The Quadratic Formula

[Sooga]

For those of you who don't know, I'm a high school math teacher.

Yesterday we had a brief math department meeting, where we all talked about the latest goings-on. Our math department chairperson was complaining about one of her Calculus kids who had the equation c^2 - 5 = 0, and solved it using the quadratic formula. This was met with laughs and chuckles from the rest of the teachers. But not me.

I asked her, 'Is that really so bad?'

She looked at me incredulously, dumbstruck that another teacher wouldn't find that horrible. Look, this is the way I see it. It is a researched fact that some people are simply born with a predisposition for math and think more logically than others. For these people, they see c^2 - 5 = 0, they know to add the 5 over and root both sides. Easy.

For the rest, who just don't 'see' math as well as the others, they can make up for the difference by simply memorizing rules and formulas and steps. The fact that the student above used the quadratic formula probably puts him in this camp. Perhaps he forgot what to do when you have a 2nd degree equation with no 1st degree term. But he did remember what always works for any 2nd degree equation: the quadratic formula. So he used that, and got the answer right. Is that so bad? In fact, when I go over the quadratic formula in my algebra 1 classes, I tell them specifically that certain methods have their limitations (factoring, completing the square, etc.), but quadratic formula will always work (yes I know completing the square will always work, but when you have to deal with fractions, it can get pretty ugly for them).

Basically it comes down to this:

Is the quadratic formula the best way to solve c^2 - 5 = 0? Obviously not.
Will you get the mathematically correct answer using the quadratic formula? Yes.
Would I mark this problem wrong if a student did it this way? No.
Would he get credit on any AP test doing this way? Yes, I would imagine so.

So I don't see what the problem is.

[HedonismBot]

I don't see this student getting much farther than algebra 2. If someone uses the quadratic formula here it seems that they don't understand the "why" behind the math they are doing and are just following formulas.

[Sooga]

Perhaps you didn't catch it in the post, but this kid was in calculus, which was why the teacher was so horrified.

[HedonismBot]

Wow that's surprising coming from somone in Calc. It's possible he just had a brain fart, or maybe he's one of those kids that spends 10X as long on the homework than everyone else.

[chezlaw]

If you think the idea of maths/education is to pass exams then its ok.

If the idea is to understand anything then its a concern unless they're being a smartarse.

chez

[TomCollins]

There's more than one way to solve a problem. Maybe this was the first thing that came to the students head. It was legal math, and the answer was correct. It's an unusual to have solved it this way, but it works.

I've solved things in much more complex ways than needed sometimes, sometimes you just don't see the easiest way when you get stuck in a certain mindset. I don't see the big deal.

No offense, I know your story and you are pretty knowledgeable, but a lot of high school math teachers are jokes (not saying you are!). I really wonder how they would do taking some of these tests.

[guesswest]

I don't see how this is any kind of problem. He did 'understand' the question he just didn't see the quickest solution, but he knew a solution that worked and applied it.

It'd probably be a poor teacher that didn't mention to him he could have solved the equation more efficiently, but inefficient is very different from wrong.

Although I recognize how simple that equation is (must be if I can solve it), I wouldn't rush to judgement on him not 'getting' math. Math is a huge area and most people who do get it have 'blind spots' - he may intuitively understand much more complicated problems but just for one reason or another didn't see this one. We've all missed some incredibly obvious stuff in our time.

[Copernicus]

No big deal, in fact I could see my son doing it this way, especially if it were in a section that included other quadratic formula problems.

[cambraceres]

I'm just curious, how old are you, and what part of the world are you in?

[chezlaw]

[ QUOTE ]
I don't see how this is any kind of problem. He did 'understand' the question he just didn't see the quickest solution, but he knew a solution that worked and applied it.

[/ QUOTE ]
I think its very possible that he doesn't understand. Being able to recognise that its a polynomial with max index 2 and matching it with the tool that solves same isn't understanding anything at all.

Maybe he understood more that that but I think it is cause for concern.

chez

[guesswest]

Well, I think all math is is identifying the meaning of problems and matching tools, the brightest student in the world wouldn't have the first idea how to solve that equation if he didn't have that background knowledge.

Your point is well taken, and it's probably a sufficient red flag that his teacher should mention to him the quicker solution, and try and gauge his understanding. But I don't think it's a huge deal or a necessary indication that he's a mathematical retard.

Most likely he was in a groove in terms of applying the quadratic formula to problems and just continued doing so, he maybe even saw a quicker solution and thought it wasn't what his teacher wanted to see. I certainly remember being in school and 'seeing' the answers to problems then trying to reverse engineer some calculations I didn't actually use (or at least couldn't articulate) so I'd get my work marked as correct.

