Should we make students memorize the quadratic formula?

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Should we make students memorize the quadratic formula?

Postby skullturf » Tue Sep 20, 2011 5:20 pm UTC

It can be a useful time-saver to be able to quickly write it down without thinking. And in about Grade 9, I remember the teacher drilling the formula into us, having us recite it in words: "the opposite of b, plus or minus the square root of, b squared minus 4ac, all over 2a."

However, when solving particular quadratic equations, especially with nice coefficients, such as

x^2 + 2x + 2 = 0

I much prefer the method of completing the square.

x^2 + 2x + 1 = -1

(x+1)^2 = -1

x+1 = ±i

x = -1 ± i

What do you all think?

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Re: Should we make students memorize the quadratic formula?

Postby LolTheForce » Tue Sep 20, 2011 6:05 pm UTC

Er I might be wrong here but I really think that (x+1)² = -1 has no real roots :S

(x+1)² >= 0 for all real values of x, right?

I make x out to be the complex number z = -1 + i and it's complex conjugate.

Anyway, I guess that's quite beside the point. When I attended the equivalent of high school here in Sweden we used the pq-formula (that's the name of it in English too, right? Kind of new to "international maths") which I kind of have no problem with, only we never got to see the proof of why it works. Completing the square and all that jazz. Now that I learned it later on I feel I get a much better understanding of the concept. It's no longer just filling in random numbers, hoping I don't forget something or mess something up. Now I actually do the math.

I guess what I'm saying is, I'd rather you show the students WHY a formula works before you make them use it blindly for three years without ever knowing why.

EDIT: First post on here by the way, hello all! :)
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Re: Should we make students memorize the quadratic formula?

Postby mike-l » Tue Sep 20, 2011 6:05 pm UTC

Complete the squares on ax^2 + bx + c
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Re: Should we make students memorize the quadratic formula?

Postby AMathematician » Tue Sep 20, 2011 6:09 pm UTC

With that in mind, I think it's important that students are taught what the quadratic formula is doing. When I was first introduced to it there was no explanation of where it came from. It was just "You know that thing you learned during the lesson today? Take the coefficients from these twenty quadratic equations and use that thing to find the zeroes." No explanation at all.

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Re: Should we make students memorize the quadratic formula?

Postby Anonymously Famous » Tue Sep 20, 2011 6:10 pm UTC

I don't think there's any harm in having students memorize the quadratic formula. I love the way my teacher did it, too, to the tune of "Pop Goes the Weasel."

"X equals negative b, plus or minus the square root of b squared minus four a c, all over two a."

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Re: Should we make students memorize the quadratic formula?

Postby Dason » Tue Sep 20, 2011 6:12 pm UTC

Anonymously Famous wrote:I don't think there's any harm in having students memorize the quadratic formula. I love the way my teacher did it, too, to the tune of "Pop Goes the Weasel."

"X equals negative b, plus or minus the square root of b squared minus four a c, all over two a."

It's not a bad formula to know but skullturf definitely has a point. It was until I was tutoring college precalc that I took the time to derive the quadratic formula. I wish I would have done that earlier - it would have made a lot more sense.
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Re: Should we make students memorize the quadratic formula?

Postby gmalivuk » Tue Sep 20, 2011 6:24 pm UTC

skullturf wrote:I much prefer the method of completing the square.
Which gives you the quadratic formula in the general case.

Yes, the general case isn't optimal for solving every quadratic formula. x^2 - 1 = 0 should be solvable at a glance. But the strength of the formula is that you can use it for all quadratic formulas, even if the coefficients aren't nice and pretty.
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Re: Should we make students memorize the quadratic formula?

Postby Tirian » Tue Sep 20, 2011 6:54 pm UTC

This question reminds me of my intensive four-semester course in undergraduate math that was intended to teach us everything from foundations through differential topology in abstract manifolds in one unified presentation. As the final weeks of the fourth-semester wound down, we students were looking at each other curious that we had yet to learn differential equations. At the penultimate class, the professor finally explained that what we think of as a DiffEq class is completely without value. Parlor tricks that only solve 1% of the differential equations in the world, and curiously not the 1% that are encountered in the field. If we were planning on going into engineering, we were told to borrow a textbook from a friend, read it over a weekend, and go straight to Numerical Methods to find the real way to get the answers we'll be asked to provide in our careers. (Having followed this advice, I can neither confirm nor deny that this plan is sound or right for everyone. My professors were quite opinionated on some subjects of mathematical philosophy as you can guess, but I will say that some professors I've talked to since then have had sympathy for this point of view.)

