What Are the Best Tools for Teaching Calculus to Children?

[SteveRives]

Dear List,

I am preparing a summer school course for children who have basic knowledge of powers, volume, and area. The course is "Calculus for Children," where we will study the integral and derivative using various ideas that make the subject easy to grasp.

I wonder if anyone on this list knows who has done work in this area, and what sorts of tools are already available? Perhaps some of you have ideas that communicate some of the aspects in new or easy to understand ways?

By the way: I am aware of "Caclulus by and for Children", which I will not be following.

Regards,

Steve Rives

 

[HallsofIvy]

What age are these "children"? And how much mathematics are they expected to already know? I think trying to teach calculus to people who are not already VERY comfortable with algebra and trigonometry would be a gread disservice.

 

[SteveRives]

First, let me introduce myself, my name is Steve Rives, I teach Calculus in Kansas City. I write software for Reuters, I am finishing a degree in Archaeology. More about me can be found at www.ugrt.com.

To your question: Calculus for all ages is just that, all ages. The requirements for the children to learn Calculus are these: be able to compute (and understand) area and volume, be able to understand basic algebraic equations (including the use of powers). The age cut-off looks like age 11 right now -- though I have tested some of the material on a seven year old and two ten year olds (enough to make me satisfied with the curriculum and material).

Regarding Trig: With my methods, trigonometry is not needed for Calculus (besides, what is trig except ratios -- and how hard is that?). I feel capable of demystifying the entire subject of Calculus. So that one can learn and understand derivative and integral at an early age. Think of it this way, we learn addition and subtraction before we learn number theory. We learn multiplication and division before we read Burtrand Russel’s Philosophy of Math. We often don’t know the depths of a thing before we know the thing at some level.

There are pairs of operators we know earl on:

+ and -
* and \

I am simply going to add two more:

Integral and Derivative.

It's really not that hard. In fact, the hardest part of Calculus is algebra! And, so long as we don't worry about integrating 1/x, or something like that, we can stay away from the difficult algebra.

Certainly there are things you have found in teaching Calculus where you have thought: "Really, this is a simple idea that even a child can understand!" Perhaps you have even encouraged your class with that sentiment. I suspect there are also parts of Calc that you teach that you wish were taught to you when you were being taught. Those are the ideas I am hunting for.

Now, I must admit, I have never taught Calculus to a class of children, so you may be spot on that this will not work. My target class is mostly 11-12 year old kids (and I expect around 15 to take the class). The parents where I teach are interested in this, so I hope to report back the results (not here necessarily, but in a paper I am presenting in a couple of months).

Okay, I am caving to the pressure to preach just a little... my basis for teaching to children the Calculus is threefole 1) The integral stands on its own independent of the derivative! 2) the derivative works (intellectually and in a satisfying way) without the limit, and 3) both work without the idea of a tangent line.

Who can imagine such a calculus? Well, my students can, because I teach it to them, and all the standard formulas hold perfectly! Nothing is lost. I could teach Calculus starting with the integral, followed by the derivative and ending with the limit! In fact, I like to use the derivative to solve those "perplexing" e-d problems. Imagine a student who learns calculus before getting caught-up by the limit and the e-d thing. It really works! I start day one with the derivative.

Well, I have revealed my secrets, now it is up to you to work out the details. But I still have a question to the list: has anyone come across material on this subject of Calculus for children?

Regards,

Steve

P.S. My best Calc student this year was a 10th grader who had not had pre-calc or geometry. Things worked out just fine, as he was the student of the year and was awarded a wonderful electronics kit (the Basic Stamp) to celebrate his work! The text we use is a standard Calculus text that would be used in a college. I just augment it (don’t we all?) and mix the order of the chapters.

 

[James R]

2) the derivative works (intellectually and in a satisfying way) without the limit

I am interested to hear how you define a derivative without limits. Can you briefly explain, please?

