Why Are Infinite Concepts Misunderstood in Math Education?

[loom91]

I was wondering about something. Has anyone noticed the sheer volume of meaningless math questions that tend to be inspired by the concept of infinite in calculus? These range from the intelligent but misguided to the utterly illegible. One recent example was a poster who considered every point to contain more points, apparently because he was taught that points are what you get when you make straight lines smaller.

I can also see this state of education first-hand. When our physics teacher introduced the concept of limit by saying things like "very small" and "unimaginably small", I didn't mind thinking that he is just seeking to convey a working idea so that we can utilize the tools of calculus. But I was shocked when our maths teacher used the same language! I opened my textbook and found an explanation of limit saying that as the limiting variable deviates from actual value by a very small amount, we can neglect this difference. The actual definition was hidden in a corner with very little importance, and our teacher didn't even mention the epsilon-deltas, immediately moving over to computing limits of indeterminate forms by factorisation and standard limits.

Such a handling of one of the most fundamentally important and useful branches of mathematics and an overwhelming emphasis on getting the right answer with 'infinite' disregard of the underlying theoretical basis is destroying the future of students and contributing to the general resentment of mathematics by students. As someone particularly fond of calculus, I find this to be heart-breaking. What can be done to improve matters?

Molu

 

[radou]

Nothing can be done. There is a difference between calculus courses which can be taken in engineering courses, and between courses for students of mathematics, which, for obvious reasons, contain far more mathematical consistency. Engeneering math is 'packaged' in order to serve as a tool which is to be applyed while solving routine basic problems and understanding general physical concepts. Furthermore, people don't like math in general, so it's reasonable to assume that the more math you give them, the worse they'll do. Also, people like to think math is hard, which it isn't, at least not to some point. If you ask me, I'd blame the whole educational system for teaching math in a wrong way. But if you'd ask me which the right one is, I wouldn't have a clue.

 

[MeJennifer]

It is not completely fair to blame it on education IMHO.

Infinity is simply not an intuitive concept (we have a few hundred years of proof with "struggling smart mathematicians" on the topic).
Yes, when you know the right approach you can say it is easy and simple but that is really hindsight.

 

[matt grime]

I
(we have a few hundred years of proof with "struggling smart mathematicians" on the topic)

would you mind rewriting that so it makes sense as a sentence, and perhaps then you might illustrate it with some examples as to what you mean? It doesn't seem to me like a reasonable statement. Mathematicians don't appear to have any problem with 'infinity', whatever it may be.

 

[MeJennifer]

I simply meant that in the history of mathemathics infinity was not as straightforward even by established mathematicians of their time.
Only after Cantor was it relatively straightforward. You can see a similarity with imaginary numbers.

 

[matt grime]

Actually, no I can't see the similarity.

I can't see what cardinals have to do with a failure to understand that something is not finite, which is all infinity alludes to. After all, we figured out why Xeno's (or Zeno's) paradox was nonsensical well before Cantor came along.

It wasn't as though people were attempting to do arithmetic with 'infinity' in the same way that the need to deal with 'imaginary' numbers came up.

 

[franznietzsche]

Actually, no I can't see the similarity.

I can't see what cardinals have to do with a failure to understand that something is not finite, which is all infinity alludes to. After all, we figured out why Xeno's (or Zeno's) paradox was nonsensical well before Cantor came along.

It wasn't as though people were attempting to do arithmetic with 'infinity' in the same way that the need to deal with 'imaginary' numbers came up.

No, but historically the development of the concept of imaginary numbers was muddled and confused, right up through hamilton's quaternions.

 

[matt grime]

Who is denying that? It was the similarity with the alleged muddled notion of infinity that I am asking about.

 

[loom91]

Nothing can be done. There is a difference between calculus courses which can be taken in engineering courses, and between courses for students of mathematics, which, for obvious reasons, contain far more mathematical consistency. Engeneering math is 'packaged' in order to serve as a tool which is to be applyed while solving routine basic problems and understanding general physical concepts. Furthermore, people don't like math in general, so it's reasonable to assume that the more math you give them, the worse they'll do. Also, people like to think math is hard, which it isn't, at least not to some point. If you ask me, I'd blame the whole educational system for teaching math in a wrong way. But if you'd ask me which the right one is, I wouldn't have a clue.