[cts]

You shouldn't be in calculus if you're using the quadratic formula to solve x^2-5=0.

[maddog2030]

Wow. I can't even believe someone would even bother pointing this out, much less a teacher. The kid most likely was busy worrying about how to solve the bigger problem and just used the first method that popped into his head to solve the quad equation.

The fact that someone would bring this up later and laugh about it often indicates they think they are a lot smarter than they really are, and that they like to stroke their own ego at the expense of others.

[chezlaw]

[ QUOTE ]
Well, I think all math is is identifying the meaning of problems and matching tools, the brightest student in the world wouldn't have the first idea how to solve that equation if he didn't have that background knowledge.

Your point is well taken, and it's probably a sufficient red flag that his teacher should mention to him the quicker solution, and try and gauge his understanding. But I don't think it's a huge deal or a necessary indication that he's a mathematical retard.

Most likely he was in a groove in terms of applying the quadratic formula to problems and just continued doing so, he maybe even saw a quicker solution and thought it wasn't what his teacher wanted to see. I certainly remember being in school and 'seeing' the answers to problems then trying to reverse engineer some calculations I didn't actually use (or at least couldn't articulate) so I'd get my work marked as correct.

[/ QUOTE ]
I agree with most of that but classifying a problem correctly is not the same as understanding what it means.

If its an engineering student just interested in methods to get answers then I'd be less concerned then if it was a maths student.

chez

[chezlaw]

Poker analogy. Someone correctly raises pre-flop and you ask then why they raised.

If they answer 'that's what DS says to do' then it may be that they don't understand why raising is correct.

If you're teaching them Poker II then there is cause for concern.

chez

[guesswest]

Yes, I accept your point and the basic principle you're referring to, I'm just not sure this case is an example of it.

I correctly raise pre-flop frequently without articulating to myself the reasons why it's the correct play. That doesn't mean I couldn't identify those reasons if pressed to deconstruct the situation - it's just auto-pilot a lot of the time.

I often raise incorrectly pre-flop, but lets not think about that /images/graemlins/grin.gif

I don't think this student's application of the quadratic formula is necessarily an indication that he doesn't understand or couldn't deconstruct the underlying principles, it's more likely just auto-pilot. Like I say, it's sufficient that it's worth his teacher mentioning the quicker route and trying to establish his level of understanding, but I don't think it necessarily means much by itself.

[chezlaw]

[ QUOTE ]
Yes, I accept your point and the basic principle you're referring to, I'm just not sure this case is an example of it.

I correctly raise pre-flop frequently without articulating to myself the reasons why it's the correct play. That doesn't mean I couldn't identify those reasons if pressed to deconstruct the situation - it's just auto-pilot a lot of the time.

I often raise incorrectly pre-flop, but lets not think about that /images/graemlins/grin.gif

I don't think this student's application of the quadratic formula is necessarily an indication that he doesn't understand or couldn't deconstruct the underlying principles, it's more likely just auto-pilot. Like I say, it's sufficient that it's worth his teacher mentioning the quicker route and trying to establish his level of understanding, but I don't think it necessarily means much by itself.

[/ QUOTE ]
Yeah, its only cause for concern. I wouldn't have him shot /images/graemlins/smile.gif

chez

[Sooga]

[ QUOTE ]
I'm just curious, how old are you, and what part of the world are you in?

[/ QUOTE ]

I'm 26, I teach in a wealthy suburban area about 30 miles east of downtown Los Angeles

[jason_t]

Next time those teachers laugh just ask them if they know why the fundamental theorem of calculus is true, or why the reals are complete, or how we know that the solution to this problem (the square root of 5) even exists, etc.

[Sooga]

[ QUOTE ]
Wow. I can't even believe someone would even bother pointing this out, much less a teacher. The kid most likely was busy worrying about how to solve the bigger problem and just used the first method that popped into his head to solve the quad equation.

The fact that someone would bring this up later and laugh about it often indicates they think they are a lot smarter than they really are, and that they like to stroke their own ego at the expense of others.

[/ QUOTE ]

This was pretty much my opinion as well. I really doubt that this student had no idea that you could add 5 to both sides and root it, but perhaps he just forgot. Whatever. The important thing is that he DID find a way to solve this equation correctly.

cts's point that he shouldn't be in calculus if he can't do x^2 - 5 = 0 without quad. formula would I think be the teacher's opinion as well. I don't really agree. Yea, this kid pulled out the heavy artillery to solve a very simple problem, but I don't see anything inherently 'wrong' about it. Is it a cause for concern? Maybe... I guess it's like, what if the student had factored x^2 - 5 into (x - sqrt5)(x + sqrt5) = 0 and solved it that way instead? Would that be less or more 'bad' than quadratic equation? Would that be less or more 'bad' than adding 5 to both sides? Personally, I have no idea. That's the beauty of mathematics. There are always so many different ways to solve the same problem that expecting all students to solve problems the same way is kind of dangerous. I dunno, that's just what I feel.

