The B/2 term places the center of the parabola, which you can verify by either inspecting the derivative or fiddling with a graphing calculator. Then, because the parabola is symmetric, the zeros must be a pair of points mirrored across the center, with the distance from the center determined by the vertical offset (the C term) versus the narrowness (the A term). Which I definitely agree is an easier way to deal with the problem! But I think that, if you're going to do it that way, it might be easier to work from geometric intuition instead of an algebra trick that represents but obscures the same relationship.
It's only obvious that x = -B/2A is the parabola's axis of symmetry if you complete the square in the traditional way, or use calculus to locate the extremum. So it seems to me that we don't need a new method, we need better teaching of the tried-and-true method.
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.
Yes, and it has a coordinate geometry interpretation too. Completing the square represents the original quadratic function as a basic x^2 parabola that's been scaled vertically, shifted vertically, then shifted horizontally. If you understand what each of these three operations do to the roots of the parabola, you can derive the quadratic formula. (It's basically just the algebraic way with diagrams, but may help the geometry-loving/algebra-hating types out there.)
I think, like the OP, this just dances around completing the square without actually eliminating the need for it. How do we know that all parabolas can be expressed as the product of two symmetrical straight lines?
I just don't get why making some unsupported postulate about how parabolas can be generated from straight lines is better pedagogically than teaching completing the square and providing a geometric interpretation for completing the square.
That's ok. I can't explain it here in a few lines. It's rather a pedagogic principle that permeates most of my teaching.
It's not just a postulate. It's demonstrated. A style used from the start of algebra.
Pedagogically there's more to it than just the algebraic manipulations.
The algebra here is basically as dull as all other algebra. It's the teacher's job to make it come alive.
Your previous comment just suggested that there's something pedagogically superior about this approach. If you don't believe that, you shouldn't have made the comment.
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 algebra students [who] have painful memories of struggling to memorize the quadratic formula."
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 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.
No he should not. The words invokes the right connotations in a pedagogically minded reader. This can be the basis for better teaching. It depends of course on implementation.
Math is not an easy subject by any means. I see the same thing when people give programming tutorials and call it “easy” and “simple”. However, once the student starts to encounter difficulties with the subject later down the road, they feel stupid and lied to, since they were promised by their teacher that what they were learning was “easy”.
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.
I'm a "pedagogically minded" reader, and I find these words unnecessary at best, and off putting at worst. Math needs better marketing based on actual science, not egotism
Teachers & the educational system in general should give examples of why stuff needs to be learnt instead of being dictated to, because they have one of the largest influences on the future direction of a country's success. Teaching was a cop out, a cushy number for those who couldn't hack it in the real world and liked to take their failure out on pupils. Many teachers couldn't run a business the way they treated kids in the classroom so when they go on strike because they want more pay they are just demonstrating that bullying works, and hard work doesn't pay. Hypocrites!
One of the biggest source of bad ideas in education is that people think it would be better if it catered to the kind of person they thought they were when they went to school.
"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 not just a "kind of person" who wants personal relevance to what they're learning.
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 was taught stuff at Primary School which I then got taught at Secondary school. Admittedly I moved around due to parents jobs but you could not get a consistence curriculum in the UK. AQA the main exam board was setup because private schools sat their pupils with exams from the easiest exam boards unlike state schools who just went with exam boards that suited them. From an employers perspective, a private school pupil with an A was like a state school C, but that's the sort of privilege having money can buy you and is perhaps why Uni's and some employers here in the UK are now waking up too.
The class system in the UK is a very repressive system of control that infects every aspect of life to the detriment of the UK and the wider world.
I just hand-derived it both ways to make a comparison, the standard way was from memory.
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 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 December 16, 2019)
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.
This is neither new, nor easier. It can be interesting to think about and deriving the same results through different methods can help with understanding. But I really doubt that there will be any students who can understand and use this who at the same would struggle with the regular formula for the quadratic.
Leaving aside the fact that this isn't new -- it's how I was taught to solve quadratic equations in the 80s -- if you look closely he's still completing the square.
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?
> 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.
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.
Isn’t the whole point that it’s the same thing with a slightly different derivation?
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.
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 + bx + b^2/4
and then I figure out what I need to add to both sides of that equation to make the RHS equal the original quadratic, which gives me:
(x + b/2)^2 - b^2/4 + c = x^2 + bx + c
and I then find the roots of the RHS by instead finding them for the LHS. We say that the LHS is the result of "completing the square" on the RHS.
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.
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 subtracting the same amount bothers you: Yes, this is done all the time.
> 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.
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.
> 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.
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.
> 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.
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.
I learned this technique while reviewing high school math to take the GRE, and really enjoyed it. I find it elegant and easier to derive than the usual formula - I prefer not to memorize equations. And with just a little practice, it was easy to work through manually on my whiteboard (no scratch paper allowed).
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.
I have absolutely horrible memory, which I hold has served me fairly well academically. The need to occasionally re-derive basic formulae serves as a nice refresher -- at least in engineering (probably the weight of redoing everything would get too heavy if I'd ended up as a mathematician). Remembering too many rules makes math feel arcane. I tutored math for a bit, and as far as I can tell the quadratic formula serves as significant marker along the path of math as an exercise in just remembering things, unfortunately.
And students need not memorise the equation, instead they can derive it from another equally unintuitive equation then go through many logical steps of multiplication, simplification and refactoring - each being locations where students are likely to introduce errors.
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.
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.
Yeah, that part of the explaination felt strange to me as it's not intuitive and the student is taking the author's word for it. There's no difference between that and completing the square without understanding how it works - or just blinding memorizing the quadratic equation.
"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 :) .
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.
Yup, agreed. But this the standard formula everyone here memorizes, so I thought it might be interesting since it wasn't mentioned anywhere in the article.
I did refer to the "result" rather than the derivation.
Nb. He's just using the N = 2 DFT to solve a quadratic equation. This approach is well-understand in the context of Galois theory, eg. the N = 3 DFT reduces the general cubic to a set of at most quadratic equations, and in general, you can DFT your way down a Galois Group's composition series whenever it's solvable.
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. )
I must be stupid because I did not find the explanation easy. In fact after I read it I thought to myself: I think I’ll just rely on memorization for this (not that it’s actually needed in my day to day life and I’ll have to look it up if I ever actually need it).
“Completing the square” may seem like an arbitrary and hard to remember math trick but it is the algebraic analogue of something that is very intuitive in geometry. The Greeks solved quadratic equations by visually completing the square.
Completing the square teaches students something beyond just how to find the roots of quadratics. It teaches them that one approach to solving problems is to transform them into a problem they already know how to solve.
As very much a non-mathematician, his method does not seem remotely intuitive to me. Whether that opinion is invalid because of my lack of skill or more valid because I'm his intended demographic I'll leave up to you.
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.
