Both education researchers and mathematicians discuss the challenge of (epsilon, delta) type definitions in real analysis and the student reception of them. My impression has been that mathematicians often hold an upbeat opinion on the success of student reception of this, whereas education researchers often stress difficulties and their "baffling" and "inhibitive" effect (see below).

Paul Dawkins recently expressed a typical educational perspective on this:

2.3. Student difficulties with real analysis definitions. The concepts of limit and continuity have posed well-documented difficulties for students both at the calculus and analysis level of instructions. Researchers identified difficulties stemming from a number of issues: the language of limits, multiple quantification in the formal definition, implicit dependencies among quantities in the definition, and persistent notions pertaining to the existence of infinitesimal quantities. Limits and continuity are often couched as formalizations of approaching and connectedness respectively.

However, the standard, formal definitions display much more subtlety and complexity. That complexity often baffles students who cannot perceive the necessity for so many moving parts. Thus learning the concepts and formal definitions in real analysis are fraught both with need to acquire proficiency with conceptual tools such as quantification and to help students perceive conceptual necessity for these tools. This means students often cannot coordinate their concept image with the concept definition, inhibiting their acculturation to advanced mathematical practice, which emphasizes concept definitions.

[See http://dx.doi.org/10.1016/j.jmathb.2013.10.002 for the entire article; the online article provides links to references for several of the comments above.]

To summarize, in the field of education, researchers decidedly have not come to the conclusion that epsilon, delta definitions are either "simple", "clear", or "common sense". Meanwhile, mathematicians often express contrary sentiments. Two examples are given below.

...one cannot teach the concept of limit without using the epsilon-delta definition. Teaching such ideas intuitively does not make it easier for the student it makes it harder to understand. Bertrand Russell has called the rigorous definition of limit and convergence the greatest achievement of the human intellect in 2000 years! The Greeks were puzzled by paradoxes involving motion; now they all become clear, because we have complete understanding of limits and convergence. Without the proper definition, things are difficult. With the definition, they are simple and clear.

[See Kleinfeld, Margaret; Calculus: Reformed or Deformed? Amer. Math. Monthly 103 (1996), no. 3, 230-232.

I always tell my calculus students that mathematics is not esoteric: It is common sense. (Even the notorious epsilon, delta definition of limit is common sense, and moreover is central to the important practical problems of approximation and estimation.)

[See Bishop, Errett; Book Review: Elementary calculus. Bull. Amer. Math. Soc. 83 (1977), no. 2, 205--208.]

When I compare the upbeat assessment common in the mathematics community and the somber assessments common in the education community, sometimes I wonder whether they are talking about the same thing. How does one bridge the gap between the two assessments? Are they perhaps dealing with distinct student populations? Are there perhaps education studies providing more upbeat assessments than Dawkins' article would suggest?

There is a parallel thread with a bounty offered at https://math.stackexchange.com/questions/683513/evaluating-the-reception-of-epsilon-delta-definitions so you can respond there (or here if you prefer).

Note 1. Several approaches have been proposed to account for this difference of perception between the education community and the math community:

(a) sample bias: mathematicians tend to base their appraisal of the effectiveness of these definitions in terms of the most active students in their classes, which are often the best students;

(b) student/professor gap: mathematicians base their appraisal on their own scientific appreciation of these definitions as the "right" ones, arrived at after a considerable investment of time and removed from the original experience of actually learning those definitions.

(c) misguided educators: educators are not sufficiently concerned with rigor and cause horrible confusion to the students.

Most of these approaches sound plausible, but it would be instructive to have field research in support of these approaches.

We recently published an article reporting the result of student polling concerning the comparative educational merits of epsilon-delta definitions and infinitesimal definitions of key concepts like continuity and convergence, with students favoring the infinitesimal definitions by large margins.


4 Answers 4


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 representativeness of the quotes of Kleinfeld and of Bishop among mathematicians: see the postscript, for one contrasting data point (and the addenda for two others); also, both pieces were written in explicit reaction against directions in college-level math education that were being championed by other mathematicians (as the first addendum illustrates). But putting that aside, not even these quotes themselves are asserting that students find the epsilon-delta proof easy. More specifically:

Kleinfeld is asserting not that the definition is easy to master, but that once mastered, it provides clarity that is otherwise unavailable. "Without the proper definition, things are difficult. With the definition, they are simple and clear." This sentence asserts that the definition, once it has been fully understood, clarifies and illuminates the matters that the notion of "limit" is intended to deal with. This does not imply that the definition itself is easy. Indeed, its difficulty is implicit in her celebration of it, quoting Russell, as "the greatest achievement of the human intellect in 2000 years."