 

[neurocomp2003]

use Maple

and he could prolly teach derivatives without limits by not dealing with the concept of derivative but how to process a given derivative x^2->2*x...after all they are children so theory at that level may not be necessary unless they ask for it. Its like teaching engineerings vector spaces without going through all the proofs.

 

[HallsofIvy]

"2) the derivative works (intellectually and in a satisfying way) without the limit, and 3) both work without the idea of a tangent line."

In other words, you are going to teach a "cookbook"- memorize this formula- course. As I said before, you would be doing the students a dis-service in making them think they have learned calculus when they are only learning formulas.

 

[SteveRives]

"2) the derivative works (intellectually and in a satisfying way) without the limit, and 3) both work without the idea of a tangent line."

In other words, you are going to teach a "cookbook"- memorize this formula- course. As I said before, you would be doing the students a dis-service in making them think they have learned calculus when they are only learning formulas.

No, not at all, this is not a formula course. In fact, it is a meaning course. I want them to know and answer this: What is the meaning of the derivative and what is the meaning of the integral? I could explain that to anyone without the use of tangent lines or limits (and still be exactly right). I can "show" that the derivate of x^2 is 2x, and I can do it without the parabola. And I can show the inverse as well! In fact, for all kinds of common objects one can abandon the Cartesian coordinate system and explain how they grow and do what they do (and from that derive the exact-same formulas as one gets with the limit method).

As an aside: We are all clear that Calculus is a tool that is uselful, and it is not owned by a mathematical priesthood, right? It is really a simple subject when it is rightly explained. If it is not easily grasped in our own minds, then perhaps we can re-think it till it is boiled down to its simplicity. What keeps us from that may be found in answering the question: How was it explained to us? If we only know one explanation of the subject (and most explanations are only variations on a theme), then I suppose it would seem like I am looting a temple by taking the high ideas to children; but there is another way. And simply put: that way is to study the growing of objects (and not the tangent to curves).

It has come to be the case that we relate both integral and derivatice to the limit for the demands of rigor. The Limit allows us to prove things in a certain way -- but the things were true before the limit! Many things are true and demonstrable in ways that might allude one who has not looked at it a different way, this may be one of those.

I am purposely being vague because of the tenor of the conversation ("doing a disservice," "cookbook approach", "only teaching formulas", ..., seriously, I have not said enough for that kind of analysis to be made. If you want to insult me, please, call me what it is that those words clearly imply I am).

I was primarily looking for pre-existing knowledge (pedagogical) on the subject in relationship to teaching children Calculus (in an understandable way). I wouldn't ask how to make kids memorize formulas, I don't need advise on how to do that (nor do I want to it).

Perhaps there is someone out there who has come across some original ideas along these lines? It may be that while you were learning the subject, you had to make yourself understand the integral or derivative in a way beyond the textbooks. That is the field of thought I am trying to walk in.

Regards,

Steve Rives

 

[arildno]

Are you using some sort of functional approach, then?

I must admit, right now you sound like a crackpot who does not know what he's talking about, but have cooked up for yourself some "explanation" you mistakenly believe is the "true" explanation.

In other words, show us your wonderful new way of teaching kids about derivatives and integrals.

 

[SteveRives]

I am interested to hear how you define a derivative without limits. Can you briefly explain, please?

To get you started: Silvanus and Gardner, "Calculus Made Easy", p.42-44 (one can read those pages online at amazon by clicking on the "Excerpt")

Sorry, but Silvanus does not reverse the Ease of the derivative and give a simple integral that maps to his derivative.

In fact, the integral without the derivative is not so easy to come by in the sources (I don't know any source). I know the answer, but I can't point you to a publication. The answer is mostly pictures, so I'd have to draw them, not tell them.

-SR

 

[SteveRives]

Are you using some sort of functional approach, then?

I must admit, right now you sound like a crackpot who does not know what he's talking about, but have cooked up for yourself some "explanation" you mistakenly believe is the "true" explanation.

In other words, show us your wonderful new way of teaching kids about derivatives and integrals.

