Home > education, math education > Guest post: What is the goal of a college calculus course?

Guest post: What is the goal of a college calculus course?

June 23, 2014

This is a guest post by Nathan, who recently finished graduate school in math, and will begin a post-doc in the fall. He loves teaching young kids, but is still figuring out how to motivate undergraduates.

The question

Like most mathematicians in academia, I’m teaching calculus in the fall. I taught in grad school, but the syllabus and assignments were already set. This time I’ll be in charge, so I need to make some design decisions, like the following:

  1. Are calculators/computers/notes allowed on the exams?
  2. Which purely technical skills must students master (by a technical skill I mean something like expanding rational functions into partial fractions: a task which is deterministic but possibly intricate)?
  3. Will students need to write explanations and/or proofs?

I have some angst about decisions like these, because it seems like each one can go in very different directions depending on what I hope the students are supposed to get from the course. If I’m listing the pros and cons of permitting calculators, I need some yardstick to measure these pros and cons.

My question is: what is the goal of a college calculus course?

I’d love to have an answer that is specific enough that I can use it to make concrete decisions like the ones above. Part of my angst is that I’ve asked many people this question, including people I respect enormously for their teaching, but often end up with a muddled answer. And there are a couple stock answers that come to mind, but each one doesn’t satisfy me for one reason or another. Here’s what I have so far.

The contenders.

To teach specific tasks that are necessary for other subjects.

These tasks would include computing integrals and derivatives, converting functions to power series or Fourier series, and so forth.

Intuitive understanding of functions and their behavior.

This is vague, so here’s an example: a couple years ago, a friend in medical school showed me a page from his textbook. The page concerned whether a certain drug would affect heart function in one way or in the opposite way (it caused two opposite effects), and it showed a curve relating two involved parameters. It turned out that the essential feature was that this curve was concave down. The book did not use the phrase “concave down,” though, and had a rather wordy explanation of the behavior. In this situation, a student who has a good grasp of what concavity is and what its implications are is better equipped to understand the effect described in the book. So if a student has really learned how to think about concavity of functions and its implications, then she can more quickly grasp the essential parts of this medical situation.

To practice communicating with precision.

I’m taking “communication” in a very wide sense here: carefully showing the steps in an integral calculation would count.

Not Satisfied

I have issues with each of these as written. I don’t buy number 1, because the bread and butter of calculus class, like computing integrals, isn’t something most doctors or scientists will ever do again. Number 2 is a noble goal, but it’s overly idealistic; if this is the goal, then our success rate is less than 10%. Number 3 also seems like a great goal, relevant for most of the students, but I think we’d have to write very different sorts of assignments than we currently do if we really want to aim for it.

I would love to have a clear and realistic answer to this question. What do you think?

Categories: education, math education
  1. Jerry
    June 23, 2014 at 8:28 am

    I would say that a course like this has the goal of preparing students for the next course. I’m not saying that flippantly – it’s a big deal. You should consider a single calculus course in the context of a program.

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  2. Priscilla Bremser
    June 23, 2014 at 8:43 am

    Several friends have told me, “I took calculus in college but I couldn’t tell you what calculus is about.” So now one of my goals for calc I is to have my students be able, years from now, to give a reasonable description of what calculus is about. What’s special, mathematically, about calculus, as compared to students’ previous experiences? Why do math professors keep referring to it as a major intellectual achievement in the history of mathematics? Of course this doesn’t satisfy your “realistic” requirement, since I have no way of knowing, really, whether it’s working. (Then again, I do have some evidence that focusing on tasks one semester doesn’t mean students will recall how to do those tasks a year later.) But at a friend’s suggestion, I’ve just started reading _Make It Stick: The Science of Successful Learning_ (Brown, Roediger, and McDaniel), which is helping me make the design decisions. For example, from page 4: “Trying to solve a problem before being taught the solution leads to better learning, even when errors are made in the attempt.”

