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I have just started to teach Calculus to freshmans and sophomores who study non-mathematical subjects, e.g., international relations, psychology. They have to take few mathematics classes -including Calculus- in order to graduate.

The course is comprehensive, we gave the $\epsilon-\delta$ definiton of a limit and find the derivative of functions using the definition of derivative. So they can not just memorize the 'formulas' and solve problems, instead, they have to think and spend some effort.

I am trying to give motivation in the beginning. Before giving the $\epsilon-\delta$ definition of a limit, I tell a story about adding some salt while cooking. Like if you add $8$ mg. of salt, the food becomes the most delicious according to your taste. I played with the amount of salt, e.g., $(7.9,8.1)-\{8\}$, and after I see the light in some of the students eyes, I gave the formal definiton.

I know how hard for some of the students to take this class. Maybe, they have chosen their departments because of maths. Does not matter how much I try so hard to explain in a more simple way, sometimes I feel the emptiness in their minds, the pain and suffer they have. This bothers me a lot and instead of being reckless, I want to help them to understand or at least to pass the class.

What can I do, so that they can understand the subject and pass the class with a preferably high grade? Thank you.

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  • $\begingroup$ My impression is similar to KCd. I did it in AP class because it was required. People hated it. And the teacher said, just bear (bare?) with it. But did give the motivation of a couple sentences to the effect that this would just be run through for one lecture and a few HW problems and was useful as some theory but in rest of course, we wouldn't bother with it. (this is motivation, tells you a reason to at least see it once.) $\endgroup$ – guest Jan 29 '18 at 3:26
  • $\begingroup$ FWIW, I did just fine on it, since I was a glutton for detailed algebra. And think I got the concept (but never took real analysis, so maybe I didn't...but felt I did.) And then rest of time in chem, engineering, math, physics (some to grad school level) never needed the epsilon delta. $\endgroup$ – guest Jan 29 '18 at 3:27
  • $\begingroup$ So I would just expose them but not worry or kill yourself if they don't get it. Since it's not important. $\endgroup$ – guest Jan 29 '18 at 3:28
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I know that this is a difficult issue that many of us struggle with. By the time a typical student reaches college, they have had over a decade of math classes and teachers. As such, most of your students will have likely decided whether or not they're a "math person" by the time they're sitting in your classroom. You have to confront that an individual might be capable of calculation and dislike math, be poor at computation and appreciate math, or be poor at computation and dislike math (I would argue that there is a correlation). Regardless of the specifics, you're fighting against phobias.

Try to maintain a level of empathy for your students---as a lecturer in math, I would wager that math was always an easy/easier subject for you throughout school. Even if you didn't necessarily fall in love with math until you took a class from Prof. X or read a book by [famous mathematician], you've likely always been good at pushing symbols around and solving for $x$. Don't take these skills for granted in your students. Try not to combine algebraic steps on the board, give examples like $\sqrt{3^2 + 4^2} \overset{?}{=} \sqrt{3^2} + \sqrt{4^2}$, try to write questions that don't hinge on a clever simplification step. Furthermore, encouraging students to work and study in groups can be supremely beneficial: hopefully they'll be able to correct each other's mistakes.

As for the motivation question, why do you think that undergraduates studying liberal arts should learn calculus? Personally, I would argue that the discovery of calculus is perhaps the greatest intellectual achievement of humanity over the last millennium. What do you want your students to know about calculus at the end of the course? Once they graduate? In ten years? Try to strike a balance between needing to learn rote computation, motivation, and learning the sorts of problems that mathematics can solve. For example, having your students learn enough mathematics to recognize that there might be a mathematics question lurking somewhere and hint hint hire a mathematician to help them out can be a useful outcome. Try to keep them from unnecessarily reinventing the trapezoid rule.

The history of calculus is rife with stories and anecdotes that can be incredibly compelling to students of history, literature, anthropology, and other liberal arts. Consider, e.g., the historical rivalry between Newton and Leibniz, the dysfunctional Bernoulli family, the tragic lives of Abel & Galois, the autodidactism of Ramanujan, the prejudice that Sophie Germain (Monsieur LeBlanc) had to fight. You don't need to provide an entire history of mathematics for every topic, but even just an offhand mention can help engage students: "Did you know that a mathematician named Evariste Galois proved in the 1800s that there isn't a thing like the quadratic equation [point to where you just used this on the board] doesn't exist for polynomials of degree five or higher? The legend goes like this: after challenging another man to a duel, he realized that he was probably going to die in the morning (again, he was a mathematician) and stayed up all night producing about ten pages of mathematics that graduate students still study to this day. He died at the age of twenty." Give them a reason to see calculus as a human endeavor and not just an edifice of facts. I have even gone as far as having calculus students write essays about historical figures.

