A new way to make quadratic equations easy (2019)
technologyreview.com
technologyreview.com
If children were exposed to this new thing, even more of them would be traumatized and develop a lifelong loathing for quadratic equations.
The ideal method, IMO, is to complete the square AND provide a geometric interpretation of the expression that results.
+1 for this. This is the approach I would recommend too, see Figure 2.2 for the picture behind the completing the square trick: https://minireference.com/static/excerpts/noBSmath_v5_previe...
"Imagine if, instead of wasting time on phonics, we inspired white upper middle class students like myself by giving them real literature!"
"Imagine if they inspired students with the kind of problems I think I would have found interesting!"
"Imagine if they taught to my learning style" (learning styles are mostly bunk, but everyone thinks they would have done better if only the teacher catered to them better).
"Imagine if a class of 20+ kids who are forced to be there was managed the way a team of 5 adult professionals was managed!"
It's human nature for learning to take place naturally and effortlessly when there is a felt purpose and relevance. For all humans.
The problem with this in schools, is that you can't actually generate this sense of purpose (which is a feeling, emotional) for arbitrary subject-student pairings. What do when it fails?
The strategy is, they TRY to generate the sense of purpose (maybe not try that hard, after years of failure and burnout), but fall back to the emotion of fear as primary coordinator/motivator for everything.
They make students afraid to look stupid in the short term, and afraid of an abstract "bad future" ("flipping burgers") in the long term. They introduce the system of grades for 3 whole years of middle school, when the grades serve no purpose (they're not used by colleges) but to generate the fear-based emotional economy of the school.
Fear is the primary emotional mechanism of the school. For every kind of person, this is worse educationally than felt purposeful engagement.
I agree that depending on the implementation, a subject can feel easy, but let’s not pretend that there are no difficulties in learning and teaching these concepts, and that it continues to feel “easy” later down the road.
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 well-taught and well-practiced traditional methods? Probably not, but I like it. It does seem slightly better, but then I'll never have to crank through a worksheet of these.
like this: https://www.youtube.com/watch?v=Z5MnP9da4EM&t=371s
It does involve writing down and checking combinations. It's roughly how I was taught back in the day and I think it's very popular even now.
So. The standard way to complete the square uses a trick to rearrange the equation so that one can use the formula (a+b)^2 = a^2 + 2ab + b^2.
This uses a different trick to rearrange the equation to make use of (a+b)(a-b) = a^2 - b^2.
Frankly, I don’t see how this is any easier. If anything, I think it’s harder to do from memory because you need to introduce a new variable in a very specific way to make the simplification work. The standard way feels a lot more systematic.
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 December 16, 2019)
[1]: https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=%22A...
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.
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]: https://fliptomato.wordpress.com/2007/03/19/medical-research... (It's 459 citations now according to Google Scholar, btw :-))
See section 3.3 "Brief historical context" of the paper: https://arxiv.org/abs/1910.06709v2 — he also mentions on his website of many attempts to find this exact derivation somewhere earlier: https://www.poshenloh.com/quadratic/
> In summary, the author has not yet found a previously-existing book or paper which states the same pedagogical method as this present article and precisely justifies the steps, but there exist independent references that contain the key ideas and can be adapted to achieve this. That said, it is entirely possible that the method in this present article was previously observed by people who did not share their findings.
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.
When I saw the title I was expecting some kind of geometric approach which was worrying to me because I think people are even worse at understanding geometric proofs of anything like this than they are at understanding algebraic proofs. Indeed, completing the square is a geometric technique originally and that is, I think, why it has that name (which makes no sense to the high school students who are taught it).
I’m also struggling to see the innovation in the article, or rather why it is any better, but I think a lot of my problem is that I’m basically fine at all the abstract manipulation of symbols required for school algebra and was when learning about completing the square yet I think it is one of the fundamentals that people really struggle with. When you look at the derivation of the quadratic formula, you need relatively complex expressions and I think only the strongest high school students at algebra (at least from my school) will be able to cope with the derivation. But I think this new method doesn’t help with what I think are the most common confusions: not understanding what a variable is (or variable notation) and not understanding what a function is (or function notation; though I think in this case it maybe doesn’t help that the function is implicit). I feel like I’m just a bad person to judge this method (but I think most people capable of talking about it would be poor judges). I’d be interested in the results of an experiment that tried different methods of teaching but the problem with such experiments is that the teacher in the experiment is likely to understand the method much better than a random teacher outside of the experiment.
x^2 + bx + c
x^2 + bx + c + b^2/4 - b^2/4
(x + k)^2 = x^2 + 2kx + k^2
(x + b/2)^2 = x^2 + bx + b^2/4
(x + b/2)^2 - b^2/4 + c = x^2 + bx + c
Question: Does anyone seriously learn to solve quadratics by rewriting x^2 + bx + c to x^2 + bx + c + b^2/4 - b^2/4. Because that seems like a bad way of learning it.
> 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.
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.