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  3. June 23, 2014 at 8:47 am

    Excellent question and I agree completely with your analysis. I struggled with the same issues whenever I’ve taught calculus, and I think it’s a symptom of the fact that the standard calculus curriculum hasn’t changed on paper in generations. Reason 1 made sense before graphing calculators and Wolfram alpha came along. The other two reasons may have made sense when college admission standards were higher and students came prepared for rigorous thought (if such a time existed).

    I think you have an excellent argument for overhauling the calculus curriculum.

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  4. Jane
    June 23, 2014 at 8:57 am

    I’ve always thought calculus was about measuring change and given that everything changes it’s a good thing to understand regardless of where your life takes you (although learning calculus is still on my ‘to do’ list so I may be wrong about its purpose).

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  5. June 23, 2014 at 9:52 am

    Part of the answer depends on your institution: calculus as a terminal course at St. %%%% College, full of sociology majors, is very different than calculus as a not-quite-terminal course for architecture students at C%%%%% University, which is different than calculus as the first course of many for would-be engineers at an Institute of Technology. Some of the frustration people find in teaching/taking calculus is in teaching/taking the wrong class for the students.

    So, for the engineers who’d go up through multi and DE, I did make sure to hit some of those mechanical skills really hard. They’re going to need power series expansion and partial fractions. On the other hand, when I taught terminal calc to sociology students, I concentrated most on understanding functions and communication. They will need to interpret the results of studies about changing rates of drug addiction over time. They will need to understand concavity and optimization. They will need to be unafraid of numbers and equations and they will need to be able to write a powerful data/math-backed sentence justifying continued public funding of (something). (I tell them all to take stats as well!)

    Whatever the student focus, these emphases are placed in the context of modern college calculus as the study of rates of change of quantities, net change of quantities, and mathematical relationships between rates of change, net change, and the quantity itself.

    Liked by 1 person

    • Min
      June 23, 2014 at 2:37 pm

      Martin Buber said that the teacher should build a bridge to the students. How can you answer your questions without knowing about your students? Are they taking calculus to satisfy the math requirement, so that it will probably be their last math course? Are they taking it to satisfy the prerequisites for becoming a math major? Are they taking it because will need it for a different field? Different students will need different things from the course. Besides, of course, having different backgrounds and abilities.

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    • June 23, 2014 at 9:21 pm

      Regardless of whether this is a first or the final college maths the students take, it seems like the goal is scaffolding the students into their future studies – and even humanities such as history are more data oriented than ever before. Hence, while providing the foundation for further maths – and physical sciences – study is important for engineers and sciences, mostly implying #1 and the beginnings of #2,as Kaisa says, being unafraid of data is a reasonable goal for other kinds of students, which implies #2 and #3 more than #1

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    • June 23, 2014 at 11:46 pm

      Amen to Kaisa and to Jason Starr (for their different statements). I just wanted to add that when I was teaching a dead-end calculus class, I decided that what my students really needed was tools to understand and study graphs. Calculus provides some of the best ones, but sometimes one can say good things about a graph without jumping to calculus. (I got some inspiration from “How to Lie with Statistics”.)

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  6. June 23, 2014 at 9:55 am

    Almost certainly the curriculum committee of your university has already answered all of these questions for you (lucky you!). Other departments communicate to that committee the curricular needs of their own majors. It is healthy to reflect whether that need is valid, but, IMHO, an individual calculus instructor should not “overrule” the experiences of entire departments that depend on the math department to meet the calculus needs of their majors. So, whatever you decide, I advise you to start with the syllabus for the course as the foundation for your course plan.

