Why is calculus considered so complex?
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Why is calculus considered so complex?
This may be an odd bit of cultural osmosis but I've always found TV to consider calculus a really difficult bit of maths. Maybe it's because it's the last bit of maths people do in school, but am i missing something here? So far it's been mainly formulas. And quite simple ones too.
Of course, counterexamples to the calculusisformulas theory are welcome too.
Of course, counterexamples to the calculusisformulas theory are welcome too.
 lordofnarf
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Re: Why is calculus considered so complex?
Calc is often considered a difficult math because it's often the last or near the last Math people ever take, and many people have a difficult time with it because it can tend toward greater conceptual requirements than Algebra and Geometry, etc. I don't find Calculus dificult at all, mostly for that same reason.
Another reason is that the educational philosophy in math, and especially Calc for many years was a response based one that never evaluated process, but only relied on producing the 'correct' answer, since many calc problems have numerous places where it can be easy for beginning students to make basic mistakes, not to mention that they often have algebra components to them lead to people performing poorly on calculus. This philosophy has been supplanted in most fields of Mathematics over the years with one that focuses on Calc problems, and then only on the process, not on the answer, or less on the answer.
But that's just my take.
Another reason is that the educational philosophy in math, and especially Calc for many years was a response based one that never evaluated process, but only relied on producing the 'correct' answer, since many calc problems have numerous places where it can be easy for beginning students to make basic mistakes, not to mention that they often have algebra components to them lead to people performing poorly on calculus. This philosophy has been supplanted in most fields of Mathematics over the years with one that focuses on Calc problems, and then only on the process, not on the answer, or less on the answer.
But that's just my take.
Re: Why is calculus considered so complex?
Yeah, I think you hit it on the mark with "it's the last bit of maths people do in school", and a lot of the general public can't even make it that far. Even though Calculus is rather simple for most people with a serious interest in math or science (with the obvious disclaimer that this depends on what you define as Calculus, because Real Analysis, Complex Analysis, PDEs are all Calculus and can be highly nontrivial), for the average person it really is hard stuff.
Re: Why is calculus considered so complex?
I think it's also one of the first times a strictly physical intuition stops working. Through the algebra and geometry covered in high school, one is always able to get a solid picture. The idea of an infinite number of tiny things adding up to a finite volume tends to blow people's minds.
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 Sir_Elderberry
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Re: Why is calculus considered so complex?
Math in general is considered complex. I've seen TV shows reference algebra as something hard. Now, suppose you want to reach an audience that's mathilliterate at best and outright hates the subject at most. You want to establish a given character as knowing Complicated Math. Saying that they've been taking analysis or topology, and you don't really get anywhere. "Analysis" is a general word, and "topology" is something to do with maps. Nothing too evocative there. But calculus! Most of your audience never took calculus, but the people who were better at math than them did. Maybe they saw some once, but they could never make sense of it. Those who did take calculus remember it as difficult, mostly, it was the highest math they ever had. Those who didn't find it difficult are math nerds and too small a demographic to worry about.
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Re: Why is calculus considered so complex?
If you don't want calculus to seem complex then do some real analysis
Re: Why is calculus considered so complex?
teacupthesauceror wrote:Of course, counterexamples to the calculusisformulas theory are welcome too.
Calculus by Tom Apostol
Re: Why is calculus considered so complex?
TheQntty wrote:If you don't want calculus to seem complex then do some real analysis
Alternatively, if you want calculus to seem complex then take some Complex Analysis.
 Yakk
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Re: Why is calculus considered so complex?
TheQntty wrote:teacupthesauceror wrote:Of course, counterexamples to the calculusisformulas theory are welcome too.
Calculus by Tom Apostol
Did a quick glance on amazon preview. That looks like a not crappy book.
Spivak is also not crappy.
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Re: Why is calculus considered so complex?
Yakk wrote:TheQntty wrote:teacupthesauceror wrote:Of course, counterexamples to the calculusisformulas theory are welcome too.
Calculus by Tom Apostol
Did a quick glance on amazon preview. That looks like a not crappy book.
Spivak is also not crappy.
Yes Spivak and Apostol both write for the same audience, Apostol is more dated than Spivak because he includes some linear algebra which isn't necessary because people take it right after calculus II or III nowadays.
Re: Why is calculus considered so complex?
Algebra is indeed hard, especially when mixed with graph theory, geometry, combinatorics, biology (don't ask me how, I saw a poster on algebraic biology in my math department), etc.Sir_Elderberry wrote:Math in general is considered complex. I've seen TV shows reference algebra as something hard. Now, suppose you want to reach an audience that's mathilliterate at best and outright hates the subject at most. You want to establish a given character as knowing Complicated Math. Saying that they've been taking analysis or topology, and you don't really get anywhere. "Analysis" is a general word, and "topology" is something to do with maps. Nothing too evocative there. But calculus! Most of your audience never took calculus, but the people who were better at math than them did. Maybe they saw some once, but they could never make sense of it. Those who did take calculus remember it as difficult, mostly, it was the highest math they ever had. Those who didn't find it difficult are math nerds and too small a demographic to worry about.
Anyways, I agree with this statement. Most of the general public do not know higher math, so they don't see the true "horror" of it. To them, the only thing harder than calculus is probably multivariable calculus.
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Re: Why is calculus considered so complex?
Well, "calculus" + "scary adjective" > "calculus".
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 headprogrammingczar
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Re: Why is calculus considered so complex?
You mean vector calculus? I consider myself very good at calculus, but god that class was horrible.
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Re: Why is calculus considered so complex?
headprogrammingczar wrote:You mean vector calculus? I consider myself very good at calculus, but god that class was horrible.
That was pretty much my experience. Multivariable was tough, especially when you get to all the different fundamental theorems.
