Anyone tasked with moving Calculus I from a textbook to students sooner or later asks her/himself that question and I am curious as to what people are thinking of when they do. But to make it more explicit, I would like to make a few assumptions.

  1. Let us assume that your students are the kind to be found in Two-Year Colleges----which, these days, include about 50% of the student population in the US, so that:
    • Their "conceptual maturity" is practically inexistent. For example, they decide what calculation to embark on solely on the basis of template examples. While they can compute to an extent, they have no idea on how to verify their result and always want to be told whether the result of their computation is right or wrong.
    • While they can usually read the template examples, they cannot read a text to extract conceptual information.
    • They are totally unused to the idea that the very text of a problem will often give them an idea on how to attack it.
    • They are totally unaware that, in mathematics---as in law, one has to make a case for whatever one asserts, be willing to present one's case to "peers", and be ready to defend one's case.
  2. Your students are not necessarily declared pre-engineering students, but can also be students who, based on what their experience in calculus 1 will have been, will decide on what to major in or, perhaps, students who have been persuaded to give it a try by a former instructor. In any case, a major issue is that many of the students will likely not continue into Calculus 2.
  3. You are not bound to use any particular textbook and in fact let us assume that you have facilities to develop the materials you need to help you reach your goal. Also, let us assume that you have the blessing of your department.

At this point, the question of what you would want your students to learn might be, for example and certainly not to limit things, whether you primarily want:

  • To remedy some of the shortcomings listed above using as means some or all of the standard topics,
  • To improve / consolidate / extend whatever computational skills they already have using as means some or all of the standard topics,
  • To let the students acquire a "coherent view" of the calculus even at the cost of not covering all the "standard" topics,
  • To let the students acquire the mathematical techniques they will be expected to use in following courses,
  • etc
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    $\begingroup$ It is crucially important to understand that the lack of "conceptual maturity" should never be viewed as a detriment to the student but instead a space that can be filled by new knowledge and extensive connections and a quintessential potential. Give them time, be patient, stress the fundamentals and the concepts and not necessarily the formulas (I give all students formula sheets for all assessments, mathematics is not about memorization, save that for biology), allow lots of chance for self-assessment, reflection, and discussion and I think you will find that they will respond positively :) $\endgroup$ – celeriko Apr 9 '15 at 3:07
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    $\begingroup$ I am very familiar with the students you describe in #1. A question that I am grappling with in my own practice is, why are these students taking Calculus I in the first place? Ideally, I would want these students in a course designed to remedy/redirect the habits described in your bullets under #1, perhaps with mathematical content that does not require so much (unreliable) prior knowledge. Unfortunately, it seems that "memorizing differentiation procedures" links more explicitly to graduation requirements than does "learning to think mathematically," so, what to do... $\endgroup$ – Xi Yu Apr 9 '15 at 3:07
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    $\begingroup$ This is primarily a rant. I'm not seeing a clear question here. $\endgroup$ – Ben Crowell Apr 9 '15 at 5:08
  • $\begingroup$ @BenCrowell The question is as stated in the title: When Dr Crowell stands in front of his Calculus 1 students, exactly what does he want them to learn? $\endgroup$ – schremmer Apr 9 '15 at 11:02
  • $\begingroup$ @BenCrowel (as you happen to be first on the list of those who put the question on hold.) I do not agree that the question is unclear. Please read it again. The motivation was that having taught calculus for nearly 50 years. I have long been aware that "what we want" can range over a large number of goals. If, in spite of the examples I gave of what these could be, I failed to make that clear, please remove the question as I certainly did not mean to offend anyone. (The help center did not help. ) $\endgroup$ – schremmer Apr 9 '15 at 21:49

I feel what needs to be said does not fit into a comment space.

You seem to have a handle on the situation when you give a list of alternatives. I have the feeling you want us to say which one of these, or two or three, are the ones to prioritize, given the scenario conditions.

The real answer is that it depends on the student. All of them need to learn how to learn, whether it be to help themselves think, or find what they need, or actually do calculus computations, or understand the material well enough to explain it to their children. Some of them will have the maturity or ability to know what specifically to get out of the course; others will be wondering and floundering at the (usually new) experience of learning something outside of high school.

You, as teacher, have to deal with all of this, usually by delegation, sometimes by ignoring, and hopefully, by understanding and encouraging. One of the universal gifts you can give the students is self-assessment, another that you can try to give is curiosity. Go for trying to spark the imagination of some, and see how many catch fire (figuratively).

If I were in your shoes, I would want them to get the high-level perspective, and learn how to use their text to solve problems they are likely to encounter, or at least need to learn as a foundation for problems they are likely to encounter. I would not have them memorize the definition of derivative so much as have them imagine what a rate of change means to them well enough to explain to someone else and have that someone else get it. I would have them appreciate that given a problem where finite differences are natural to use, it is ok to shift domains and phrase an analogous problem in terms of smooth functions, and have an idea how to interpret an answer in that domain back to the problem domain which involves discrete quantities. I would be thrilled if they could solve problems using only what they remembered; I would be satisfied if, six months after the class, they could take the textbook and use it quickly to solve problems given on an exam.

Gerhard "It's Different For Online Classes" Paseman, 2015.04.08

  • $\begingroup$ I was not asking for advice about what I should do. I specifically asked "what you would want your students to learn might be, for example and certainly not to limit things, whether you primarily want ..." And since, as you said, it "depends on the student", I made a "few assumptions". $\endgroup$ – schremmer Apr 10 '15 at 21:04
  • $\begingroup$ Indeed. Even with the assumptions, it still depends on the student. You have my priorities based on the assumptions you gave, plus a few more unstated assumptions. I still think there is more room for opinion as the question is presented than stackexchange questions normally allow. Hopefully you find some benefit in my saying what I would prioritize. Gerhard "Assumes Prioritized Assumptions Are Priority" Paseman, 2015.04.10 $\endgroup$ – Gerhard Paseman Apr 11 '15 at 5:24
  • $\begingroup$ As mentioned above, I am quitting this exchange but, if only out of courtesy, I owe you an answer. First, let me say that I absolutely and totally agree with you (and Michael Shermer's The Moral Arc) that dealing exclusively with the "set" and ignoring the "elements" is immoral and, in fact, counter-productive. Second, I do not think that such a discussion is possible here given the atmosphere set by the vigilante but, should you wish, you can reach me at schremmer.alain at the website mentioned in my user profile. $\endgroup$ – schremmer Apr 12 '15 at 13:38

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