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 binomial and trinomial factoring.
To me it is obvious that the method in the article is far superior than teaching completing the square. I’m teaching a pre-calculus course this semester and many of my students still can’t complete the square. Pre-calculus is 3 math courses beyond elementary algebra.
All of this is just my opinion of course and I have no data or studies to back up my opinion. I will be using the method described in the article from now on in my elementary algebra courses.
I’m surprised that these students learn to solve quadratic equations before they learn what parabolas are.
If I were teaching quadratics, I would probably start with squares and square roots, and I would draw pictures. I would present some motivating examples (from kinematics, maybe? just the pictures may be okay, especially if the fun game where you try to zap the targets by hitting them with graphs is still around.). Then I would teach translations. After that comes the distributive law and polynomials written ax^2 + bx + c.
And now you can solve them! As far as I’m concerned, solving equations (polynomials, systems of equations, integrals, ODEs, PDEs, etc) is a puzzle, and learning a bag of tricks to solve them is just that: learning a bag of tricks. Quadratics are nice because the tricks always work. In more complicated math, it’s important to understand that the tricks can be very hard or even probably nonexistent, and accepting that is important.
But I don’t see why we should teach people to solve quadratics before teaching what they are.
edit: it’s not clear to me that the method in the article is dramatically different from completing the square. Assume a=1 for simplicity. Given the knowledge the the average of the roots is -b/2 (which one can deduce by any number of means), you can solve the equation in quite a few ways. One is the way in the article. Another is to say “the average is -b/2, so the vertex of the parabola is at x=-b/2, so the polynomial can be written (x+b/2)^2 + something”. Another is to just write down the solution x = -b/2 ± something and solve for “something” (which is more or less the same thing as in the article).
In high school, I used to have fun solving quadratics in my head by seeing which technique gave a quick answer.
> 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 teaching by the teacher. Completing the square has uses beyond proving the quadratic theorem; this does not.
I should say: I find this an incredibly cool and level-appropriate proof of the quadratic equation, but I think its merits as an improvement in pedagogy are dubious.
I doubt I can convince you. I’m just going by my experience teaching the topic. At the time students first learn solving such equations they have just been taught factoring and what it means to factor a trinomial. They know the product of the constant terms in the binomials must be c. It’s also easy to explain that the average of two numbers is the midpoint. And thus if I start with the midpoint then to get to the numbers I took the average of I add and then subtract some number from the midpoint. The geometry makes this easier to explain over using completing the square.
I’ve seen a shocking number of calculus students struggle with completing the square. The merits of the approach in the article are entirely obvious to me but like everyone else I’ve had my share of obvious beliefs turn out to be false.
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)
All of the world’s best scientists didn’t know that projectiles moved on parabolic paths until Galileo’s experiments on inclined planes in the 17th century. 21st century schoolchildren who haven’t been taught about it likely don’t either.
I deduce from your post that you have very little experience in teaching people at the level of beginning algebra. And while one might know geometrically what a parabola is there is a lot one must know before dealing with parabolas algebraically. I suggest that these things appear easy and obvious because you already know them and that you no longer remember what is hard for people learning this stuff for the first time.
I don’t show conic sections in elementary algebra. One typically really mentions the phrase “conic section” is pre-calculus which is 3 courses after elementary algebra. Over the past few centuries the order in which concepts are introduced has been developed. It’s not perfect but one should not be so quick to discount the way things are done without knowledge/experience in presenting these ideas to beginning students.
You've got me. I wish I could delete the comment, or at least edit it.
It's a bit worse than just having no experience teaching algebra, I
didn't have the same experience as most kids trying to learn it. I was
kind of a freak. I was the weird quiet kid in the back of the class who
always knew the answer to every question. (Other kids tended to not like
that, but I'm also very disarming (in person) and so I did alright.) One
year, I was misplaced into a basic geometry class, and the teacher very
kindly let me pick out some calculus textbooks and sit in the back of the
class teaching myself calculus. (Some bureaucratic reason for why I
couldn't transfer, or the calc classes were full, or we didn't have any,
or something, I forget.) Learning math for me feels like remembering
things I always knew.
So, yeah, maybe I should keep my mouth shut when it comes to teaching normal
people how to do math.
Or maybe we should try to figure out what
my brain is doing and how to teach people to do that too?
Maybe I have a
normal brain and I'm just using it differently than most people?
I like that idea better, because then, instead of a freak, I'm a
front-runner. And, in addition, there's hope for a great improvement in
didactic technique and humanity's general "numeracy" level, eh? If we could teach people to math like i do we could compress basic math education (up to calculus) into just a year or so.
"Something self-referential and hopefully unsnarky about how this is a
thread on a how-to-math-better article." ~me, failing at rhetoric.
And for kicks, here's Iconic Math: http://iconicmath.com/ (No
affiliation with me. I'm just trying to end on an upbeat, constructive note.)
I think a big part is that many (most?) children find school – i.e. lectures, textbooks, homework exercises, exams – unmotivating/boring at best, and often extremely stressful/frightening, which means that they aren’t fully focused on it and can easily miss important details. Beyond that, schools often fail to provide meaningful feedback or support when people suffer serious misconceptions or are missing fundamental prerequisite knowledge/skills, which makes it easy for students to fall behind and have great difficulty recovering. Someone who spends the exact same amount of time on academic work but for whatever reason (external help outside school, internal motivation, some insightful introspection, ...) manages to pay closer attention, stay more relaxed, think about things ahead of where the class is expected to be, connect new learning to material learned before and build a better-connected mental map, etc. can end up pulling far ahead.
A whole lot of this has to do with level of preparation before ever arriving at school. Some kids read with their parents for hours every day from age 1–5+, learn to play a variety of strategic games (and games involving basic arithmetic practice), build structures or mechanisms or electronics, practice making art, work through books of logic puzzles, etc. Other kids are left alone and bored without learning materials at a reasonable level, plonked down in front of developmentally inappropriate or just badly produced TV, or handed over to unthoughtful video games.
Then consider how many kids and parents have serious problems at home, with confrontational or even abusive relationships. Pile on work stress, financial stress, poor diet or even hunger, poor sleep, environmental toxins, illness, etc.
To me it is obvious that the method in the article is far superior than teaching completing the square. I’m teaching a pre-calculus course this semester and many of my students still can’t complete the square. Pre-calculus is 3 math courses beyond elementary algebra.
All of this is just my opinion of course and I have no data or studies to back up my opinion. I will be using the method described in the article from now on in my elementary algebra courses.