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  7. Albanius
    June 23, 2014 at 10:24 am

    For a few students, mostly future mathematicians, the sheer beauty of the ideas is essential.
    You might want to mention in passing sources where students who fall in love with the subject can look for deeper understanding.
    For the rest, when I think of applications, physics, chemistry, engineering, economics, and social sciences that use probability and statistics may be more relevant than biology or medicine. You may want to include examples and problems from all those fields.
    Among technical skills, the fact that rational functions can be integrated using partial fractions is relevant for the sake of completeness, but I wouldn’t test students on that technique. Among harder topics, I would rather include the intermediate value theorem and L’Hopital’s rule, as well as introducing multivariate calculus and differential equations.
    To motivate humanists, you may want to start with the paradoxes of Zeno.

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  8. June 23, 2014 at 10:26 am

    This is a tough question that I’ve also struggled with, and I don’t have an answer. But I do have something to add to #1. At my institution the professors of the higher level engineering courses have complained to the calculus teachers that not only are their students unable to solve integrals, but they can’t even *set up* integrals properly to feed them to a computer!

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  9. brian t
    June 23, 2014 at 11:34 am

    I did Calculus at university (as a mature student) and (in retrospect) the looming exams bent the courses out of shape. Whatever the lecturer said was filtered through an “exam filter”: “is this going to be on the exam?”. It made it hard to see the bigger picture. I would stop end-of-semester exams entirely if it was up to me, and stick with continuous assessment.

    Another problems was the lack of real-world relevancy – not in the subject, which I know is highly relevant, but we couldn’t tell that from the course content. We could have used the equations of motion at an example e.g. if distance is “s”, the velocity is ds/dt and acceleration is d²s/dt²

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  10. yenergy
    June 23, 2014 at 12:09 pm

    I actually think #1 is your best contender- yes, they likely won’t have to compute things again. But those doctors etc. will someday need to implement algorithms (aka decision making) that use different rules for different situations, and learning to set up integrals is a controlled situation with many different solutions to the same (correct) answer in that vein. I think it’s hard for us who teach calculus to remember that quantitative/logical reasoning isn’t “natural” for most of our students.

    Goal-wise, I’ve been teaching ESP (a calc workshop class) for a year and it does teach communication, but it’s unrealistic in a lecture setting.

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  11. Anders Schneiderman
    June 23, 2014 at 2:05 pm

    Speaking as someone who remembers hardly anything from the calculus classes I took, I’d vote for a focus on #2 with #1 and 3 in supporting roles. And given the graphing tools that are available now, I think it’d be much easier to make some headway on #1 because it’d be easier to play. And some of those tools are free & open source — you could, for example, do some pretty cool stuff using IPython Notebook, especially given the new interactive widgets that are part of IPyN 2.0.

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  12. noneya
    June 23, 2014 at 5:33 pm

    I think the goal should be to do enough of #1, with as much of doing it from ground up as possible, that #2 and #3 emerge. I don’t believe teaching #2 or #3 directly leads to long lasting effects.

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    • July 22, 2014 at 8:06 pm

      My experience suggests exactly the opposite, that #2 and #3 rarely emerge unless explicitly taught, and that those things are much more likely to stick than items from #1.

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  13. June 24, 2014 at 12:43 am

    When I taught calculus, the primary purpose of the course seemed to be to serve as a gateway / weeding process for pre-meds, as one of the many signs of commitment they were required to show while going through that system. I’m not going to argue that the candidates you list aren’t potentially important, and I’m perfectly willing to believe that my experiences were colored by my observations and the specifics of the institution I was at, but still: I’m pretty sure that the systemic forces supporting calculus (which are incredibly important in terms of situating math departments within the broader university context) don’t just depend on Fourier series or an intuitive understanding of functions.

    Not that this helps in figuring out what your personal goal might be when teaching the course, of course. Maybe one intermediate question, though, might be: what are your students’ goals when taking the course? For example, if they couldn’t take the course for some reason, what (if anything) would they fear that they would be missing out on, and how seriously would they take those fears?

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  14. Deane
    June 24, 2014 at 2:21 am

    What is the point of calculus, as well as precalculus and multivariable calculus? My answer is that it should be introducing to students to the power and usefulness of (mathematical) functions. Everyone assumes that calculus about derivatives and integrals, but I’ve decided that these should be viewed only as a means to an end.