But I'm talking about calculus that is taught at level before you specialise in engineering or math, the high-school level. Is it right to convey completely false ideas in the name of simplification (I have the same grudge against teaching the Bohr model)? Also, how much does it really simplify to be doing the thing before you know what you are supposed to be doing?

I'd say a system more right than the current one would be one which emphasised clear and unambiguous conceptions about the fundamentals before the calculations begun.

 

[MeJennifer]

Who is denying that? It was the similarity with the alleged muddled notion of infinity that I am asking about.

Well then Matt let me ask you this: Are you of the opinion that in the history of mathemathics at each era the top mathematicians had a solid grasp of infinity?

 

[matt grime]

I am of the opinion that the description of something as infinite if it is not finite is and always has been well understood by mathematicians. Mathematicians rarely use the word 'infinity', by the way in any definite sense, unless they are geometers when it, again, has never had a 'vague' meaning.

I would certainly agree that there was contention around the time Cantor attempted to introduce the notion that two infinite sets can be 'quantitatively' different, but that is almost entirely a point about philosophy. And that certainly does not imply that Euclid had some misapprehension in his proof that there are 'an infinity' of primes'. There is not a finite number of them, therefore there is an 'infinity' of them. There is no 'wobbly grasp of infinity' there at all.

 

[Hurkyl]

I simply meant that in the history of mathemathics infinity was not as straightforward even by established mathematicians of their time.

I wanted to point out that history is not an indicator of what is "intuitive". Intuition is a wholly subjective thing, and is a product of one's formal and informal education. (remember Einstein's "Common sense is the collection of prejudices acquired by age eighteen" )

Most laypersons are not formally taught about the infinite, and there's a lot of popular junk out there to confuse them... so it should not be surprising that a lot of laypeople find it counter-intuitive.

But when you study infinite things and become used to them, they can become rather intuitive.

And as mathematics progresses, new ways to look at things are discovered. For example, geometrically, there is no essential difference between the entire real line and the open interval (0, 1) -- and forming the extended real numbers by adding in the "numbers" $\pm \infty$ is exactly analogous to adding the endpoints 0 and 1 to the interval (0, 1).

 

[loom91]

I am of the opinion that the description of something as infinite if it is not finite is and always has been well understood by mathematicians. Mathematicians rarely use the word 'infinity', by the way in any definite sense, unless they are geometers when it, again, has never had a 'vague' meaning.

I would certainly agree that there was contention around the time Cantor attempted to introduce the notion that two infinite sets can be 'quantitatively' different, but that is almost entirely a point about philosophy. And that certainly does not imply that Euclid had some misapprehension in his proof that there are 'an infinity' of primes'. There is not a finite number of them, therefore there is an 'infinity' of them. There is no 'wobbly grasp of infinity' there at all.

Those who use infinity in a definite sense, such as null infinity or the extended real number [itex]\infty[/tex] have (I hope) a good grasp of what they are talking about. But a beginner encounters infinity (or the infinitesimal) as a limit in calculus, where it is not at all a definite quantity. Still our teachers write things like 1/infinity = almost 0, whatever that's supposed to mean. Any half-competent high schooler can compute the limit $$lim_{x\rightarrow 1} \frac {x^2 - 1}{x-1}$$ but how many of them know what they are computing?

They are all given vague ideas such as very small, very close, and negligible differences, which are not simplifications but simply a bag of lies invented by witless teachers who are either too ignorant or too lazy to tackle the real thing.

 

[pete5383]

Loom, this may be a stupid question, but I don't understand how the answer to that limit is something beyond the grasp of any high school student (I assume most understand what 2 is), or am I missing something deeper in that...which is likely the case. I guess using L'Hospitals rule involves an indeterminate form, but in my evaluation of that limit, I didn't even have to think the word "infinty". Ooooh, or was that just a statement about the teaching of limits in general?