I am sorry, but at this point I am going to dismiss myself from the discussion.

Next time you ask for something, try this:

"That sounds interesting, what is it?"

NOT

"You sound like an idiot, Now prove that you are not."

 

[arildno]

Well, for one, it is incorrect of you to say that Silvanus does not utilize the CONCEPT of limiting processes on pages 42-44, although he chooses not to use the WORD "limit".

 

[arildno]

BTW, I have no trouble with the idea that there might exist other interpretations of the derivative which is easier to understand than "the tangent of the curve" as you call it.

As it happens, what Gardner&Silvanus advocates seems to be Newton's original arguments concerning the behaviour of fluxions (or, possibly, Leibniz' theory of infinitesemals, although the accessible text on amazon was too short to make a firm judgment on this).

 

[Crosson]

Hi, Steve. I think your goal is achievable, but it will be difficult.

In my opinion, the biggest block to understanding calculus is a poor understanding of functions. In my high school calculus class, there were people who thought that f(x) meant "f multiplied by x" (when they bothered to think at all).

If you can get them to understand functions in general, through a series of examples (i.e. the temperature is a function of where I am and what time it is), then you can get them to understand calculus in general. If the children don't see functions all around them, then they will think of calculus as "...something you can use when you have a polynomial". (this is the general message in "calculus for business" classes.

I see the biggest danger as a failure to make the connection between polynomials and functions in general, and therefore a tendency to view calculus as manipulating polynomials (rather then functions in general).

 

[mathwonk]

1) It is easy to define and discuss derivatives rigorously without limits, as Descartes did. I.e. for all polynomials f(x) the derivative at 0 is the coefficient of the linear term, which has meaning in terms of the best linear approximation to f near x=0.

for x = a, one simply makes the substitution x = (x-a)+a, re expands f, and takes the coefficient of (x-a) as the derivative at x=a.

2) this has more intrinsic meaning in terms of tangency, as "double intersection multiplicity". I.e. two polynomials are tangent at a point x=a, if and only if (x-a)^2 divides their difference. then the derivative of f at x=a is the unique number c such that f(x) is tangent to the linear function f(a) + c(x-a) at x=a.

3) one can also easily show this definition is equivalent to saying that the derivative of the polynomial f at x=a, is the value of the polynomial g(x) = [f(x)-f(a)]/(x-a), at the point x=a. This g is of course a polynomial by the root factor theorem.

Hence derivatives have nothing at all necessarily to do with derivatives, or with evaluating them, until one reaches the two transcendental function sin(x) and a^x. Then they give a tool for evaluating the expression sin(x)/x at x= 0, when this expression cannot be simplified, due to ones not knowing the infinite series expansion for sin. for [a^x-1]/x, at x=0, even this technique does not work, since there is no useful way to evaluate the limit. hence one appeals to intuitiona nd guessing, or the inverse function theorem, and integral calculus and treats 1/x instead.


having said this much, i have a comment on the hypothesis for steve's project, i.e. to take only children who "understand powers, areas and volumes". Frankly this would not include most members of a freshman college non honors incoming calculus class today, at many, many schools.

Hence a group of children like this, aged less than 12, is very special, and far more likely to succeed than one might think, or it may not exist, or may not be easy to find.

But let the guy try. anything works if they are willing and enjoy it.


archimedes apparently did integral calculus in a few important special cases, i.e. area under parabolas, volume of spheres, essentially as limits, by using the formula for the sum of the first n squares: 1^2 + 2^2 + 3^2 +...+n^2 = (1/6)(n+1)(2n+1)(n).

the only limit needed to do the problems above, is that 1/n^k goes to zero as n goes to infinity when k > 0.

there exist books on calculus for polynomials, presumably these take advantage of the restriction, perhaps as i did above. i also have notes on derivatives this way, including motivating the derivatives of sin and cos via the squeeze law.

 

[mathwonk]

i am sorry we ran mr. rives off. He did come across a bit like a thin skinned crackpot, but hey, he is actually out there teaching children. if we think we could do better i hope we will try.

