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There is a set of related topics in a freshman calc course that includes the completeness axiom for the reals, the intermediate value theorem, extreme value theorem, Rolle's theorem, and mean value theorem.

How can one interest a freshman calc student in these topics, when they almost certainly have zero interest in foundational issues?

This material seems to me to have a fairly tenuous connection to the mainstream of a freshman calc course. I'm not an expert in the history of mathematics, but it seems that Dedekind cuts and such were not invented until about 1860. That leaves a period of about 150 years during which the world's greatest mathematicians happily did calculus without feeling a need for these foundational issues to be addressed. They would probably have felt that a result like the extreme value theorem was geometrically obvious. (Some of the theorems were stated earlier, e.g., Rolle's theorem dates to 1691, but I don't see how Rolle could have given a meaningful proof 150 years before the arithmetization of the reals.)

Further adding to the difficulties in selling this to a class of students are the facts that:

  1. These theorems are not particularly useful in solving problems (because in most problems where they were needed, the theorems' claims would have been intuitively obvious anyway).

  2. In a freshman calculus course we don't really attempt to present this material in detail. E.g., textbooks tend to omit most of the proofs.

(Just to put this in perspective, I recently asked a class of students who had all already had a year of calculus to describe in general terms how the derivative was defined -- I wasn't asking them to regurgitate the definition, just to describe its flavor. Despite a lot of poking, prodding, and hinting, not a single student volunteered that it was defined using a limit.)

The best idea I've been able to come up with so far is just to give the class a series of tasks in which they come up with counterexamples to the various theorems when one of the assumptions is omitted, e.g., give an example of a function that is continuous everywhere in its domain but that never attains a maximum value. This might at least help to convince them that the results make sense and have a geometrically visualizable interpretation.

Related: https://mathoverflow.net/questions/184358/what-did-rolle-prove-when-he-proved-rolles-theorem

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    $\begingroup$ I think the selling point of completeness (though it may not be so in a freshman calculus class) of any metric space is the contraction mapping principle, which I don't think is "intuitively obvious". It comes up in every field that interacts with differential equations. $\endgroup$ – PVAL Oct 14 '14 at 1:51
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    $\begingroup$ Perhaps the opposite question would be worthwhile too: How can we persuade the people designing curricula to remove these topics from the class? $\endgroup$ – user173 Oct 14 '14 at 2:25
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    $\begingroup$ @MattF.: Math majors should know this stuff. The problem is that we lump math majors in with physical scientists, engineers, and biology majors. $\endgroup$ – Ben Crowell Oct 14 '14 at 2:38
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    $\begingroup$ @ben, don't the math majors all take analysis? Why not delay these topics until then? $\endgroup$ – Sue VanHattum Oct 14 '14 at 19:38
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    $\begingroup$ @SueVanHattum: That might be reasonable, but I just don't think it's ever going to be optimal to teach calc the same way to a biology major and a math major. $\endgroup$ – Ben Crowell Oct 15 '14 at 0:03
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In the preamble to your question you gave several great arguments against dwelling on foundational issues in a freshman calculus class, and indeed I think it is inappropriate to focus too much on these issues. In my view, the point of a calculus class is to teach students how to use a particular set of tools to solve problems, and we should give them time to understand how to use the tools before expecting them to appreciate why the tools work. It's better to teach someone how to ride a bike before teaching them how to build one.

That said, you were asking how, not if, you should sell foundational concepts to your students. My suggestion is to integrate them into your examples as much as possible. If you go over the intermediate value theorem in one class toward the beginning of the semester and then never revisit it, the theorem will appear as an esoteric curiosity that they can safely dismiss. But suppose that later on you do an example such as:

Let $f(x) = e^x + 5x^2 + 3x$. Is there a point on the graph of $f$ where the tangent line has slope $2$?

This comes down to solving the equation $f'(x) = e^x + 10x + 3 = 2$. This can't be done algebraically, but you can observe that $f'(0) = 4 > 2$ and $f'(-1) = e^{-1} - 7 < 2$, so indeed $f'(x) = 2$ for some $x$ between $-1$ and $0$ by the intermediate value theorem. This makes the intermediate value theorem feel more like a technique than just an idle fact.

With a little thought, you can find ways to sneak other theorems into your examples. For instance, after you've done a bit of integration you can give the following example:

Find the average value of $f(x) = x \sin x$ on the interval $[-\pi, \pi]$. Does $f$ attain its average at some point in the interval?

After you find that the average is $1$, you can remark that the equation $x \sin x = 1$ can't be solved algebraically, and a direct application of the intermediate value theorem is no help since $f(-\pi) = f(\pi) = 0$. But $f$ attains its minimum and maximum values at points $x_{min}$ and $x_{max}$ by the extreme value theorem, and you can observe that $$f(x_{min}) \leq \frac{1}{2\pi} \int_{-\pi}^\pi f(x)\, dx \leq f(x_{max})$$ So by the intermediate value theorem $f$ has to attain its average. (Notice that this actually sneaks the proof of the mean value theorem for integrals into an example. The word "proof" shuts off students' brains, but anything you say after the word "example" commands their full attention.)

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  • $\begingroup$ I also have a suggestion for how to sell the idea of completeness to your students. Toward the beginning of the semester when you are reviewing exponential functions, ask the class "What does $2^\pi$ mean?" Eventually you will be able to guide them to the notion that $2^{3}$, $2^{3.1}$, $2^{3.14}$, etc. all make sense and are converging to something. But this shows that you need a careful understanding of the real numbers to even define the function $x \mapsto 2^x$, a function most students think they understand. $\endgroup$ – Paul Siegel Oct 14 '14 at 8:59
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    $\begingroup$ I feel that an example like your first one is relatively weak as motivation, since I would have expected students to be able to do it without ever being "officially" taught the intermediate value theorem. In an example like this, the intermediate value theorem simply confirms our intuition that a curve can't cut across another curve without there being a point in common. I think the motivation for the extreme value theorem is particularly weak in a concrete example like this where a student could put $e^x+10x+3$ in a graphing calculator and say, "See, it crosses y=2!" $\endgroup$ – Ben Crowell Oct 15 '14 at 0:29
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    $\begingroup$ @BenCrowell Well, I suppose this depends on what you want to sell. The examples above are intended to convince students that the intermediate/extreme value theorem are useful tools for solving problems, and to this end the fact that students find them so intuitive is a feature rather than a bug. But if your goal is to convince your students that these theorems are deep facts about the structure of $\mathbb{R}$ then you probably have to provide context: if your students don't know how to prove anything then every fact you tell them is deep! $\endgroup$ – Paul Siegel Oct 15 '14 at 1:42
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    $\begingroup$ When I first teach the intermediate and extreme value theorems I often point out that $x^2 - 2$ and $1/(x^2 - 2)$, respectively, are counterexamples if you replace $\mathbb{R}$ with $\mathbb{Q}$. I also comment (with little explanation) that most of the rest of the theory of limits and continuity works over $\mathbb{Q}$, so the IVT and EVT are serious business. I think some of the brighter students appreciate this point, but I am content if the rest at least understand how to apply the theorems to examples. $\endgroup$ – Paul Siegel Oct 15 '14 at 1:48
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    $\begingroup$ Yes, what you're describing is pretty much what I'm doing, and in fact, in the great-minds-work-alike department, the $x^2-2$ example is one that I do. I'm just not under any illusions that it's meaningful to 98% of students. $\endgroup$ – Ben Crowell Oct 15 '14 at 15:59

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