As a math major at my university, I always get questions about why I bother to study such a horrible subject. Maybe people are so afraid of math, and especially calculus, because figuring it out just takes repetition. You get better by doing more of it. Not to say that this isn't true for other subjects, but other subjects get fun faster. At least in physics, music, English, and petty much everything else you get to do stuff (often with your hands) other than stare at a chalk board.
 Yakk
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Re: Why is calculus considered so complex?
If you define fun as "doing physical stuff with your hands", possibly the Queen of the Sciences is not the field for you. You rarely pick up mathematical theories with your hands. (although, you can do lots of mathematics using physical analogues  mirrorbased geometry can be quite physical in inspiration. And people do knit projective planes.)
If you define fun as "imagining 7 dimensional space by starting with ndimensional space, then setting n to 7", then ...
If you define fun as "imagining 7 dimensional space by starting with ndimensional space, then setting n to 7", then ...
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Re: Why is calculus considered so complex?
I think that one of the funnest things about maths is that we have no bounds, we don't just start with what we know and build on it(although it's certainly important), we start with what things could be and see where they take us. It's precisely the explorative properties of mathematics that draws me to the subject and I think I would have been too enclosed in another, there's a certain freedom about maths that's unique and I wouldn't trade that for the chance to touch what I'm studying.
Re: Why is calculus considered so complex?
I think twentythirty years ago people probably spoke of "algebra" or "trigonometry" as the dreaded endall beall math course.
I suspect calculus pedagogy has vastly improved over the last few decades and calculus is becoming more "mainstream" as time passes, and it will eventually be the norm that it is taught in high schools to regular level students and assumed knowledge by the time students start universitylevel maths. Maybe a couple decades from now most people will talk about the dreaded "DEs" class.
I suspect calculus pedagogy has vastly improved over the last few decades and calculus is becoming more "mainstream" as time passes, and it will eventually be the norm that it is taught in high schools to regular level students and assumed knowledge by the time students start universitylevel maths. Maybe a couple decades from now most people will talk about the dreaded "DEs" class.
 keeneal
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Re: Why is calculus considered so complex?
Sir_Elderberry wrote:stuff about needing to convey "hard math" in media
This. I agree with this.
Also: Calculus is something most people never have to touch. At my high school, you not only had to be good at math to take it, you had to be a full year ahead of everyone else (and you had to get that jump in freaking 7th grade or gasp double up on math one year). To take AP Calc BC, you had to be two years ahead. At my University, you don't have to take Calculus at all unless you're a Math or Sciences major... for everyone else, there are other classes. This means that a lot of people not only are never exposed to it, they see it as something only for those absurdly good at math, even if it's not really all that complex.
I would like to be good at calculus. I understand the concepts pretty well (or did, when I took it), but for some reason was never able to succeed at it (we're talking a C for the class and a 2 on the test). Assuming I'm not the only one in this situation, a lot of people who were "good at math" in high school had trouble with it. Again, Mere Mortals saw that the Heroes of the Age were bested, and they were terrified of the Great Beasts called Integral and Derivative.
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Re: Why is calculus considered so complex?
You forgot engineering and CS. In our school, business and economics students also need to take calculus. The reason is that most economics model are based upon differential equations.
Anyways, I failed to see how calculus is a beast. It's just a tool like any other. I mean, it's like algebra, trig, Maple (a very powerful computing software that can do all the year 1 calculus problems for you, and much more), just another tool. In fact, I do not want to imagine life without calculus (and numerical methods). Without it, technology would not be as it is now.
Anyways, I failed to see how calculus is a beast. It's just a tool like any other. I mean, it's like algebra, trig, Maple (a very powerful computing software that can do all the year 1 calculus problems for you, and much more), just another tool. In fact, I do not want to imagine life without calculus (and numerical methods). Without it, technology would not be as it is now.
 keeneal
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Re: Why is calculus considered so complex?
Oh, sure, it's just a tool. You're absolutely right. (no sarcasm, I promise) That doesn't make it easier, though.
Just because to you, there's nothing to doing calc or whatever type of math you do, doesn't mean that it's easy for other people. Take me: I like math. I'd like to be good at it. I even find I understand a great deal of it, but I fail when I get to the "actually using it" part.
I think people in a math community forget how complex the idea of math actually is. You're good at it; it makes sense to you. The comments in this thread (and other that I've read) give me the idea that maybe you don't understand why other people might have to struggle with it like it was a bear. Let me try to explain it. I'm going to start off by talking about languages, but I have a point... really. Just in case, I'm gonna spoiler some of it for length.
tl;dr:
Language has a set of basic elements (parts of speech), and the differences in languages in how those elements are manipulated (rules such as conjugation, word order, &c). All you need to do to learn a language is train yourself to apply those elements according to different rules than you're used to. These rules have been created by the human brain to be easy to learn, and some people are good at it, but a lot some aren't.
tl;dr: learning math is like learning a new language, but one that isn't designed to fit nicely into the human way of dealing with things
Math is often called "the universal language", and that title is fitting, is it not? But here's the thing: To learn the language of Calculus, you need to first be fluent in Algebra, which requires Arithmetic. I'm sure you've seen people struggle with something as simple as figuring out what a 15% tip would be at a restaurant... that's just arithmetic. Just as some minds can handle learning language easily, so some can handle learning new kinds of math. I argue that anyone can learn a language given enough time, patience, and effort, and I believe the same can be said of any mathematical discipline. But you shouldn't expect that it's easy for everyone, just as learning a new language is hard for many people.
(yes, I know that's kind of offtopic, but maybe it explains (or reminds) for some of you why math is looked at as being hard, and therefore why anyone would be confused by any of it, even something so "simple" as calculus. combine this understanding of why math is, in fact, hard with my previous argument (which is really Sir_Elderberry's), and you have my full answer to the stated question.)