One thing to keep in mind is that it is very hard to describe in words what you plan to do in a class. I have virtually never heard people agree in conversation about teaching philosophy, and yet people seldom disagree with what they see someone actually demonstrate in his class.


So when Steve says he is going to teach "calculus" to 11 year olds, we respond using our own meaning of the word "calculus", but if we watch him at work, we may find out that what he means by that in practice is very different and much more workable.


Anyone actually in the classroom is forced to do something that more or less works, or the students go to sleep or fail to return next time.


.

 

[HackaB]

i am sorry we ran mr. rives off. He did come across a bit like a thin skinned crackpot, but hey, he is actually out there teaching children. if we think we could do better i hope we will try.

I agree completely. Couldn't we at least hear his approach before one of the know-it-alls asserts that their "math member" is bigger than his, frightening him off. So what if he was completely bogus? We'll never know now.

edit: I will say this though, if he'd spent less time writing long, defensive posts and more time descibing his method, things might have gone better.

 

[mathwonk]

yes i was surprized when i read the discussion, that he cut bait after those fairly mild exchanges.

 

[HackaB]

BTW, here is a book on calculus without limits by two non-crackpots.

http://www.cds.caltech.edu/~marsden/volume/cu/CU.pdf

I haven't read it--limits suit me just fine--but it shows that one can apparently do rigorous calculus without explicit limits. Whether or not it's worth the trouble, I have no idea. The book isn't exactly a best-seller.

 

[arildno]

Before he actually chose to provide that reference to Silvanus&Gardner, his whole "project" looked extremely dubious.
If he can't handle criticism, that's his problem.

It would be nice if he returned, though, because if he is using Martin Gardner's approach on derivatives, this would be a good way to visualize them by (based on the excerpt I read).

 

[Anzas]

i don't understand why one would insist on teaching kids calculus improperly without going through limits first.

 

[mathwonk]

if you read my earlier post i have explained that limits have no role at all in differential calculus at least as long as one sticks to polynomials.

hence descartes was able to do differential calculus for polynomials before limits were invented.

the whole point is merely to evaluate the continuous function[f(x)-f(a)]/(x-a) at a.

This is easy with polynomials, first divide out, then evaluate. only with functions you cannot divide, like e^x and sin do you need limits to guess at the value.


in fact most books cheat you. they pretend limits are being used, then they evaluate those "limits" exactly as descartes did, i.e. they divide first then evaluate.


hence the presentation of derivatives in most calculus book, as far as it applies to polynomials, is entirely bogus. there is an unwarranted pretense that limits are essentially involved.

but all that is done, is to introduce limits when they are not needed, then remove them again when it comes time to evaluate an actual example.


unless you are forced to use the "squeeze" lemma, you are not doing an essential limit. many books skip exactly these cases, i.e. the proof that sin(x)/x approaches 1 as x approaches zero, as too hard for beginners. and no books at all to my knowledge, treat the derivative of e^x directly as a limit. [they all use integrals and the inverse function theorem, but without proving the latter, in the case of standard non honors books.]

that just means they think limits are too hard, and are not really using them.

in that case one might ask why they confuse the beginner with the concept at all?

one possible hypothesis is that almost everyone teaches the way they were taught, without thinking about the material or researching the historical approach.

integral calculus on the other hand, seems to use limits essentially, either to do it via "riemann sums" i.e. archimedes method, or via antiderivatives, which does use the limit approach to derivatives.

i.e. if one does not have a formula for the area, and one wants to show its derivative nonetheless is the height, one needs a conceptual definition of the derivative which is not dependent on the polynomial formula. one also seems to need the squeeze law. hence one needs the limit approach to derivatives for use later, to do the fundamental theorem of calculus, not to do derivatives themselves.

of course if one does not prove the FTC, where is the benefit?

i have thought about all this and tried teaching it, discussing it, and writing notes on it, for at least 35 years, ever since grappling with the notion of "zariski cotangent space" in algebraic geometry, i.e. descartes definition of derivative.

