. [1] This kinda reminds me of that medical researcher who rediscovered trapezoid rule in 1994, and called it "Tai's Model". [2] To be fair, this is slightly better as it is supposed to be pedagogical, but it is still quite dishonest to pretend that it is new. [1]: https://en.wikipedia.org/wiki/Po-Shen_Loh [2]:…

If this is indeed not new, one should be able to find a published source anywhere in the world's literature from before 2019 that does it this way, correct? The author mentions in his blog post / paper that he has looked for this exact method in many sources and not found it anywhere: can someone find it? See section 3.3 "Brief historical context" of the paper: https://arxiv.org/abs/1910.06709v2 — he also mentions on…

The problem is, with simple ideas like this, they typically don't get published. The only place they would be published is in introductory textbooks, or books/papers about pedagogy; and both of those are relativly rare.

In contrast, the amount of people who would be in a posistion to discover this method is huge. Imagine looking through every algebra test and homework math students have submitted that involves solving quadratic equations. Do you really believe that not a single student stumbled into this method?

The existance of this method as a pedalogical tool is a bit more interedting. But even there, most pedegogical knowledge exists only in the minds of experienced practitioners. Some of that knowledge gets written down in their lesson plans. Some gets shared with other teachers nearby. Some gets passed on by osmosis to students who will become teachers.

A tiny minority of pedagogical knowledge actually finds its way to being published.

Re: A new way to make quadratic equations easy (2019)

#64

Post by ogogmad »

HarryHirsch wrote: post #59
ogogmad wrote: post #40

I briefly looked at the author's paper. I find the account of the traditional method there questionable. He suggests it starts by going from: x^2 + bx + c to x^2 + bx + c + b^2/4 - b^2/4 This step is freaking weird, I've never seen it before, and I don't think this way. The process I follow is more like I observe that for general k: (x + k)^2 = x^2 + 2kx + k^2 so I observe that for k = b/2 I get (x + b/2)^2 = x^2 + b…

We did in school, but the original equation x^2 + bx + c was rewritten as x^2 + bx + (b/2)^2 - (b/2)^2 + c, and now you see the binomial. Our teacher said it felt like pulling a rabbit out of the hat.

Where did you go to school? I'm from the UK.

> Our teacher said it felt like pulling a rabbit out of the hat.

This is something you do when you prove something. It's the shortest way but it's also totally lacking in motivation. It's a Deus Ex Machina, which is a terrible way of explaining mathematics to people. When I was taught "completing the square", it was presented as another method for solving a quadratic equation, and not just a proof of the quadratic formula. If people in the US are taught this way, that's not good for US-ians. And this whole discussion is weirdly US-centric.

Re: A new way to make quadratic equations easy (2019)

#65

Post by ogogmad »

ogogmad wrote: post #40

I briefly looked at the author's paper. I find the account of the traditional method there questionable. He suggests it starts by going from: x^2 + bx + c to x^2 + bx + c + b^2/4 - b^2/4 This step is freaking weird, I've never seen it before, and I don't think this way. The process I follow is more like I observe that for general k: (x + k)^2 = x^2 + 2kx + k^2 so I observe that for k = b/2 I get (x + b/2)^2 = x^2 + b…

For clarity: The geometric view of "completing the square" by splitting a parabola in two is not really necessary here. In my view, the algebra is well-motivated on its own.

Re: A new way to make quadratic equations easy (2019)

#66

Post by ogogmad »

BeetleB wrote: post #60
ogogmad wrote: post #40

I briefly looked at the author's paper. I find the account of the traditional method there questionable. He suggests it starts by going from: x^2 + bx + c to x^2 + bx + c + b^2/4 - b^2/4 This step is freaking weird, I've never seen it before, and I don't think this way. The process I follow is more like I observe that for general k: (x + k)^2 = x^2 + 2kx + k^2 so I observe that for k = b/2 I get (x + b/2)^2 = x^2 + b…

It's only as bad as memorizing the usual quadratic formula. While most people are taught the derivation of the quadratic formula, few remember it, opting instead to memorize it. Likewise, you've just derived the b^2/4 aspect. Most people don't want to rederive it every time they solve it, so they just memorize that they should add and subtract b^2/4 and complete the square. BTW, if the notion of adding and subtractin…

> BTW, if the notion of adding and subtracting the same amount bothers you: Yes, this is done all the time.

