Should we make students memorize the quadratic formula?

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gorcee
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Re: Should we make students memorize the quadratic formula?

Postby gorcee » Mon Oct 03, 2011 2:13 pm UTC

Memorization and learning are not mutually exclusive.

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Re: Should we make students memorize the quadratic formula?

Postby skullturf » Mon Oct 03, 2011 2:37 pm UTC

Yesila wrote: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.


I'm with you there. I *hate* the FOIL acronym.

As you point out, "FOIL" suggests that there's something special about that order. It could just as well be FIOL or LIOF or OLIF.

And as you also point out, "FOIL" doesn't explain what to do if your sums have more than two terms. Over-reliance on the "FOIL" acronym might cause students not to know how to expand (a+b)(x+y+z) or (a+b+c)(w+x+y+z). (Even worse, what if they "foil" the latter to get aw+az+cw+cz??)

I realize that this, like some other things, is partly an aesthetic judgment, but to me, "FOIL" does a very BAD job of capturing the "essence" of what we do when we distribute. In my opinion, we should stress: "you have to multiply every term in the first brackets times every term in the second brackets."

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Re: Should we make students memorize the quadratic formula?

Postby Nat » Mon Oct 03, 2011 10:37 pm UTC

Probably not. We shouldn't even spend so much time on quadratic equations, they're neither interesting nor useful. To go with the differential equations example, quadratic equations are a very specific kind of equation that you almost never actually see, so there's no point in teaching tricks to solve it.

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Re: Should we make students memorize the quadratic formula?

Postby gorcee » Tue Oct 04, 2011 12:15 am UTC

Nat wrote:Probably not. We shouldn't even spend so much time on quadratic equations, they're neither interesting nor useful. To go with the differential equations example, quadratic equations are a very specific kind of equation that you almost never actually see, so there's no point in teaching tricks to solve it.


Actually, linear second order equations are used to model quite a great number of structural dynamics problems.

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Re: Should we make students memorize the quadratic formula?

Postby Marbas » Tue Oct 04, 2011 12:25 am UTC

I realize that this, like some other things, is partly an aesthetic judgment, but to me, "FOIL" does a very BAD job of capturing the "essence" of what we do when we distribute. In my opinion, we should stress: "you have to multiply every term in the first brackets times every term in the second brackets."


I very much preferred what my Algebra 2 teacher did instead of FOIL. I believe it was called the box method. Basically, you'd make a box, give it one row for every element in one of the monomials, one column for each element in the other, and then you'd write each entry above one of the rows or columns. Then you'd multiply by the edges of each box, then write the result down inside. Then you'd add all these resulting numbers/things up.

I am very bad at explaining things.
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Re: Should we make students memorize the quadratic formula?

Postby PM 2Ring » Tue Oct 04, 2011 12:40 am UTC

Marbas wrote:
I realize that this, like some other things, is partly an aesthetic judgment, but to me, "FOIL" does a very BAD job of capturing the "essence" of what we do when we distribute. In my opinion, we should stress: "you have to multiply every term in the first brackets times every term in the second brackets."


I very much preferred what my Algebra 2 teacher did instead of FOIL. I believe it was called the box method. Basically, you'd make a box, give it one row for every element in one of the monomials, one column for each element in the other, and then you'd write each entry above one of the rows or columns. Then you'd multiply by the edges of each box, then write the result down inside. Then you'd add all these resulting numbers/things up.

I am very bad at explaining things.

That sounds sensible.

Your explanation was a little vague, but I'm sure most of us reading the thread understood what you meant. FWIW, the technical name of finding a box of products like that is the Cartesian product or direct product.

http://en.wikipedia.org/wiki/Cartesian_product

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Re: Should we make students memorize the quadratic formula?

Postby Nat » Wed Oct 05, 2011 11:06 am UTC

gorcee wrote:
Nat wrote:Probably not. We shouldn't even spend so much time on quadratic equations, they're neither interesting nor useful. To go with the differential equations example, quadratic equations are a very specific kind of equation that you almost never actually see, so there's no point in teaching tricks to solve it.


Actually, linear second order equations are used to model quite a great number of structural dynamics problems.

Yeah, they do come up, and they're important for engineers (and maybe even high school physics), but they're not a subject in themselves. The point of high school math should be to show kids how to think, not force them to memorize formulas.