this esoteric object is M/M^2, where M is the maximal ideal of functions vanishing at the given point. I.e. this is the linear part, the part left after setting the higher order parts, i.e. M^2, of a function equal to zero.

there is a basic lemma in differential geometry that every smooth function can be written as its value at a point, plus its derivative plus the sum of products of pairs of smooth functions, with each factor vanishing at the given point, so this zariski approach also applies to non polynomials, i.e. M/M^2 also gives the zariski cotangent space for smooth manifolds. this lies behind many abstract looking definitions of the tangent space of a manifold, i.e. as derivations on the local smooth functions.

This is also silvanus p thompsons approach to derivative, when he says he only cares about the first order change of a function, dx, whereas higher order degrees of smallness, like (dx)^2 will be negelcted.

i loved silvanus p thompson, especially before martin gardner got his hands on it. i had a leather bound copy for years, which i gave to a favorite student, but his book is primarily recommended for its humor and light hearted approach, i.e. for sayings such as: "what one fool can do, another can".

[why would one call the book of thompson and gardner, as silvanus and gardner? not read it? or was silvanus the actual name and thompson a pseudonym?]

I was first introduced to it at Harvard by a student in the elite honors calculus course. But I must say that the actual insight provided by such delightful phrases as; "also fleas have fleas to bite em, and so on ad infinitum", was extremely minimal.


I never understood why his statements were actually correct until years later as he does not even attempt to justify them toi the curious mind.

Since chidren do not fear calculus as adults do, the value of his fear - lessening rhymes is also lessened.

i would prefer showing smart children the actual correct approaches of someone like descartes.

but i am reminded that my own children, who were very strong math types at an early age, were easily challenged and tired out by such difficult concepts. these ideas really are hard, and that was partly what made Steve's posts come across as almost ludicrously misguided: with statements like: "well children learn addition and multiplication, why not integration and differentiation?"

But I stand by my original claim that intrinsically, and historically, limits have nothing at all to do with differential calculus of polynomials.

 

[mathwonk]

it just dawned on me afterw riting those remarks, that eprhaps they explain why no one thought of the fundamental theorem of calculus until the 17th century. but now i am again puzzled, as the limit idea is attributed to Newton, but i thought isaac barrow? discovered the FTC, presumably earlier.

maybe barrow just noticed it for special cases. and then conjectured the theorem. i.e. one cannot strictly prove a theorem without having a definition of the concepts in that theorem. perhaps the internet contains some information albeit third hand,


a quote:

The Lectiones Geometricae probably represent work which Barrow studied while at Gresham. They contain the important work on tangents which was to form the starting point of Newton's work on the calculus. Barrow stated the main two aims were [15]:-


... first, to investigate tangents without the trouble or wearisomeness of calculation and, second, to determine the dimensions of many magnitudes quite quickly by means of their tangents ...


? whoops, to get it first hand, i would have to recall my old latin, and read 17th century english "fcript" [script].

I find it interesting that the english, who aspired to colonize the world, did not realize that one of their main contributions would be the spread of their language. i.e. why did they assume latin would be a more universal language for science?

 

[jma2001]

Wow, I've just been reviewing this thread to see where it went wrong. I would agree that Steve could have been more receptive to criticism, however I think arildno really lit the fuse by labeling him a crackpot. "Crackpot" is a pejorative term and should be reserved for those who have truly earned it, like the TimeCube guy.

 

[HackaB]

Before he actually chose to provide that reference to Silvanus&Gardner, his whole "project" looked extremely dubious.

Absolutely.

If he can't handle criticism, that's his problem.

True enough, but this is a "junior-high-school" attitude on your part. The object of a useful discussion is not to prove the other guy wrong, but to see if you can learn something from him. You should have encouraged him to say more about his methods before attacking him. Once he proves he's a crackpot--that's the time to pounce on him, not before. You smart people too often let your big egos get in the way of possible learning opportunities.