[chezlaw]

[ QUOTE ]
There are always so many different ways to solve the same problem that expecting all students to solve problems the same way is kind of dangerous. I dunno, that's just what I feel.

[/ QUOTE ]
I don't think that's the issue. The cause for concern is that the method he chose may indicate a lack of understanding.

chez

[Sooga]

[ QUOTE ]
[ QUOTE ]
There are always so many different ways to solve the same problem that expecting all students to solve problems the same way is kind of dangerous. I dunno, that's just what I feel.

[/ QUOTE ]
I don't think that's the issue. The cause for concern is that the method he chose may indicate a lack of understanding.

chez

[/ QUOTE ]

It's hard to imagine that a student in calculate wouldn't understand that adding 5 to both sides would work. But you're right, if he doesn't even know how to do that, that is a definite cause for concern.

[atrifix]

I look at c^2 - 5 = 0 and see a second degree polynomial. I wouldn't use the quadratic equation to solve it, but then I also wouldn't use the quadratic equation to solve anything. If you wrote it as c = +/- sqrt(5), I'd see a value for c. But that's just me. I seriously doubt that a student in calculus doesn't know basic arithmetic.

I've often wondered how teachers can find such terrible things to talk about amongst their peers. You'd think they'd talk about research opportunities and teaching methods, but most stories I hear are about kids who ask how long papers have to be and solve problems in a way that's not in the text.

Ask your teachers if they know what "=" means. I doubt most can even provide you with a definition. If they can, ask them if a statue and the sum of its atoms are numerically identical. That ought to shut them up.

[chezlaw]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
There are always so many different ways to solve the same problem that expecting all students to solve problems the same way is kind of dangerous. I dunno, that's just what I feel.

[/ QUOTE ]
I don't think that's the issue. The cause for concern is that the method he chose may indicate a lack of understanding.

chez

[/ QUOTE ]

It's hard to imagine that a student in calculate wouldn't understand that adding 5 to both sides would work. But you're right, if he doesn't even know how to do that, that is a definite cause for concern.

[/ QUOTE ]

This worries me a bit - from your OP

[ QUOTE ]
For the rest, who just don't 'see' math as well as the others, they can make up for the difference by simply memorizing rules and formulas and steps. The fact that the student above used the quadratic formula probably puts him in this camp.

[/ QUOTE ]

Many fall in-between and can be helped to 'see' maths better. It's not a matter for ridule but I think not following it up could be doing the student a disservice.

chez

[Borodog]

[ QUOTE ]
Many fall in-between and can be helped to 'see' maths better. It's not a matter for ridule but I think not following it up could be doing the student a disservice.

[/ QUOTE ]

I agree completely. Problem solving skills can be taught, learned, honed. It's not a matter of you're either born with it or you're not.

[felson]

I would be concerned about this student. A calculus student should have the intuition to solve this problem without resorting to the QF. If this approach is a typical one for him, then his homework will take him a lot longer than is necessary, and his Advanced Placement score will suffer too. It sounds like he would profit from a review of algebra.

Then again, anyone could have a flash of absent-mindedness, so maybe this is an isolated case. And of course ridicule is not warranted.

[Sooga]

Well there's a difference between understanding that you can add 5 to both sides and not remebering to do it, and simply not knowing it altogether. If the student is the former, then i wouldn't be worried. If the student is the latter and in calculus, that's a huge problem.

Plus, what I brought up at the meeting yesterday is I don't know how possible it is to be taught to 'see' math better. I really do think it's one of those things that's inborn, or something that you just need to reach a certain mathematical maturity to where it all just 'clicks'. Some people either don't have that innate ability, or never reach that maturity.

[Sooga]

[ QUOTE ]
[ QUOTE ]
Many fall in-between and can be helped to 'see' maths better. It's not a matter for ridule but I think not following it up could be doing the student a disservice.

[/ QUOTE ]

I agree completely. Problem solving skills can be taught, learned, honed. It's not a matter of you're either born with it or you're not.

[/ QUOTE ]

Absolutely. But let's say you got two kids, one mathemiatcally inclined, student A, and one not so much, student B. Let's say they've had the exact same math teachers. Student A will see that problem and just automatically know to add 5 to both sides. In fact, they might take a few minutes to think of even an alternate way to do it. But student B, even though he'd been taught problem solving strategies, perhaps is not mathematically mature enough to just 'see' math, and instead has to remember the quadratic formula to solve the problem.

So while I agree that problem solving CAN be taught, I don't think that it will have the same effectiveness to everyone.