I tend to feel the same way about completing the square. It's an interesting trick, and certainly has a valuable application in deriving the quadratic formula, but by the time you're trying to find the solution of 2x^2 - 3x + 4 = 0 I think you're just reinventing the wheel by not applying the QF directly.

I suppose the other question is whether students should pick up the QF by memorization or by intuition from having derived it themselves. Perhaps I had a particularly bad teacher that year in high school, but ISTR repressing the day that that derivation was done in class, so I think that this is one of those cases where the tool is within reach of the students a year or two before the proof of it is. My lone regret is that I had to memorize it without the pleasure of watching my teacher singing the Pop Goes the Weasel song. :(

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Re: Should we make students memorize the quadratic formula?

Postby Yakk » Tue Sep 20, 2011 6:59 pm UTC

mike-l wrote:Complete the squares on ax^2 + bx + c

Sure!

ax^2 + bx + c = 0
= a * (x^2 + b/a x + c/a)
= a * ((x +b/2a)^2 - b^2/4a^2 + c/a)
a (x+b/2a)^2 = a*( b^2/4a^2 - c/a )
(x+b/2a)^2 = ( b^2/4a^2 - c/a )
x+b/2a = +/- sqrt( b^2/4a^2 - c/a )
x = - b/2a +/- sqrt( b^2/4a^2 - c/a )
x = (- b/2a +/- sqrt( b^2/4a^2 - c/a ) ) * 2a/2a
x = (- b +/- 2a sqrt( b^2/4a^2 - c/a ) ) /2a
x = (- b +/- sqrt(4a^2( b^2/4a^2 - c/a )) ) /2a
x = (- b +/- sqrt( b^2 - 4a^2 c/a ) ) /2a
x = (- b +/- sqrt( b^2 - 4ac ) ) /2a
2a x = (- b +/- sqrt( b^2 - 4ac ) )
2a x +b = +/- sqrt( b^2 - 4ac )
(2a x +b)^2 = b^2 - 4ac
4a^2 x^2 +4abx + b^2 = b^2 - 4ac
4a^2 x^2 +4abx + 4ac = 0
a x^2 +bx + c = 0/4a
a x^2 +bx + c = 0
So 0 = 0
That isn't interesting.

;)

And yes, learning the quadratic formula is useful, because my teacher sang it to me (and danced as well).
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Re: Should we make students memorize the quadratic formula?

Postby LaserGuy » Tue Sep 20, 2011 8:44 pm UTC

skullturf wrote:It can be a useful time-saver to be able to quickly write it down without thinking. And in about Grade 9, I remember the teacher drilling the formula into us, having us recite it in words: "the opposite of b, plus or minus the square root of, b squared minus 4ac, all over 2a."

However, when solving particular quadratic equations, especially with nice coefficients, such as

x^2 + 2x + 2 = 0

I much prefer the method of completing the square.

x^2 + 2x + 1 = -1

(x+1)^2 = -1

x+1 = ±i

x = -1 ± i

What do you all think?


Completing the square gets you the same result as quadratic formula, but takes longer and it's easier to make a mistake... Pedagogically, it's nice to teach it and show that the quadratic formula doesn't just come out of nowhere, but the whole point of deriving a general result is that it is easier to use the general result rather than having to derive it from scratch every time. To me, this is like asking why we shouldn't just use the definition of the derivative for all problems. Sure you can do it, and you should always get the right answers if you're careful, but it's a complete waste of your time.

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Re: Should we make students memorize the quadratic formula?

Postby Cleverbeans » Tue Sep 20, 2011 9:29 pm UTC

LaserGuy wrote: Pedagogically, it's nice to teach it and show that the quadratic formula doesn't just come out of nowhere, but the whole point of deriving a general result is that it is easier to use the general result rather than having to derive it from scratch every time. To me, this is like asking why we shouldn't just use the definition of the derivative for all problems. Sure you can do it, and you should always get the right answers if you're careful, but it's a complete waste of your time.


I disagree, pedagogically the purpose of the exercise is to help them discover strategies to solve problems involving quadratics, and familiarize them with the properties of quadratic equations. I consider the only value in the quadratic formula to be in defining the discriminant, beyond that it's trivia and has little value beyond being quick at standardized tests. In fact, I cannot recall using it at any time after high school where I wasn't either allowed to write it down, or use a calculator. However I wouldn't advocate memorization of this method either, simply that as an exercise it would provide the student with more insight than tenfold plug-n-chugs with the quadratic formula would.
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Re: Should we make students memorize the quadratic formula?

