27

I'm primarily a physicist, but I also teach first-semester freshman calc once in a while. Your characterization of a cultural divide between physicists and mathematicians on this subject does not seem at all accurate to me. If anything, I think the characterizations should be reversed, at least on the average -- but it would only be an average, because ...


18

This might be taking things too far, but Kiesler's book (available free online) does everything using infinitesimals, which make differentials literally immediate. The rigorous underpinning for infinitesimals is nonstandard analysis, but this book doesn't dwell on that. It just teaches how to use them correctly. I'm guessing this isn't exactly what you were ...


12

A faculty member in my department is a former student of Dray's and 2-3 years ago showed me the differentials approach. He and I both use it quite extensively in our first semester calculus courses. Here's the pros and cons (in no particular order) and comments on them. Pro: 1) It's generally awesome. Students in my class struggle a lot less with chain ...


11

I'd like to know why textbooks cover limits first. I don't think there's any big mystery as to why commercial textbooks tend to be similar. It's a market mechanism known as the network effect, the same mechanism that makes Microsoft Windows so popular. Once people start to see something as a standard, anything different becomes non-viable in the marketplace....


10

The apparent conflict between points of view expressed in the OP is illusory. There is no real conflict. The mathematics education researcher quoted in the OP is arguing that students find the definition difficult to appreciate and master. The mathematicians quoted in the OP are arguing that, once mastered, it provides clarity. I have questions about the ...


10

As @BenCrowell mentioned, the transfer principle proves that direct algebraic manipulation of infinitesimals in single variable calculus is allowed. But I stumbled upon this post in theshapeofmath.com, which shows how things can break when switching to multiple variables. It provides the following basic example of a possible error when handling partial ...


10

Teaching calculus using infinitesimals from nonstandard analysis has been tried, Keisler's text is considered a staple in this regard. But I am curious as to what is so "terrible" about epsilon-delta approach? Using infinitesimals invites us to envision a hidden world of numbers beyond the real numbers, into which all elementary functions miraculously extend,...


9

It has become almost a dogma that the math curriculum should teach technical prerequisites to what will be covered later. The consequence is that zillions of high-school students learn algorithms for doing partial-fractions decompositions and will never take later courses in which that is used. In effect the broad public learns that mathematics consists of ...


9

A great quote from Dray and Manogue: . . . many mathematicians think in terms of infinitesimal quantities: apparently, however, real mathematicians would never allow themselves to write down such thinking, at least not in front of the children. —Bill McCallum [16] I think in this type of discussion it is important to separate carefully between two issues: (...


9

My currently preferred approach is to start the course with an introductory lecture explaining the difference between average velocity over a time interval (something we can always in principle measure using a stopwatch) and instantaneous velocity at a given instant of time (which we do not have a means to measure). This motivates the question of what ...


9

If I may add my 2¢ as a former student, in no way involved in teaching this stuff, my own initial confusion came from discussions on "moving towards $x_0$" and "for very small ..." and other such. When I came to grasp $\epsilon$ - $\delta$ as a static situation, no "moving," no "very small" required, things became clear. I had an exceptionally gifted teacher ...


8

Alain Robert's book on non-standard analysis (following Edward Nelson's IST approach) was what finally convinced me that non-standard analysis could be packaged in an effective form. It is a small book, and the narrative is very unpretentious and informal, yet touches several further topics. "Nonstandard analysis in practice", edited by Diener and Diener is ...


8

The essence of your question is why do mathematicians and math educators come to different conclusions about teaching the epsilon-delta definition. May I submit that we have different primary allegiances: math educators: do all the students understand what is being taught? Typical response to difficult situation, take the technical and replace it with a ...


7

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 ...


7

This has certainly been tried before. See for example, H. Jerome Keisler. Elementary Calculus: An Infinitesimal Approach. On-line Edition. This has also been published in print by Dover. Kathleen Sullivan. The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach. The American Mathematical Monthly Vol. 83, No. 5 (May, 1976), pp. 370-...


6

In general, can one be led astray by reasoning directly with differentials in this way (as opposed to defining a Riemann sum and taking a limit)? If so, are such examples difficult to contrive or are they typical enough that arguments like the one I presented should be avoided? People used differentials for centuries before the calculus was reformulated in ...


6

What you are referring to, I believe, is students who are actively engaged with, thinking about, and emotionally connected to the material rather than being told to passively absorb it. This distinction between Active and Passive learning modes is a hot topic in education research these days. I don't know about mathematics specifically, but physics ...


6

First of all, I am not happy with putting rote learning against fun, since one of them is more cognitive and the other more affective. Moreover, fun or not, there is something called "cognitive obstacle" which basically implies that it doesn't matter whether you enjoy learning what you are supposed to learn or not, there are certain fundamental obstacles ...


6

I had the chance to read Dray & Minogue's online stuff shortly before I was first assigned to teach a Calculus course, an Applied Calculus course intended for biology and economics majors. As it was a terminal math course, I felt justified in taking an unusual approach; as it was applied, I didn't worry about rigour (as long as I knew that it could be ...


6

This is an interesting question... I think there is a volatile bifurcation at the very outset: certainly students who will (one way or another) be filtered/tested on the Cauchy-Weierstrass viewpoint would not immediately benefit from being made aware that there were other viable viewpoints. "Might confuse them"?!? Yet, ironically, many physics and ...


5

I believe St. Johns college does. They are a liberal arts school that focuses on historical texts. From their sylabus: Junior Mathematics: Calculus and its Foundations Junior Mathematics concerns itself with questions about the continuity of motion, the infinite, and the infinitesimal, which lead to a new form of mathematics, the calculus. The ...


5

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 ...


5

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 ...


5

Yes, Karl-Dieter is referring to myself and a colleague. My current long-term project is to produce a modern full-color calculus textbook using infinitesimals, making use of my own definitions, notation, etc. I currently plan to classroom-test the first-semester portion of the book this fall, although I have already taught the infinitesimal material in all ...


5

My understanding of this question is that it proposes the idea of a "monolingual" freshman calc course in which students mostly learn the language of NSA, and limits are largely or completely neglected. This makes it different from this question by Mikhail Katz, which asks whether it's a good idea for students to be "bilingual." I have some experience ...


4

This does not directly address the question about where nonstandard-analysis-based Calculus is taught, nor the request for educational studies comparing the two approaches, but it is one of the few (maybe only?) studies that take an epistemological perspective on nonstandard analysis: Ely, Robert. (2010) Nonstandard student conceptions about ...


4

Since the rearranging part of your strategy involves dividing $dy$ by $dx$ this post on MO is quite related to your concern. As far as the arc length is concerned, I usually avoid your strategy as a teaching tool at the outset, but sometimes mention it as a way to memorize the formula. Why do I avoid this? I have no conceptual way to distinguish between $...


4

One can be misled by anything and everything. However mathematicians have written things like $$ ds^2 = E\, du^2 + 2F\, du\, dv + G\, dv^2 $$ for a long time without problems. LINK


4

Regarding your parenthetical comment "And it turns out that limits are not the only way to do so. Non-standard analysis uses infinitesimals in a logically rigorous way" I would like to comment that the opposition limits vs infinitesimals implied here is not entirely accurate. Limits are present in both approaches; the true opposition is between epsilon-...


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