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There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in real analysis and the student reception of it. My impression has been that the mathematical community often holds an upbeat opinion on the success of student reception of this, whereas the education community often stresses difficulties and their "baffling" and "inhibitive" effect (see below). A typical educational perspective on this was recently expressed by Paul Dawkins in the following terms:

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 (e.g. Cornu, 1991; Cottrill et al., 1996; Ferrini-Mundy & Graham, 1994; Tall & Vinner, 1981; Williams, 1991). Researchers identified difficulties stemming from a number of issues: the language of limits (Cornu, 1991; Williams, 1991), multiple quantification in the formal definition (Dubinsky, Elderman, & Gong, 1988; Dubinsky & Yiparaki, 2000; Swinyard & Lockwood, 2007), implicit dependencies among quantities in the definition (Roh & Lee, 2011a, 2011b), and persistent notions pertaining to the existence of infinitesimal quantities (Ely, 2010). 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 (note that the online article provides links to the papers cited 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 one compares the upbeat assessment common in the mathematics community and the somber assessments common in the education community, sometimes one wonders 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?

Note 1. See also http://mathoverflow.net/questions/158145/assessing-effectiveness-of-epsilon-delta-definitions

Note 2. Two 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. Both of these sound plausible, but it would be instructive to have field research in support of these approaches.

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I think there is a wide misconception that mathematics is something you understand in a first read. It might very well happen, but it is usually not the case. Modern definitions are the result of many years of investigation and thinking of people, and this "good" definitions might sometimes be hard to grasp or nonintuitive. A definition is like a jawbraker, you cannot take it in one bite, rather, you have to keep it in your mouth and wait to get to the center. Like so. –  Pedro Tamaroff Apr 24 at 14:22
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Hi @Pedro, I appreciate your insight. I recall liking the challenge of reading Lang's algebra book when I was an undergraduate. One of my peers pointed out that van der waerden's Algebra was more accessible. When I pointed out that Lang makes you think, he retorted that van der Waerden makes you understand. With time I came to appreciate his viewpoint. I think there are two genuinely different approaches to learning at work here, and each may be more appropriate than the other depending on the particular circumstances of the course in question. Incidentally your image was blocked by K9. –  user72694 Apr 24 at 14:29
    
Oh, it is merely a cartoon. –  Pedro Tamaroff Apr 24 at 14:32
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The thing I hated the most in textbooks was when they introduced concepts with absolutely zero motivation or explanation. I find that one needs to look very hard to find books that are actually designed to teach you something, rather than just a collection of 'theorem/proof' blocks meant for people who are already familiar with the material. –  DepeHb May 2 at 12:34
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@PedroTamaroff I must respectfully disagree. A definition must be no less understandable than the concept it is trying to describe. That is the way of true logic, from the generally accepted and understood idea towards the harder to understand and accept. Most difficult definitions should actually be theorems, but authors don't want to be bothered to include the reasoning behind a definition. Too often definitions are used as a crutch to avoid difficult but logically necessary mathematics. –  DanielV May 2 at 15:24

6 Answers 6

up vote 2 down vote accepted
+50

In his answer, Paramanand Singh suggests that freshman students are unfamiliar with certain concepts and methods that are prerequisites for understanding $\varepsilon-\delta$. On the other hand Singh suggests, that once these concepts and methods have been succesfully placed in someones mind, they become part of that persons intuition on the subject. Here intuition is a word I substituted for Singh's use of the word natural. I hope this is a fair account!

This suggestion, perhaps, fits very well with the perspective suggested in the article "On The Dual Nature of Mathematical Conceptions" by Anna Sfard (published in Educational Studies in Mathematics 22, 1-36, 1991). She argues that the process of doing algorithmic operations leads through stages of gradually maturing perceptions, ultimately identifying new objects. Maybe freshman regards $\varepsilon,\delta$ as heavy algorithmic processes, whereas the matured view is to see it as a whole concept, an object.