Postby gmalivuk » Tue Sep 20, 2011 9:38 pm UTC

I see it as similar to memorizing multiplication tables or tables of basic integrals or the mean and variance of common distributions. Sure, it's possible to compute all those things from first principles every time, but whenever you actually have to use it, you're probably not going to have time to do so.
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Re: Should we make students memorize the quadratic formula?

Postby skullturf » Tue Sep 20, 2011 10:04 pm UTC

This is subjective and may vary with the individual, but in my opinion, in my example given above, completing the square takes almost exactly the same amount of time as using the quadratic formula (I admit that using the quadratic formula is probably marginally faster) and I also believe one is less likely to make a mistake when doing it by completing the square.

Admittedly, though, that might no longer be true with an example such as 2x^2 - 3x + 4 = 0, given above by Tirian.

I agree with gmalivuk that it probably is a good idea to memorize the quadratic formula, and that it's on a par with things like memorizing common antiderivatives, or means/variances of common distributions.

But I sometimes wonder how much sense it makes to make students memorize the quadratic formula when they are about 14. I wonder if it would make more sense to focus on specific examples where one can complete the square, and then also mention that one can do this in general and get the quadratic formula.

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Re: Should we make students memorize the quadratic formula?

Postby gmalivuk » Tue Sep 20, 2011 10:19 pm UTC

skullturf wrote:But I sometimes wonder how much sense it makes to make students memorize the quadratic formula when they are about 14. I wonder if it would make more sense to focus on specific examples where one can complete the square, and then also mention that one can do this in general and get the quadratic formula.
Can that not all also be done when they are about 14?

In any case, I could see arguments either way. Either teach and practice the formula for a bit first, so students see that it works, and then derive it by completing the square so they can see *why* it works. Or teach and practice completing the square step-by-step first, and then do it with a generalized quadratic equation to derive the quadratic formula.
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Re: Should we make students memorize the quadratic formula?

Postby ConMan » Wed Sep 21, 2011 12:26 am UTC

We went from solving simple quadratic equations by what I think LolTheForce was calling the pq method (find numbers that multiply to give c and sum/difference to give b, yes?), then we did completing the square, then finally we completed the square on a generic quadratic and found the formula - which we then practically memorised by applying it to pages and pages of exercises. I think this is a fairly good way of doing it, particularly the last couple of steps - the things I need to remember usually worked best when I've both proved them and used them a lot. What's more, often when teachers said "you need to know the proof of X and how it works" it was because in the exam they asked for a proof of X', which is mostly like the proof for X but slightly different, and in working out what the difference was I often understood X and the proof for it better too.
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Re: Should we make students memorize the quadratic formula?

Postby PM 2Ring » Wed Sep 21, 2011 2:03 am UTC

ConMan wrote:We went from solving simple quadratic equations by what I think LolTheForce was calling the pq method (find numbers that multiply to give c and sum/difference to give b, yes?), then we did completing the square, then finally we completed the square on a generic quadratic and found the formula - which we then practically memorised by applying it to pages and pages of exercises. I think this is a fairly good way of doing it, particularly the last couple of steps - the things I need to remember usually worked best when I've both proved them and used them a lot. What's more, often when teachers said "you need to know the proof of X and how it works" it was because in the exam they asked for a proof of X', which is mostly like the proof for X but slightly different, and in working out what the difference was I often understood X and the proof for it better too.

What ConMan said, which is probably not a coincidence, since we both learned mathematics in Australia.

I stumbled across the quadratic formula in the Time/Life Mathematics book when I was around 10 or 11. That book gave no derivation or explanation for the formula, so for a few years I treated the quadratic formula as a magical incantation. I was very happy when I was taught how to derive it. :)

These days, I rarely use the quadratic formula. For simple quadratics, I first quickly check to see if it's factorisable, and if it's not I usually complete the square. I agree with skullturf that it's less error-prone than plugging numbers (or expressions) into the quadratic formula, but YMMV. And if your quadratic has various algebraic expressions for its coefficients you're more likely to spot a factorization or other useful pattern via completing the square than by use of the quadratic formula.

FWIW, completing the square is discussed in this old thread: Is it just me, or does the average guy really suck at math?

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Re: Should we make students memorize the quadratic formula?

Postby mister k » Wed Sep 21, 2011 10:21 am UTC

I really disliked the way I was taught it, given a formula from on high and only much later using completing the square to derive it. Doing things like presenting a magic formula encourages children to stop thinking, and assume mathematics comes from a weird place that makes little sense. Now while its probably impossible to prove every result that is used (some trignometric results might be a bit too hard), completing the square is a relatively simple algebraic rearrangment, and shows that this magical result isn't magical at all.
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Re: Should we make students memorize the quadratic formula?