Just to expand on this thought... in logic, "cut elimination" is a desirable property to have in a logical system. The "cut" rule states that, if you've proved "A -> B" (A implies B) and "B -> C", then you're licensed to deduce "A -> C". But in the statement "A -> C", B doesn't ever have to appear on its own, so if you're working backward, the "B" that splits the problem into two pieces can appear deeply unmotivated and appear out of nowhere.
A deductive system with the "cut elimination" property means that, whenever you have a proof that uses "cut", there's always a way to transform the proof into one that doesn't use "cut". Put less precisely, there's always a proof that doesn't use any unmotivated variables. (In computer science terms, "cut" is like implementing a function in terms of an existing function; you can always inline the original and then simplify the resulting code.)
"Cut elimination" is desirable because it makes automatic reasoning tractable. If you want to prove "A -> C", rather than trying to discover some arbitrary "B" that produces more tractable goals "A -> B" and "B -> C", there are (usually) only finitely many deductive steps that could terminate in "A -> C", and you can simply try each one.
Although humans aren't machines, creativity is still difficult in fields we're not accustomed to. While adding and subtracting a term formally doesn't change the result, there are infinitely many choices of term that produce this result. Why that one? It's entirely unmotivated; it clearly works, but there's no obvious way you could have plucked that particular term out of thin air yourself.
> The process I follow is more like I observe that for general k:
Your derivation is awesome! Rather than deducing forward from the raw materials (the given quadratic), you deduce backward from the goal. This lets you leverage known information to pick values that get you visibly closer to the starting point, term by term.
In the classic case, the goal here is to "complete the square", which is still of the unmotivated "cut" flavor -- it's not clear to the unskilled eye why that ought to be helpful.
I don't think the article's approach is any better motivated in that respect; we still have to introduce the idea that all quadratics can be factored into two first-order terms. This is not a trivial theorem, and it assumes access to the entire complex field (otherwise it's simply untrue in general).
But I think both approaches are valuable, as reliant on some "unmotivated" magic as they are. They all lend some insight into what's going on. Minimizing those moments is definitely beneficial though.
That explains everything :-)
You probably don't solve quadratic equations anywhere near as much as an engineer/physicist would, and so prefer less memorization and more "being able to derive when I need it". The latter is just too time consuming to do it every time.
I'm guessing you don't recall off the top of your head what the integral of 1/sqrt(a^2-x^2) or of 1/(a^2+x^2) is (unless you happen to teach Calc II), but most physicists likely do as they encounter it often and just memorized it. At the very least, if they had to derive it every time it showed up in a problem in an exam, they would run out of time.
Of course it is not some big discovery in fundamental knowledge - but it is a helpful pedagogical advance and I am happy Po-Shen Loh has advertised it.
This is not simpler for the student. Its is, at best, a simpler way to derive the equation, a thing which most people will never do.
And, dare I say it, still a waste of time. In my not-so-humble opinion you either know enough algebra to understand that, for non-zero a, 0 = a x^2 + b x + c <=> 0 = x^2 + b/a x + c/a <=> 0 = (x + b/2a)^2 - b^2/4a^2 + c/a <=> (x + b/2a)^2 = (b^2 - 4 a c) / 4a^2 or you need to improve your basic algebra skills - not search for a better derivation of this particular formula.
https://en.etudes.ru/models/conic-sections-cavalieri-parabol...
This may make all of the variables more "real" - but note that the equation graphed is really A(X - B)^2 = Y - C, where (B, C) is the vertex and A is the "setting" (clear if you examine the device. This is relatively similar to the complete-the-square formalism but with a straightforward geometric and "hands-on" interpretation.
Is knowing to take advantage of that really any more intuitive than completing the square is?
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 :) .
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:
I did refer to the "result" rather than the derivation.
I need to solve quadratic equations. It is much harder. Like
a1 x y + b1 x + c1 y + d1 = 0
a2 y z + b2 y + c2 z + d2 = 0
a3 z x + b3 z + c3 x + d3 = 0
Given a,b,c,d, solve it for x,y,z.
And in the general case where there are more than 3 equation or more than 3 x,y,z variables.
We can make a undirected graph with the variables as nodes and an edge if there is an equation containing those two variables. Like (x,y), (y,z), (x,z) here. If there is a cycle in that graph, we can combine the equations to obtain a single quadratic equation, which has two solutions for x.
I can search for cycles, and sometimes two cycles restrict each other, such that only one solution for x remains. Sometimes two solutions remain. I do not understand when that happens, and if there is a better way than searching cycles.
(also I can have an additional list of restrictions between the parameters. Like a1 = -c2. This sometimes leads to a cycle canceling itself out, revealing no information about x at all. )
If this is the new standard for the web, I want to get off the ride.
For this very specific use case, I would call it sub-optimal, for the simple reason it takes all of five minutes to teach a class the formula by singing it to the tune of Frère Jacques. I haven't used the formula in over a decade but can still sing it.