Just because to you, there's nothing to doing calc or whatever type of math you do, doesn't mean that it's easy for other people. Take me: I like math. I'd like to be good at it. I even find I understand a great deal of it, but I fail when I get to the "actually using it" part.
I think people in a math community forget how complex the idea of math actually is. You're good at it; it makes sense to you. The comments in this thread (and other that I've read) give me the idea that maybe you don't understand why other people might have to struggle with it like it was a bear. Let me try to explain it. I'm going to start off by talking about languages, but I have a point... really. Just in case, I'm gonna spoiler some of it for length.
Spoiler:
Language has a set of basic elements (parts of speech), and the differences in languages in how those elements are manipulated (rules such as conjugation, word order, &c). All you need to do to learn a language is train yourself to apply those elements according to different rules than you're used to. These rules have been created by the human brain to be easy to learn, and some people are good at it, but a lot some aren't.
Spoiler:
Math is often called "the universal language", and that title is fitting, is it not? But here's the thing: To learn the language of Calculus, you need to first be fluent in Algebra, which requires Arithmetic. I'm sure you've seen people struggle with something as simple as figuring out what a 15% tip would be at a restaurant... that's just arithmetic. Just as some minds can handle learning language easily, so some can handle learning new kinds of math. I argue that anyone can learn a language given enough time, patience, and effort, and I believe the same can be said of any mathematical discipline. But you shouldn't expect that it's easy for everyone, just as learning a new language is hard for many people.
(yes, I know that's kind of offtopic, but maybe it explains (or reminds) for some of you why math is looked at as being hard, and therefore why anyone would be confused by any of it, even something so "simple" as calculus. combine this understanding of why math is, in fact, hard with my previous argument (which is really Sir_Elderberry's), and you have my full answer to the stated question.)
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Re: Why is calculus considered so complex?
In a way, I still do. In fact, I usually only estimate by dividing by 20 then multiplying by 3, using the nearest dollar.keeneal wrote:Math is often called "the universal language", and that title is fitting, is it not? But here's the thing: To learn the language of Calculus, you need to first be fluent in Algebra, which requires Arithmetic. I'm sure you've seen people struggle with something as simple as figuring out what a 15% tip would be at a restaurant... that's just arithmetic.
Truth be told, (some) professors are notoriously bad at arithmetic and algebra. Anyways, for most of my experience, (I did tutoring before and am currently a TA) it isn't that people are not capable of learning it, or that there are too many rules to remember, but instead they are trying to remember too many rules. With so many different rules, of course one is bound to get confused. If they can only remember less, (but remember the important bits) their results will improve. In fact, I personally never remembered the quotient rule, nor the derivative of arcsin/arccos/arctan, and have to rederive them all the time. But seeing how it only takes 2 minutes to derive these, it was no great loss of time to me, but it eased my memory a lot. In fact, one can probably summarize the whole course in a dozen or so pages, which is probably impossible for a course like biology.
Edit: If only the rules for language is as simple as that of mathematics...... (English is particularly bad, given by the number of exceptions.)
Re: Why is calculus considered so complex?
achan1058: I totally agree with you. In the past, I've run into trouble trying to remember the different formulas, such as for trig and geometry. This year I followed someones advice and just paid attention to how they derived the different formulas. Tests take a little bit longer, but it makes things much less stressful.
 Yakk
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Re: Why is calculus considered so complex?
Take me: I like math. I'd like to be good at it.
I'd argue that it's more essential to human life than math is (not to say that math is unimportant; just that communication is a higher priority).
Clearly you don't like math enough.
Some formula can be reasoned out if you have to, but I think it's fair to say that a lot of them are pretty much arbitrary.
I'm not aware of an arbitrary formula in any math I know sufficiently much about. Math formulas are only arbitrary when you do not understand them.
For example, take a look at the quadratic equation. When you're memorizing it, it's a pain in the butt. I'm sure if I got good enough at math I'd be able to derive it eventually (someone did, after all), but when I'm learning it, it's just a string to be memorized. For people to whom math doesn't come naturally, every single concept is like that, and it may never "click" into an intuitive understanding of the principle at work.
Exactly. The quadratic formula isn't an arbitrary formula. It is derivable from the question being asked. Many of its properties fall out from symmetry and both higher and lower level concepts in mathematics.
Let's look at it, and look at what properties come from symmetry:
[b +/ sqrt(b^2  4ac)] / 2a
So the factor of a you divide by  that is because ax^2 + bx + c has the same solutions as x^2 + b/a x + c/a.
The negative b at the front  as the b term goes up, the 'extreme' point (min/max) of the equation slides left. In particular, 2ax + b = 0 when x = b/2a.
The roots of a quadratic end up being symmetrically on each side of this extreme point. This gives you the +/ blah part of the equation.
Now look at what happens when you pass the a into the sqrt:
sqrt((b/a)^2  4c/a)
you'll see that the "without a" balance remains. Let b` = b/a, and c` = c/a, giving us:
sqrt(b`^2  4c`)
from now on we'll use the "aless" version of the equation.
If c is zero, then the equation reduces to x^2 + xb, which has its extrema at x = b/2, and roots at 0 and b. If you go back to the equation, this gives you the +/sqrt(b^2)/2 (which is actually +/b/2 when c = 0).
When c does not equal zero, things change. Let's imagine that b is zero. Then we have x^2 + c as our polynomial.
The roots end up being +/ sqrt(c). As we want that 'global divided by 2' to be there, this makes it +/ sqrt(4c)/2.
Now the only tricky part becomes what happens when both b and c are nonzero. But the point is that the equation isn't arbitrary  you can derive properties of the equation without even going through the steps of deriving the equation.
At this point, we have [b +/ sqrt(b^2)]/2a when c = 0, and [ +/ sqrt(4ac)]/2a when b = 0 derived. All from looking at much simpler cases.