    The concept of a function is a fundamental yet extremely powerful one that is used in all aspects of advanced mathematics and its applications. Ideally, the full sequence of courses (precalculus, single and multi variable calculus) should be devoted to making sure students understand deeply what the abstract notion of a function is, specific types (different domains and ranges, different dimensions of domains and ranges), how functions can be found in the wild, what are useful questions to ask and answer about functions, and how to go about answering these questions. Derivatives and integrals are arguably the most important and powerful tools available, but they should be introduced only after the student understands what functions are and what questions derivatives and integrals can help answer.

    Unfortunately, the current courses don’t come within a million miles of doing that. If a student learns any of the above, it’s despite and not because of what he or she was taught.

    Instead, we take for granted that the concept of a function can be taught and learned in one lecture during a precalculus or calculus course. Precalculus then becomes just a course in doing calculations involving algebraic, trigonometric, exponential, and logarithmic formulas, and calculus a course on how to compute derivatives and integrals of formulas. In no way does the student really learn clearly what a function is. Everything is taught using formulas, and students can usually do perfectly well assuming that “function” is a synonym for “formula”.

    I know it’s an unpopular view, but as far as I’m concerned the only calculus textbooks I’ve ever seen that follow the view espoused above (except perhaps 19th century texts on functions) are the Harvard Calculus texts. I’m especially fond of the first chapter of the calculus text and the precalculus text (Functions Modeling Change) that arose out of it. It restores the concept of a function to the central focus of a calculus course.

    As long as students think they are differentiating and integrating formulas, rather than functions, our calculus courses are bad jokes.

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    • June 24, 2014 at 1:29 pm

      I think I disagree with the spirit of this comment. Of course I can’t argue that the idea of a function is important and belongs in a calculus class, nor can I contest that many students remain confused about basic things like what “one-to-one” means even after having taken several semesters of calculus. But I disagree that calculus class (at least for the first few semesters) is the right place for students to master the subtleties of the abstract notion of a function.

      When you think about why the abstract idea of a function is so important, what comes to mind? I would think about how useful it is to probe the structure of an algebraic or geometric object by constructing functions into and out of it. Or I’d think about the subtle obstructions that arise when trying to extend the domain of a holomorphic function and the sophisticated theory that one needs to understand them. What definitely does not come to mind is the importance of recognizing that f(x) = sqrt(x) is onto if you restrict its codomain to the set of nonnegative real numbers. I might think about how some of the subtleties associated with the abstract notion of a function enter into the proof of, say, the extreme value theorem, but this does not belong in a first course on calculus.

      Note that the notion of a function as we understand it today did not really emerge until over a century after calculus was invented. Before then functions were thought of largely as our students think of them: either as formulas, or as one quantity varying with another quantity (often pictured as a plane curve). This was good enough for most of the people who invented calculus in the first place – Newton, Euler, Clairaut, Bernoulli, etc. – because for them calculus was first and foremost about geometry. I think it wasn’t until the 19th century that mathematicians were forced to come to terms with phenomena that required a better language, and we certainly don’t teach calculus students about those phenomena.

      So in my opinion calculus is fundamentally about the relationship between geometry and motion. The notion of a function is useful because it helps clarify that relationship, but we needn’t obsess over how well our students understand the subtleties of the modern foundations of that notion.

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      • Deane
        June 24, 2014 at 3:53 pm

        I think you’re thinking about an overly formal definition of a function. When I teach calculus, I explain simply that a function is a box that takes an input (or in multivariable calculus, possibly more than one input) and returns an output (or maybe more than one). It is a consistently behaving box (if you feed it the same input, it always returns the same output). That’s it. If the students know any programming languages, then it’s really easy, because it’s just a function in that sense too. Then what’s calculus? It’s the study of how the output changes as the input changes. In particular, it’s about the sensitivity of the output with respect to changes in the input. The first functions a student should understand are ones where the sensitivity is always the same, i.e., constant and linear functions. After that, nonlinear functions are simply functions where the sensitivity changes depending on the initial input. The notion of a derivative as a function itself now appears naturally. The concept of concave up and down functions also arise naturally. Notice that all of this can and should be taught without relying on formulas to define the functions. That’s all I mean by the word “abstract”; we’re not trying to teach real analysis here.