Postby mfb » Wed Sep 21, 2011 4:39 pm UTC

mister k wrote:completing the square is a relatively simple algebraic rearrangment, and shows that this magical result isn't magical at all.

... and should be done before using the formula. Square roots must be available to show the formula anyway so all concepts necessary for the derivation are available.

ax^2 + bx + c = 0
[...]
x = (- b +/- sqrt( b^2 - 4ac ) ) /2a

That is the way my teacher first presented the formula (with filled [...] of course) back in school, so there is no magic and it is clear that it is a useful - but not the only - way to get the solutions.


>> In fact, I cannot recall using it at any time after high school where I wasn't either allowed to write it down, or use a calculator.
I used it several times. On paper, with numbers and with variables, sometimes for science sometimes for other stuff. And it is always quicker than asking a computer about solutions for equations like [imath]R_D + \sqrt{R_D}\cdot y \cdot t + \frac{x^2+y^2}{4}\cdot t^2=c[/imath].

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Re: Should we make students memorize the quadratic formula?

Postby Eagle Eye » Thu Sep 22, 2011 12:40 am UTC

Apparently I'm one of the few in this thread who have actually had the quadratic formula proved to them in school. It's not a useless formula and some people have a preference for one method over another, so it's useful to teach kids both methods and let them choose which one they want to use. One benefit of learning the quadratic formula in middle school for me was being able to use the discriminant to quickly determine if a problem has any real roots at all, so on a worksheet, instead of doing the whole process of completing the square, I was able to almost immediately say that there is not real solution in those situations, which saved time.

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Re: Should we make students memorize the quadratic formula?

Postby ++$_ » Fri Sep 23, 2011 4:07 am UTC

I don't think kids should be required to memorize the quadratic formula. However, if you show them the quadratic formula (which you should), then they will memorize it all by themselves. It is the kind of thing that middle- and high-schoolers love more than anything else -- a mechanical formula that solves their homework and exam problems for them.

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Re: Should we make students memorize the quadratic formula?

Postby Krealr » Fri Sep 23, 2011 3:11 pm UTC

++$_ wrote:It is the kind of thing that middle- and high-schoolers love more than anything else -- a mechanical formula that solves their homework and exam problems for them.


The bizarre thing for me is that this is not what I see at the high school I work at. Kids HATE the quadratic formula they would rather solve by factoring (which of course is not always practical).

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Re: Should we make students memorize the quadratic formula?

Postby skullturf » Fri Sep 23, 2011 4:30 pm UTC

My memory of my classmates, when we were about 14 or so, is that students like the idea of a "magic" formula where they can just "plug and chug", but they didn't like being required to memorize the quadratic formula. It's unwieldy and hard to remember at first.

However, some things can be a little bit ugly or unwieldy, but still worth memorizing.

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Re: Should we make students memorize the quadratic formula?

Postby cphite » Fri Sep 23, 2011 7:20 pm UTC

skullturf wrote:It can be a useful time-saver to be able to quickly write it down without thinking. And in about Grade 9, I remember the teacher drilling the formula into us, having us recite it in words: "the opposite of b, plus or minus the square root of, b squared minus 4ac, all over 2a."


Students in the maths or sciences, sure; for the rest of us, why bother?

In the fifteen years since I graduated from college, not once have I had any use for the quadratic formula. I would wager that the vast majority of the population can say the same. I understand that it's an important thing for folks who use math in their work; but for me, it's utterly irrelevant. I cannot imagine a situation where I would need to use it, much less one where I'd actually need to have it memorized.

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Re: Should we make students memorize the quadratic formula?

Postby gorcee » Fri Sep 23, 2011 7:25 pm UTC

cphite wrote:
skullturf wrote:It can be a useful time-saver to be able to quickly write it down without thinking. And in about Grade 9, I remember the teacher drilling the formula into us, having us recite it in words: "the opposite of b, plus or minus the square root of, b squared minus 4ac, all over 2a."


Students in the maths or sciences, sure; for the rest of us, why bother?

In the fifteen years since I graduated from college, not once have I had any use for the quadratic formula. I would wager that the vast majority of the population can say the same. I understand that it's an important thing for folks who use math in their work; but for me, it's utterly irrelevant. I cannot imagine a situation where I would need to use it, much less one where I'd actually need to have it memorized.