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Re: Should we make students memorize the quadratic formula?

Postby mister k » Wed Oct 05, 2011 11:50 am UTC

Nat wrote:
gorcee wrote:
Nat wrote:Probably not. We shouldn't even spend so much time on quadratic equations, they're neither interesting nor useful. To go with the differential equations example, quadratic equations are a very specific kind of equation that you almost never actually see, so there's no point in teaching tricks to solve it.


Actually, linear second order equations are used to model quite a great number of structural dynamics problems.

Yeah, they do come up, and they're important for engineers (and maybe even high school physics), but they're not a subject in themselves. The point of high school math should be to show kids how to think, not force them to memorize formulas.



Should it? Should a maths class be about teaching children to think rather than teaching them maths? Ideally I'd go for both.
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Re: Should we make students memorize the quadratic formula?

Postby doogly » Wed Oct 05, 2011 12:30 pm UTC

"Thinking" is an extremely vague thing to say is your goal. Maybe you want them to think about things, maybe? Things that are algebraic and geometric? You need to think with some tools.
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Re: Should we make students memorize the quadratic formula?

Postby WarDaft » Wed Oct 05, 2011 12:33 pm UTC

You're not teaching them math if you're not teaching them to think. You're just teaching them assorted useful formulas, and trying to make a batch of (largely ineffectual) human calculators.

I do agree that 'thinking' is vague, though in this context we could safely interpret it to mean that we are teaching them how to look for solutions to problems that they don't actually know how to solve yet.
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Re: Should we make students memorize the quadratic formula?

Postby gorcee » Wed Oct 05, 2011 1:33 pm UTC

Holy fuck, people, you can do both.

Memorization is not done to the exclusion of critical thinking. Memorization can aid critical thinking because it reduces the complexity of solving many types of problems.

You can teach process all you want, but if it takes a student 10 minutes to complete the square, that's not solving anything.

You can teach memorization all you want, but if you're just blindly throwing formulae around, that's not solving anything either.

And in the end, remembering the process to, say, complete the square requires just as much memorization as the quadratic formula does. I deal with second order approximations all the time and never once have I completed the square to do a back of the envelope calculation. Why not? I don't remember how to do it. I could figure it out, sure. Or, I could just use the quadratic formula, which takes up far less bar napkin space. The quadratic formula is remarkably useful in quickly answering relevant questions (does this 2x2 block have eigenvalues with real parts? Do I need to code for complex numbers given my expected parameter space?)

So, this thread has a really simple answer: Yes, we should make students memorize it... and no, that should not be the only way the material is presented. It's really not that complicated.

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Re: Should we make students memorize the quadratic formula?

Postby yurell » Wed Oct 05, 2011 1:44 pm UTC

I had to look up 'FOIL' ... you guys are really taught that as an acronym? I found we stopped with those after BODMAS in grade 6, and from then on we were told to use the distributive law (although we didn't call it that for some obscure reason).
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Re: Should we make students memorize the quadratic formula?

Postby gorcee » Wed Oct 05, 2011 2:23 pm UTC

yurell wrote:I had to look up 'FOIL' ... you guys are really taught that as an acronym? I found we stopped with those after BODMAS in grade 6, and from then on we were told to use the distributive law (although we didn't call it that for some obscure reason).


This is an example where the overuse of mnemonics certainly hurts. Just teach the distributive law, people! And the best way to teach the distributive law is to show basic 2-digit multiplication.

78 * 64? (70 + 8)*(60 + 4).

I mean, we already teach this in 3rd grade when learning how to multiply two numbers. We just arrange it differently.

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Re: Should we make students memorize the quadratic formula?

Postby gmalivuk » Wed Oct 05, 2011 2:39 pm UTC

gorcee wrote:I mean, we already teach this in 3rd grade when learning how to multiply two numbers. We just arrange it differently.
I think this is a good way to present "new" material: show how students have already been using the algorithm.

Then again, some of the people objecting to learning the quadratic formula likely also object to "forcing" students to remember long division and multiplication in the first place...
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Re: Should we make students memorize the quadratic formula?

Postby Qaanol » Wed Oct 05, 2011 9:35 pm UTC

Indeed, the simplest way to introduce algebra is to inform students it is nothing more than arithmetic with letters.
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Re: Should we make students memorize the quadratic formula?

