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I am teaching Calculus I and will be teaching it again. To me, the $\varepsilon$-$\delta$ definition of limit is one of the key ideas of Calculus; learning calculus without learning $\varepsilon$-$\delta$ well is like building a tall building without a solid foundation.

My problem is, most students in my class have never received any formal training in logic, whether in Pre-Calculus or in other courses. They are unfamiliar with the quantifiers and some basic ideas in logic, which makes learning the $\varepsilon$-$\delta$ definition very difficult.

One possible solution that I have never tried is: Spending one or two or three classes on logic in my Calculus class, before teaching the $\varepsilon$-$\delta$ definition. After some training in logic, it is my hope that my students will understand the concept better.

Questions:

  1. Is having a few classes on logic in a Calculus class a good idea?
  2. If yes, any recommendation on materials? My goal in to cover the very basics of logic in no more than say four classes.
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    $\begingroup$ I disagree that calculus (not real analysis!) can't be taught without spending time on the $\varepsilon$-$\delta$ definition of a limit. Most people learn how to drive a car competently without understanding how the internal combustion engine or transmission on a car works. It is unnecessary to build rigorous logical foundations in calculus for the defintion (vs. the intuitive meaning) of limits in order for students to learn how to use calculus in a practical way. $\endgroup$ – KCd May 3 '17 at 9:52
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    $\begingroup$ @KCd, thanks for your comment! I personally feel that teaching calculus without $\varepsilon-\delta$ is teaching applied maths instead of teaching maths, or teaching calculus at high school level instead of at college level. The students not only lost an excellent opportunity of training on logic, but also, they only learn how to apply Theorems instead of why theorems work. Perhaps I have been wrong on this? $\endgroup$ – Zuriel May 3 '17 at 12:42
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    $\begingroup$ That's why I distinguished between calculus and analysis. If you want to teach students logic, there are more effective ways than through having them "learn" the $\varepsilon$-$\delta$ definition of a limit, which is very sophisticated because of the nested quantifiers. I put the word learn in quotes because nearly all students will not understand this definition as they are simply not psychologically prepared to internalize it. Look at the history: mathematicians made great creative use of calculus for almost 150 years before anything like a clear definition of a limit was proposed. $\endgroup$ – KCd May 3 '17 at 13:03
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    $\begingroup$ So if your course is calculus, aimed at students who are not math majors, make sure they learn well what a limit is supposed to mean intuitively. If you are are teaching real analysis, for math majors, then give them the whole works of $\varepsilon$ and $\delta$, compactness, and so on. This is why I made the analogy with learning to drive a car. Nearly everyone learns how to drive a car quite well without having to learn how a car actually works. It is not necessary to know the mechanical foundations of automobiles in order to make good practical use of them. $\endgroup$ – KCd May 3 '17 at 13:06
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    $\begingroup$ I agree that the epsilon-delta definition is important. Every calculus book I've seen has it -- it is not reserved for books on "analysis". Calculus is already a very late time to start sharing real math with proper foundations; delaying it any further is borderline criminal. $\endgroup$ – Daniel R. Collins May 4 '17 at 2:29
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Your assumption that teaching calculus needs to be backed by the $\varepsilon$-$\delta$ definitions could be challenged, but since it is not your question I won't do that here.

My recent experience about a few logic classes first has been disappointing. It took much hard work, and the outcome seemed good at first, but vanished as soon as we got to the main topics. While I don't have much more to build on than my experience, I tend to think that it is in fact preferable to insert small bits of logic and reasoning principles at the point of use, rather than put them aside at the beginning of the course. The student tend to keep things apart in their mind, and to fight this it seems that not separating them in lecture is a sound strategy.