        Integration is now simply the process of reconstructing the function from knowing its sensitivity over a range of inputs.

        I believe calculus taught this way creates a natural connection between the subject itself and its applications. And that’s exactly what the Harvard Calculus texts do. It pains me to see so many mathematicians dismiss the books because they are too “imprecise” and not “rigorous” enough.

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        • June 26, 2014 at 2:25 pm

          I thought I would put my scheme for removing almost all math (certainly calculus) from statistics.

          This link provides some background in the bottom comment http://magazine.amstat.org/blog/2013/10/01/algebra-and-statistics/

          Essentially it works because fully Bayesian analyses can be implemented using just simple two stage simulation on a computer for toy problems and with a few tricks realistic problems until they get too complex. A machine analogy is used to initially get around simulation as a black box

          Then the likelihood can be mechanically approximated from the output and things like Frequentist and well as Bayesian sufficiency, ancillarity, convergence in distribution, interval coverage, confounding (causal inference) can be demonstrated and investigated.

          There is a problem (and the reason for “almost all” clause above). It is lack of abstract thinking and appreciation of how _reality_ is being represented, those representations manipulated and then taken to learn something “unknown from what was known”. With those representations fully understood to not be fully accurate (always wrong in some sense). Your comment seems closest to addressing this.

          Also, I don’t teach anymore, but would expect a lot of push back from students and other faculty – so one really needs to be up to it.

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  15. Savonarola
    June 24, 2014 at 4:31 am

    I’m kind of a math idiot – but took microeconomics and econometrics at the graduate level, so perhaps I’m a good person to speak to this question. I never took calculus, although it was a prerequisite for a lot of the courses I took towards the end of my undergraduate education. Someone showed me how to do a derivative, and that pretty much took care of the prereq. If you are good at algebra, you can fake your way through most anything economics throws at you.

    But from the outside, calculus appears to be to math what music theory is to playing the piano. You learn the tools that make it easier to figure out what you are seeing, rather than merely memorizing what you need to know to read the music. It seems to mean unpacking the concepts that underlie other things you will be using. For me, it seemed unnecessary once I understood enough about the matrix algebra underlying my regression analyses that I could figure out when I was asking the computer to do something pointless. But there is definite appeal in being conversant with what makes math tick.

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  16. June 24, 2014 at 5:44 am

    I’m going to suggest you back up even further, and put some thought into answering:

    What is the goal of a college education (and how does learning calculus fit into that)?

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  17. Larry Headlund
    June 24, 2014 at 8:23 am

    I would say the tools you see to achieve 1 (To teach specific tasks that are necessary for other subjects.) as you describe them:”These tasks would include computing integrals and derivatives, converting functions to power series or Fourier series, and so forth.” are the neccesary means to your second goal “Intuitive understanding of functions and their behavior. ”

    Maybe there is a way to get an intuitive understanding without doing the nuts and bolts work to get a familiarity, maybe there is a royal road but I don’t know what it is. Tools like those from Wolfram are grat but they are magic boxes unless the user has some notion of what is going on.

    I don’t know about your school but where I taught Calculus I it was never a terminal course. I did teach some of the those, they had titles like Statistics for the Health Sciences.

