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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 http://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.

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3 Answers 3

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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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. –  katz May 6 at 11:49
    
@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. –  vonbrand May 6 at 12:08
    
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. –  katz May 6 at 12:09
    
@katz, check this question –  vonbrand May 6 at 12:13
    
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. –  katz May 6 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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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. –  katz May 5 at 16:14
    
@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? –  Matt F. May 7 at 14:22
    
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 :-) –  katz May 7 at 14:23
    
@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. –  Matt F. May 7 at 14:38
    
Matt, there is an interesting response by editor String that in my opinion adds productively to the discussion here: math.stackexchange.com/questions/683513/… –  katz May 7 at 14:40

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

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