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  • $\begingroup$ Thanks for your answer! This is exactly what I was considering: "insert small bits of logic and reasoning principles at the point of use". When I am designing the course, I think I can afford up to say 4 class periods on this, not sure how much time on logic is sufficient; but what they learn about logic will not only be used in teaching definitions and proofs in Calculus, but also likely to benefit the students for the rest of their life. $\endgroup$ – Zuriel May 3 '17 at 12:46
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    $\begingroup$ @Zuriel: there is always some ambiguity, don't overestimate the squarness of what we do (o you start mathematics with ZFC axioms? Do you tell your student what a real number is? What do you think a real number is?) We always work from axioms, sometimes implicitly. One can do a lot of calculus accepting here is such thing as limits, and that it satisfies some axioms, and deducing further properties from there. I do that with integral calculus for freshmen: no Riemann integral, just the notation and the fundamental theorem as an axiom. $\endgroup$ – Benoît Kloeckner May 3 '17 at 15:43
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    $\begingroup$ I agree with this. When I taught Calc 2 I found it very helpful to briefly talk about logic, specifically with truth values of a statement and its contrapositive, converse, and inverse. I think it's very important that students understand these concepts so that they correctly use the fact that $\displaystyle \lim_{n\to+\infty} a_n \ne 0$ implies that $\displaystyle \sum_{n=1}^{+\infty}a_n$ does not converge. Too many students think $\displaystyle \sum_{n=1}^{+\infty} a_n$ converges just because $\displaystyle \lim_{n\to+\infty} a_n = 0$ :\ $\endgroup$ – tilper May 3 '17 at 19:30
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    $\begingroup$ @BenoîtKloeckner We must have very different ideas about what the "important narrative" is for calculus. In my mind, the most important thing for students to realize is the linear approximation $f(a+h) \approx f(a)+f'(a)h$ for small values of $h$. Then to approximate $f(b)-f(a)$, you can think of this as repeated linear approximation: $f(b) \approx f(a)+f'(a)h+f'(a+h)h+f'(a+2h)h+...+f'(b-h)h$, where $h=\frac{b-a}{n}$. We can then interpret these multiplications as areas of thin rectangles. Taking FTOC as definition eliminates the whole story I want to tell. I wonder what your story is? $\endgroup$ – Steven Gubkin May 5 '17 at 12:27
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    $\begingroup$ @StevenGubkin: we are departing from what comments are for, but while I do explain the idea behind the FTOC with a drawing (which don't really need a Riemann sum, but some understanding of limits), I don't insist on proving it formally until later on. $\endgroup$ – Benoît Kloeckner May 6 '17 at 12:54
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I had the same thought this year. My suspicion was that many students get anxious about suddenly dealing with quantifiers and they also struggle with understanding how the ordering of them can affect the logical statement. To try to remedy this, I spent a short amount of time (less than 2 classes) introducing quantifiers and practicing parsing simple logical statements about real numbers starting from 1 variable (e.g. $\forall x,\thinspace x^{2}+1>x$) to 3 variables (e.g. $\forall x\exists y\forall z,\thinspace x-\left|y\right|\leq\left|1-z\right|$). I will say that it was helpful in that I was able to catch some misunderstandings before we got to the calculus concepts and it was very easy to resolve them at that stage. As for this approach's effectiveness and the ideal materials to use, I am not sure and I am also interested in hearing others' experiences.

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  • $\begingroup$ Thank you for your answer! For teaching the basics on logic, which material did you use? $\endgroup$ – Zuriel May 5 '17 at 21:20
  • $\begingroup$ @Zuriel I didn't have anything appropriate so I just made some hand-out sheets. $\endgroup$ – cocoahomology May 8 '17 at 0:30
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Generally speaking, it would be nice to have a foundations class at the initiation of the Math major. Some of my colleagues envision this course centered around teaching college algebra. Well, to be more precise, algebra done with care, algebra beyond symbol pushing and problem solving.

The Foundations course we currently run in our major requires Calculus II as a prerequisite, but, this is mostly for maturity. Logically, it ought to go at the start. If the proofs course was first then more students would have a better view to what major course work looks like in Math. Calculus is still very much a general education requirement, so, the inculcation of students into native math culture really only begins in the proofs course at our institution.

But, it's not really this simple. In fact, we do have epsilon delta proofs in our first semester calculus. Indeed, we also study convergence and divergence of sequences, sometimes the least upper bound property and other bits a pieces of analysis in the second semester. However, these are not the true flavor of the calculus sequence. They're deviations from the main story of calculation and application. In my personal opinion, the real analysis needed for second semester calculus is far more sophisticated than mere epsilon delta proofs. For example, the proof that the derivative of a power series is the derivative of a power series is quite beyond the students. I'm certain of this as none of them ever complained the many years I failed to appreciate the need of the proof.

We introduce integrals in calculus I in terms of limits of sequences! Many texts do this. So, this is after we cover sequential limits, right? Wrong. The limit appearing in the Riemann integral is merely intuitively framed by justifying approximations. Now, don't misunderstand, I understand the place of providing an intuitive version of calculus before rigor sets in. But, I am aware of the hypocrisy of carefully covering epsilon deltas in the same semester I fail to properly define limits of sequences, much less the integral itself.

So, why study epsilon deltas in calculus I? It doesn't seem practical to maintain the level of detail throughout the calculus sequence. We must admit there are tools we wish to use, but, not construct; how many major theorems do we have time to present the proof? I have a lot of time here, but, even so, it's not possible to cover everything. I can tell you from experience, the students only appreciate so much honesty in a given semester. So, why epsilon deltas?