 

[arildno]

Wow, I've just been reviewing this thread to see where it went wrong. I would agree that Steve could have been more receptive to criticism, however I think arildno really lit the fuse by labeling him a crackpot. "Crackpot" is a pejorative term and should be reserved for those who have truly earned it, like the TimeCube guy.

I never said he was a crackpot, I said he sounded like one.
Note for example his sentence about a "mathematical priesthood" who has no right on monopolizing maths.
This is a very typical example of comments most often made by actual crackpots.
That Steve Rives is not a crackpot does not invalidate this.

This was just one example.

 

[arildno]

The object of a useful discussion is not to prove the other guy wrong.. .

I didn't try to; that was impossible, since he refused to give any details until that reference.
This is another typical example of how crackpots tend to couch their ideas, by keeping them on a vague&vacuous level.

Once he gave that reference, my response and appraisal of him changed.
That does not mean my initial response wasn't well-founded; rather, the information I had been given led naturally to that conclusion, however false it is when more information had to be taken into account.

 

[HackaB]

Fair enough. I see now that before you posted, Mr. Rives had been given ample opportunity to explain himself, and he didn't. It's a shame. I wanted to hear his approach.


mathwonk: I believe Vladimir Arnold refers to the fundamental theorem of calculus as Barrow's theorem in his book on ODEs. I don't remember if he said Barrow proved it, but he did mention that Barrow was Newton's teacher. And the latest copy of "Calculus Made Easy" that I've seen had the author's name as Sylvanus (or Silvanus?) P. Thompson, like you said.

 

[arildno]

Fair enough. I see now that before you posted, Mr. Rives had been given ample opportunity to explain himself, and he didn't. It's a shame. I wanted to hear his approach.
.

So would I!

 

[The Reverend BigBoa]

Well hopefully he'll check back and see that he's actually welcome in order that this discussion could continue. I agree though, if he's going to bail on account of THAT exchange, he must not get along well on too many web boards! Hopefully he'll realize that a lot of times, people say something in print, and it just doesn't mean anything nearly as confrontational as what it might sound like.

 

[mathwonk]

actually i think it is hopeless to have an online dialog with anyone that sensitive to criticism.

everyone should realize that in this milieu, one should simply ignore comments one does not like, rather than expect them not to be offered.

i.e. there is no way to prevent everyone from saying whatever they think.

I am rather amazed in the other direction on this forum, at the saint - like patience of many people who try to post useful advice here, even in the face of know it all comments from people who would apparently not recognize the objects they pontificate about if they were bitten in the backside by them.

 

[The Reverend BigBoa]

Can't argue with that. You're quite correct in making the statement that too many people need to ignore things they don't like, rather than whining and crying and always taking any comment too personal. I'm not suggesting he was quite that bad, though he did seem to bail quickly. You're also quite right stating that it could prove fruitless to try and have a discussion if he is going to be "offended" everytime someone disagrees with him. Some people just need to lighten up!

 

[mathwonk]

well criticism is always tricky. people ask for comments and then they don't like the ones they get. someone asks for help passing their course two days before the final, or 2 hours before, and they don't like being advised to start sooner next time, or to go back to their book and do the work themselves. they do not appreciate that this comment is actually an attempt to be genuinely helpful. i.e. being told to stand on your own feet is actually better advice than being carried.

someone asks for comments on a set of notes or book they have written and when you point out logical errors or uninsightful arguments, or much easier and more complete proofs, they say, well i didn't mean that, i meant factors of pi or wrong minus signs, i did not want to be told my proof was wrong, or my assumptions were inadequate, or my motivational discussion misguided. i was just being informal there.

still many people here are very skillful at being helpful and diplomatic at the same time.

 

[The Reverend BigBoa]

Skillful and patient indeed. I was quite impressed the first time I checked this board out. It is certainly loaded with some very sharp minds. And yes, a good number of them are indeed very patient and helpful. I definitely consider it to be one of the more useful places on the net.