In all the years since I've graduated from high school, I've never once needed to apply the history of Mesopotamia to my life. Why, then, did I learn about it in 6th grade, 8th grade, and 9th grade? I've never once had to make use of the fact that the stages of cell division are Interphase, Metaphase, Prophase, Anaphase and Telophase, so why do I still remember "I Pack My Apples Tightly?" I've never once found any practical use for the ceramic work I did in art class, and I don't even sing Karaoke while drunk, so why did I have mandatory choir through middle school?

Why bother?

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Re: Should we make students memorize the quadratic formula?

Postby Talith » Sun Sep 25, 2011 1:04 am UTC

gorcee wrote:[snip]

In all the years since I've graduated from high school, I've never once needed to apply the history of Mesopotamia to my life. Why, then, did I learn about it in 6th grade, 8th grade, and 9th grade? I've never once had to make use of the fact that the stages of cell division are Interphase, Metaphase, Prophase, Anaphase and Telophase, so why do I still remember "I Pack My Apples Tightly?" I've never once found any practical use for the ceramic work I did in art class, and I don't even sing Karaoke while drunk, so why did I have mandatory choir through middle school?

Why bother?

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Re: Should we make students memorize the quadratic formula?

Postby Superisis » Sun Sep 25, 2011 8:02 pm UTC

Why memorize it? I mean, sure if you want to, but if you know completing the square you can always derive it by completing x^2+ax+b = 0 (switch a and b for p and q, etc). I know it sorta, but I can always get it if I need it, the same goes for the quotient rule when doing derivativies. If you can derive f(x) = g(x)*h(x), why memorize the short cut for f(x) = g(x)/h(x). I'm not saying it's bad if you want to, but it shouldn't be forced on students who might feel they're learning too many new things already.

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Re: Should we make students memorize the quadratic formula?

Postby gmalivuk » Sun Sep 25, 2011 8:04 pm UTC

Superisis wrote:Why memorize it? I mean, sure if you want to, but if you know completing the square you can always derive it by completing x^2+ax+b = 0 (switch a and b for p and q, etc). I know it sorta, but I can always get it if I need it, the same goes for the quotient rule when doing derivativies. If you can derive f(x) = g(x)*h(x), why memorize the short cut for f(x) = g(x)/h(x).
gmalivuk wrote:I see it as similar to memorizing multiplication tables or tables of basic integrals or the mean and variance of common distributions. Sure, it's possible to compute all those things from first principles every time, but whenever you actually have to use it, you're probably not going to have time to do so.
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Re: Should we make students memorize the quadratic formula?

Postby Superisis » Sun Sep 25, 2011 8:27 pm UTC

gmalivuk wrote:I see it as similar to memorizing multiplication tables or tables of basic integrals or the mean and variance of common distributions. Sure, it's possible to compute all those things from first principles every time, but whenever you actually have to use it, you're probably not going to have time to do so.


And thus it depends on usage. I'd wager that the multiplication table is used a lot more often than most other things, and I consider myself lucky that I memorized it at an early age (before I started school, which made math classes for the first 5-6 years really easy and confidence building). whenever I've had to do a lot of intergrals or quadratic equations, etc I've sorta guessed the formula, derived it (or looked it up) once and remembered it for the rest of the test/period/class. But after that it fades, and I frankly don't see the problem with it. I don't feel the need to have it at my finger tips (compared to other things, and my brain capacity being limited), nor do I see why it should be at the average students (compared to other concepts in maths).

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Re: Should we make students memorize the quadratic formula?

Postby cphite » Fri Sep 30, 2011 5:36 pm UTC

gorcee wrote:
cphite wrote:
skullturf wrote:It can be a useful time-saver to be able to quickly write it down without thinking. And in about Grade 9, I remember the teacher drilling the formula into us, having us recite it in words: "the opposite of b, plus or minus the square root of, b squared minus 4ac, all over 2a."


Students in the maths or sciences, sure; for the rest of us, why bother?

In the fifteen years since I graduated from college, not once have I had any use for the quadratic formula. I would wager that the vast majority of the population can say the same. I understand that it's an important thing for folks who use math in their work; but for me, it's utterly irrelevant. I cannot imagine a situation where I would need to use it, much less one where I'd actually need to have it memorized.


In all the years since I've graduated from high school, I've never once needed to apply the history of Mesopotamia to my life. Why, then, did I learn about it in 6th grade, 8th grade, and 9th grade? I've never once had to make use of the fact that the stages of cell division are Interphase, Metaphase, Prophase, Anaphase and Telophase, so why do I still remember "I Pack My Apples Tightly?" I've never once found any practical use for the ceramic work I did in art class, and I don't even sing Karaoke while drunk, so why did I have mandatory choir through middle school?

