26 votes

Why don’t we teach a topological view of continuity instead of epsilon-delta?

Yes. You are crazy to be spending time on another effort to bring real analysis style rigor into a calc 1 course. In this case, even off brand real analysis. Instead of a "different" way ...
guest troll's user avatar
15 votes

Why don’t we teach a topological view of continuity instead of epsilon-delta?

You aren't crazy because Leonard Gillman wasn't crazy. See Emphasizing order relations rather than metric concepts in real analysis The family of all open intervals (A, B) about a point L assuredly ...
Marc Shelikoff's user avatar
12 votes

Evaluating the reception of (epsilon, delta) definitions

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 ...
benblumsmith's user avatar
  • 1,926
12 votes

Why don’t we teach a topological view of continuity instead of epsilon-delta?

If you want the comment on the approach yourself rather than on whether you should try to implement it, the key words in your post are Given epsilon I just need to show that the set of solutions to $L−...
fedja's user avatar
  • 3,831
11 votes

Why don’t we teach a topological view of continuity instead of epsilon-delta?

If you want to teach a topological view of continuity, then you have to teach topology first. The set of open intervals is a basis of a topology, but it's not itself a topology, and the topology ...
Acccumulation's user avatar
9 votes

When should we get into limits in introductory calculus courses?

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 ...
Michael Hardy's user avatar
9 votes

Why don’t we teach a topological view of continuity instead of epsilon-delta?

A positive attribute of the standard $\epsilon$-$\delta$ definition continuity is that the same formalism can be mutated to define similar things such as limits at infinity or the limit of a sequence $...
user52817's user avatar
  • 10.5k
8 votes

Why don’t we teach a topological view of continuity instead of epsilon-delta?

One approach that has been advocated several times is to significantly decrease the focus the focus on “pointwise continuity” and “pointwise differentiability” notions (or entirely omit them) and ...
Dave L Renfro's user avatar
8 votes
Accepted

A calculus book that uses differentials?

My book Calculus from the Ground Up focuses on differentials, and uses it to provide a unification of process and simplification of understanding of a lot of different parts of calculus. To read ...
johnnyb's user avatar
  • 1,237
7 votes

Would teaching nonstandard calculus in an introduction calculus course make it easier to learn?

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. ...
Brian Borchers's user avatar
7 votes

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

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 ...
paul garrett's user avatar
  • 14.3k
6 votes
Accepted

Is it to the students' advantage to learn the language of infinitesimals?

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 ...
paul garrett's user avatar
  • 14.3k
5 votes
Accepted

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

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 ...
Mikhail Katz's user avatar
  • 2,142
5 votes

Would teaching nonstandard calculus in an introduction calculus course make it easier to learn?

Speaking as a former student, though an engineering one . . . It was hard enough learning to integrate tricky expressions and solve differential equations, without having to learn a new number system ...
timtfj's user avatar
  • 571
5 votes

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

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 ...
Paul Burchett's user avatar
5 votes

Is it to the students' advantage to learn the language of infinitesimals?

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, ...
Bryan Dawson's user avatar
4 votes

When should we get into limits in introductory calculus courses?

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 ...
Mikhail Katz's user avatar
  • 2,142
4 votes

Non-Rigorous Use of Differentials

It has long been a puzzle to historians how Leibniz could not have been misled by using differentials, given that to him "differential" meant "infinitesimal" and infinitesimals were thought to be ...
Mikhail Katz's user avatar
  • 2,142
4 votes

Should we "program" calculus students, like the physicists seem to want us to?

There is evidence that both a computational and conceptual approach are needed: https://www.jstor.org/stable/3482237 The paper of Sfard linked to does seem to agree that the scale must tip first ...
Jon Bannon's user avatar
  • 6,173
3 votes

Why don’t we teach a topological view of continuity instead of epsilon-delta?

Introduction I take it that the OP, now departed from the site, was exploring a pedagogical way to introduce the concept of continuity. I do not think they were trying to put continuity on a new ...
user1815's user avatar
  • 5,475
3 votes

Would teaching nonstandard calculus in an introduction calculus course make it easier to learn?

It's not really that relevant since the bulk of a normal calculus course (e.g. AP BC, Thomas Finney, Stewart) just does a small amount of epsilon-delta (so student is exposed to it) and then moves to "...
guest's user avatar
  • 39
3 votes

Should we "program" calculus students, like the physicists seem to want us to?

With regard to your comment "talking about 'infinitesimal changes' like an 18th century mathematician or physicist". The implication of this comment is that we should not teach students this way, ...
Mikhail Katz's user avatar
  • 2,142
3 votes

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

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 ...
Alecos Papadopoulos's user avatar
3 votes

The 'epsilon-delta' method for teaching limits

The idea that historical infinitesimals were self-contradictory is prevalent among historians (see e.g., the accepted answer above) and also many mathematicians, but it has been challenged in the ...
Mikhail Katz's user avatar
  • 2,142
2 votes

When should we get into limits in introductory calculus courses?

You ask why to cover limits prior the derivatives when it would be easier to cover derivatives first and limits later; to show why are limits good to know. Why shall we cover derivatives prior ...
Crowley's user avatar
  • 219
2 votes

Teaching Infinitesmals and Non-Standard Analysis

"Infinitesimal Calculus" by James Henle A short but complete introduction to Calculus which starts by defining a hyper-real number system. Very enjoyable. I had the pleasure of taking a course using ...
Justsalt's user avatar
  • 121
2 votes

A calculus book that uses differentials?

Edward's "Advanced Calculus: A Differential Forms Approach" does this, writing everything in terms of differential forms and getting all the way up to Stokes' theorem, while giving lots of ...
David E Speyer's user avatar
2 votes

A calculus book that uses differentials?

Here: https://archive.org/details/traitlment00qu/page/6/mode/2up?view=theater "Traité élémentaire du Calcul differentiel et de Calcul integral" Modern textbooks can learn a thing or two from ...
Mariano Cifuentes's user avatar

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