In her article, A. Sfard is also referring to Miller, G. A.: 1956, "The magic number seven plus minus two" suggesting that one can only juggle about seven chunks of information in the "working memory" at the time. So for the trained $\varepsilon,\delta$-scholar the concept of $\varepsilon,\delta$ is just one object, one chunk of information, whereas for the untrained person each symbol, each quantifier, occupies space in the "working memory" thus rendering the understanding nearly impossible at that stage?

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This line of thought was pursued even earlier by David Tall in terms of his notion of a procept. The point is that a mathematical idea is both a PROcess and a conCEPT. –  user72694 May 6 at 12:59
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@user72694: Yes, Sfard refers to Tall & Vinner as well! What I like about Sfard's approach in that article is that it comprises of references to historical concept development, research about abilities of the human brain, a synthetic view of concept and process like you noted, and, not least, a very clear and concise terminology. That said, I have heard that she has mostly been criticized for the rigidity of the devolopmental hierarchy she is suggesting. I am rather new to that subject, so these are merely references and catious interpretations. –  String May 6 at 13:15

My feeling is that the biggest problem with the epsilon-delta definition is that this is the first time students have ever seen the universal and existential quantifiers. By the time you say, "For every epsilon there exists a delta," you have already lost 95% of your audience before you even get to the business end of the proposition.

And of course the other problem is with the lower-case Greek letters. Students have been seeing x, y, z, and t all their lives; and out of nowhere you show them epsilon and delta.

In other words it's the basic form of the definition that's intimidating and confusing to students; not so much the actual idea, which is simply that you can arbitrarily constrain the output by suitably constraining the input.

Perhaps if instructors started with the conceptual understanding and then spent time explaining "for all" and "there exists" and giving them a gentle introduction to Greek letters used as variables, things would get better.

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I agree, also I think the $\epsilon-\delta$ definition has a major difference from pre-calculus concepts: this definition is in the form of If (...) Then(...) which is not in the form as (something) IS (formulas), and I think many students having trouble understand this difference. –  TTY May 3 at 21:46
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+1 Especially the difference between $ \forall\exists$ and $\exists\forall$ seems to be an obstacle, in spite of illustrative examples from everyday world –  Hagen von Eitzen May 5 at 6:30

First let me focus on the reasons behind the difficulty in assimilating the $\epsilon, \delta$ definitions.

For any beginner in calculus, assimilating the $\epsilon, \delta$ definition is a challenge. I have rarely seen any student for whom this definition seems natural. I don't think anyone would dispute that given the fact that these definitions were arrived at after a long long time Newton invented calculus.

However the reasons for the difficulty in assimilating these definitions is not so much related to the definitions, but rather to the approach of presenting them to students. A student who is learning calculus for the first time normally has experience of algebraical manipulation but has very less interaction with order relations or inequalities. And another block is the understanding of "infinite". A student needs to be trained first in order relations and some understanding of "infinite". I can illustrate my point with two examples:

1) A student of 13 yrs of age would find it very easy to solve $x + 5 = 3$ and at the same time find it bit difficult to solve $|x - 5| < 3$.

2) A student of 16 yrs of age would find it easy to show that there is no rational number whose square is $2$. But at the same time he will be hard pressed to show that we can find as good rational approximation to $\sqrt{2}$ as we want especially if you don't allow him the square root extraction method to find decimal approximation of $\sqrt{2}$ to any number of digits.

I would say that there is a huge gap between "algebraical manipulation of expressions" and "appreciation of inequalities and infinite nature of integers and rationals" in terms of problem solving techniques and related conceptual framework. Unless this gap is bridged by the student himself or through his teachers, it is natural to expect that the student would find it challenging to accept the $\epsilon, \delta$ definitions.

Next I come to question asked here. Mathematics community in general feels that these definitions of calculus are the most appropriate and natural and are hugely successful in teaching huge amount of further "mathematical analysis". This is simply because once you have understood these definitions you can't think of any more natural choice of any other definition. After the initial fight with $\epsilon, \delta$ is over, the general feeling is that these definitions are the simplest and most powerful tools to teach these topics. My own view is the same but I can't forget my days when I was fighting with $\epsilon, \delta$ and crossed the chasm with help of Hardy's Pure Mathematics.