Why bother?


I'm not saying people shouldn't learn the formula - just that it's not all that valuable to memorize it, especially for those of us who aren't going to actually use it. If I need the quadratic formula, I can look it up online, or grab a math book. I don't need to know it from memory. It just seems to me far more valuable to spend the time learning how to actually apply formulas than memorizing them.

The same sort of thing applies to a subject like history. If all you gained from your history class was the ability to repeat a bunch of facts about Mesopotamia from memory, then yeah; why bother. If, on the other hand, you actually took the time to learn about Mesopotamia and the influence that the various cultures from that region have influenced the world, that may be more worthwhile.

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Re: Should we make students memorize the quadratic formula?

Postby dean.menezes » Fri Sep 30, 2011 11:34 pm UTC

What about solving quadratics using this method?
[imath]a x^2 + bx + c = 0[/imath]
[imath](x - z_1) (x - z_2) = 0[/imath]
[imath]z_1 + z_2 = -b/a[/imath]
[imath]z_1 * z_2 = c/a[/imath]
[imath]z_1^2 + z_2 ^2 + 2 z_1 z_2 = b^2 / a^2[/imath]
[imath]z_1^2 + z_2^2 - 2 z_1 z_2 = (b^2 - 4ac) / a^2[/imath]
[imath](z_1 - z_2)^2 = (b^2 - 4ac) / a^2[/imath]
[imath]z_1 - z_2 = \sqrt{b^2 - 4ac} / a[/imath] (arbitrarily chosen root)
so
[imath]z_1 = (-b + \sqrt{b^2 - 4ac}) / (2a)[/imath]
[imath]z_2 = (-b - \sqrt{b^2 - 4ac}) / (2a)[/imath]

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Re: Should we make students memorize the quadratic formula?

Postby Tirian » Fri Sep 30, 2011 11:42 pm UTC

That's a sweet derivation. I think you owe the reader an a>0 and a z1>=z2 to make that square root work out without absolute values, but it's still so much more direct than the completing the squares derivation. If I'm ever forced to do this for a class (and I suppose I will be someday :? ) I'm taking this path.

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Re: Should we make students memorize the quadratic formula?

Postby gorcee » Sat Oct 01, 2011 12:26 am UTC

dean.menezes wrote:What about solving quadratics using this method?
[imath]a x^2 + bx + c = 0[/imath]
[imath](x - z_1) (x - z_2) = 0[/imath]
[imath]z_1 + z_2 = -b/a[/imath]
[imath]z_1 * z_2 = c/a[/imath]
[imath]z_1^2 + z_2 ^2 + 2 z_1 z_2 = b^2 / a^2[/imath]
[imath]z_1^2 + z_2^2 - 2 z_1 z_2 = (b^2 - 4ac) / a^2[/imath]
[imath](z_1 - z_2)^2 = (b^2 - 4ac) / a^2[/imath]
[imath]z_1 - z_2 = \sqrt{b^2 - 4ac} / a[/imath] (arbitrarily chosen root)
so
[imath]z_1 = (-b + \sqrt{b^2 - 4ac}) / (2a)[/imath]
[imath]z_2 = (-b - \sqrt{b^2 - 4ac}) / (2a)[/imath]


Sure that's way easier to compute than memorizing the formula.

The arguments saying that memorization precludes understanding are silly. There is really only one concept that relates to the quadratic formula: the notion of roots of a function. That's the important thing. That's what the QF is for, and it's not really useful for a whole lot else.

So we should be teaching students about the importance of roots of a function. And then we should teach them about quadratic functions. And then we can teach them how to derive the roots of a quadratic, ending with memorization of the QF.

If you memorize the QF and you have no idea what it's for, it does you no good. If you know what roots are but don't know how to get the roots of a quadratic, you're missing out, too. All of the algebraic trickery is fine and good for more advanced problem solving -- and it should be taught, absolutely.

But the students who are "never going to use" the QF similarly are never going to have to complete the square. And the notion of "set this to an arbitrary root" is a concept that makes sense to people who study math, but it is basically black fucking magic to a 13 year old.

Either way, algebraic derivation of the roots of polynomials is a little silly. Quadratics make sense because there are so many real-world quadratic problems. But I dare you to memorize the general closed form formula for the roots of a cubic.

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Re: Should we make students memorize the quadratic formula?

