9 ms·

This is bizarre. It’s no less a mathematical trick than completing the square, and it doesn’t seem to be any simpler to use. The example given is to find the r

This is bizarre. It’s no less a mathematical trick than completing the square, and it doesn’t seem to be any simpler to use. The example given is to find the roots of x² - 2x + 4 = 0. Completing the square gives (x - 1)² + 3 = 0, from which you can immediately see that the roots are 1 ± √3 i. If anything this seems easier than the method of the article. Am I missing something? Added: The argument seems to be that young students will find this method easier to understand than completing the square. I have no experience of teaching mathematics to children, so this may be true for all I know. It would be interesting to test this hypothesis experimentally, because I don’t think it’s obviously true.

banachtarski 7y ago
Was going to reply precisely this. Completing the square is, I believe, already taught in standard curricula (at least, it was when I was in grade school), and is a trivial way to derive the quadratic formula.
- pflats 7y ago
  You are correct. It is generally expected to be part of an Algebra I curriculum (as per Common Core Appendix A's Traditional Pathway). > Common Core High School: Algebra » Reasoning with Equations & Inequalities » Solve equations and inequalities in one variable. » 4 » a > Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)^2 = q that has the same solutions. Derive the quadratic formula from this form. http://www.corestandards.org/Math/Content/HSA/REI/B/4/a/ http://www.corestandards.org/Math/Content/HSA/REI/B/4/a/
- leggomylibro 7y ago
  Depends. I took a mid-level calculus class recently because it had been awhile since I'd integrated anything, and the professor had to spend a day reviewing how to factor and divide polynomials when they got to integrals which used trig identities. "Completing the square" was a new phrase to most students. Everyone knew the quadratic formula, but most people don't pay attention to how formulas are derived, much less remember that sort of thing a few years later. If you want to guess whether an average student knows something, ask yourself, "would it be on the test?"
ken 7y ago
Programming is full of these. Every day I see some new technique or style appear. Is it any good? We've got easier access to the biggest collection of free source code, the biggest and most diverse set of programmers, and the fastest automated testing of all time. Just do an experiment. You don't need to sell me on how cool it is. Show me how it's better. - Does it lead to smaller programs? Don't show me a 10-line example. (I've heard medical researchers say: "Anyone can cure cancer in mice.") Convert a 10,000- or 100,000-line program. - Does it lead to humans writing fewer bugs? Have a bunch of them try it, and measure their speed and defect rates. Everyone has designed a system that they themselves love, and nobody else can understand. If you promote a new method which is logically equivalent to old ones, and not obviously much better, and without any data showing specific metrics that it improves, I'm forced to assume your metric is simply "I happen to like it better". Experimentation is cool, but it doesn't cause lasting long-term shifts.
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.
jerf 7y ago
I believe the objection is that deriving the formula itself from completing the square is the challenge, not the direct use of completing the square as you show in your example. (Although I wonder where you learned that, because I was certainly never taught that so directly.) The problem with deriving the quadratic equation that way is that for most people, that is a lot of symbols to keep track of, and you have to not only do an unintuitive "completing the square" step but you have to unintuitively do it fully generically. For a professional mathematician this is barely a warmup, of course, but for the average person in middle school or early high school, this is distinctly nontrivial work. (I'm not taking a side here, just trying to describe what the issue is, since you asked. I'm ambivalent. Neither of my kids are quite this far yet, so I'm not quite here yet.)
- pflats 7y ago
  > The problem with deriving the quadratic equation that way is that for most people, that is a lot of symbols to keep track of, and you have to not only do an unintuitive "completing the square" step but you have to unintuitively do it fully generically. "Unintuitive" depends entirely on your introduction to the topic. If you're already completing the square, using it to solve quadractic equations you cannot factor is not unintuitive. As far as doing it fully generically, well, how else do you get a generic formula? When teaching this, we would do some completing the square to solve quadractics, and then tell our students: "You know, this is annoying to have to complete the square EVERY TIME. What if we just decided to solve it the really hard way once, with A, B, and C in the equation instead of the numbers, and see what we get?"
  - jerf 7y ago
    I think you're speaking from the perspective of someone very casually comfortable with symbol manipulation. This does not describe the average middle school student. When I said "unintuitive", I was speaking from the perspective of an average middle school student, for whom this is all either at the edge of the ability, or, often, a bit past it, and for a non-trivial number of them, way past it. In high school, I was a tutor for a non-accelerated, non-honors class that was about at this level in high school. After years of being in the accelerated course, it was a bit of an eye-opening experience. There's a lot of people who are just passed through this stuff with a C-, and I'm not even sure that's wrong, because there's a lot of people who just aren't ever going to get to the point where they can fluidly derive any of these equations. What you, and probably a great deal of the HN commetariat experience as "average" is actually way above average. (And the students I was tutoring for, in the parlance of the day, would still mostly be considered "privileged". I would still not be calibrated for the mathematical skill of the truly disadvantaged.)
WhompingWindows 7y ago
I thought the article was trying to compare to the "typical" fractional formula, not to completing the squares?