Edit: Spelling fixes.
Math is tricky when you think it is arbitrary. When you realize that almost none of it is arbitrary that it gets actually difficult.
You can derive the quadratic formula from first principles: but because mathematics is a huge web of interrelated information, you can approach it from lots of angles and get hints about it without having to do the derivation.
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Re: Why is calculus considered so complex?
I saw the quadratic formula in a TimeLife book when I was about 10. I memorized it because I thought it looked cool. But I had no idea at the time how it worked: it was just an arbitrary magic formula as far as I was concerned. Several years later, when I was actually taught about quadratics in school, I learned about factoring polynomials, and completing the square, and how the quadratic formula was actually derived. A mystery was solved!
These days, I rarely bother to use the quadratic formula. I normally just complete the square. (IIRC, the last time I used the formula was because the coefficients were fairly unwieldy expressions involving several parameters).
I taught myself some introductory calculus in junior high school (which was very helpful a few years later when I learned the same material in class). So I actually knew some calculus before I understood quadratics properly.
These days, I rarely bother to use the quadratic formula. I normally just complete the square. (IIRC, the last time I used the formula was because the coefficients were fairly unwieldy expressions involving several parameters).
I taught myself some introductory calculus in junior high school (which was very helpful a few years later when I learned the same material in class). So I actually knew some calculus before I understood quadratics properly.
 keeneal
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Re: Why is calculus considered so complex?
Okay, so maybe I picked a bad example
Yes, I realize that to people who either a) "just get" math or b) have been using a given piece of knowledge long enough slash advanced beyond the item in question far enough, these things are easy to derive or puzzle out if you don't feel like memorizing them. In this case, it doesn't seem arbitrary; it just "makes sense". A place for everything, and every thing in its place, right? The trouble comes if you fall in to neither of those categories, and then you're just stuck memorizing it, and that can be hard. And to these people, there's no reason behind the formula. Sure, maybe they can explain it, or if they've been working with the concept long enough derive it themselves, but at the outset, it's just a jumble of operations and numbers and letters in their heads.
PS: Last time I tried to explain how I see this this, some people got almost angry about it; I'm pleased that it worked better this time. Everyone in the thread gets a cookie.
Yes, I realize that to people who either a) "just get" math or b) have been using a given piece of knowledge long enough slash advanced beyond the item in question far enough, these things are easy to derive or puzzle out if you don't feel like memorizing them. In this case, it doesn't seem arbitrary; it just "makes sense". A place for everything, and every thing in its place, right? The trouble comes if you fall in to neither of those categories, and then you're just stuck memorizing it, and that can be hard. And to these people, there's no reason behind the formula. Sure, maybe they can explain it, or if they've been working with the concept long enough derive it themselves, but at the outset, it's just a jumble of operations and numbers and letters in their heads.
I feel like they are, they're just arranged differently. Like, for example, the who/whom thing. Not complicated, really, but something a lot of people get wrong. The trick is to know if the pronoun needs to be the subject or the object, and that's something that I intuitively understand. A lot (most?) people have to work through "okay, what's the verb? okay... now is the pronoun the one doing it? (IF "yes" > who)... okay, is the pronoun having the verb "done to" it? (IF "yes"> whom)" So, a lot of people have to go through a sort of algorithm to get at the answer (parallel in my mind to memorizing a formula), but others understand the underlying principle (being able to go "well, that's the right way because it just sounds right"). The same thing happens in math... it just matters which type of intuition you're better with (unless you're one of the lucky ones who are good at everything, of course )achan1058 wrote:Edit: If only the rules for language is as simple as that of mathematics...... (English is particularly bad, given by the number of exceptions.)
Not only did I walk right into that, I saw it coming when I posted it .Yakk wrote: Take me: I like math. I'd like to be good at it.Clearly you don't like math enough.keeneal wrote:I'd argue that it's more essential to human life than math is (not to say that math is unimportant; just that communication is a higher priority).
PS: Last time I tried to explain how I see this this, some people got almost angry about it; I'm pleased that it worked better this time. Everyone in the thread gets a cookie.
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Re: Why is calculus considered so complex?
keeneal wrote:The trouble comes if you fall in to neither of those categories, and then you're just stuck memorizing it, and that can be hard. And to these people, there's no reason behind the formula.
I haven't read through most of this thread, but I'm reasonably certain no one disagrees with you on this point.
keeneal wrote:So, a lot of people have to go through a sort of algorithm to get at the answer (parallel in my mind to memorizing a formula), but others understand the underlying principle (being able to go "well, that's the right way because it just sounds right").
The difference between good mathematical intuition and the word "intuition" as it's commonly used is that good mathematical intuition comes side by side with the ability to translate your intuition into proof whenever it is necessary. In other words, if your intuition (in the mathematical sense) is good enough, you should be able to write down a proof justifying it. For example, if you have good intuition about integrals, you should be able to convert that intuition into an algorithm for solving a wide class of integrals and therefore a proof that a wide class of integrals are doable.
Language is completely different. Addition is one of the few pieces of mathematics hardwired into our brains; everything else, starting with multiplication, is abstract. (Times tables are memorized verbally, whereas addition activates a separate part of your brain.) A significant portion of language, on the other hand, is hardwired. (There's even a fun little CS "proof" of this fact: if babies learn languages by recognizing them on the basis of no other input, then babies can break RSA.) As another example of the difference between the two, languages are best learned during a critical period at a very young age, whereas abstract mathematics is best learned after the cerebral cortex finishes developing. There's really no comparison.
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Re: Why is calculus considered so complex?
