Is there research for or against such an approach in teaching calculus?

Question

Copying from Calculus Made Easy by Silvanus Thompson (2nd ed., 1914):

CHAPTER I:TO DELIVER YOU FROM THE PRELIMINARY TERRORS

The preliminary terror, which chokes off most fifth-form boys from even attempting to learn how to calculate, can be abolished once for all by simply stating what is the meaning -in common-sense terms-of the two principal symbols that are used in calculating. These dreadful symbols are: (1) $d$ which merely means "a little bit of." Thus $dx$ means a little bit of $x$; or $du$ means a little bit of $u$. Ordinary mathematicians think it more polite to say "an element of," instead of "a little bit of." Just as you please. But you will find that these little bits (or elements) may be considered to be indefinitely small. (2), $\int$ which is merely a long $S$, and may be called (if you like) "the sum of." Thus $\int dx$ means the sum of all the little bits of $x$; or $\int dt$ means the sum of all the little bits of t. Ordinary mathematicians call this symbol "the integral of." Now any fool can see that if $x$ is considered as made up of a lot of little bits, each of which is called $dx$, if you add them all up together you get the sum of all the $dx$'s, (which is the same thing as the whole of $x$). The word "integral" simply means "the whole." If you think of the duration of time for one hour, you may (if you like) think of it as cut up into $3600$ little bits called seconds. The whole of the $3600$ little bits added up together make one hour. When you see an expression that begins with this terrifying symbol, you will henceforth know that it is put there merely to give you instructions that you are now to perform the operation (if you can) of totalling up all the little bits that are indicated by the symbols that follow.

That's all.

I was wondering whether there has been (specific) research as regards the educational merits (or demerits) of such an approach to calculus.

Considering that basic computational and calculation skills are necessary for the reproduction of society, I can understand why one could approach arithmetic like that (in an attempt to make all citizens literate on arithmetic). But once we are into differential and integral calculus, it is not about basic computational skills anymore.

So does such an approach make for perhaps a fabulous ride in the beginning, only to be regretted later when symbols become more abstract in content and their manipulation must become more rigorous?

Or does it instill in the students an attitude like:

What? The integral should not necessarily be thought of as a sum? Great, because we were getting bored – let’s see what else we can do with it

Again, I am not asking for opinions (I find opinions interesting and useful, since they usually come with arguments – at least on this site –, but they are not allowed here as far as I know), but for any research results on the matter.

Answer 1

The answer to your question whether there is such research is affirmative.

In the approach adopted at my university and used to train over 400 students over the past three years, the role of infinitesimals is not to replace epsilon-delta definitions, but rather to prepare the students for such definitions.

Students react positively to such an approach, as reported in this article in Journal of Humanistic Mathematics.

We use a modification of Keisler's approach in his book Elementary Calculus. We strengthen the epsilon-delta component but only after the students are already familiar with the basic concepts of the calculus like derivative and continuity via more intuitive infinitesimal definitions a la Cauchy. Once the students understand the concepts, it is easier for them to relate to the epsilon-delta paraphrases of their definitions.

Answer 2

Ideas, devices, methods, etc., under the name "method of exhaustion" were the effective form of "calculus" for 1500+ years, successfully answering many questions both within mathematics and in continuum models of the world. I myself am very fond of "finding out what is true", even if I recognize that there are "gaps" or ambiguities. Then one looks at the latter, to resolve them, if there's any doubt, or if the conclusion is sufficiently scandalous.

As I have often said, and will say again, if the informal (pre-Robinson, pre-Skolem, et-al) idea of infinitesimals had not been so wildly successful, the epsilon-delta episodes in the 19th century would have been of interest to specialists only. (As it is, despite the stories we often tell students, being super-careful with calculus was by far not the main point of either Cauchy's or Weierstrass' work.)

So, when I have taught calculus intermittently over the last 40+ years, my first goal is to encourage people to use their physical intuition (and middle-school algebra, and basic geometry) to first-of-all get to an answer, or else discover serious obstacles/dangers. After all, which is better, having an understanding/answer of which one may be only 65% sure, or having no answer at all? Refining a flawed understanding/answer iteratively is what happens in real life, and in real mathematics.

A very poignant, and distressing, anecdote about "rigor" in calculus occurred in the mid-1980's when I was teaching a "rigorous math experience for future grade-school teachers". One could imagine that there was virtue in this. At the outset, a diagnostic "exam" showed that everyone in the room could carry out the usual operations of basic calculus, especially with polynomials and such. Long-story-short, after a quarter's discussion of epsilons and deltas, about half the population came to doubt themselves... to the amazing extent that they balked at doing things they could do a few months earlier. (Sure, maybe I did a bad job talking about things, but I don't think that's the dominant feature, since I was as much a pragmatist then as now...) This stunned me, and not in a good way.

Further, after all, the epsilon-delta biz is just one possible rule-set, with nothing about it more sacred than its current quasi-popularity.

Yet further, one could ask why lower-level mathematics (e.g., through undergrad and basic grad) is so rule-based, so prohibition-based, with quite a few of the rules nearly unguessable to non-initiates. I really do not like the "secret club" aspects... since the genuine difficulties are great enough.

Answer 3

From what I know of Nonstandard Analysis this seems to be similar. Your notion of "a little bit of" seems very close to infinitesimals. There were attempts to ground Calculus with NSA, and teach it that way, but this way of teaching has not caught on, for whatever reason. Below is some research on it.

H. Jerome Keisler, Elementary Calculus: An Infinitesimal Approach. First edition 1976; 2nd edition 1986.

Edward Nelson: Radically Elementary Probability Theory, Princeton University Press, 1987.

Answer 4

I am posting this as an answer to not overburden my question, but also to add content here that I consider useful. I found the following paper

Schwarzenberger, R. L. E. (1980). Why calculus cannot be made easy. The Mathematical Gazette, 64(429), 158-166.

(some info and links on the author can be found here).

Despite the title, the paper is not a polemic against the book. Rather it starts by noting that two years earlier "A Course of Pure Mathematics" by G.H. Hardy had been published, and the two books represented the end-points of the spectrum as regards the approach to mathematics as a teaching subject, and (quote) "both books shared a tone of evangelical enthusiasm".

The author summarizes his argument as follows:

"(i) The set $\mathbb R$ of real numbers possesses simultaneously many different structures.
(ii) Any attempt to explain calculus "easily" will use some of these structures but not others, and so will make certain aspects of calculus more easy but other aspects more difficult.
(iii) This applies particularily to those aspects of calculus which depend on the completeness of $\mathbb R$, and any attempt to avoid explicit mention of completeness puts a heavy burden on subjective and intuitive ideas about $\mathbb R$.
(iv) Small differences in the way different students think about sets and functions may grow into large -and even irreconcilable- differences in intuitive ideas about calculus."

He then discusses the concepts of "Instrumental understanding" vs "relational understanding", (see Skemp, R. R. (1976). "Relational understanding and instrumental understanding", Mathematics Teaching 77, 20-26), cautions about their use, and ends by arguing for

"(...)the freedom to teach a particular group of students, with particular intuitive ideas and particular applications of calculus in mind, in a manner appropriate to their particular needs."

Well, certainly such a "flexibility ideal" is indeed only an ideal in any human activity that must be done en masse, structured as a system, standardized (some would even raise issues of "equitable treatment" of students if "freedom to adapt" was granted to teachers). Still, some degree of flexibility must exist (and I suspect it indeed does, at least in practice and unofficially, if not officially sanctioned).

But to an outsider like me, points $(i)$ and $(ii)$ above strike an abstract structural chord as to why "things like calculus" can not be made "universally easy".