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  18. June 24, 2014 at 9:36 am

    The realistic (i.e. cynical) answer to your question is that universities use calculus classes to weed out engineering, business, and premed majors. These departments’ incentives are organized around the ratio (number of students who declare major X)/(number of students who graduate with degree in X), and by requiring two or more semesters of calculus to declare X, the ratio can be improved. Math departments play ball because engineering and business schools periodically threaten to teach their own calculus classes, which would be devastating to the math department’s funding stream.
    But there is another reason which might actually help answer your question: by making precise phrases like “local optimum”, “asymptote”, “stable/unstable equilibrium”, etc. and providing tools for manipulating these phrases, calculus provides the language that a scientist needs to do one of his or her most basic jobs: tell a story about data. Many calculus students will never again calculate an integral or maximize a polynomial over an interval, but they may very well see mathematical models written in scientific papers and be expected to understand the stories that those models are trying to tell. Here are the consequences that this has on your actual questions:
    1) Design your exams (and your class in general) so that students have to think geometrically and dynamically about functions. If you can do this in such a way that technology won’t replace the thinking, then allow the technology. For instance, you don’t want to allow a graphing calculator on an exam if there are curve sketching problems, but it might make sense if the exam focuses on related rates or optimization problems.
    2) I can think of almost no particular technical skill which accurately measures how well a student understands calculus. Consider problems like: “Find the length of the longest ladder which can be rotated around a corner in a hallway” or “Find the rate at which the water level in a spherical bowl rises if water is poured in at a constant rate”. These problems have two compoents: first, you must think geometrically to write down a model and understand what to do with the model. Second, you must do some technical calculations to actually get an answer. In my view, a student who nails the first part but fumbles the second still understands calculus (though a student who executes both perfectly understands it even better). This can be the basis for a good philosophy on how to grade calculus exams.
    3) In the first few calculus classes, proofs (and even precise definitions) are unnecessary and possibly even detrimental. There were about two centuries between the invention of calculus and Cauchy’s precise definition of a limit, and this is because it took mathematicians a long time to pose problems for which basic geometric intuition was inadequate. Unless you seriously think that you are teaching your students about that sort of problem (and you probably shouldn’t in the first few semesters of calculus) it is disingenuous to teach lots of theory.

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    • June 24, 2014 at 11:25 am

      There is truth in your “cynical” explanation, but obviously calc is only one of the “gates”. For instance, engineering students need to pass intro physics courses, not only calc. In my experience, intro physics has fail rates that are as high or higher than intro calc. So calc is not the most selective gate in the process. More optimistically, the university wants solid calc courses so that students who pass calc and enroll in physics are more likely to have learned what they need to pass physics. There is nothing wrong with that.

      For pre-med, my experience is that intro chem is a more selective “gate” than intro calc (LABS!). At any rate, even if we (calc instructors) do not respect the use of calc as a gate, this is part of what “service” teaching is about. My opinion is that we should test our speculative pedagogy theories first on our own majors, before we roll the dice in our service courses.

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    • Deane
      June 24, 2014 at 4:04 pm

      I agree with the “cynical” view expressed here. Liberal arts students who are not hard science majors but are taking calculus are almost always taking it as a degree or premed requirement. However, no one really expects them to remember or use the knowledge in the future. This allows for a rather cynical approach to teaching and learning calculus, where professors go through the motions and curve the grades carefully to meet expectations and the students figure out how to do well enough with a minimum amount of effort. Actual understanding and learning are not needed at all. This is all evident to anyone who has ever taught such calculus courses.

      As for science and engineering students, there is some need for students to know at least some of the formulas and procedures for differentiation and integration. However, when I’ve discussed this with my engineering colleagues, they make it clear that it’s more important that the students know what a derivative or integral (and that means a *definite integral*) is and how it can be useful. Usually, the most important aspect is how a derivative measures the sensitivity of the output to changes to the input. Although we get students to chant “rate of change”, we simply don’t teach the students to really understand this point “in their gut”. So they are easily confused when they encounter this in engineering courses. This causes great frustration in my engineering colleagues. It’s easy enough for students to review rules of differentiation and integration, but if they didn’t learn what derivatives and integrals are in the first place, well, what was the point of having them take calculus in the first place?