[mindless]

I taught HS physics for three years before selling out, so I've seen this a lot before as well. I tried to give each student a toolbox of problem solving methods and teach them how to pick the right tool. You can pound a nail into a wall using the base of a screwdriver, and get points for realizing that this approach works, but it's much easier to simply use a hammer. The vast majority of students that left my physics class/your coworkers calculus class will enver use Maxwell's Equations/Green's Theorem/whatever, but hopefully will learn how to solve problems and think clearly from a holistic standpoint.

I'm sure that you run into students that want a cookie-cutter solution to each problem; the quadratic equation is a good example of this for specific students. Occasionally the only way that students can learn the material is by reducing the tools in their box, classifying a problem, and picking one of the few methods they know how to solve it. I feel that those students who are reduced to that method are limiting themselves in the classroom, and more importantly, in the real world when they have to solve more complex problems.

[TomCollins]

I always broke people into two groups - memorizers and problem solvers. It sounds like the student is most likely to be a memorizer. Memorizers rarely understand the concepts behind something, and only can repeat back what they know. Problem solvers know the big concepts, and can figure out things based on these concepts, sometimes reinventing the things the memorizers memorize. Memorizers tend to work a lot harder, but put them in a brand new situation and they are utterly lost.

On the other hand, if the problem was in the middle of the following set of problems:

2c^2 - 5c + 5 = 0
c^2 +3c -2 = 0
c^2 -5 = 0

It makes a lot of sense that they would use the quadratic formula. Once the mind has been conditioned to think a certain way through repitition, it's pretty easy to keep going on that train of thought.

[Sooga]

[ QUOTE ]
I always broke people into two groups - memorizers and problem solvers. It sounds like the student is most likely to be a memorizer. Memorizers rarely understand the concepts behind something, and only can repeat back what they know. Problem solvers know the big concepts, and can figure out things based on these concepts, sometimes reinventing the things the memorizers memorize. Memorizers tend to work a lot harder, but put them in a brand new situation and they are utterly lost.

[/ QUOTE ]

This is exactly what I wanted to say, but couldn't find the words to say it as clearly. I remember a few days ago a friend of mine and I were doing some things on probability, and we got stuck on a problem. We found out later that it was simply an application of the Binomial Theorem, but we had forgotten it. However, we ended up deriving it on our own through other methods. Is it good that we were able to problem solve on our own, and get back what we forgot? Or was it bad that we forgot it in the first place? I'm not sure. I think one could argue either point.

I guess the bottom line though is that I'd prefer to be a problem solver than a memorizer. You're exactly right though - Memorizers usually end up doing a LOT more work than problem solvers.

[JaBlue]

i think its the math department's attitude that is horrifying

[Carl_William]

The Quadratic Formula:

Personally -- I have solved many many math type problems or problems requiring math during my lifetime, and many times after getting a solution I realize that I did it the hard way. I (later on) visualize the problem definition more clearly and realize that there was a better more efficient way to come up with the solution. Usually things like this happen to me because I dive head first into a problem without proper planning or forethought. So if I was the teacher in a situation like this; I would be sympathetic to a young person who did something the hard way. If I were the teacher I would try to explain things so that the student would do better in the future, and explain same in a manner which did not lower the self esteem of the student.

This was a simple problem “C times C minus 5 = 0” – it does not even seem like a problem to me (just that the solution contains two answers). But most of the real life problems are more complex and require proper planning before going off on for the solution. They say there is no such thing as a stupid question; (and sometimes some smart joker replies; “only stupid people who ask them.”) Well, maybe sometimes teachers (stupid or whatever) put stupid questions on their tests. They just want to make the test a little bigger.

[Metric]

R. P. Feynman, on "memorizing formulas" vs. "understanding."


I have a few moments left, so I'd like to make a little speech about the relation of the mathematics to the physics -- which, in fact, was well illustrated by this little example. It will not do to memorize the formulas, and to say to yourself, "I know all the formulas; all I gotta do is figure out how to put 'em in the problem!"

Now, you may succed with this for a while, and the more you work on memorizing the formulas, the longer you'll go on with this method -- but it doesn't work in the end.

You might say, "I'm not gonna believe him, because I've always been successful: that's the way I've always done it; I'm always gonna do it that way."

You are not always going to do it that way: you're going to flunk -- not this year, not next year, but eventually, when you get your job, or something -- you're going to lose along the line somewhere, because physics is an enormously extended thing; there are millions of formulas! It's impossible to remember all the formulas -- it's impossible!

[Aukai]

The student probably just brain farted. Unless there was some known history of this student not understanding things, he probably had just been practicing these problems over and over again, saw c^2 and reverted to the formula without taking a mental step back and "seeing" the problem. It seems like a really easy oversight to make if you've been drilling yourself on the formula.