  1. it gives students a much needed chance to study how inequalities work.
  2. it can be a great chance to better understand how graphs can organize analytical thought
  3. the logic isn't that complicated if you do a half-dozen examples. Keep it simple, basic functions like linear, quadratic, reciprocal all are helpful.
  4. it allows you to be precise in the meaning of a limit not existing, what we mean when we say the limit point need not be considered in the evaluation of a limit
  5. it does make introduction of sequential limits a natural continuation of an existing discussion
  6. the students who can think can do it. It's not that big a deal.
  7. for students who are math majors at heart, it's a breath of fresh air in an otherwise logically questionable development. I have had the topic gain the major new students. In fact, my best student to date came to Math from a different major in large part because I was overly formal in Calculus I. It can make a difference.

Ok, so, how many students don't get it? Lots. Just like lots of students don't get a number of topics in calculus. Convergence, divergence, the chain rule, the idea that you plug the number into the derivative to find the equation of a tangent line, this list is endless really. There is a risk and a possible reward, you have to decide on the basis of your audience whether it's worth the risk.

That said, do you need a logic class beforehand? No, not in my experience.

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    $\begingroup$ +1. But: Is there a typo in "the proof that the derivative of a power series is the derivative of a power series"? $\endgroup$ – Daniel R. Collins May 8 '17 at 2:09
  • $\begingroup$ @DanielR.Collins it is easy to prove the term-by-term derivative of a power series converges on the open interval of convergence, however, the proof that the term-by-term derivative series is in fact the derivative of the given series is subtle. Perhaps I should edit that sentence, I will ponder it. $\endgroup$ – James S. Cook May 8 '17 at 3:23
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    $\begingroup$ +1. I'm feeling that at the first-year calculus level in the US, the students' trouble with inequalities eclipses the logical difficulties of the definition. (So I like your point 1.) $\endgroup$ – user1527 May 23 '17 at 17:36
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It is well known that learning epsilon-delta definitions is difficult and is the intellectual equivalent of jumping over a tall wall in order to join the enlighted ones on the other side, a feat never accomplished by at least three-quarters of the student population in calculus (a conservative estimate).

It is clear that students need some preparation that would serve as a ladder to help them over the proverbial wall. The use of alternating quantifiers needs to be prepared. This can be done either by giving a series of examples of formulas of increasing quantifier complexity, or by providing a more intuitive approach to the key concepts like the derivative and continuity first.

One recent proposal in the literature is to teach them rigorous infinitesimals as a way of providing a ladder so as to climb the Epsilontik wall. Once the students understand the basic concepts like derivative and continuity via their intuitively appealing definitions exploiting infinitesimals (essentially following Cauchy's approach), they are in a better position to scale the long-winded epsilon-delta paraphrases of those definitions.

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No. This is the same kind of pedagogical fallacy that led to the "new math" of the 1960s, when they tried to teach elementary school students deep concepts of abstract algebra as an introduction to arithmetic.

Instead, think of epsilon-delta proofs as an introduction to mathematical logic. If you're worried that students can't understand calculus without them, consider Newton, Leibniz, the Bernoullis, Euler, and Fourier. They all did pretty well with calculus despite not knowing a single epsilon-delta proof. There's a reason why epsilon-delta proofs didn't emerge until the 19th century: before then, they weren't needed—but at that time, the need for them became pressing.

The pedagogical fallacy is thinking that teaching a deeper, more abstract, more refined concept first makes it easier for the students to learn a concept that can be mathematically deduced from it. That's not how learning works—or even logic in the everyday sense. To grasp an idea, you need to anchor it in some way. For example, we learn basic concepts of quantity by comparing tangible quantities, like longer and shorter things, and heavier and lighter things, before we learn real analysis. We learn numbers by counting things, not by deducing from the real-number axioms. We start with one, not zero. We start with the concrete and build progressively more abstract ideas upon it.

A frequent complaint that chemistry professors make about math professors is that students who've been taught linear algebra can't solve stoichiometric equations. This is because the students were taught linear algebra as a bunch of axioms and derivations unanchored in anything more concrete and understandable. That way of teaching leads the student to see mathematics as a symbol game disconnected from reality. Of course they can't see how linear algebra applies to stoichiometric equations—who thinks of chemistry when they're told the rank-nullity theorem? This manner of teaching teaches most students to hate linear algebra. They have no idea of its vast applicability. That's what math teachers do to students when they try to "front-load" ever more abstract ideas.