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  19. Anselm
    June 24, 2014 at 2:36 pm

    I think this is a really difficult question to answer because there are two somewhat conflicting goals: (conceptual) understanding an technical skill.
    Only dealing with concept is like talking about carpentry without ever touching a saw or a hammer or even a piece of wood.
    Too much technical stuff will finally drown any conceptual understanding in a sea of weired epsilons and magical deltas.

    Lee Lady wrote some nice stuff about these problems.
    http://www.math.hawaii.edu/~lee/calculus/index.html
    His summary regarding calculus was
    “What is worthwhile for students to gain from a calculus course is the ability to read books that use the language of calculus and, at least to some extent, follow the derivations in those books. Unfortunately, being being proficient at the sort of chickenshit skills required to get a good grade in a calculus course is not a lot of help in this respect. ”
    He also wrote a more detailed text about “what is Math” –
    http://www.math.hawaii.edu/~lee/education/whatis.html
    were he talks a bit about “math as language”. I think this idea applies very much to every calculus course.

    As a physicist (even if not working as such any more) I agree very much with his attitude. The very concept of a Riemann integral as a method to set up an equation or to derive something (like e.g. the gravitational potential of a sphere) was extremely useful.
    And this kind of “application” is much more than a simple exercise – in the real world people really use this kind of approcah, even if it is not considered as exact by the mathematicians.
    The technical stuff is quite often of secondary importance only. OK its surely important to know some technicalities – partial integration, interchange of differentiation and integration etc – but who cares about the actual evaluation of a non trivial integral? That’s what books like Gradshteyn-Ryzhik or software as Maple, Mathematica, etc. are for.
    (And by the way – this is one reason why the riemann integral is so useful: the lebesgue integral is totally useless for this this kind of reasoning. )

    So in practice I think a mix of both will be needed.
    Clearly, some skills are really necessary and must be trained again and again (rote exercises can be important), but I think it is important to tell this to the students in advance.
    And all technical issues should be “motivated” by the concepts. But keep it simple, straightforward and relevant. Mathematicians like proofs and generalizations and abstractions, they might even find it interesting to prove the continuity af a function in three different ways. This is a fast lane if you want annoy someone else. The same applies to “elegant” proofs or demonstrations. Most mathematicians forget that elegance is a very individual idea and usually by no means interesting or helpful to a beginner. Keep it simple, even if it needs five lines more.
    Also take care of your examples: the typical ones, hmmh, most of the time they are either of interest to a pure mathematician only or just a dumb & completely artificial fake that dresses up as “applied” but will piss off nearly everybody. If an example is artificial, than say so and give a reason (e. g. to illustrate a point or a technical issue), but do not pretend that it is important or realistic.

    In this regard I think # 2 schould be a first focus and a foundation to built up other skills, in particular regarding #1.
    I feel quite uneasy regarding #3 – sounds too good to make any sense. Its pure theory only, just the typical way a mathematician thinks but which is unintelligible to the rest of the world.
    I think your example in #2 (the concave curve) is a particular bad example. Concavity (or convexity) is just a word! I know the idea and the geometrical picture (and that its derivative is monotonous), but I never remember when a function is calles concav (convex): when it is bend down – or upwards. So what, if i need it I look it up.
    Ths geometrical idea is surely important, but that’s it. The first part of the exact definition (line between to points) mirrors this idea und is mandatory to learn. But the second part – that this line is below (or above?) the function – is exactly this kind of thing that annoys people because its too easy to screw it up. Even worse: this is exactly this kind of definition that is so easy to learn for idiots who dont care about understanindg. The more interested and inelligent students become the loosers instead – really great teaching!

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  20. June 25, 2014 at 10:58 am

    Echoing some of the comments above, my view would be that unless you have some success with your goal #2, any attainment of #1, #3 (or any other objective for that matter) isn’t worth much. I’d also think that this applies no matter whether you’re ultimate goal is equipping students for life or next semester’s follow-on course.