 

[SteveRives]

I notice this thread has been read by many (815 as I now type). I take that to mean that this subject is of interest to many people (perhaps there are a lot of teachers out there looking for ideas!).

Therefore, and despite the state of the conversation as it now stands, I think it would be right to direct folks to a resource on the limit . You will recall, I have already drawn attention to an underused way to view the derivative (please find my earlier post in this thread).

On the Limit:

Keith Devlin, in my mind, is doing wonderful work in this area (see his latest book, Math Instinct). Dr. Devlin is the National Public Radio "Math Guy", his work is recognized as outstanding. He has a short article on the limit where he investigates (challenges!) the way it has historically been used and taught:

http://www.maa.org/devlin/devlin_5_00.html

For those who can stomach a thicker exploration of Dr. Devlin's methodology, I suggest this site: http://www.cogsci.ucsd.edu/~nunez/web/

-SR

 

[mathwonk]

I have tried to explain, perhaps too succintly, in posts 14 and 21 how one does derivatives quite rigorously without limits, indeed as they were done by Descartes, Fermat, and other pioneers before the introduction of limits. I have taught this approach off and on for several decades.

This topic is interesting, although not news to some of us, (indeed the citation of Decartes would seem to prove it is pretty old). I first learned the algebraic approach and wrote notes on it in 1967 at Brandeis, while studying the Zariski tangent space, and also in a solicited "book review" of a calculus book in 1978, called Lectures on Freshman Calculus, by Cruse and Granberg, in which Descartes' method is used only for quadratics, and the authoirs imply it does not work otherwise.

[I carefully explained to the publishers in my review that the method works for all polynomials, if understood properly, but they ignored me and published the misleading version anyway (in an otherwise excellent book, now long out of print). I realized later that like many people who ask for comments, they did not want corrections to their errors, but only wanted praise they could use in advertising copy.]

As to Mr Devlin's commments on the contrast between intuitive continuity and the epsilon - delta definition of it, he is quite right, but again this is hardly shocking news. The point I routinely make to my classes, and presumably many others do as well, is that the intuitive property we want for continuity of real functions, captured in Euler's "freely hand drawn graph" description is the statement of the intermediate value theorem. Indeed this was the definition of continuity given by some 19th century mathematicians, perhaps Dirichlet.

However today we realize that examples, such as f(x) = sin(1/x), for x not zero, and f(0) = 0, have the IVT property (near zero) but not the other intuitive property desirable for physics, namely that when the data we input into the experiment is approximately what is desired, then so should be the data output from the experiment, i.e. the epsilon - delta definition.

So we make the epsilon - delta definition for several very good reasons.

1) it is precise, and can be actually verified easily in specific cases such as polynomials and all elementary functions, so as to conclusively prove they satisfy it. It has an appropriate intuitive meaning, namely when x is near a, then f(x) is also near f(a).

2) it has as a CONSEQUENCE, i.e. as a provable result, the intuitive intermediate value property, in the case of functions defined on the real line.

3) it also embodies the desirable "physics" property above, i.e. if the measurements are approximate correct, then the result should be approximately correct. this epsilon - delta continuity of physical phenomena in the large, is assumed in all laboratory physics experiments, else they would be useless in the presence of any error at all.

4) it also applies to cases where the domain space is not "continuous", i.e. we can also speak of functions on the rationals being continuous in the epsilon delta sense, where we do not expect the intermediate value property to hold.


The news that what is taught in many current textbooks seems not quite all it should be, is only a remark on the lack of scholarship of some textbook authors, (or their desire to please their publishers) not on that of actual mathematicians. People who get their education from better sources, e.g. original works by great mathematicians, or from better textbooks, are not as limited by these misconceptions.

So I suggest that some, perhaps all, of the problems being posed, have been considered, even answered, hundreds of years ago; and one may profit from studying the historical development of the idea of derivatives by the old masters.


Or it may be more fun to rediscover it all over again for oneself.


Best wishes.