Postby doogly » Sun Oct 02, 2011 3:45 am UTC

cphite wrote:I'm not saying people shouldn't learn the formula - just that it's not all that valuable to memorize it, especially for those of us who aren't going to actually use it. If I need the quadratic formula, I can look it up online, or grab a math book. I don't need to know it from memory. It just seems to me far more valuable to spend the time learning how to actually apply formulas than memorizing them.

The same sort of thing applies to a subject like history. If all you gained from your history class was the ability to repeat a bunch of facts about Mesopotamia from memory, then yeah; why bother. If, on the other hand, you actually took the time to learn about Mesopotamia and the influence that the various cultures from that region have influenced the world, that may be more worthwhile.

If you think you know how the various Mesopotamian cultures influenced the world, but you can't actually locate them in space and time, you are probably not going to say anything intelligent. If you think you can solve problems with the quadratic formula without knowing what it is, you are not going to solve anything difficult or interesting. The student who brings a formula sheet to an exam of mine will do worse than one who does not have one, even if they exam is exactly the same.
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Re: Should we make students memorize the quadratic formula?

Postby Qaanol » Sun Oct 02, 2011 5:50 pm UTC

gorcee wrote:But I dare you to memorize the general closed form formula for the roots of a cubic.

That’s actually pretty easy once you see it written out. Just notice all the symmetries among the terms.
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Re: Should we make students memorize the quadratic formula?

Postby Eastwinn » Sun Oct 02, 2011 11:49 pm UTC

I think it's a bit much to make students memorize it. Just have it written at the beginning of every test for reference. It's a quick Google to find it, if they should ever need it later in life.
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Re: Should we make students memorize the quadratic formula?

Postby Yesila » Mon Oct 03, 2011 2:41 am UTC

Eastwinn wrote:I think it's a bit much to make students memorize it.


A bit much!?! It's in the neighborhood of 20 words if you speak it. I remember being asked to memorize mush much longer things in my Middle school and High school English classes, History classes, Biology... Heck I even had to memorize rules for field hockey in PE class... Calling this "A bit much" (to memorize) is ridiculous.

Eastwinn wrote:Just have it written at the beginning of every test for reference.


And lets include a copy of Moby Dick at the top of my literature test, a copy of the periodic table for my chemistry class, The constitution for my Government class, A french-english dictionary for my french class, A labeled map in Geography and while we're at it lets have some one else do my push-ups in gym.

Eastwinn wrote:It's a quick Google to find it, if they should ever need it later in life.


As is most of Jr High and Sr High Math. But why bother looking up a formula on line? Sites like wolfram alpha can be used to solve most (if not all) math problems that happen in those classes. Let's just not teach math at all anymore to non-scientists, after all they won't ever need it.

For that matter let's clean up the rest of the school system too. I've never needed my knowledge of Dostoyevsky, Poe, Whitman, Joyce or Homer so English class can go. I could give a rat's ass about... Rat's asses and the rest of biology so out that goes too. My knowledge of the rules of field hockey, rugby, football and golf certainly aren't being used in my everyday life, so send Gym class out the window. The level at which a couple of years of language classes gets you to isn't enough to even remotely make myself understood --- Besides I think my iphone has an app for that -- Goodbye high school Spanish, french, mandarin, and Russian! If only I'd taken a Home Eq class, there is finally one that's useful, cause I eat, so I have to cook. And I wear clothes and sometimes I need to stitch up a hole or sew on a button... Oh wait, I have all those skills without that class! We don't need that one either!!! Our whole education system is Bunk! Buncha' worthless classes, what a waste. I can look up anything I need online.... should I ever need it later in life.

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Re: Should we make students memorize the quadratic formula?

Postby Dopefish » Mon Oct 03, 2011 3:20 am UTC

Should it be memorised? Yes.

Should you make them memorise it? No.

As with most of the things mentioned here, they're all potentially useful things that people should learn, even if they might not actually need it later on. However, they're largely things that should be memorised inadvertantly through regular use on assignments and the like, not something that should be blindly drilled in solely for the sake of memorisation.

Yesila wrote:And lets include a copy of Moby Dick at the top of my literature test, a copy of the periodic table for my chemistry class, The constitution for my Government class, A french-english dictionary for my french class, A labeled map in Geography and while we're at it lets have some one else do my push-ups in gym.


With the exception of the last one, I actually feel like those should be fair game. School should be about learning, not about memorisation, and I strongly feel like a great number of students who are required to memorise material for a test only retain it for a relatively short period following the actual test. If someone really has learned the material, they'll require relatively light use of those resources, and when they do need to check something, they'll likely know exactly where in the resource to look. An unprepared student can have all those resources, and will still do quite poorly since they'll need to look up every little thing and may not even do so efficiently, and will run out of time before the test is completed.

