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


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.

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    $\begingroup$ What exactly do you mean by "such an approach to calculus"? E.g., the use of the term "indefinitely small" is a hard indicator that the approach here concerns the infinitesimal calculus, which turns out to be the case: Calculus Made Easy (wikipage). At that link you will also find that the book "ignores the use of limits with its epsilon-delta definition" (etc). All of this is a bit nonstandard, and I cannot tell what precisely you are looking for research on... $\endgroup$ – Benjamin Dickman May 4 '16 at 18:18
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    $\begingroup$ The "Harvard Calculus" pioneered by Andrew Gleeson takes a similar approach, without epsilon/delta limits. $\endgroup$ – Nat Stahl May 4 '16 at 20:03
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    $\begingroup$ @NatStahl, pioneered by Andrew Gleeson? I had always thought of it as pioneered by Deborah Hughes-Hallett. Perhaps it was both of them? $\endgroup$ – Sue VanHattum May 5 '16 at 13:49
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    $\begingroup$ It seems to me that this isn't teaching calculus at all, but only describing calculus so that a person might solve problems from calculus without really understanding them. I don't expect that a person who was taught with this approach would be very effective in applications of calculus, butt this is my opinion. $\endgroup$ – Andrew May 6 '16 at 16:45
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    $\begingroup$ @MikhailKatz I forgot that the epsilon delta definition was given by Weierstrass, not Cauchy. $\endgroup$ – Andrew May 12 '16 at 17:53

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.

  • $\begingroup$ The role of infinitesimals is not to replace epsilon-delta definitions, but rather to prepare the students for such definitions. Are you saying that it must be done this way? Is done this way? That you prefer it this way? It certainly hasn't always been done this way throughout history. This doesn't really seem like an answer, more like a comment. $\endgroup$ – Ben Crowell Apr 20 '17 at 0:32
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    $\begingroup$ @TommiBrander, OK, I tried to. $\endgroup$ – Mikhail Katz Apr 20 '17 at 7:12
  • $\begingroup$ @BenCrowell, I clarified this in my answer. $\endgroup$ – Mikhail Katz Apr 20 '17 at 7:12

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.

  • $\begingroup$ Thank you for these bits of experience (hard-earned I suspect). But please elaborate more on one point: You write that the success of "informal infinitesimals" is exactly the reason why the "epsilon-delta" approach became an issue of wide interest. I am particularly interested in the sociology of sciences, so if you could explain the connection a bit more, I would be grateful. $\endgroup$ – Alecos Papadopoulos May 5 '16 at 23:03
  • $\begingroup$ Although I do not pretend to be an expert historian of mathematics, I have tried to understand... At least as definite examples, the happy success of Abel's theorem about convergence and term-wise differentiation of convergent power series set ... what turns out to be ... a high bar for behavior of other "expansions". E.g., Fourier's ideas about expansions in trig functions/exponentials immediately ran into trouble (enough to lead Cantor to set theory 50 years later...) Sturm-Liouville eigenfunction expansions' convergence were ... [cont'd] $\endgroup$ – paul garrett May 5 '16 at 23:13
  • $\begingroup$ [cont'd] evidently problemmatical enough that it waited for Bocher and Steklov to really "prove" everything... (see J. Lutzen's wonderful discussion of this and other history...) Cauchy himself over-shot the mark on a few occasions, leading to awareness of the notion of uniform convergence, etc. But, as far as I can tell, this was all secondary to the discovery of new phenomena!!!!!! Sure, flawed proofs of true theorems are disquieting, but, hey, discovery is more significant than "propriety". Another line of examples is Dirac's quantum mechanics, "versus" von Neumann's ... [cont'd] $\endgroup$ – paul garrett May 5 '16 at 23:17
  • $\begingroup$ [cont'd] precisification of "self-adjoint", versus "symmetric" [earlier typo], and Schwartz' generalized functions... and Heaviside and the transatlantic telegraph... and, yes, too much to say... $\endgroup$ – paul garrett May 5 '16 at 23:49
  • $\begingroup$ Paul, rule-set is a nice way of describing the epsilon-delta definitions; another is paraphrase of the infinitesimal definitions that preceded them. $\endgroup$ – Mikhail Katz May 12 '16 at 16:21

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.

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    $\begingroup$ Thank you for your answer. I happen to possess both books. I wouldn't see Nelson's book as similar in spirit with "Calculus made easy", but indeed Keisler's use of non-standard-Analysis-founded infinitesimals does have the same, let's say, "carefree aura". $\endgroup$ – Alecos Papadopoulos May 4 '16 at 22:51
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    $\begingroup$ More information can be found in Vinsonhaler's article in the Journal of Humanistic Mathematics: "Teaching Calculus with Infinitesimals". $\endgroup$ – Joel Reyes Noche May 6 '16 at 3:19
  • $\begingroup$ @JoelReyesNoche Thanks for the reference this was a useful paper to read. $\endgroup$ – Alecos Papadopoulos May 6 '16 at 17:35
  • $\begingroup$ Specifically for Non-Standard Analysis, this thread has some thoughtful answers, math.stackexchange.com/q/51453/87400 $\endgroup$ – Alecos Papadopoulos May 6 '16 at 17:50

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".

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    $\begingroup$ As example of (ii.), if your primary interest is studying the continuity of functions over $\mathbb{R}$ then the infinitesimal approach is probably not so helpful. On the flipside, continuity is the cornerstone of the standard approach. Conversely, differentiation becomes mere algebra in the infinitesimal approach, so if you only care about differentiation without regard to its connections to continuity then the "easy" way gains merit. Personally, I take a two-fold approach, infinitesimal emphasis for engineering applications and continuity with care for epsilonics for math majors. $\endgroup$ – James S. Cook May 5 '16 at 13:44
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    $\begingroup$ To be clear, I do not teach nonstandard analysis in my introductory calculus sequence, but, every so often I'll give it a mention for future reading. Kids read these days, at least the ones for whom the comment is given. $\endgroup$ – James S. Cook May 5 '16 at 13:45
  • $\begingroup$ @JamesS.Cook Thanks, this appears to be an example of flexibility applied in actual teaching practice. By the way, the issue I questioned about is not Robinson's Non-Standard Analysis usage in teaching calculus, but rather the "let's be no more formal and rigorous than necessary" that seems to be implied by Thompson's book. $\endgroup$ – Alecos Papadopoulos May 5 '16 at 13:57
  • $\begingroup$ Although Hardy was a very good mathematician, his "Course of Pure Math" might be my least favorite of his stuff. $\endgroup$ – paul garrett May 5 '16 at 22:23
  • $\begingroup$ The link is not working. $\endgroup$ – Sue VanHattum May 6 '16 at 15:36

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