banachtarski 7y ago

Was going to reply precisely this. Completing the square is, I believe, already taught in standard curricula (at least, it was when I was in grade school), and is a trivial way to derive the quadratic formula.

pflats 7y ago
You are correct. It is generally expected to be part of an Algebra I curriculum (as per Common Core Appendix A's Traditional Pathway). > Common Core High School: Algebra » Reasoning with Equations & Inequalities » Solve equations and inequalities in one variable. » 4 » a > Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)^2 = q that has the same solutions. Derive the quadratic formula from this form. http://www.corestandards.org/Math/Content/HSA/REI/B/4/a/ http://www.corestandards.org/Math/Content/HSA/REI/B/4/a/
leggomylibro 7y ago
Depends. I took a mid-level calculus class recently because it had been awhile since I'd integrated anything, and the professor had to spend a day reviewing how to factor and divide polynomials when they got to integrals which used trig identities. "Completing the square" was a new phrase to most students. Everyone knew the quadratic formula, but most people don't pay attention to how formulas are derived, much less remember that sort of thing a few years later. If you want to guess whether an average student knows something, ask yourself, "would it be on the test?"

ken 7y ago

Programming is full of these. Every day I see some new technique or style appear. Is it any good? We've got easier access to the biggest collection of free source code, the biggest and most diverse set of programmers, and the fastest automated testing of all time. Just do an experiment. You don't need to sell me on how cool it is. Show me how it's better. - Does it lead to smaller programs? Don't show me a 10-line example. (I've heard medical researchers say: "Anyone can cure cancer in mice.") Convert a 10,000- or 100,000-line program. - Does it lead to humans writing fewer bugs? Have a bunch of them try it, and measure their speed and defect rates. Everyone has designed a system that they themselves love, and nobody else can understand. If you promote a new method which is logically equivalent to old ones, and not obviously much better, and without any data showing specific metrics that it improves, I'm forced to assume your metric is simply "I happen to like it better". Experimentation is cool, but it doesn't cause lasting long-term shifts.

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.

jerf 7y ago

I believe the objection is that deriving the formula itself from completing the square is the challenge, not the direct use of completing the square as you show in your example. (Although I wonder where you learned that, because I was certainly never taught that so directly.) The problem with deriving the quadratic equation that way is that for most people, that is a lot of symbols to keep track of, and you have to not only do an unintuitive "completing the square" step but you have to unintuitively do it fully generically. For a professional mathematician this is barely a warmup, of course, but for the average person in middle school or early high school, this is distinctly nontrivial work. (I'm not taking a side here, just trying to describe what the issue is, since you asked. I'm ambivalent. Neither of my kids are quite this far yet, so I'm not quite here yet.)

pflats 7y ago
> The problem with deriving the quadratic equation that way is that for most people, that is a lot of symbols to keep track of, and you have to not only do an unintuitive "completing the square" step but you have to unintuitively do it fully generically. "Unintuitive" depends entirely on your introduction to the topic. If you're already completing the square, using it to solve quadractic equations you cannot factor is not unintuitive. As far as doing it fully generically, well, how else do you get a generic formula? When teaching this, we would do some completing the square to solve quadractics, and then tell our students: "You know, this is annoying to have to complete the square EVERY TIME. What if we just decided to solve it the really hard way once, with A, B, and C in the equation instead of the numbers, and see what we get?"
- jerf 7y ago
  I think you're speaking from the perspective of someone very casually comfortable with symbol manipulation. This does not describe the average middle school student. When I said "unintuitive", I was speaking from the perspective of an average middle school student, for whom this is all either at the edge of the ability, or, often, a bit past it, and for a non-trivial number of them, way past it. In high school, I was a tutor for a non-accelerated, non-honors class that was about at this level in high school. After years of being in the accelerated course, it was a bit of an eye-opening experience. There's a lot of people who are just passed through this stuff with a C-, and I'm not even sure that's wrong, because there's a lot of people who just aren't ever going to get to the point where they can fluidly derive any of these equations. What you, and probably a great deal of the HN commetariat experience as "average" is actually way above average. (And the students I was tutoring for, in the parlance of the day, would still mostly be considered "privileged". I would still not be calibrated for the mathematical skill of the truly disadvantaged.)

WhompingWindows 7y ago

I thought the article was trying to compare to the "typical" fractional formula, not to completing the squares?