If you're not going into higher-level, proof-based mathematics courses, then it can be argued that it isn't necessary to know the rigorous definition of a limit.
An interesting point. If you choose to view courses in a basic Calculus sequence (Calc I, II, possibly III and Diff Eq) as general-ed requirements, and if you will never venture into higher-level pure mathematics (where logical arguments and geometric ideas are more important than computational methods) then perhaps there is little to no need for such rigor as the ε-δ definition of a limit.
What bothers me is whether we demand reductions in other general-ed courses.
- Take a freshman writing class - if you're not going to be a Lit or humanities major, then does it mean that you don't have to analyze your readings as deeply? Does it mean that your essays don't have to be as circumspect and readable?
- Consider a first course in programming, say in C or in JAVA - does it mean that you don't have to learn to comment your code?
The ε-δ definition isn't just a description of a limit; that's what a limit is in mathematics. It's not a pretty graph or a table of values, but such methods help us to understand the nature of a limit. A math teacher should emphasize that: if you choose to learn mathematics, then as an important mathematical idea, you should learn this.
If all you need is how to use a limit in practical, day to day life, such as understanding instantaneous velocity or approximating marginal revenue, then that's fine. You don't need to learn it, but if you take a Traditional Calculus course, don't expect a math teacher to cater exclusively to your non-mathematical needs. Its purpose is to teach mathematics and to emphasize mathematical thinking.
Maybe what is needed is "Calculus I for Scientists and Engineers" or "Calculus I for Economists and Sociologists" or "Calculus I for Mathematicians," and this notion of Reformed Calculus is a step towards that direction. Maybe it is "watered down," but if planned correctly, a Reformed Calculus class teaches you what you need in order to work in your own field. It doesn't teach you what mathematics is like, because that is not its purpose.
I say: if you choose to allow Reformed Calculus classes, then you should also keep a few alternative Traditional Calculus classes. It's a specialization of needs: students are different and their studies demand different approaches. The coursework should reflect that.
9 comments:
The ε-δ definition isn't just a description of a limit; that's what a limit is in mathematics.
I'm not sure I can agree with you, unless you clarify the meaning of is. Since there are limits with respect to a filter, and also infinitesimals of nonstandard analysis, one can argue that ε-δ is just that - one of possible descriptions of a limits.
I introduced limits in my class just a week ago, without putting much emphasis on the rigorous definition. One reason is that I am teaching Calculus for Engineers. Another is that I would not use the definition myself to solve any of the [Stewart's] textbook problems.
That's a good point, and thanks for the catch. I wasn't being very precise, nor did I choose my words appropriately. For instance, this ε-δ definition describes continuity and not limits in general.
Of course one could discuss continuity from the topological viewpoint, but if one knows no topology and works with real-valued functions on the real line, then the ε-δ definition is enough.
Nonetheless, in the context of sequences of real numbers, the notion of a limit of a sequence has a standard definition and is taught in, say, a first course in real analysis.
Maybe I'm being too demanding in asking Calculus students to learn mathematics or think mathematically, whatever that means. I suppose that's what happens when I start forming opinions about mathematics and education.
There's an easy way to find out if you are being too demanding: just try teaching your calculus class to the tune of \forall \exists. If you can pull it off, then you're not too demanding. If not, then you can try some other approach next semester... there is no shortage of experimental subjects. :)
How Clintonian (Clintonesque?) that one must ask what the meaning of "is" is. In the context, "limit" refers to limits of real-valued functions at a point, not necesarily a point in the domain. Epsilon-delta is not a description, it is the DEFINITION of a limit as described above, not a description of a limit. If one chooses another definition for limit, then one must show that these differing definitions are indeed equivalent, or else give an example to show that they are not. Once a student has been taught the definition, one then proves theorems, lemmas, etc, that are natural consequences of the definition. The proved result may make using the definition to verify limits impractical. (e.g., Theorem: A function f(x) continuous at point x=a has limit f(a). This theorem is used to find the limit of, say, log(3+x^5-e^x) at x=0 without an appeal to the definition. In fact, using the definition would be a horrible way to work this problem, for finding the right delta, given epsilon, would be an exercise in pain.) However, an instructor must properly define the concepts the students are introduced to. Failure to do so results in a failure to understand. From personal experience, I have encountered many calculus students who will tell me that the double integral of a function is, by definition, the iterated integral. And they proceed on their merry way. All it takes is one of the examples where an iterated integral exists but the double fails to, and the students are baffled, annoyed, and confused.
This is because they were not given the proper definition of a double integral and then the theorem that in certain cases, the iterated equals the double. Instead, they were given the theorem as a definition and now have no understanding of what a double integral really means.
Theorem: A function f(x) continuous at point x=a has limit f(a).
Apparently, understanding the difference between theorems and definitions is not a prerequisite for having strong opinions about math education.
Behave yourselves, commentators.
You've made some good and useful points in light of my written inaccuracies, but I didn't start this weblog so that others could make it a pissing contest ..
.. so behave yourselves, will you?
Jasun, go get some Elton John's mp3s, will you? :)
Don't give us none of your aggravation
We had it with your discipline
Saturday night's alright for fighting
Get a little action in
This theorem is found in Bartle "The Elements of Real Analysis, Second Ed., 1976. Section 20 for those following along. The definition of limit and the definition of continuity are topological, involving neighborhoods. Thm. 20.2 then shows the epsilon-delta, topological and "limit of f(x) =f(a)" definitions of continuity are equivalent. Since the context of the previous posts was concerning epsilon-delta definitions, the point is that an epsilon-delta definition of continuity is hard to apply to problems, but can be replaced by the easier (for students) to grasp technique of computing limits without appealing to epsilon-delta at all. Granted, one may present the theorem as a definition, but not all textbooks do. If one is using this textbook, it would be a crime to skip this theorem for the sake of ease of computation.
Thanks for the Bartle & Sherbert reference. To be honest, I didn't realise they discussed continuity in terms of neighborhoods.
I guess I'm biased from my own "upbringing." In my first analysis course, we used Apostol's Mathematical Analysis book and covered metric topology on the real line: as a result, continuity was defined with δ's and ε's, since those seem the easiest ways to form neighborhoods in 1 dimension.
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