What Are the Best Tools for Teaching Calculus to Children?

[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.

 

[calculus1967]

I think this is a good idea. I am 12, and I taught myself calculus, for the most part. It would be much easier for the students to learn with someone to actually teach and help.

 

[mathwonk]

you sound like the perfect person to answer the question of how to teach calculus to a 12 year old. how did you do it?

 

[Icebreaker]

Most calculus courses that I've taken simply skimmed through limits and rushed to differentiation and integration. Cal II and III were simply expanding "techniques of integration", which seemed to me like a fancy term for "memorizing the formulas", which is what we pretty much did.

 

[mathwonk]

the low level of preparation of most students today in algebra, geometry, and logic, forces most schools to reduce calc courses to rote formulas and their applications. many however preserve a few courses called honors or elite honors, where the material is actually explained and explored somewhat. (from books such as spivak, apostol, courant, kitchen).

I indulge myself on this forum by frequently explaining as if i am talking to honors classes, to the dismay of some students. but this is a public forum, and i am not getting paid here, so I do what I think is right.

 

[Stingray]

I learned calculus when I was about 12 (from community college courses), and have always thought that it was something that could and should be taught much earlier than it usually is. Calculus (/analysis) can be made very difficult indeed, but the essence of the subject is really quite intuitive. It is also much more interesting than any other (traditionally taught) type of elementary mathematics. I think teaching these concepts early on can go a long way towards improving a student's overall interest in mathematics.

Despite all of that, my own (limited) teaching experience has been disappointing. I've TA'd a few introductory calculus-based physics courses, and the students' lack of motivation, knowledge, and ability is just depressing. And these were mostly 18-20 year old engineering majors. A lot of them seemed to have been told by their peers that math and physics were impossible and ultimately pointless (!) subjects, so it was almost fashionable to do poorly. I wonder if a younger group of students who haven't had (much of) this kind of exposure would be easier to deal with?

Anyway, I'm glad that somebody has the courage to try this out. Good luck.

 

[maverickmathematics]

In the UK calculus used to be taught at 15-16, this is a while ago now - probably when the GCSE was still an O-level, now it is taught @ 17-18.

I believe that the single greatest hurdle in the teaching of calculus is the symbols - it can appear to be a different language that is only understandable by mathematicians - there are many good books out there which teach maths by bringing in these symbols as time goes on in the book - and I think this should always be the method for it is true that the symbols were only created to serve to describe something that had already been known.

Regards,

M

 

[mathwonk]

As has been pointed out here clearly, the opinion people have about when calculus should and could be taught changes radically when they start trying to do it.

the key point is not the age of the students but their interest, work ethic, and preparation. what you and I may think about how deprived we were as students and how much more we would have liked to be taught, has essentially no bearing on the experience of unmotivated students we may wind up teaching.

so today even teaching calc to most college students is no easy job. by contrast teaching it to brilliant 12 year olds who actually know their algebra is a snap.

all successful teaching is a process of adaptation to the needs and problems of the students. there is no a priori method that is guaranteed to work in any given class. everyone with teaching experience knows that a class with more than one person in it already presents problems of different paced learning.

I had an advanced class recently with less than 10 people in which 2 students claimed we were going so slow they were completely bored, and they demanded we speed up. But at least 2 others asserted that they would drop out if we accelerated any at all.

The difference here was not in degree of understanding as the slower stdunts actually mastered more material than the "fast paced" complainers; it was a matter of wilingness to accept careful presentation, complete proofs, and and in depth examination and generalization of topics.

 

[HackaB]

The difference here was not in degree of understanding as the slower stdunts actually mastered more material than the "fast paced" complainers; it was a matter of wilingness to accept careful presentation, complete proofs, and and in depth examination and generalization of topics.

This is a crucial point. You can get as much or as little out of a course as you want to. Well, at least "as little". And I must say that I am a little skeptical of claims that calculus is so "intuitive" and "simple" that it ought to be taught much earlier. It depends on how well you want to understand it. I expect (or hope, anyway) that my understanding of the subject will continue to grow and deepen as long as I use it. You don't just sit down and learn calculus once and "that's it". You learn it over and over, and eventually get used to it. So I guess it doesn't matter that much when someone *starts* learning the basic concepts of calculus. But it is unreasonable to expect a 12 year old to absorb everything in some short summer course for wiz kids, just as it is unreasonable to expect a 17-19 year old to get anything more than the very basics out of a first course in high school or college.

 

[mathwonk]

well calculus mis about the reamtion between the height and the area of a figure, and that can be taught at many levels of difficulty short of full blown calculus.

e.g. the discovgery by metrictensor that the ratio of area to volume is the same for a square and a circle is basically explaiend by calculus, but can be elarned and notioced much earlier.

in the same vein, i agree it makes little sense to push calculus on someone who does not aprpreciate the areas and volumes of figures.

but my experience teaches that it is a mistake to crtoticize what someone says he is going to teach in a class without seeing him teach it. usually the etacyher ahs pout ebnough thought into it to have arrived at something that will work. it just usually is not what we would have described by the same names.

so whether or not what Steve Rives teaches would qualify as calculus seems questionable to me, but i'll give him the benefit of the doubt that he has something figured out that will be fun and instructional to his charges.

 

[arcarsenal1]

Sorry to reawaken a long dead thread.

I felt I need to add my 2 cents.

While I don't think SteveReis or whoever the original poster is is going to be teaching TRUE inDEPTH calculus, I absolutely admire his goal and have thought the same myself.

If one teaches children the method of computing integrals and derivitives as opposites and complements, in the same way that addition and subtraction and multiplication and division are opposite complements, they may not know CALCULUS, but they will be more than ready to accept the rigorous proofs and theorems later in life.

X - Y, X + y

X * Y, X / Y

These are simply procedures.

I think it is absolutely logical, feasible, and admirable to teach younger kids derivitives and integrals as PROCEDURES. they don't need to know the theory, or why they are doing it just yet. but if they understand and can perform the procedures, and are given simple procedural practice problems, they will simply be using their multiplication, division, addition, and subtraction in a procedure.

to know this procedure and understand it as a simple set of opposite complements, like the procedures of addition and subtraction and multiplication and division, will greatly enhance their ability to learn and understand formal calculus later in their academic career.

The first respondants to this thread did a GREAT disservice to the goal of the teachers who started this thread. their mathematical righteousness, and this inability to separate the simple procedural components of calculus from their theoretical context, is exactly what turns kids who arent math majors off from ANY math at all. quit being so self-righteous and acknowledge that some people will be done a great service with a procedural introduction early in life.

 

[mathwonk]

i disagree. i think it was the OP who killed the thread. the criticisms were no more than is usual, and he actually tried to provoke them, then recoiled rather suddenly and defensively.

His posts were all pie in the sky philosophy and complaints, no substance at all.

what is your point in resurrecting the thread only to heap further criticism on people who were actually participating in the discussion?

talk about self righteousness.

I got trashed far worse than this when I logged onto to a rabid BMW site and asked innocently, but in true frustration, whether others had also found their BMW less reliable than a Honda.

 

[qspeechc]

Mathwonk, why haven't you written several maths textbooks by now? I think they would be brilliant.

 

[Diffy]

I wonder how the course went...

 

[mathwonk]

in have written them but not published them. and i admit they ain't that great. but i like them.

my best one is my algebra text, the longer one, free on my webpage. (math 843-4-5 notes)

and it is far easier to make one short inspired entry, than to keep it up for 400-800 pages of a whole book.

but thank you! any apparent modesty on my part is entirely a pose.me too, diffy. (your last name is Q, i presume.)