Just be glad you teach kids that understand that much.

[Sooga]

Well that's the thing. This was a calculus student, so I only assume that the c^2 - 5 = 0 came at the end of some long calculation, and that it wasn't some part of a quadratic formula drill.

[LCposter]

I was thinking this too. After algebra, the student has completed a year of geometry, a year of trigonometry, a year of analysis/pre-calculus, and is now in a calculus class. Algebra has been the basis of everything he/she has done in math class for the last 4+ years, along with heavy usage in other classes the student has likely taken such as chemistry, physics, economics, etc.

"Forgetting" how to solve for a single unknown is not unlike seeing a 5th grader calculate 3 * 4 as 3 + 3 = 6 + 3 = 9 + 3 = 12.

Agreed it should not be a source of amusement, but as a teacher I would definitely be a little worried.

[dc_publius]

[ QUOTE ]
So I don't see what the problem is.

[/ QUOTE ]

The problem is that you fail to see the humor of using such a long way to solve such a quick problem and are trying to find deeper meaning in a joke.

[guesswest]

I personally find it frustrating (and not especially amusing) that teachers would be spending their time ridiculing students behind their backs. Especially when they seem to be missing the point that if their kids don't 'get it' it's a reflection on their teaching skills, not their comparitive genius.

I don't think anyone with that mindset has a place in the school system.

[HLMencken]

[ QUOTE ]
i think its the math department's attitude that is horrifying

[/ QUOTE ]

or your lack of humor...

[HLMencken]

[ QUOTE ]

This was pretty much my opinion as well. I really doubt that this student had no idea that you could add 5 to both sides and root it, but perhaps he just forgot. Whatever. The important thing is that he DID find a way to solve this equation correctly.

[/ QUOTE ]

I totally disagree. A calc senior "forgot" how to quickly solve c^2-5=0? This is far more disturbing than taking comfort that he found a way to solve it.

Frankly, I think you are the one who is at most fault in this story. You fail to see the humor and the warning flag that someone resorted to the quadratic formula for this problem. Assuming it wasn't a joke by the kid, it is right in front of your nose that his basic concepts at math are sorely lacking, yet you seem to almost applaud his perseverence. I agree its great he solved it by remembering the formula. But guess what--he'll forget that formula in a year or two, and then be left unable to solve the most basic of problems with ease.

[surftheiop]

Here is my opinion as an 11th grader in AP chemistry.
I think it most likey happened in a setting were the entire test/worksheet/homework assignment had problems that all followed a very similar format. After working each problem the student was probaly left with something that took the quadratic formula to solve except for this case. Because the student was forced into doing something repitevlt he probaly wasnt even thinking about why he was doing stuff and was just doing it because thats how the example happened on the board and the last 10 problems had been.
I seriously doubt this is a cause for concern as far as the student goes. Stuff like this happens all the time but usaully we manage to catch ourselves once we start doing it the hard way, might show carelessness but definately not a fundamental misunderstanding.

[John Feeney]

Here's a nice interview with Feynman which touches on some related points:

Feynman video (http://video.google.com/videoplay?docid=6586235597476141009&q=feynman&pl=t rue)

[TheHusky]

While writing a calculus exam in university. After correctly solving a somewhat tricky integral I finished the problem with "2+2=2"

so this stuff happens.

[wmspringer]

[ QUOTE ]
I personally find it frustrating (and not especially amusing) that teachers would be spending their time ridiculing students behind their backs. Especially when they seem to be missing the point that if their kids don't 'get it' it's a reflection on their teaching skills, not their comparitive genius.

I don't think anyone with that mindset has a place in the school system.

[/ QUOTE ]

Some things are just funny, regardless of who does them.

http://i75.photobucket.com/albums/i300/wmspringer/geometry.jpg

[Copernicus]

[ QUOTE ]
[ QUOTE ]
I personally find it frustrating (and not especially amusing) that teachers would be spending their time ridiculing students behind their backs. Especially when they seem to be missing the point that if their kids don't 'get it' it's a reflection on their teaching skills, not their comparitive genius.

I don't think anyone with that mindset has a place in the school system.

[/ QUOTE ]

Some things are just funny, regardless of who does them.

http://i75.photobucket.com/albums/i300/wmspringer/geometry.jpg

[/ QUOTE ]

now thats funny

[lastchance]

Well, us Calc students make mistakes in the heat of the test. Sometimes, you just don't see it because you're tired. It happens. (And yes, I should be doing practice problems for the AP Test instead of this, but so be it).

And yeah, when you point out to the kid that he/she should not have used the Quadratic Formula, he'll probably realize it and wonder WTF he was doing in the first place.

