Calculus for All Ages

1. May 16, 2005

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

2. May 16, 2005

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.

3. May 18, 2005

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.

4. May 18, 2005

James R

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

5. May 18, 2005

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.

6. May 18, 2005

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.

7. May 18, 2005

SteveRives

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

8. May 18, 2005

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.

9. May 18, 2005

SteveRives

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

10. May 18, 2005

SteveRives

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."

11. May 18, 2005

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".

12. May 18, 2005

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).

Last edited: May 18, 2005
13. May 18, 2005

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).

14. May 18, 2005

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.

Last edited: May 18, 2005
15. May 19, 2005

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.

.

Last edited: May 19, 2005
16. May 20, 2005

HackaB

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.

Last edited: May 20, 2005
17. May 20, 2005

mathwonk

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

18. May 20, 2005

HackaB

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

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

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.

Last edited by a moderator: May 2, 2017
19. May 20, 2005

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).

Last edited: May 20, 2005
20. May 20, 2005

Anzas

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