Evaluating the reception of (epsilon, delta) definitions

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.

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?

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.