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.
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.
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.
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.
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?"
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.
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.)
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?"
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.)
I teach mathematics and write math curriculum professionally, and have literally taught the lesson I describe above for the better part of a decade. I say "we" in that post because in more than half of those classes, I did not teach it alone, but in a co-taught inclusion class for special education and general education students together. None of the classes where I used this lesson was an "honors"/"accelerated"/"pre-AP"/etc. class.
Every single one of those students was able to derive the quadratic equation by completing the square. It was not easy for some of them, but every single one did it.
Your parenthetical also implies to me that you think that the "truly disadvantaged" have less "mathematical skill". I would encourage you to reflect on that.
Comments
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.
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.
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.
Refreshing candor! Wish it held true that more people saw it through to discover their obvious truths didn't hold up.
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)
Apologies in advance for the harshness of my tone, but I nearly spit up my coffee...
The mind boggles. Have you never shown them a sliced cone? ( 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.
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_librar...
And here’s another similar diagram tossed up by image search: 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
Maybe Tartaglia just didn’t like quadratics. He did, after all, find the cubic formula. ;)
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.
I couldn't agree more.
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.
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.
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?"
You are correct. It is generally expected to be part of an Algebra I curriculum (as per Common Core Appendix A's Traditional Pathway).
http://www.corestandards.org/Math/Content/HSA/REI/B/4/a/
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.)
"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?"
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.)
I teach mathematics and write math curriculum professionally, and have literally taught the lesson I describe above for the better part of a decade. I say "we" in that post because in more than half of those classes, I did not teach it alone, but in a co-taught inclusion class for special education and general education students together. None of the classes where I used this lesson was an "honors"/"accelerated"/"pre-AP"/etc. class.
Every single one of those students was able to derive the quadratic equation by completing the square. It was not easy for some of them, but every single one did it.
Your parenthetical also implies to me that you think that the "truly disadvantaged" have less "mathematical skill". I would encourage you to reflect on that.
I thought the article was trying to compare to the "typical" fractional formula, not to completing the squares?