[guesswest]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
I personally find it frustrating (and not especially amusing) that teachers would be spending their time ridiculing students behind their backs. Especially when they seem to be missing the point that if their kids don't 'get it' it's a reflection on their teaching skills, not their comparitive genius.

I don't think anyone with that mindset has a place in the school system.

[/ QUOTE ]

Some things are just funny, regardless of who does them.

http://i75.photobucket.com/albums/i300/wmspringer/geometry.jpg

[/ QUOTE ]

now thats funny

[/ QUOTE ]

I agree that one's funny /images/graemlins/grin.gif

[rJ_]

Unrelated but curious.

You are referring to high school and then saying calculus 1, algebra 1, trig 1. In Canada we dont break all of math up and teach it as an entire block, but instead mix it all up and teach it a little each year.

Grade 10 math covers radicals, sequences/series, polynomials, relations, finance, coordinate geometry.

Grade 11 math covers trigonometry, functions, linear systems, quadratics, geometry.

Grade 12 math covers logs, sequences/series, calculus, geometry (unguided proofs), more function, probability.

Im sure each has its advantages and disadvantages, but i wonder which is really better.

rJ_ (also a high school math teacher)

[Marko Schmarko]

Is there room for the more gifted to thrive/skip ahead?

[sweetjazz]

Sooga,

To me the main problem with using the quadratic formula to solve c^2 - 5 = 0 would be that the method does not generalize to equations such as c^3 - 5 = 0. If the student can indeed solve the latter equation as well, then there is not much of a problem.

Nevertheless, being successful at mathematics, or more generally critical thinking, requires more than an ability to "get the right answer." Efficiency and generality are also important.

A further concern is that students who show a tendency to rely on formulas whenever possible tend to struggle very mightily if and when they reach a level when problems that rely on conceptual understanding become more frequent.

For example, if the student were asked whether, for the function f(x) = x^2 - 5, there were two distinct x values that had the same f(x) value, would s/he recognize that his previous solution immediately answers this question affirmatively?

[Sooga]

[ QUOTE ]

I totally disagree. A calc senior "forgot" how to quickly solve c^2-5=0? This is far more disturbing than taking comfort that he found a way to solve it.

Frankly, I think you are the one who is at most fault in this story. You fail to see the humor and the warning flag that someone resorted to the quadratic formula for this problem. Assuming it wasn't a joke by the kid, it is right in front of your nose that his basic concepts at math are sorely lacking, yet you seem to almost applaud his perseverence. I agree its great he solved it by remembering the formula. But guess what--he'll forget that formula in a year or two, and then be left unable to solve the most basic of problems with ease.

[/ QUOTE ]

I disagree with your disagreement. There is a huge difference between simply forgetting to add 5 to both sides in the heat of a test or something, and completely not understanding that you could solve it that way. The former is not a huge deal, and I have a feeling it is what happened here. The latter is a huge deal. However, being that this student is in Calculus, I find the latter possibility to be very improbable.

Plus, the fact that he remembered the quadratic formula all the way since Alg1 makes it very unlikely that he'll forget it in a year or two.

[Sooga]

[ QUOTE ]

A further concern is that students who show a tendency to rely on formulas whenever possible tend to struggle very mightily if and when they reach a level when problems that rely on conceptual understanding become more frequent.

[/ QUOTE ]

You are absolutely right. In fact, I brought up this very point to the District superintendent when we had a meeting about how our district wants our students to think 'outside of the box'. I told him basically that given the standards that need to be met in the curriculum, I don't think it's possible to teach outside-the-boxness. Math standards (for CA anyway) are based mostly on knowledge of the concepts, not necessarily being able to apply them to general questions. For example, in Algebra 1 classes, you ask them to solve a quadratic equation, they can do that all day. You ask them to find the values of x for which the graph of a quadratic equation crosses the x-axis, they'll have no idea what you're talking about. Exact same question, yet in on case students can do it, and in another they're stumped. It's sad, but application and deep understanding are simply is not taught nearly as much as it should be.

I remember reading somewhere that if driving a car were a class offered in high school, they would accomplish this by spending 1 day turning the ignition on and off, then another day pressing the accelerator, then another day turning blinkers on, and so on. By the end of the course, you should surely have all the knowledge necessary to drive a car, right? I really like that analogy, and I think it's perfect for how math is handled in most public schools.

[Nottom]

[ QUOTE ]

I remember reading somewhere that if driving a car were a class offered in high school, they would accomplish this by spending 1 day turning the ignition on and off, then another day pressing the accelerator, then another day turning blinkers on, and so on. By the end of the course, you should surely have all the knowledge necessary to drive a car, right? I really like that analogy, and I think it's perfect for how math is handled in most public schools.