Yeah, I talked about language being hardwired into our brains and math not being so hardwired.t0rajir0u wrote:Language is completely different. Addition is one of the few pieces of mathematics hardwired into our brains; everything else, starting with multiplication, is abstract. (Times tables are memorized verbally, whereas addition activates a separate part of your brain.) A significant portion of language, on the other hand, is hardwired. (There's even a fun little CS "proof" of this fact: if babies learn languages by recognizing them on the basis of no other input, then babies can break RSA.) As another example of the difference between the two, languages are best learned during a critical period at a very young age, whereas abstract mathematics is best learned after the cerebral cortex finishes developing. There's really no comparison.
t0rajir0u wrote:The difference between good mathematical intuition and the word "intuition" as it's commonly used is that good mathematical intuition comes side by side with the ability to translate your intuition into proof whenever it is necessary. In other words, if your intuition (in the mathematical sense) is good enough, you should be able to write down a proof justifying it. For example, if you have good intuition about integrals, you should be able to convert that intuition into an algorithm for solving a wide class of integrals and therefore a proof that a wide class of integrals are doable.
This is true; and it's why people who don't have this mathematical intuition have trouble with proofs. That's why I brought it up. However, we weren't talking about normal intuition vs. mathematical inuition, we were talking about linguistic intuition, and the same thing applies there; I can take my inuitive grasp of the nominative and accusative cases and work backwards towards an algorithm for figuring out why something "just sounds right".
I certainly hope so, but I don't think I was clear enough the first time I brought it up; people focused too much on my calling forumlas "arbitrary". Obviously, they aren't, they just seem that way to some people. This is what I was getting at.t0rajir0u wrote:I haven't read through most of this thread, but I'm reasonably certain no one disagrees with you on this point.
Since you didn't read most of the thread, a lot of this conversation is based off of this post, wherein I compare learning math to learning a foreign language. But feel free to go back ontopic if you wish
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Re: Why is calculus considered so complex?
First, there is no such thing as "just get" math, except for the minority of geniuses. Most people, mathematicians included, needs practice and familiarity. Using it more does make it easier. It's just like how nobody can be a professional sports player without any training. Now, in most cases the tools for deriving the formula are taught to you as well, and if your teacher is 1/2 decent, they will also teach you the derivation method. (If not, don't blame math, blame your teacher.) For example, the quadratic formula is simply completing the square, and completing the square is usually taught before quadratic formula. Another example are trig derivatives. You definitely would want to remember sin and cos, but since tan = sin/cos, sec=1/cos, csc=1/sin, and cot=cos/sin, you do not need to remember these derivatives at all, and simply use product rule. Since you need to know the definition for these terms anyways, there is no extra things you need to remember. In fact, I wonder why they put the derivatives of these functions on the table. The main thing here is to convince yourself that mathematics is not arbitrary, and that there are always connections. If you are willing to find them, you will find them, regardless of how smart/dumb you are. Furthermore, this technique of reducing the amount to memorize not only works in math, but in CS/physics/chemistry (and probably other fields) as well.keeneal wrote:Okay, so maybe I picked a bad example
Yes, I realize that to people who either a) "just get" math or b) have been using a given piece of knowledge long enough slash advanced beyond the item in question far enough, these things are easy to derive or puzzle out if you don't feel like memorizing them. In this case, it doesn't seem arbitrary; it just "makes sense". A place for everything, and every thing in its place, right? The trouble comes if you fall in to neither of those categories, and then you're just stuck memorizing it, and that can be hard. And to these people, there's no reason behind the formula. Sure, maybe they can explain it, or if they've been working with the concept long enough derive it themselves, but at the outset, it's just a jumble of operations and numbers and letters in their heads.
No, English is definitely more messy. Who/whom might be logical, but the 6.02*10^23 tense exceptions are not. You won't find this in mathematics.keeneal wrote:I feel like they are, they're just arranged differently. Like, for example, the who/whom thing. Not complicated, really, but something a lot of people get wrong. The trick is to know if the pronoun needs to be the subject or the object, and that's something that I intuitively understand. A lot (most?) people have to work through "okay, what's the verb? okay... now is the pronoun the one doing it? (IF "yes" > who)... okay, is the pronoun having the verb "done to" it? (IF "yes"> whom)" So, a lot of people have to go through a sort of algorithm to get at the answer (parallel in my mind to memorizing a formula), but others understand the underlying principle (being able to go "well, that's the right way because it just sounds right"). The same thing happens in math... it just matters which type of intuition you're better with (unless you're one of the lucky ones who are good at everything, of course )achan1058 wrote:Edit: If only the rules for language is as simple as that of mathematics...... (English is particularly bad, given by the number of exceptions.)
A lot of mathematics is about communications, so it is natural that communication is a higher priority. In fact, if nobody can understand your discovery, how are you going to publish it? At least, that's my view on mathematics anyways.keeneal wrote:Not only did I walk right into that, I saw it coming when I posted it .Yaak wrote:Clearly you don't like math enough.keeneal wrote:I'd argue that it's more essential to human life than math is (not to say that math is unimportant; just that communication is a higher priority).
No, we are not only saying that formula are not arbitrary, but that they should not even seem arbitrary to most people, given competent teachers.keeneal wrote:I certainly hope so, but I don't think I was clear enough the first time I brought it up; people focused too much on my calling forumlas "arbitrary". Obviously, they aren't, they just seem that way to some people. This is what I was getting at.t0rajir0u wrote:I haven't read through most of this thread, but I'm reasonably certain no one disagrees with you on this point.
Re: Why is calculus considered so complex?
achan1058 wrote:First, there is no such thing as "just get" math, except for the minority of geniuses. Most people, mathematicians included, needs practice and familiarity. Using it more does make it easier. It's just like how nobody can be a professional sports player without any training.