    I’m an engineer by training, with a focus on computational math (some 25 years ago), but have never worked in that field after leaving university (got side tracked into so called ‘business’). Nevertheless I consider the math education I received at university (four semesters of undergrad math in Europe plus grad school in Europe and the US) the most important and relevant ‘life skill’ school and universities equipped me with. True: I haven’t (at least explicitly) tackled any differential equations and not even integrated a simple function in the past 25 years. But having an intuitive feel of such basic concepts like a gradient or what the integral for some underlying data series will look like is something I’ve benefited from almost daily over the past 25 years. Having ‘a feel’ for how a differential component makes a feedback loop (in economics, social sciences or your car’s ABS) behave ‘more edgy’ while an integral term has the same effect on these systems as a few beers have on most people is valuable in any field of work.

    I don’t think the chances of achieving your #2 are as bad as you make them appear. I’d think it’ll take a little bit of #1 to have the basic tools and terminology (in the same sense that you need to have mastered multiplication tables a few years earlier) and #3 may even pop out as a desirable side effect. I remember that one of the biggest ‘intuition builders’ in this context was to write and play with simple algorithms (on programmable calculators back then) for discrete differentiation and integration of functions and data series. It doesn’t require prior programming knowledge for students to implement these say in an iPython notebook as mentioned above.

    I wish you luck and success with your teaching. I do believe it’s important and relevant!

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  21. June 25, 2014 at 10:21 pm

    I’m an engineer. We took a lot of math and then most of our courses were taught using math as the primary language. A strange thing happened to me when I was exposed to so much math – it became part of my normal speaking and thinking patterns. I don’t think about a derivative as something I have to compute, rather it is an abstract concept that can be used in speech to rapidly explain an idea.

    I think this relates to the question above in the following ways. While math is not a perfect model for the world, it can be close, or at least give us upper and lower bounds. And unlike the world, given the right set of assumptions, the mathematical bounds are proven. The idea of modeling a system and knowing if you are too high or too low is an incredibly powerful tool. Similarly math, and especially calculus, gives us the ability to predict the future. It might only be the future of where a marble might hit the floor after rolling off a table. But again, this is an incredibly powerful concept.

    Calculus isn’t about computing areas under curves, it’s about communication and prediction and finding boundaries.

    What should your students know? They should know that these tools exist and how unbelievably powerful they are. And even if they never compute anything again, they should be able to find someone who can and then trust their answer. Also, if this will be their last math class, please throw in a day or two about reading stats and the utility/non-utility of things like p-values and how it all applies to research. If you can get your students to question the math in the things they read, you’ve accomplished more than most teachers.

    Lastly, there is a beauty to all this. To e and pi and how they relate to each other and to integers in unexpected ways. Engineering math tends not to be pretty, but when people might not be interested in the subject, aesthetics might draw them in.

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    • Deane
      June 25, 2014 at 10:27 pm

      I like the way you describe how you use derivatives as an engineer. It is in fact not so different from how mathematicians use derivatives, too. Although we do try to teach this to the students, our tests tend to focus on the computations, so students focus their energy accordingly.

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  22. Sawgill
    June 26, 2014 at 3:31 am

    Many great and detailed comments above. But as a rookie instructor you need to get to know the culture of your institution.(love your idealism/enthusiasm). So, as advised above, go with the curriculum designed by the department.

    Secondly, get to know your students. They will be evaluating you as you are evaluating them. Expectations of the student and school trumps all for a beginner. Tread lightly.

    As you develop your reputation you can modify your curriculum and experiment with your areas of concern as expressed in your questions. Teaching is great fun when you connect with students and help them appreciate the beauty of math.

    Your three concerns should be viewed as long term goals that you can decide for yourself through trial and error with your classes. You get to learn about yourself as an instructor as you guide your students through the subject. Enjoy the journey!

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