Epsilon-delta proofs are a clever way to put calculus into a framework that avoids various paradoxes of infinity. They're one of the great intellectual achievements of human history, and calculus students should get some exposure to them. But they're nearly impossible to appreciate until you've seen the troubles that arise without them. See A Radical Approach to Real Analysis by David Bressoud for a nice explanation of the need for them and how the approach was worked out. Basically, it's hard to tell if trigonometric series converge or not without them. I don't think you need such a hard example to motivate epsilon-delta proofs, but you need some conceptual difficulty that's meaningful to the students. Maybe just a couple series for which it's hard to tell whether they diverge or converge. Before you can lead them to such an impasse, they need a fair amount of experience with plain ol' calculus.

Starting them off with negations of double quantifiers will just pile confusion on top of confusion. But once they've struggled with epsilon-delta proofs a while, oh will they be ready to learn about negating quantifiers!

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  • $\begingroup$ You got your New Math comparison backwards. In fact throwing the epsilon-delta monster without preparation at unsuspecting students can be put in remotely the same category as the new math debacle. They are both an educational debacle. $\endgroup$ – Mikhail Katz Jul 14 '17 at 7:41
  • $\begingroup$ @MikhailKatz Regardless of which topic one introduces first, I think the main thing is to let the students know why it's important. Historically, epsilon-delta proofs were important because they resolved paradoxes about infinite series. I think the value of all that quantifier stuff only became apparent later, suggesting that it's unwise to introduce it first. However, I've heard that the quantifier stuff, or something like it, was first invented in the Middle Ages, suggesting that it actually can be motivated before epsilon-delta proofs (contra my answer above). $\endgroup$ – Ben Kovitz Jul 16 '17 at 0:37
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    $\begingroup$ @MikhailKatz There is, however, a natural and intuitive way to explain delta-epsilon proofs which makes direct use of quantitative insight rather than abstract symbol-twiddling: "How small a range do you want to constrain $f(x)$ to? There is a range in which to constrain $x$ that will keep $f(x)$ 'corralled' as narrowly as you like." If teachers explain this rather than giving a pure meaningless symbol-twiddling explanation, whether quantifier-reversal or (unmotivated) epsilon-delta proofs, I think most students will "get it". $\endgroup$ – Ben Kovitz Jul 16 '17 at 0:42
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    $\begingroup$ Ben Kovitz, you may want to consult this post for further thoughts regarding this issue. $\endgroup$ – Mikhail Katz Jul 16 '17 at 9:12
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Just as a supplement to other answers: of course, in the first place, configuring "a calculus course" depends on the larger purpose of the thing. I do think it is too naive to say that it is literally to teach exactly the mathematical content of the course... since, as the other answers show, there seem to be larger philosophical issues in play. At the same time (e.g., in the U.S., most familiar to me) there are dominant sociological issues in the situation.

After those platitudes, my main point would be that I don't think that mathematics is fundamentally about "logic", nor that calculus (or even "analysis") is fundamentally (!) about epsilons and deltas (and certainly not about order of quantifiers). That is, yes, while certain immediate issues can be formalized as being those things, first, attempts to change things by directly addressing "(first-order predicate) logic" will mostly fail, because FOPL is not reflected in the ambient culture, and, second, there is some artificiality (and convention) involved in deciding that first-order predicate logic is "the true logic", and deciding that epsilon-delta is "the true analysis". Neither of these is clearly true, although simple, popular choices.

I myself do like genuine proofs of interesting things, but I am less enthusiastic about implicit meta-hypotheses that force me/you/us into technically awkward/ponderous arguments. My fave is the intermediate value theorem: obviously true, without "definition" of continuity or the real numbers. Why does it even need proof? (Yes, I know, ...)

Also, the syntax of legitimate mathematical argument should (for stability) resemble ordinary language as much as possible. Yes, ordinary language has many (sometimes unfortunate) assumptions that are contrary to the "truths" of first-order predicate logic. There's the point: do we (people explaining/teaching mathematics) want to tell students that the ambient culture (their culture) is "wrong" ("kewl", sure, but silly...), or, rather make a higher-level transition from the not-easily-changed vernacular, and explain mathematics in those terms !?!

That is, I don't think we all have to learn Esperanto before we can do the amazing things that calculus (and other good mathematics) allows us. :)

(I know I have a bit-extreme viewpoint... )

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  • $\begingroup$ "Also, the syntax of legitimate mathematical argument should (for stability) resemble ordinary language as much as possible." I believe this is not possible. From my learning experience, the language of mathematics differs from natural languages more than they differ from each other. Only when I abandoned attempts to think about mathematics in my native language, I was able to progress. I would call the language of mathematics Martian, alien, ☺ not Esperanto, so different it is. $\endgroup$ – beroal Aug 18 '17 at 6:20

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