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Oddly enough, my HS actually offer Driver's Ed as a class. We spent the first couple weeks in the class room learning all the driving laws and basic stuff, then spent most of the rest of the semester driving around on a small little driving course. Looking back, I'm shocked that my crappy HS had any of this stuff available yet am quite happy they did.

[wmspringer]

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Unrelated but curious.

You are referring to high school and then saying calculus 1, algebra 1, trig 1. In Canada we dont break all of math up and teach it as an entire block, but instead mix it all up and teach it a little each year.

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Some parts of the US are doing that now as well.
As a 7th grade math teacher this year, I taught probability, linear equations, basic geometry, (very) basic topology, etc

[rJ_]

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Is there room for the more gifted to thrive/skip ahead?

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Not really like you could in your system i think.

Almost every school is on a semester system, so what some do is take math 10 first semester, then math 11 in the second semester of their grade 10 year.

Many do that so they dont have such a long gap between math classes, and not to skip ahead.

After grade 12 math there is a grade 12 calculus with an AP exam.

rJ_

[atrifix]

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Sooga,

To me the main problem with using the quadratic formula to solve c^2 - 5 = 0 would be that the method does not generalize to equations such as c^3 - 5 = 0. If the student can indeed solve the latter equation as well, then there is not much of a problem.

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The quadratic formula can be derived using only basic arithmetic, so this will generalize to higher-order polynomials. However, the solution becomes increasingly complex as the order increases. Deriving the quadratic formula is a routine matter, but deriving a cubic formula is fairly difficult, and deriving formulas for higher-order polynomials may be best left to machines.

[sweetjazz]

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Sooga,

To me the main problem with using the quadratic formula to solve c^2 - 5 = 0 would be that the method does not generalize to equations such as c^3 - 5 = 0. If the student can indeed solve the latter equation as well, then there is not much of a problem.

[/ QUOTE ]

The quadratic formula can be derived using only basic arithmetic, so this will generalize to higher-order polynomials. However, the solution becomes increasingly complex as the order increases. Deriving the quadratic formula is a routine matter, but deriving a cubic formula is fairly difficult, and deriving formulas for higher-order polynomials may be best left to machines.

[/ QUOTE ]

It has been known for over 200 years that there is no formula (involving the four basic operations of arithmetic and taking radicals) for the general polynomial equation in one variable of degree 5 or higher. /images/graemlins/blush.gif

[atrifix]

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Sooga,

To me the main problem with using the quadratic formula to solve c^2 - 5 = 0 would be that the method does not generalize to equations such as c^3 - 5 = 0. If the student can indeed solve the latter equation as well, then there is not much of a problem.

[/ QUOTE ]

The quadratic formula can be derived using only basic arithmetic, so this will generalize to higher-order polynomials. However, the solution becomes increasingly complex as the order increases. Deriving the quadratic formula is a routine matter, but deriving a cubic formula is fairly difficult, and deriving formulas for higher-order polynomials may be best left to machines.

[/ QUOTE ]

It has been known for over 200 years that there is no formula (involving the four basic operations of arithmetic and taking radicals) for the general polynomial equation in one variable of degree 5 or higher. /images/graemlins/blush.gif

[/ QUOTE ]

Sorry, math is not my specialty. I know that there is a general formula for solving cubic equations, though, that can be derived using only basic arithmetic, and that some people have developed some rather complicated algorithms for solving the general higher-order polynomials. Can you tell me any more about the proof that there is no general equation for the roots of the polynomials of degree 5 or higher?

[sweetjazz]

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Sooga,

To me the main problem with using the quadratic formula to solve c^2 - 5 = 0 would be that the method does not generalize to equations such as c^3 - 5 = 0. If the student can indeed solve the latter equation as well, then there is not much of a problem.

[/ QUOTE ]

The quadratic formula can be derived using only basic arithmetic, so this will generalize to higher-order polynomials. However, the solution becomes increasingly complex as the order increases. Deriving the quadratic formula is a routine matter, but deriving a cubic formula is fairly difficult, and deriving formulas for higher-order polynomials may be best left to machines.

[/ QUOTE ]

It has been known for over 200 years that there is no formula (involving the four basic operations of arithmetic and taking radicals) for the general polynomial equation in one variable of degree 5 or higher. /images/graemlins/blush.gif

[/ QUOTE ]

Sorry, math is not my specialty. I know that there is a general formula for solving cubic equations, though, that can be derived using only basic arithmetic, and that some people have developed some rather complicated algorithms for solving the general higher-order polynomials. Can you tell me any more about the proof that there is no general equation for the roots of the polynomials of degree 5 or higher?

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The proof is rather complicated. Evariste Galois gave the first proof. There are algorithms for solving polynomial equations approximately, such as Newton's method.