This is a very important point. Mathematics, more than any other field, has a reputation for being something that some people "just get": it's an easy way to convince yourself that you don't need to try harder because you're just not one of "those people." You become a virtuoso of mathematics the same way you become a virtuoso of anything else: 10,000 hours of practice.
achan1058 wrote:Another example are trig derivatives. You definitely would want to remember sin and cos, but since tan = sin/cos, sec=1/cos, csc=1/sin, and cot=cos/sin, you do not need to remember these derivatives at all, and simply use product rule. Since you need to know the definition for these terms anyways, there is no extra things you need to remember.
And as yet another step of compression, if you remember the derivative of the exponential function and Euler's formula, you not only don't need to remember any other trigonometric derivatives, you don't need to remember any other trigonometric identities
Re: Why is calculus considered so complex?
But that is going to far for most students. Besides, having a very long chain of derivation is not helpful, since you need to remember the chain as well, while in a short chain you can trial and error.t0rajir0u wrote:And as yet another step of compression, if you remember the derivative of the exponential function and Euler's formula, you not only don't need to remember any other trigonometric derivatives, you don't need to remember any other trigonometric identities
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Re: Why is calculus considered so complex?
achan1058 wrote:First, there is no such thing as "just get" math, except for the minority of geniuses. Most people, mathematicians included, needs practice and familiarity. Using it more does make it easier. It's just like how nobody can be a professional sports player without any training
I would still argue that there are people in the world who are predisposed to being good at math, and to whom understanding the underlying principles comes more easily that it does to others. Surely you're not arguing that the idea of being naturally good at something doesn't exist? This is my argument: that individuals are, in fact, predisposed to being good at different things. Math can be one of those things, as can language, writing, singing, or playing an instrument. However, I am not saying that not being one of these people is an excuse to not try to learn the material, and I am willing to agree that the natural deficiency can be overcome with good teachers and enough time in which to practice.t0rajir0u wrote:This is a very important point. Mathematics, more than any other field, has a reputation for being something that some people "just get": it's an easy way to convince yourself that you don't need to try harder because you're just not one of "those people." You become a virtuoso of mathematics the same way you become a virtuoso of anything else: 10,000 hours of practice.
But here's the thing: to teach the same amount of material to predisposed mathematicians and everyone else in the same amount of time (as all teachers must do, at least in any country that uses standardized tests), the logical thing to do is to not take the extra time to get the concepts down, and just teach the algorithms and how to apply them. Yes, this is bad, but it's the reality of what happens in schools. As an example, there are no "honors" math courses at my high school  if you wanted an honorslevel math course, you took "theoretical" algebra, geometry, trig, etc. instead of the usual class. The primary difference here is that the theo classes (as they were called) dealt with why certain things worked, and included deriving your own proofs, whereas the traditional classes did not. So, I say don't blame the teacher; blame the organization of the educational system in the West.
Yes, it's messy, but it's not illogical. A very is either regular or irregular. If it's regular, it follows the default pattern. If it's irregular, it follows a different pattern, or one of a set of different patterns. For example, PRINT is regular: print >printed>have printed. SING is irregular: sing>sang>(have) sung in the same way as SWIM: swim>swam>(have)swum. A different rule is that for RUN: run>ran>(have) ran and SPEAK: speak>spoke>(have) spoke(n). If I gave you a madeup verb that fit one of these rules and asked you to guess at the past tense formation and the participle, you'd almost certainly give me the ones conforming to the given rule, and not the apply the "regular" rule to it. The likelihood that you'd give me the "wrong" irregular rule is also highly unlikely. (There's a topic in the linguistics forum that's currently locked where gmal. talks a bit about this... I'll find the link if you're interested).achan1058 wrote:No, English is definitely more messy. Who/whom might be logical, but the 6.02*10^23 tense exceptions are not. You won't find this in mathematics.
The point I'm trying to make is that "exceptions" in language are almost always a rule in and of itself. For example, "i" before "e" except after "c" when the sound is "ee". The "except after 'c'" bit isn't an exception; it's a rule of its own. When you look at it that way, you *do* see the same thing in math. For example, 1/X is the reciprocal of X, unless X=0, in which case it's undefined. 1/0 isn't an exception, it's an instance of a different rule at work.
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Re: Why is calculus considered so complex?
To a certain extent, yes, there are people who are a bit better at math, but that's not my point here. My point is except for a minority of students, most can do reasonably well in math, just like how except for a minority of students, most can learn English reasonably well. Furthermore, nobody will go home and say "singing is hard" like they do with math, do they, and that's what I have an issue of.keeneal wrote:I would still argue that there are people in the world who are predisposed to being good at math, and to whom understanding the underlying principles comes more easily that it does to others. Surely you're not arguing that the idea of being naturally good at something doesn't exist? This is my argument: that individuals are, in fact, predisposed to being good at different things. Math can be one of those things, as can language, writing, singing, or playing an instrument. However, I am not saying that not being one of these people is an excuse to not try to learn the material, and I am willing to agree that the natural deficiency can be overcome with good teachers and enough time in which to practice.
But there aren't nearly as many exception rules in mathematics as there is in English. Just the various tenses of am/are/is is a lot of new exception/rules there, as for he/him/his.keeneal wrote:Yes, it's messy, but it's not illogical. A very is either regular or irregular. If it's regular, it follows the default pattern. If it's irregular, it follows a different pattern, or one of a set of different patterns. For example, PRINT is regular: print >printed>have printed. SING is irregular: sing>sang>(have) sung in the same way as SWIM: swim>swam>(have)swum. A different rule is that for RUN: run>ran>(have) ran and SPEAK: speak>spoke>(have) spoke(n). If I gave you a madeup verb that fit one of these rules and asked you to guess at the past tense formation and the participle, you'd almost certainly give me the ones conforming to the given rule, and not the apply the "regular" rule to it. The likelihood that you'd give me the "wrong" irregular rule is also highly unlikely. (There's a topic in the linguistics forum that's currently locked where gmal. talks a bit about this... I'll find the link if you're interested).