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Your basic contentions seems to be that once you have understood these definitions you can't think of any more natural choice of any other definition. After the initial fight with $\epsilon, \delta$ is over, the general feeling is that these definitions are the simplest and most powerful tools to teach these topics. In other words, what you are arguing is that mathematicians are making a psychological error here which has to do with the student/professor gap. This is an interesting hypothesis, but has this been studied in the literature? How would one go about testing this hypothesis? –  user72694 May 4 at 15:55
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I am not sure about any specific studies based on this hypothesis. I believe the only way to check this hypothesis is to ask a sample of students who are doing some real-analysis courses in their post-grad studies. We should ask them to compare their views on epsilon-delta when they first met this concept and their current views on this. –  Paramanand Singh May 4 at 17:56
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This suggestion, perhaps, fits very well with the perspective suggested in the article On The Dual Nature of Mathematical Conceptions by Anna Sfard, published in Educational Studies in Mathematics 22, 1-36, 1991. She argues that the process of doing algorithmic operations leads through stages of gradually maturing perceptions, ultimately identifying new objects. Maybe freshman regards $\varepsilon-\delta$ as heavy algorithmic processes, whereas the matured view is to see it as a whole concept, an object? –  String May 6 at 8:16
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In her article, A. Sfard is also referring to Miller, G. A.: 1956, 'The magic number seven plus minus two...' suggesting that one can only juggle about seven chunks of information in the "working memory" at the time. So for the trained $\varepsilon,\delta$-scholar the concept of $\varepsilon,\delta$ is just one object, one chunk of information, whereas for the untrained person each symbol, each quantifier, occupies space in the "working memory" thus rendering the understanding nearly impossible at that stage? –  String May 6 at 8:22
    
@String, thanks for these interesting comments. I suggest you format them as an "answer". Otherwise nobody notices them (I didn't even get notification of a comment even though I am the question poser). –  user72694 May 6 at 12:29

This is most likely a non-answer, but my (personal, strong, heavily biased, another math culture infused) opinion is that the confusion stems from things being presented in calculus classes in a bizarre illogical order, starting with complicated things (limits of change in functions, i.e., derivatives) and then, as a hindsight, dropping back to simper things (limits of sequences).

In teaching math to the elementary school students the basic arithmetics, we don't throw $\pi$ and $\sqrt{2}$ and ${\rm e}^\pi$ at them. Instead, we talk about 1, 2, 3, then 1+2=3, then $3 \times 4=12$, then introduce division... well, you all know. Natural numbers is a simpler set to digest than rational numbers, which are in turn easier to digest than real numbers. Now, think about limits: are limits on natural numbers easier to digest than limits on real numbers?

While you are thinking about it, take a look at Rudin's book:

  1. Real and complex numbers
  2. Elements of set theory
  3. Sequences and series
  4. Continuity
  5. Differentiation
  6. The Riemann-Stieltjes integral

etc. He does go in a logical order, from simpler objects to more complex: real line first, then mappings from natural numbers to real line (sequences), and limits for these; then mappings from reals to reals (functions) and limits on these (continuity). All of the calculus books I have been exposed to in my... uhm... childhood (this was in Soviet Union, so the books were in Russian) went in this order. No author tried to jump ahead of the train engine, and try to excite students with derivatives. Studying sequences first help establishing the concept of a limit. Fewer pathologies are possible with these: you cannot have jumps at infinity, unlike say what $\mathop{\rm sign} x$ does at zero. Once students learn to operate with limits on sequences (what should $N$ be so that $1/n^2$ is less than $10^{-6} \, \forall n\ge N$?), and understand quantifiers, you can start pushing into the world of functions.

To convert my non-answer to semi-answer, I'd be curious to see whether there are differences in reception of the Rudin's sequence of material with the Stewart's sequence.

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This is an intersting comment (+1) but the relation to the question seems a bit remote. One relation I do see is the challenge to the idea that epsilon, delta techniques have to be introduced early on so as to achieve "rigor"; instead, your answer suggests that one proceed from the simple to the complicated. On the other hand, Rudin's book is not a calculus text but rather an analysis text. I haven't looked at it but from the summary you presented it seems as though he is assuming familiarity with epsilon, delta already in chapter 1. Otherwise how is one to define the real numbers? –  user72694 May 8 at 13:15

I'm really just putting forth my opinions on $\epsilon$-$\delta$, and how I think it should be introduced to people who have not seen it before.