[HajiShirazu]

To give the opposite perspective, I got a 5 on both AB and BC calc AP exams and I didn't know/remember the quadratic formula when I took them. I was a horrible student too, but those AP tests were all easy for anyone who could think at all, plus I went to a HS where they knew everybody in the top classes wanted to get into a good college and would never give less than a B in a weighted class.

[srjunkacct]

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Sooga,

To me the main problem with using the quadratic formula to solve c^2 - 5 = 0 would be that the method does not generalize to equations such as c^3 - 5 = 0. If the student can indeed solve the latter equation as well, then there is not much of a problem.

[/ QUOTE ]

The quadratic formula can be derived using only basic arithmetic, so this will generalize to higher-order polynomials. However, the solution becomes increasingly complex as the order increases. Deriving the quadratic formula is a routine matter, but deriving a cubic formula is fairly difficult, and deriving formulas for higher-order polynomials may be best left to machines.

[/ QUOTE ]

It has been known for over 200 years that there is no formula (involving the four basic operations of arithmetic and taking radicals) for the general polynomial equation in one variable of degree 5 or higher. /images/graemlins/blush.gif

[/ QUOTE ]

Sorry, math is not my specialty. I know that there is a general formula for solving cubic equations, though, that can be derived using only basic arithmetic, and that some people have developed some rather complicated algorithms for solving the general higher-order polynomials. Can you tell me any more about the proof that there is no general equation for the roots of the polynomials of degree 5 or higher?

[/ QUOTE ]

The proof is rather complicated. Evariste Galois gave the first proof. There are algorithms for solving polynomial equations approximately, such as Newton's method.

[/ QUOTE ]

The basic idea of Galois' proof is to look at the group of "symmetries" of the roots of the equation which preserves the rational numbers (or more precisely, the splitting field of the equation over the field of rational numbers). E.g. for the equation x^2 + 1 = 0, the group of symmetries has size 2 -- there's the symmetry which does nothing, and the symmetry which exchanges i and -i. In fact, for any degree 2 polynomial which doesn't factor over the rationals, the symmetry group has order 2 (the identity, plus changing the sign of the square root in the roots obtained from the quadratic formula). If it does factor over the rationals, then the symmetry group has order 1 -- only the identity is allowed. (Since the roots are rational, they cannot be moved around.)

The idea of Galois' proof is to show that the symmetry group of a general quintic doesn't break down into a series of "cyclic" groups -- the cyclic groups correspond to the ability to take radicals.

Also, Newton's method sometimes fails to converge to a root (if one chooses the initial seed really, really carefully) and also exhibits chaos (initial seeds which are very close can yield different roots). There are more sophisticated algorithms such as this one (http://www.math.harvard.edu/~ctm/papers/home/text/papers/icos/icos.pdf). (The second author is the owner of a Fields medal.)

[CrayZee]

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Wow. I can't even believe someone would even bother pointing this out, much less a teacher. The kid most likely was busy worrying about how to solve the bigger problem and just used the first method that popped into his head to solve the quad equation.

The fact that someone would bring this up later and laugh about it often indicates they think they are a lot smarter than they really are, and that they like to stroke their own ego at the expense of others.

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This was pretty much my opinion as well. I really doubt that this student had no idea that you could add 5 to both sides and root it, but perhaps he just forgot. Whatever. The important thing is that he DID find a way to solve this equation correctly.

cts's point that he shouldn't be in calculus if he can't do x^2 - 5 = 0 without quad. formula would I think be the teacher's opinion as well. I don't really agree. Yea, this kid pulled out the heavy artillery to solve a very simple problem, but I don't see anything inherently 'wrong' about it. Is it a cause for concern? Maybe... I guess it's like, what if the student had factored x^2 - 5 into (x - sqrt5)(x + sqrt5) = 0 and solved it that way instead? Would that be less or more 'bad' than quadratic equation? Would that be less or more 'bad' than adding 5 to both sides? Personally, I have no idea. That's the beauty of mathematics. There are always so many different ways to solve the same problem that expecting all students to solve problems the same way is kind of dangerous. I dunno, that's just what I feel.

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I ain't no fancy-pants math teacher, but teachers laughing at a student solving a simple equation in a non-traditional way is lame. Most people from time to time make simple "errors" like this. Sometimes you're on solving auto-pilot and don't care, if, say solving this is an intermediary step.

I think it should be brought to the attention of the student that there are easier ways to solve the problem than others. I've been lightly tutoring my gf in math (trig now) and I try to encourage the "math people are lazy" thing for the sake of efficiency. Solve it the easy way; briefly think through it before jotting stuff down on the paper. You don't always have to solve degree 2 polynomials w/ the quadratic, etc.