The quote from Bishop acknowledges the definition as "notorious". More importantly, it is taken out of context. Here it is with the 6 prior and 5 following words as well:

Although it seems to be futile, I always tell my calculus students that mathematics is not esoteric: It is common sense. (Even the notorious epsilon, delta definition of limit is common sense, and moreover is central to the important practical problems of approximation and estimation.) They do not believe me.


There is no assertion that students are readily assimilating to Bishop's point of view.

Postscript: Here is David Tall, writing in 1981:

During the last century the epsilon-delta method has led to fruitful advances in analysis and a clarification of the meaning of certain concepts for the professional mathematician, but it is too complex and intractable for the beginning student in calculus.

In 1981, Tall was making a transition into mathematics education research. Still, he was by then established as a mathematician. I think the quote nicely illustrates how one can simultaneously appreciate the definition's clarificatory power and the difficulty it presents to students.

Addendum 8/12: I just bumped into another mathematician (this time, one who is not also a math education researcher) whose point of view is decidedly in contrast to the quotes of Kleinfeld and Bishop given in the OP. Here is David Mumford, writing in 1997 in defense of the Gleason-Hallett Calculus Consortium's textbook's less-formal definition of continuity (namely, "the closer $x$ gets to $a$, the closer $f(x)$ gets to $f(a)$"):

If one says instead, "For any $\epsilon > 0$, there is a $\delta > 0$ such that whenever $|x-a|<\delta$, then $|f(x)-f(a)|<\epsilon$," what happens to most students? First off, since Greek letters and complex English syntax ("for any ... there is ... such that whenever ... then ...") are used, the student is convinced that something very complicated must be going on. What is worse, even if you give the simple description of the meaning afterwards, the student will be sure that something more complex is going on or else why did you put it in such an opaque way? I think it is impossible to explain to most students that we prefer the complex syntax of the $\epsilon$-$\delta$ definition because we have crafted it precisely to squeeze out all the ambiguity of normal English. The important question is: do most people learn a new concept most efficiently by being exposed to elegant definitions of this sort?

Addendum 8/21: And here is David Bressoud, writing in 2014:

The very first [pedagogical problem] is that any definition of limit that is mathematically correct makes little sense to most students. Starting with a highly abstract definition and then moving toward instances of its application is exactly the opposite of how we know people learn. This problem is compounded by the fact that first-year calculus does not really use the limit definitions of derivative or integral. Students develop many ways of understanding derivatives and integrals, but limits, especially as correctly defined, are almost never employed as a tool with which first-year calculus students tackle the problems they need to solve in either differential or integral calculus. The chapter on limits, with its attendant and rather idiosyncratic problems, is viewed as an isolated set of procedures to be mastered.

Post-postscript: Tall's quote (addendum: as well as Mumford's and Bressoud's) includes something that goes beyond an assessment either of the definition's ease of mastery or its clarificatory value -- namely, an assessment regarding its appropriate place in the curriculum. Kleinfeld also makes such an assessment -- presumably an opposing one, although she doesn't specify whether she's talking about the "beginning student" of Tall's quote. Dawkins makes no such assessment (see here) nor, as far as I can tell, does any of the research he cites on the difficulty of delta-epsilon proofs (see here, here, here, or here).

  • $\begingroup$ +1 for your first paragraph. While I still think there is something left that could be discussed, such as the tendency of one group to focus on the resulting clarity that is obtained and the tendency of the other group to focus on attaining that clarity, your point is important in that we need to make sure we don't stray into distinctions between apples and oranges when we want to compare apples to apples or oranges to oranges. $\endgroup$ Commented Jul 26, 2019 at 17:49
  • $\begingroup$ @DaveLRenfro - (Agree, and) I didn't have time to dig up mathematicians talking about how hard $\epsilon$-$\delta$ is for students (I happened to have the Tall quote on hand for a different reason), but I suspect that there's tons of that to be found. That said, given the concerns of mathematicians vs. of math education researchers, it seems intuitive to me that the former would be more focused on what $\epsilon$-$\delta$ does for all of us, and the latter on what's involved in learning it. $\endgroup$ Commented Jul 26, 2019 at 21:52
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    $\begingroup$ I'd forgotten about it until just now, but my first answer in this group (7 April 2014) has some references for anyone interested in this general topic. $\endgroup$ Commented Jul 27, 2019 at 7:41
  • $\begingroup$ @DaveLRenfro - Wow, those are great links! Thank you! (Also, those are great-looking textbooks! I taught a real analysis course a year ago for which I wish I had known about Abbott at the time! I used Rudin, which was too aseptic and unmotivated and too hard; I considered Bressoud, which was too idiosyncratic for my purposes; and I supplemented with Velleman's Calculus: A Rigorous First Course, which was perfect for $\delta$-$\epsilon$ but doesn't cover most of what I needed.) $\endgroup$ Commented Jul 28, 2019 at 19:06

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:

  1. math educators: do all the students understand what is being taught? Typical response to difficult situation, take the technical and replace it with a heuristic so more of the student population "gets" it. Cost: no students learn the true definition. Benefit: many students learn some approximate version of the true, hopefully they return to it later and seek deeper truth in later coursework.
  2. mathematicians: is the content correct? Typical response to student difficulty, face it head-on, refuse to replace true definition with a heuristic. Cost: many students do not learn the definition. Benefit: those who learn the definition correctly do not need to learn the material needlessly again in a later class, hopefully those who learned nothing get more next time they retake the course.