I believe the entire difficulty that the $\epsilon-\delta$ approach puts forth is the idea of a constant that depends on something, since before undergrad level (i.e. A level) constants are constants, and we don't look at situations where they may change (Due to a change in another constant).

The only times things seem to change (at a first glance) is when we look a functions, where the variable changes and the function changes, and this seems natural.

But then when we are approached with an idea that some constant changes, and then another one must it seems very foreign.

I am just finishing my BSc and I feel very well versed in $\epsilon-\delta$ type arguments, but to understand it I had to get there on my own, and I feel that this is the way to go, you can't completely understand things just from someone telling you. You need to explore it yourself.

But the part in the explanations that was always lacking, is the fact that $\delta$ depends on $\epsilon$, and I think the definitions should be written in the form $\forall\epsilon\gt 0,\exists \delta(\epsilon)$...

The whole scenario could be argued in a challenge formulation, so someone can really see how $\delta$ must change if $\epsilon$ changes.

For instance, say we want to prove that the function $f(x)$ is continuous at $x_0$, we must show that $\forall\epsilon\gt 0,\exists\delta(\epsilon):|x-x_0|\lt\delta(\epsilon)\Rightarrow|f(x)-f(x_0)|\lt\epsilon$.

Then we argue in this sense: someone gives us the challenge of $\epsilon=\epsilon^*$

Then we find a value of $\delta$ which depends on $\epsilon^*$ so that for this $\epsilon^*$, if $x\in(x_0-\delta(\epsilon^*),x_0+\delta(\epsilon^*))$

Then $f(x)\in(f(x_0)-\epsilon^*,f(x_0)+\epsilon^*)$.

Now summarising in non mathematical terms. We have one "$\delta$" that holds for our one "challenge" ($\epsilon^*$) that makes our condition hold, by taking that choice of "$\delta$".

So up until this point we have been very clear that $\delta$ depends on $\epsilon$.

Now we go a step further, and we ask what happens if any $\epsilon$ is given to us as a challenge? Well, clearly in many cases, the $\delta$ we found in the first instance wont work every time, so we must have to find a new $\delta$.

And in this sense again $\delta$ depends on $\epsilon$.

Now we realise that we must find a relationship between $\delta$ and $\epsilon$.

So in common sense $\delta$ is a function of $\epsilon$, but the "variable" $\epsilon$, only changes when we are handed a new "challenge" or situation, rather than over a general domain i.e. something like an interval which is much easier to imagine.

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This is a good point that was in fact mentioned by Dawkins in the passage I cited in my question: implicit dependencies among quantities in the definition (Roh & Lee, 2011a, 2011b). –  user72694 May 9 at 7:12
    
Thank you, it is good to see that peoplare thinking about this stuff, as I've always thought that the approach to teaching it has been in the wrong manner! –  ellya May 9 at 7:18

The opinions are not in conflict. Something can be simple, obvious, intuitive, etc. and a person can still fail to grok that it is simple, obvious, intuitive, etc. The notion of building intuition is an oxymoron according to a common understanding of intuition, but is in fact central to the understanding of intuition relevant to mathematical training.

A joke every student of mathematics eventually hears:

[...] our professor then formulated a theorem, wrote its statement on the board, and declared to us that "the proof is obvious". Another student raised a hand in objection. "I'm sorry but I don't see the proof immediately, could you elaborate?" Our professor stopped for a moment, and mulled over the statement. He paced back and forth in front of the board, stroking his beard in deep puzzlement, and then wandered out of the classroom. Us students sat dumbfounded for half the remaining class period, a good quarter hour in all, until our professor returned. With a large smile beaming on his face, he announced to the class "indeed, it is obvious!", and continued the lecture without further comment.

Obvious($X$) $\not\rightarrow$ Obvious(Obvious($X$)). The mathematicians are declaring Obvious($X$), while the educators are declaring $\neg$Obvious(Obvious($X$)). There is no conflict between these propositions.

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