As you mentioned with your analogy of using salt in cooking, try to motivate definitions and theorems with everyday experiences. It's difficult to imagine our modern understanding of the universe without the influence of calculus. Mean Value Theorem? Average speed over a road trip. Linearization? What was the weather like when you walked into the building...predict if it's nicer out now. Exponential functions? Account balance under interest. Derivative? How can you figure out your speed if the speedometer is broken. Integration? How can you figure out how far you've driven if the odometer is broken. One tip that I've found is to explicitly not use introductory examples from physics. While we might find the notion of a swinging pendulum or accelerating rocket intuitive, it is surprising how many students struggle with these analogies. Even the example of throwing a ball under the effect of gravity trips people up (although you should absolutely address this misconception!). Most students have poor intuition when it comes to science, e.g., this article from The Atlantic or this article from the Harvard Gazette or this list of Physics misconceptions by New York Science Teacher or this list of misconceptions on Wikipedia. Being able to correct even a few of these is also a good objective for a calculus course. I personally find that economics/money or driving a car to be the most accessible areas to pull motivating examples from, before addressing physics examples with your newfound machinery.

As a parting thought, try pondering the flipped situation: you presumably had to take some liberal arts courses as an undergraduate for your presumably math-or-related degree. What did you get out Foreign Language / Art / Music / Literature / Psychology / Economics 101? What do you remember? Why was it important that you take those courses? Did you have any particularly effective instructors?

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    $\begingroup$ I must admit that I'm not positive a strong correlation exists. I have peers studying physics who can pull any calculation out of their hat, but I find I often make simple arithmetic mistakes when rushed, though I fully understand the underlying theory. For me, this is mostly because in maths I work with either small integers or variables far more than anything else; accordingly, arithmetic on large real numbers is something I leave to my calculator so that I can focus on the principle itself and not the calculation (which means I end up doing the simple arithmetic on a calculator too...) $\endgroup$ – Brevan Ellefsen Mar 28 '17 at 17:31
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I am not here to bring hopeful news, but a harsh dose of reality. There is no simple way to make most students grasp the $\varepsilon$-$\delta$ definition of a limit in a first calculus course and I think it is a mistake to force this on them. Doing so feels to me to be analogous to requiring everyone to be able to repair an internal combustion engine before they're allowed to learn how to drive a car. The rigorous limit concept is extremely subtle (it took over 150 years after Newton/Leibniz for it to be formalized, and the need for it escaped Euler), and most people starting out in calculus don't have sufficient mathematical maturity to grasp it. Let me quote from the introduction to Serge Lang's textbook A First Course in Calculus: "Any student is ready to accept as intuitively obvious ... limits and their basic properties. Experience shows that the students do not have the proper psychological background to accept a theoretical study of limits, and resist it tremendously." I think what Lang is saying is quite accurate.

When I took calculus in high school I learned the $\varepsilon$-$\delta$ definition of a limit (it was part of the AP calculus curriculum in America) as a kind of mantra without really being able to see what it meant and could apply it to the (in retrospect, stupid) case of linear functions, but the textbook's way of applying the definition of a limit to quadratic polynomials made the whole business look horribly complicated. It all made very little sense, and I was a very good student in math; it must have been really meaningless to the majority of the class. Two years later I was a college freshman in a course based on Rudin's Principles of Mathematical Analysis and figured out how to use the $\varepsilon$-$\delta$ language comfortably, although I only truly mastered it when I took topology and learned how to express the idea of a limit without saying $\varepsilon$ and $\delta$ at all! Why was I unable to grasp the $\varepsilon$-$\delta$ definition in my first calculus course? I think, as Lang put it, I just was not psychologically ready.

In continental Europe, I've been told there is no calculus, college math courses start with real analysis, and almost everyone flunks that course. (Are you in the UK? You wrote maths rather than math.) Colleagues of mine from Spain and Greece told me that only a handful of their classmates survived the first year math program during their university education. So I think the difficulties of conveying the idea and use of the $\varepsilon$-$\delta$ definition of a limit are quite serious and not likely to be overcome anytime soon. No amount of storytelling about the personalities in the history of math is going to help anyone solve math problems, so while it may be interesting I don't see how supplementing your lectures with such anecdotes will have an impact on the grades of those in your class.

I admire your desire to pass on to the students your hard-won knowledge of limits rather than making it their fault that they can't figure this out, but I don't see any way to make most of the class understand the rigorous definition of a limit when they don't even know what calculus can do for them in a pre-rigorous form.

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