Of course, I'm kidding. The truth is some amalgam of these. Moreover, there is actually no demarcation between math educators and mathematicians. In fact, there are many individuals who are in both camps. I try to incorporate both modes because I don't want to be rude to either camp of students. I want to reach as much of the population I see in my classes as possible. However, one size never fits all.

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    $\begingroup$ My point in my answer/comments above is that one must be very careful when introducing heuristics. They may do more harm than good. $\endgroup$
    – vonbrand
    Commented May 7, 2014 at 5:27
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    $\begingroup$ Whenever I hear about heuristics doing harm my hand automatically reaches for my infinitesimal microscope. $\endgroup$ Commented May 7, 2014 at 14:25

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 (he was the dean of Math here, and taught first year calculus out of love for the subject). He very carefully avoided the "moving" and "very small" imagery, and I believe that helped enormously. Even some earnest discussion of what to do if $\epsilon = 1000$ was thrown in.

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    $\begingroup$ Thanks. While your answer is interesting, what I was asking is not whether epsilontic definitions are helpful or not, but rather to account for the apparent difference in perception between the education community and the mathematical community. One possible answer is that there is in fact no difference, but that's not what you seem to be saying. $\endgroup$ Commented May 6, 2014 at 11:49
  • $\begingroup$ @katz, as a student I found it useful if done right. I fear that much of the discussion of limits in class might be muddied with "approaches" and "very small," which in my experience don't help, more the contrary. $\endgroup$
    – vonbrand
    Commented May 6, 2014 at 12:08
  • $\begingroup$ You could try posting a question to find out whether calculus classes should start with a formal epsilon, delta definition of limit. I would certainly be interested in other editors' reactions. $\endgroup$ Commented May 6, 2014 at 12:09
  • $\begingroup$ @katz, check this question $\endgroup$
    – vonbrand
    Commented May 6, 2014 at 12:13
  • $\begingroup$ That question is indeed an excellent illustration of the point of view of the mathematical community, particularly the accepted answer. As I mentioned above, the education community seems to disagree. What I am curious about is this difference of perception between the two communities, and the possible underlying reasons for such a difference, as reflected in the literature. $\endgroup$ Commented May 6, 2014 at 12:16

As I said on MathOverflow:

I would attribute this to sample bias -- which is "distinct student populations" of a sort.

Mathematicians who teach epsilon-delta definitions rarely study student reception of those definitions in a rigorous way. If they did, they would probably be publishing in education journals, not math journals.

Math teachers evaluating the reception of these ideas probably make informal evaluations, focusing on the most obvious students: those who participated actively in class, those who stayed in touch with their teachers, those who continued with more mathematics. Those students are substantially more likely to appreciate the epsilon-delta definitions.

Education researchers are less likely to suffer from this bias, though surely they suffer from others.

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    $\begingroup$ Matt, thanks. Your answer seems plausible but I was hoping to get some support from the literature, as I indicated in the stackexchange bounty comment. $\endgroup$ Commented May 5, 2014 at 16:14
  • $\begingroup$ @katz, What study would show sample bias among mathematicians? You want someone to find a bunch of mathematicians who have positive impressions of student receptions of epsilon-delta definitions, then interview those mathematicians to find out where those impressions came from? $\endgroup$
    – user173
    Commented May 7, 2014 at 14:22
  • $\begingroup$ Designing such a study is a separate issue. I don't think it is impossible. One could for example interview the professor and then before he knows it go ahead and interview the students, as well, and compare the reactions :-) $\endgroup$ Commented May 7, 2014 at 14:23
  • $\begingroup$ @katz, I think we've already done that well enough. The education researchers have shown how students respond. Many mathematicians, for whatever reason, continue to believe otherwise, as you've documented. I don't think it's worth further conversation. $\endgroup$
    – user173
    Commented May 7, 2014 at 14:38
  • $\begingroup$ Matt, there is an interesting response by editor String that in my opinion adds productively to the discussion here: math.stackexchange.com/questions/683513/… $\endgroup$ Commented May 7, 2014 at 14:40

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