The point I'm trying to make is that "exceptions" in language are almost always a rule in and of itself. For example, "i" before "e" except after "c" when the sound is "ee". The "except after 'c'" bit isn't an exception; it's a rule of its own. When you look at it that way, you *do* see the same thing in math. For example, 1/X is the reciprocal of X, unless X=0, in which case it's undefined. 1/0 isn't an exception, it's an instance of a different rule at work.
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Re: Why is calculus considered so complex?
achan1058 wrote:My point is except for a minority of students, most can do reasonably well in math, just like how except for a minority of students, most can learn English reasonably well. Furthermore, nobody will go home and say "singing is hard" like they do with math, do they, and that's what I have an issue of.
However, people aren't compelled to sing and then be graded on how well they sang in a class that's considered part of the core of the educational system. People don't complain about singing being hard (and it is, if you're trying to do it well) because it's not important that they be good at it, and they aren't required to be.
People don't complain about math because they think it's inordinately harder than anything else, they complain about it because it's hard. They complain more loudly about math than they do about other things because they don't like it and they have to be good at it to succeed in school. If you made Singing Class just as important to the educational system as Math Class, people would complain about it as well, I guarantee you.
By the way, if you look at my first post, I agree with you; I say there that learning math is equally as difficult as learning a foreign language. I was just addressing myself to people who think math is easier, and you're addressing people who think it's harder.
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Re: Why is calculus considered so complex?
achan1058 wrote:But that is going to far for most students. Besides, having a very long chain of derivation is not helpful, since you need to remember the chain as well, while in a short chain you can trial and error.t0rajir0u wrote:And as yet another step of compression, if you remember the derivative of the exponential function and Euler's formula, you not only don't need to remember any other trigonometric derivatives, you don't need to remember any other trigonometric identities
"Long chain"? The angle addition formula follows from Euler's formula in one step and so do the derivatives of the sine and cosine (simultaneously). If you learn the connection to rotation matrices and polar notation, it's much more intuitive than learning all the trig identities separately.
keeneal wrote:don't blame the teacher; blame the organization of the educational system in the West.
I don't think anyone was blaming the teacher. The educational system is responsible for the teachers being poorly trained.
keeneal wrote:The point I'm trying to make is that "exceptions" in language are almost always a rule in and of itself.
Exceptions in language are a consequence of either linguistic drift or unusual linguistic origins. Certainly there's some kind of rhyme or reason to them, but I think that the structure of language has much more in common with, say, the structure of DNA (a naturally evolved structure that isn't perfect but has a lot of regularity) than with the structure of mathematical thought.
keeneal wrote:people aren't compelled to sing and then be graded on how well they sang in a class that's considered part of the core of the educational system. People don't complain about singing being hard (and it is, if you're trying to do it well) because it's not important that they be good at it, and they aren't required to be.
Didn't you ever see "Happy Feet"?
keeneal wrote:People don't complain about math because they think it's inordinately harder than anything else, they complain about it because it's hard. They complain more loudly about math than they do about other things because they don't like it and they have to be good at it to succeed in school. If you made Singing Class just as important to the educational system as Math Class, people would complain about it as well, I guarantee you.
I actually disagree. Mathematics is special in that it is highly hierarchical: to understand one subject thoroughly you must first understand the subjects on which it is based. On the other hand, it is much easier to get away with "skipping" parts of, say, world history or literature; you don't have to be familiar with the literature of the Middle Ages to understand and appreciate the literature of 18thcentury America. So there's a "weakest link" problem: most people are only as good at mathematics as their worst mathematics teacher, and for most people that teacher was extremely bad.
On the other hand, I think that the "standard" mathematical curriculum in America, which I imagine is at least a century and a half old, is focused on the wrong things. What I think would be much more useful today is a huge emphasis on statistics, as much as I personally dislike the subject. If you're going to go into anything except engineering, physics, or pure mathematics, statistics is what's going to help you out the most. Statistics will help you understand important issues facing the world today. Honestly, as much as it pains me to say this, for the majority of Americans it would be fine if they didn't learn anything else.
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Re: Why is calculus considered so complex?
t0rajir0u wrote:On the other hand, I think that the "standard" mathematical curriculum in America, which I imagine is at least a century and a half old, is focused on the wrong things.
I agree with everything else you said, but man... if you have access to a good university library, see if you can track down some textbooks from the 1860s. They're very, very different. The modern "standard" american math curriculum is, AFAIK, no more than 50 or so years old.
I really couldn't possibly agree more with your opinion re: statistics. It's criminal that it's not a required class to graduate from most high schools.
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Re: Why is calculus considered so complex?
stephentyrone wrote:The modern "standard" american math curriculum is, AFAIK, no more than 50 or so years old.
Right, I forgot about the New Math business. I guess it just feels that way to me because all of the actual material is that old
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Re: Why is calculus considered so complex?
Should we teach people a math curriculum aimed at making their own lives personally better, or a math curriculum that if the master they will be able to make everyone else's lives better?
Our western high school math curriculum is currently aimed at making engineers, basically. It is aimed strait at calculus. The idea being that is you master that, you can shape the world, and everyone benefits.
Our western high school math curriculum is currently aimed at making engineers, basically. It is aimed strait at calculus. The idea being that is you master that, you can shape the world, and everyone benefits.
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Re: Why is calculus considered so complex?
The role of education should serve both goals, both inside and outside the mathematics curriculum. For example, I don't think anyone really believes that we teach history because people who know who won the Battle of Agincourt are particularly happier than people who don't. Perhaps the most important reason (to my mind) for teaching history is because people who know history will be better able to make decisions about what the United States should be doing in the world of today, and in a democracy it's important that everyone be as able as possible of making informed political decisions.