I don't think a students memory should be a significant factor is their test scores, but rather their understanding of the material. Sure, there are some situations where instant memory recall is the point and so it may be appropriate to drill on memory, but I think in the vast majority of cases memory recall isn't the point.

To take the chemistry example since I'm most familiar with it, I haven't a clue what the atomic number of Technetium is, or where exactly it is on the periodic table, and so if you ask me a chem question about the properties of Technetium then I'm screwed. On the other hand, if you ask me the propeties of mystery element X which has 43 protons and is in the the 7th group (i.e. in the transition metals; I know some folk call the halogens the 7th group instead, but not the point), then I'd be able to infer plenty of properties and be able to answer a fairly wide range of chemistry questions about X. It happens that X is Technetium, but my knowing that is unimportant to my ability to do chemistry. Someone else might spend their time memorising the table and thus be able to recall that Technetium is element 43, but not be able to do anything with it. So while I'm probably the more capable chemist by most standards there, on a test that only refered to Technetium and assumed a memory of the table, I would do worse then the mostly incompetent person who simply knew the periodic table by heart but without any meaning behind it.

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Re: Should we make students memorize the quadratic formula?

Postby Yesila » Mon Oct 03, 2011 4:18 am UTC

Dopefish wrote:
With the exception of the last one, I actually feel like those should be fair game.


I actually agree with you too. I was just going to a bit of an absurdest level with a number of things we are asked to memorize (or partially memorize-- I hope no one had to memorize moby dick.) in school.

My little rant above though did get me to a spot where I can finely have my own opinion on whether or not we should ask students to memorize the quadratic formula. And I think we should not. At the same time we should defiantly not give it to them on the top of a test. It shouldn't be part of the "common" curriculum. The QF is a wonderful example of a piece of math that a student that who is sick and tired of factoring, and completing the square, can come up with on their own if they want a "time saving/effort saving" device for solving routine problems. And for those students that are not creatively inclined to come up with it on their own there is a place to teach that to them, but not in 8th grade algebra or in their high school algebra 2 class... maybe not in high school at all.

Spoiler:
I've seen too many students in remedial college courses that learned tricks in high school, and time-savers, and silly songs and acronyms... and those can be great... but not too early in their mathematical lives, not before they've "earned the right" to use them. They couldn't do the basics reliably, they "learned" the tricks, which they then couldn't use correctly most of the time -- BUT it felt easier so that is all they want to use now.
*They "remember" FOIL, but they can't distribute so they mess up the simple products, they get confused and "stuck" on products of non-binomials, and when I ask them if OLIF works just as accurately as FOIL they say no.
*They try to "cross multiply" when they have equal ratios but don't know how to multiply both sides of an equation by a common denominator to handle such things as 4/x = 17+x since the right hand side "isn't" a fraction and they "can't" cross-multiply.
*They use the quadratic formula to "solve" every quadratic equation, so solving (x+1)(3x-5)=0 becomes a multi-step process involving "multiplying out" and "pluging-in". Worse yet since what you "do" when you see a quadratic is "use the QF" when asked to factor some quadratic they dig out the QF and tell me x=... is the answer.

They know of the existence of 40 or 50 such tricks and time savers... but they don't remember them correctly, or know when to use them, or even how to use them in some cases because they got these "tricks" introduced to them before they mastered a very few fundamentals.

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Re: Should we make students memorize the quadratic formula?

Postby Qaanol » Mon Oct 03, 2011 4:51 am UTC

Yesila wrote:I can finely have my own opinion

Yesila wrote:we should defiantly not give it to them


Either you are making a very subtle point, or one of us needs to spend more time memorizing vocab words.
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Re: Should we make students memorize the quadratic formula?

Postby Yesila » Mon Oct 03, 2011 5:57 am UTC

Qaanol wrote:
Yesila wrote:I can finely have my own opinion

Yesila wrote:we should defiantly not give it to them


Either you are making a very subtle point, or one of us needs to spend more time memorizing vocab words.



Oh the later for sure. I'm "definitely" as bad at memorizing vocab as my students are at memorizing the quadratic formula -- and just as "defiantly" I'll still (try to) use it even though I know I shouldn't.

Spoiler:
Damn spell check not giving me the choice I wanted at the top of the list -- at least with definitely, I know I spelled that really wrong and then clicked a stupid choice. I think on finally I just used the homophone on my own.


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