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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

by sykick 7y ago

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.

pflats 7y ago
> 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.
- sykick 7y ago
  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.
  - Isamu 7y ago
    >but like everyone else I’ve had my share of obvious beliefs turn out to be false. Refreshing candor! Wish it held true that more people saw it through to discover their obvious truths didn't hold up.
- Anon84 7y ago
  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)
carapace 7y ago
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 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.
- jacobolus 7y ago
  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. Here’s Tartaglia’s picture from the 16th century, drawn from his experimental research: https://www.maa.org/sites/default/files/images/upload_library/46/Swetz_2012_Math_Treasures/Smithsonian-Dibner-else/SIL28-191-05.png https://www.maa.org/sites/default/files/images/upload_librar... And here’s another similar diagram tossed up by image search: http://ej.iop.org/images/0143-0807/33/1/149/Full/ejp405251f1_online.jpg http://ej.iop.org/images/0143-0807/33/1/149/Full/ejp405251f1... If you look for even earlier diagrams they show a projectile moving in a straight line for some distance and then suddenly dropping straight down. Cf. https://en.wikipedia.org/wiki/Aristotelian_physics https://en.wikipedia.org/wiki/Aristotelian_physics
  - DavidSJ 7y ago
    Maybe Tartaglia just didn’t like quadratics. He did, after all, find the cubic formula. ;)
- sykick 7y ago
  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.
amluto 7y ago
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.

by sykick 7y ago

pflats 7y ago

> 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.

sykick 7y ago
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.
- Isamu 7y ago
  >but like everyone else I’ve had my share of obvious beliefs turn out to be false. Refreshing candor! Wish it held true that more people saw it through to discover their obvious truths didn't hold up.
Anon84 7y ago
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)

carapace 7y ago

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 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.

jacobolus 7y ago
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. Here’s Tartaglia’s picture from the 16th century, drawn from his experimental research: https://www.maa.org/sites/default/files/images/upload_library/46/Swetz_2012_Math_Treasures/Smithsonian-Dibner-else/SIL28-191-05.png https://www.maa.org/sites/default/files/images/upload_librar... And here’s another similar diagram tossed up by image search: http://ej.iop.org/images/0143-0807/33/1/149/Full/ejp405251f1_online.jpg http://ej.iop.org/images/0143-0807/33/1/149/Full/ejp405251f1... If you look for even earlier diagrams they show a projectile moving in a straight line for some distance and then suddenly dropping straight down. Cf. https://en.wikipedia.org/wiki/Aristotelian_physics https://en.wikipedia.org/wiki/Aristotelian_physics
- DavidSJ 7y ago
  Maybe Tartaglia just didn’t like quadratics. He did, after all, find the cubic formula. ;)
sykick 7y ago
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.

amluto 7y ago

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.