Hehe. Sorry, I kind of babble in that post.

 

[CPL.Luke]

loom your limit is undefined in the above as it simplifies to 0/0

 

[Omega_6]

loom your limit is undefined in the above as it simplifies to 0/0

No it isn't. You have to use L'Hospital's Rule (the answer is 2).

0/0 is not considered undefined such as 3/0 is anyways...it is considered an indeterminate quantity.

 

[Hurkyl]

There's a much easier way than using L'Hôpital's rule.

 

[Omega_6]

There's a much easier way than using L'Hôpital's rule.

Yeah, you could just simplify...

 

[matt grime]

Those who use infinity in a definite sense, such as null infinity

what is null infinity?

or the extended real number [itex]\infty[/tex] have (I hope) a good grasp of what they are talking about. But a beginner encounters infinity (or the infinitesimal) as a limit in calculus, where it is not at all a definite quantity.

Eh? The notions that the limit of something is 'infinity' is purely a statement about the finite. It doesn't matter what you think 'infinity' might be in any larger sense at all. 1/x tends to infinity as x tends to zero is purely a statement about the finite - that 1/x can be made larger than any Y for x suitably small. There is nothing tricky about the notion of infinity there at all.

Still our teachers write things like 1/infinity = almost 0, whatever that's supposed to mean. Any half-competent high schooler can compute the limit $$lim_{x\rightarrow 1} \frac {x^2 - 1}{x-1}$$ but how many of them know what they are computing?

if they are truly even half competent then they know what they are doing. Perhaps we have different ideas about competency. And the answer is 'draw a picture', by the way.

They are all given vague ideas such as very small, very close, and negligible differences, which are not simplifications but simply a bag of lies invented by witless teachers who are either too ignorant or too lazy to tackle the real thing.

I am not going to defend, support or deny such insults. (If teachers are witless what does that make the student, most of whom are incapable of becoming teachers?)

 

[loom91]

what is null infinity?

I do not know exactly, but it is a tool sometimes employed in general relativity, related to conformal infinity.

Eh? The notions that the limit of something is 'infinity' is purely a statement about the finite. It doesn't matter what you think 'infinity' might be in any larger sense at all. 1/x tends to infinity as x tends to zero is purely a statement about the finite - that 1/x can be made larger than any Y for x suitably small. There is nothing tricky about the notion of infinity there at all.

I'm not claiming that the limiting infinity is a tricky concept, I'm claiming that the students and teachers I've seen do not understand it.

if they are truly even half competent then they know what they are doing. Perhaps we have different ideas about competency. And the answer is 'draw a picture', by the way.

Perhaps the state of education is better where you live, but here only a select few students will be able to tell you what a limit is. Most of them will regurgitate what they were told by their teacher, which makes limit sound like a numerical approximation.

Because of this perception of limits as approximate rather than exactly defined quantities, I had a hard time convincing a student (who happens to a wizard at solving math problems) that events that are not impossible may nevertheless have exactly 0 probability. He kept claiming that they will have almost 0 probability, and I'm not sure I managed to convince him completely at the end.

I am not going to defend, support or deny such insults. (If teachers are witless what does that make the student, most of whom are incapable of becoming teachers?)

Teachers are not witless, only the witless among them are, and unfortunately these seem to constitute a depressingly large proportion among those I've met. I've found that this can not be related to their educational qualifications. There are three teachers holding doctorate in my school, and they are among the poorest teachers in the school. Exams taken to give teaching posts test only knowledge, but extent of knowledge is almost independent of capability to teach that knowledge. As for your question, a witless teacher usually makes a witless student, the point I was trying to make.

 

[loom91]

loom your limit is undefined in the above as it simplifies to 0/0

State of education.

 

[matt grime]

I had a hard time convincing a student (who happens to a wizard at solving math problems) that events that are not impossible may nevertheless have exactly 0 probability. He kept claiming that they will have almost 0 probability, and I'm not sure I managed to convince him completely at the end.