Postby gmalivuk » Wed Oct 05, 2011 9:39 pm UTC

Well I meant more in terms of specific algorithms. Distributivity made sense of the multiplication algorithm I'd been using for years, and synthetic division of polynomials is just long division with a bit more freedom of base and notation.
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Re: Should we make students memorize the quadratic formula?

Postby Yesila » Thu Oct 06, 2011 5:46 am UTC

yurell wrote:I had to look up 'FOIL' ... you guys are really taught that as an acronym? I found we stopped with those after BODMAS in grade 6, and from then on we were told to use the distributive law (although we didn't call it that for some obscure reason).



And I had to look up BODMAS :) Which it appears is just a cultural divide from PEDMAS.... which I also had to look up a few years ago when some of my students started writing that on their tests.
Spoiler:
And then I started wondering why they still found it necessary to remind themselves of this in a written fashion at the top of every test, 10+ years after they first learned it... while at the same time not instead writing down reminders of the quotient rule, or whatever topic we happened to have just covered.
These "short" acronyms I never really understood (the benefit of) anyway. Given some long complicated string of words, or numbers I can see some benefit to trying to make an easy to remember catchy phase, or words to help recall the long complicated mess... But with a string of 4-8 words... isn't just memorizing the words something that takes almost the exact same amount of effort? I don't think my primary school teachers ever tried to teach us math stuffs with acronyms.... but then again maybe I felt the same way about the topic then as I do now and just refused to pay attention to something I thought was silly and redundant (since they had already taught us the "words" and I saw no need for a memorization"trick".)

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Re: Should we make students memorize the quadratic formula?

Postby Afif_D » Thu Oct 06, 2011 11:53 am UTC

Actually its a good thing to memorize the formula as time management is a critical factor during exams. However students should know both the ways and follow wat is easy.
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Re: Should we make students memorize the quadratic formula?

Postby gorcee » Thu Oct 06, 2011 1:45 pm UTC

On a side note, I was taught FOIL (among other things; it wasn't the principal method of teaching binomial multiplication). I still use FOIL, but I extend the I. I also pretend I'm yelling it like Kirk yelled KHAAAN!

(a+b+c+d+e)*(x+y+z)?

FOIIIIL!

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Re: Should we make students memorize the quadratic formula?

Postby Tirian » Thu Oct 06, 2011 4:05 pm UTC

I don't get why people are getting such a bee in their bonnet over FOIL. I think it's nice while you're learning it to have a disciplined approach for getting all four products exactly once each, and it does reinforce the general method of multiplying each term of the second polynomial by the first term of the first polynomial and so on. I think another nice feature about FOIL is that when you get distracted in the middle of a problem (which happens to neophytes significantly more often than us, I'd wager) and come back to see that there are already two terms of the product written down, you immediately know that the next term is supposed to be the inside products without having to backtrack. Once you master it, you can leave it behind, whatevs.

As to memorizing nonsense words versus phrases, it seems to work either way depending on what you're comfortable with. At least in my region, Please Excuse My Dear Aunt Sally is far more popular than remembering the "word" PEDMAS, but most of the trig students I tutor seem to remember SOHCAHTOA where I learned it as Some Old Horse Caught Another Horse Taking Oats Away. I'll leave it to the behavioral psychologists to sort out.

My only constant is that you have to teach the fundamentals. I don't mind if you have one mnemonic or another or something totally different to remember the order of operations, but it's a crime how many of my students come in not realizing that multiplication and division have equal rank in the actual order.

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Re: Should we make students memorize the quadratic formula?

Postby skullturf » Fri Oct 07, 2011 3:15 am UTC

Tirian wrote:I don't mind if you have one mnemonic or another or something totally different to remember the order of operations, but it's a crime how many of my students come in not realizing that multiplication and division have equal rank in the actual order.


That's actually one of my complaints about PEDMAS or BODMAS. Those acronyms create the illusion that division is done "before" multiplication, or addition "before" subtraction.

It should really be PEMA or BOMA, with the understanding that division "is" multiplication, and subtraction "is" addition.

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Re: Should we make students memorize the quadratic formula?

Postby Deveno » Sat Oct 08, 2011 4:39 pm UTC

no. we should make students derive it. if after doing so, they choose to memorize it to spare themselves the agony of re-deriving it in the future, well...ok.


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