I'm a mathematician. I'm not ignorant; I have taste.

I've seen a similar technique used two prove the product rule in analysis. I always find ways of avoiding such tricks because I see them as Deus Ex Machinas. I don't think of mathematics as an exercise in ill-motivated tricks.

Re: A new way to make quadratic equations easy (2019)

#67

Post by jonsen »

sedeki wrote: post #58
neximo64 wrote: post #57

Earlier quoted context omitted.

I assume you didn't actually read the article. It is not the QP formula.

I read the article. It is called "pq formeln" _in Swedish_. The p and q refers to the traditional names of the coefficients. I cannot find any link in English referring to it, but it is part of the standard curriculum. One example of a website explaining it here below (in Swedish), but still, you can clearly see that it's the same formula OP has given: https://eddler.se/lektioner/pq-formeln/

Right, but this just describes how to use this variant of the more standard formula. What is interesting is how do you derive the formula.

Re: A new way to make quadratic equations easy (2019)

#68

Post by HarryHirsch »

ogogmad wrote: post #64
HarryHirsch wrote: post #59

Earlier quoted context omitted.

We did in school, but the original equation x^2 + bx + c was rewritten as x^2 + bx + (b/2)^2 - (b/2)^2 + c, and now you see the binomial. Our teacher said it felt like pulling a rabbit out of the hat.

Where did you go to school? I'm from the UK. > Our teacher said it felt like pulling a rabbit out of the hat. This is something you do when you prove something. It's the shortest way but it's also totally lacking in motivation. It's a Deus Ex Machina, which is a terrible way of explaining mathematics to people. When I was taught "completing the square", it was presented as another method for solving a quadratic equat…

This was in Germany, in 9th grade. Should have added that this was the proof of the quadratic formula.

Re: A new way to make quadratic equations easy (2019)

#69

Post by jonsen »

dataflow wrote: post #50

If you have to write "now here comes the clever bit", doesn't it kind of disqualify what you're talking about as "easy"?

It depends on what "easy" actually refers to. When we say it's easier to program in language X than language Y, we don't speak about the cleverness behind the X compiler. The author thinks that his way to "code" a solution to the quadratic equation is easier. He proposes an alternative "language" for teaching this.

Re: A new way to make quadratic equations easy (2019)

#70

Post by gordaco »

"An interesting question is why nobody has stumbled across and widely shared this method before"

Hmmmm... I "discovered" this method when I was like 14, but I didn't think it was a big deal, since it was equivalent to the standard formula. I'm very sure that I'm not among the first million people who "discovered" this on their own. Also, while this allows some quick calculations in your head (the example equation, x^2-2x+4=0, is certainly faster to do with this method than with the standard formula), for more complicated equations it loses it appeal very quickly, especially if the main coefficient is greater than one (think of something like 3x^2+2x+1=0, which is still not very complicated, and you will realise the usefulness of the standard formula).

As someone who has loved math since forever and who has discovered a million of "neat tricks" like this one (mostly useless, though), I've sometimes wondered whether some of them could actually be useful. Over the years it has become clear that all the "neat tricks" that are actually neat and useful are indeed widely shared everywhere, while the not so neat remain obscure despite being relatively easy to be discovered, because they don't have much application. And people like me who are willing to spend the time rediscover them again and again, only to discard them because they aren't a big deal.

An example of an actually good trick: I was so proud of my novel algorithm to find the gcd of two numbers when I was in high school... then, in my first year at the university, I learned that it was called "Euclid's algorithm" and, as the name implies, it was pretty old :) .

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