I am not surprised at this: perhaps you were thinking of different things. Certainly in probability there are non-trivial events with measure zero, such as P(number picked at random from [0,1] is rational) is zero. But if the other person is thinking of the real world then this doesn't make much sense since there is no such thing as the interval [0,1] nor a random generator of its elements in the real world.

Without wishing to get into too much philosophy, the limit in the example you gave is exactly just a definition, in my opinion. It does not have any significance other than a calculation. What you can do with it afterwards is a separate issue, and if they don't explain that then it is a short coming, but not necessary for understanding how to take limits.

Which goes someway to your first post. When you say your 'maths teacher' glossed over some of the detail, who do you mean? Are you at university here? Is this a class for mathematics majors? Is it one for general engineering people? Because that is why they'll omit epsilons and deltas: the answer is more important than understanding the method rigorously. The thing you can do is wait until you do proper courses in this case. Won't help the students who hate maths and won't do more of it, but then I doubt they'd appreciate a proper treatment of calc. Of course that kkind of assumption might be totally wrong and the cause of all the problems. But see the thread started by someone on 'proofs' who railed against mathematics being taught with boring rigor in it. You can't please all of the people all of the time.

 

[loom91]

the answer is more important than understanding the method rigorously. The thing you can do is wait until you do proper courses in this case. Won't help the students who hate maths and won't do more of it, but then I doubt they'd appreciate a proper treatment of calc. Of course that kkind of assumption might be totally wrong and the cause of all the problems. But see the thread started by someone on 'proofs' who railed against mathematics being taught with boring rigor in it. You can't please all of the people all of the time.

That is where I disagree. I don't think throwing the concepts out of the window and going for the formula memorisation approach helps students who hate math. Such an approach is precisely the cause for the pervasive resentment for math which is a global phenomena. It's quite possible to teach without sacrificing rigour yet not making it boring by filling pages with dry unmotivated derivations. The best textbooks and teachers have done this, but these are rather few.

 

[matt grime]

A pervasive resentment of mathematics throughout the world? Really? I know students who found it dull, but 'resentment' is a little strong don't you think? I can't say I know how people in Mali feel about maths, to be honest.Oh, and it is perfectly possible to teach the entireity of calculus without mentioning infinity at all (getting back to the point of the thread). People's misuse of infinity in mathematics is entirely to do with the use of it outside of mathematics.

 

[loom91]

A pervasive resentment of mathematics throughout the world? Really? I know students who found it dull, but 'resentment' is a little strong don't you think? I can't say I know how people in Mali feel about maths, to be honest.


Oh, and it is perfectly possible to teach the entireity of calculus without mentioning infinity at all (getting back to the point of the thread). People's misuse of infinity in mathematics is entirely to do with the use of it outside of mathematics.

I was not talking specifically about the abue of infinity, but rather the abuse of limit, trying to sell it using grossly inaccurate language because it would take time and thinking to come up with something better.

 

[shmoe]

Oh, and it is perfectly possible to teach the entireity of calculus without mentioning infinity at all (getting back to the point of the thread).

This is sadly lost on many students and they end up thinking their ideas on 'infintessimals' make any sense inside a standard calculus class. I'm not sure why this happens, but I suspect it's from skimping on the actual definitions of limits in less rigorous calculus classes. I wouldn't say that this skimping is necessarily a bad thing, different disciplines will need to learn the maths at different depths, but if they don't realize they are skimping it can lead to misunderstandings.

 

[HallsofIvy]

loom your limit is undefined in the above as it simplifies to 0/0

Please take a bit more math before you say things like that. The limit, as given, is 2. The fact that it "simplifies to 0/0" does not say anything about whether the limit exists or not. (Had it simplified to something with a non-zero denominator, that would have told us the limit definitely existed; had it simplified to something with 0 denominator, non-zero numerator that would have told us the limit definitely did not exist; 0/0 does not tell us either way.)

 

[matt grime]

I was not talking specifically about the abue of infinity, but rather the abuse of limit, trying to sell it using grossly inaccurate language because it would take time and thinking to come up with something better.

Abuses like 'so small as to be negligible'? Well, that usage is both useful to the non-pure mathematician, which for 99.9% of the people taking the courses is what is important, and justifiable (the space of dual numbers). It is indeed the point of teaching linearization to engineering students.