From this point of view, statistics is unquestionably more important than calculus as a subject to be taught in high school, and perhaps is the most important bit of mathematics for the average citizen to learn.
From this point of view, statistics is unquestionably more important than calculus as a subject to be taught in high school, and perhaps is the most important bit of mathematics for the average citizen to learn.
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"With math, all things are possible." —Rebecca Watson
"With math, all things are possible." —Rebecca Watson
Re: Why is calculus considered so complex?
Thank you! Yes! That's it exactly.
To my mind, the most important things you could learn in high school to be an informed citizen who can meaningfully participate in society are, roughly in order (this list is not claimed to be either consistent or complete):
 Writing (not Literature)
 History (this, in a sneaky way, subsumes important Literature) and Government
 Statistics (this, in a sneaky way, subsumes Basic Math) and Economics
 Biology (this, in a sneaky way, subsumes Chemistry) and Ecology.
Calculus definitely doesn't make it anywhere near the top of this list (as much as it pains me to say that; neither does physics, unless someone gives me a compelling practical reason). Knowing how to compute the volumes of surfaces of integration won't get the right Presidents elected or help you decide how to vote on therapeutic cloning or bailouts.
What I think would also be interesting is if schools taught history classes focusing specifically on the history and development of technology. In an increasingly hightech world, people should have some appreciation of the stuff that makes their world go 'round, even if they don't understand all of the mathematics, chemistry, or physics behind it. That can wait until college, I guess.
To my mind, the most important things you could learn in high school to be an informed citizen who can meaningfully participate in society are, roughly in order (this list is not claimed to be either consistent or complete):
 Writing (not Literature)
 History (this, in a sneaky way, subsumes important Literature) and Government
 Statistics (this, in a sneaky way, subsumes Basic Math) and Economics
 Biology (this, in a sneaky way, subsumes Chemistry) and Ecology.
Calculus definitely doesn't make it anywhere near the top of this list (as much as it pains me to say that; neither does physics, unless someone gives me a compelling practical reason). Knowing how to compute the volumes of surfaces of integration won't get the right Presidents elected or help you decide how to vote on therapeutic cloning or bailouts.
What I think would also be interesting is if schools taught history classes focusing specifically on the history and development of technology. In an increasingly hightech world, people should have some appreciation of the stuff that makes their world go 'round, even if they don't understand all of the mathematics, chemistry, or physics behind it. That can wait until college, I guess.
Re: Why is calculus considered so complex?
This conversation has sort of drifted from "why is calculus considered so complex" to "why are some people not so good at math". I'll run with that.
On one hand, I agree with t0rajir0u that a student is generally only as good as their worst math teacher. I also think in the West  in particular North America, I won't speak for Europe  we're pretty lenient in our education system and accept that students won't excel in maths. In a way, the good points of our education system is that it strives to create students who are independent, selfmotivated, and creative; the bad points is that it does not create students who are disciplined and value effort. We let students fall behind or not excel in math and sciences and excuse it, rather than push our students to exceed. It doesn't help that those same students often go on to be school teachers, perpetuating that cycle. Calculus is one thing, but when I see intelligent adults incapable of performing basic arithmetic and unit conversions (Verizon Math Fail anyone?), I shake my head in amazement. As one professor arguing in Canada's national newspaper "The Globe and Mail" put it, if a large portion of adults could not read and write at a Grade 6 level, alarm bells would be going off, and yet we accept it for maths.
On the other hand, I think the problems associated with people who have difficulties and disinterest in math go a lot deeper than lack of practice and discipline. Some people, perhaps a large majority, have a vague and even wrong viewpoint of what maths is. I once got in a twohour argument with my father that pi is not equal to 22/7. Even after showing him on a calculator, I could not convince him that "good enough" is not good enough. The attitude is if it's out of the realm of immediate practicality, it doesn't matter. Math is associated a lot with "calculations" and "formulas" and less associated with rigour and precision. It's hard to convince someone with that viewpoint otherwise. And then there are people who engage in behaviours that are completely irrational to a mathematician. People who try to come up with gambling systems based on emotion rather than logic, for example. And don't kid yourself thinking only dumb people do this. Many of the people engaging in this kind of behaviour are otherwise very rational and intelligent.
On one hand, I agree with t0rajir0u that a student is generally only as good as their worst math teacher. I also think in the West  in particular North America, I won't speak for Europe  we're pretty lenient in our education system and accept that students won't excel in maths. In a way, the good points of our education system is that it strives to create students who are independent, selfmotivated, and creative; the bad points is that it does not create students who are disciplined and value effort. We let students fall behind or not excel in math and sciences and excuse it, rather than push our students to exceed. It doesn't help that those same students often go on to be school teachers, perpetuating that cycle. Calculus is one thing, but when I see intelligent adults incapable of performing basic arithmetic and unit conversions (Verizon Math Fail anyone?), I shake my head in amazement. As one professor arguing in Canada's national newspaper "The Globe and Mail" put it, if a large portion of adults could not read and write at a Grade 6 level, alarm bells would be going off, and yet we accept it for maths.
On the other hand, I think the problems associated with people who have difficulties and disinterest in math go a lot deeper than lack of practice and discipline. Some people, perhaps a large majority, have a vague and even wrong viewpoint of what maths is. I once got in a twohour argument with my father that pi is not equal to 22/7. Even after showing him on a calculator, I could not convince him that "good enough" is not good enough. The attitude is if it's out of the realm of immediate practicality, it doesn't matter. Math is associated a lot with "calculations" and "formulas" and less associated with rigour and precision. It's hard to convince someone with that viewpoint otherwise. And then there are people who engage in behaviours that are completely irrational to a mathematician. People who try to come up with gambling systems based on emotion rather than logic, for example. And don't kid yourself thinking only dumb people do this. Many of the people engaging in this kind of behaviour are otherwise very rational and intelligent.
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