Also, there is little that goes wrong when declaring e^2=0 most of the time, and the algebraic rigor can be provided, in many cases.

Oh, and you are now contradiciting your original post which specifically labelled the abuse of infinity as a major problem.

 

[loom91]

Abuses like 'so small as to be negligible'? Well, that usage is both useful to the non-pure mathematician, which for 99.9% of the people taking the courses is what is important, and justifiable (the space of dual numbers). It is indeed the point of teaching linearization to engineering students.

Also, there is little that goes wrong when declaring e^2=0 most of the time, and the algebraic rigor can be provided, in many cases.

Oh, and you are now contradiciting your original post which specifically labelled the abuse of infinity as a major problem.

The course I'm talking about is neither for engineers nor for mathematicians, it is for students who may become either, and the presentation is making sure that it's the former rather than the latter. I don't think an explanation of limit as an approximate or subjective concept is at all useful to anyone. I'm not sure what you mean by justifying such an approach. In any case no such justification is provided.

I'm not sure what you mean by e^2=0, as far as I know e^2 is a uniquely defined quantity that, being the limit of a strictly increasing positive sequence, is necessarily non-zero.

The abuse of infinity (and infinitesimal) is only part of the problem that is abuse of calculus.

 

[matt grime]

Who said anything about justifying limits as 'approximate or subjective'? The fact that derivatives are used to linearize and approximate is what I was talking about and has nothing to do with what you think I said. Dont' get me wrong, I wish that maths were taught properly, but I don't think the problems you perceive are to do with poor understanding of the concepts by teachers. The course content is not decided by the teachers, or the mathematicians, but by the engineers who teach the output of the course. I think you're ascribing too much mathematical sophistication to the average student as well.

Further, I suggest you look up the space of dual numbers before saying much more.

This, also, makes little sense:

"as far as I know e^2 is a uniquely defined quantity that, being the limit of a strictly increasing positive sequence, is necessarily non-zero"

What limit of what strictly increasing sequence? e is just a symbol, and the ring of dual numbers is just the the ring k[e]/e^2. It is how algebraists do the algebraic version of differential geometry without taking limits which are usually meaningless in algebraic contexts.

 

[loom91]

Who said anything about justifying limits as 'approximate or subjective'? The fact that derivatives are used to linearize and approximate is what I was talking about and has nothing to do with what you think I said. Dont' get me wrong, I wish that maths were taught properly, but I don't think the problems you perceive are to do with poor understanding of the concepts by teachers. The course content is not decided by the teachers, or the mathematicians, but by the engineers who teach the output of the course. I think you're ascribing too much mathematical sophistication to the average student as well.

Further, I suggest you look up the space of dual numbers before saying much more.

This, also, makes little sense:

"as far as I know e^2 is a uniquely defined quantity that, being the limit of a strictly increasing positive sequence, is necessarily non-zero"

What limit of what strictly increasing sequence? e is just a symbol, and the ring of dual numbers is just the the ring k[e]/e^2. It is how algebraists do the algebraic version of differential geometry without taking limits which are usually meaningless in algebraic contexts.

I don't know what interpretation of e^x you are talking about and what meaning can be assigned to the statement e^2 = 0, but I don't think dual numbers have any relevance here. In the context of calculus, which we have been discussing here, e^2 is defined as the limit of a sequence of partial sums, and this limit is not 0.

I also don't think poor teaching is due to teachers not undersanding, it is due to them not teaching. I'm sure most teachers have a good understanding of what limits are, but as long as they are teaching wrong concepts to the students their internal knowledge is of little relevance.

I don't know why you are bringing engineers into this, I repeat that I'm not talking about engineering courses. It also has little to do with course content, I'm talking about teaching methods. As our chemistry teacher used to say, students lives are being destroyed because the lords of education are eternally afraid that too much stress will be placed upon the fragile minds of the students. The whole process is like trying to pull back the steering more to generat lift when the wing has already stalled: the situation goes from bad to worse.