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I'm considering creating a series of classes that explore deeper ideas in calculus without overemphasizing the various computational tricks used in integration and differentiation.

My vision is essentially akin to a "Survey of Calculus" kind of instruction. Nonetheless, i intend to cover a significant part of the basic differential equations and the conventional calculus sequence.

Are there any examples of these classes that I could look up?

Tommi
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Bilbo
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  • Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on [meta], or in [chat]. Comments continuing discussion may be removed. – Xander Henderson Dec 06 '23 at 14:38

4 Answers4

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As you undertake the journey, you should read about the Calculus Reform movement of the 1990s. A unifying idea of the efforts was to increase conceptual understanding by using symbolic computational systems (Maple, Mathematica, TI-89, etc.) to do calculations, thereby freeing up time and mental bandwidth for conceptual understanding.

An excellent summary of these efforts is found in the article by Deborah Hughes Hallett titled What Have We Learned from Calculus Reform? The Road to Conceptual Understanding. This is chapter 5 in the book A Fresh Start for Collegiate Mathematics book published by the MAA. The article is also available at this University of Arizona website.

In many ways, Calculus Reform took us three steps forward, and then we took one or two steps back. Hughes Hallett provides an excellent summary of the positive changes that persist to this day, including increased conceptual understanding. But as always with reform, there was overreach. Deprecated calculation skills come with a cost. "Calculus without calculation" borders on being an oxymoron.

Another outcome of Calculus Reform was research mathematicians teaching undergraduate mathematics coming to realize how important it is to focus not just on the usual themes of content, textbooks, and curriculum. But we learned how important it is to focus on how people learn mathematics. I think the work of Ed Dubinsky is emblematic of this lesson. He headed a Calculus Reform project at Purdue titled $C^4L$: Calculus, Concepts, Computers and Cooperative Learning. Dubinsky was a constructivist in the sense of Piaget, and gave focus to how learning mathematics involves construction of concepts and knowledge, not only socially (hence the cooperative learning part of $C^4L$) but also in their individual minds. Dubinsky referred to a process he called APOS, which stands for "Action, Process, Object, Schema." See Actions, Processes, Objects, Schemas (APOS) in Mathematics Education

Curriculum innovation alone will not guarantee deeper conceptual understanding. We must also focus on pedagogy and the messiness of how knowledge is constructed.

user52817
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    As a concrete example I suggest @Bilbo to look into the Harvard Consortium Calculus book by Hughes-Hallett et. al. It focuses on problems that differ significantly from standard calculus book problems based on "calculation tricks", but it doesn't require symbolic computation software. – Michael Bächtold Nov 29 '23 at 08:40
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    As another concrete example for @Bilbo see Calculus in Context. The Five College Calculus Project by James Callahan, et al. (1995, 2008), which incidentally I discussed briefly in this answer. – Dave L Renfro Nov 29 '23 at 13:33
  • @MichaelBächtold, I had some experiences with Harvard calculus book (although it was the version customized for Michigan State). I don't know if it is still true, but I remembered the book's missing many important things. E.g., I think the correct definition of exponent, logarithm, and even $e$ are not included. The explanation of to limits and continuity are also very shallow, as I recall. – Bilbo Nov 30 '23 at 16:37
  • @DaveLRenfro, Very helpful! Thank you! – Bilbo Nov 30 '23 at 16:37
  • @user52817, Thanks! I certainly underestimated how complicated this issue is. More lessons from history to be learned. – Bilbo Nov 30 '23 at 16:42
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    I mean, Calculus without calculation starts looking like Analysis. – Yakk Nov 30 '23 at 21:42
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    @Yakk This is interesting! I see what you mean. I was thinking of it in the other direction. "Calculus without calculation" means weakness and lack of confidence with symbolic manipulation like using the chain rule. Going through life being able to differentiate nothing more than polynomials. It's the analog of being so dependent on a calculator that you need to use one to figure out $50\times 0.1$. – user52817 Nov 30 '23 at 22:10
  • @Bilbo Those issues seem to be orthogonal to your question. – Michael Bächtold Dec 01 '23 at 06:33
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Take a look at this: https://calculus.ucmerced.edu/wwh-calculus-project/project-mission-and-goals

You may find other examples that the California Learning Lab is funding at: calearninglab.org

Bamdad Samii
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You may be interested in the approach taken in the Mathematics for Life Sciences course series (LS 30a and LS 30b) at UCLA, meant to be taken in place of Calculus I/II. It's designed to efficiently teach the calculus and linear algebra needed to understand dynamical systems and to emphasize numerical and qualitative methods over analytic methods. You can find more information on this website, made to support the textbook written for the course.

I don't think this is the "survey of calculus" that you're looking for. Nevertheless, one of the tenets of the course design is to eschew symbolic tricks, if that's what you mean by "the various computational tricks used in integration and differentiation." For example, rather than spending time on integration techniques, separation of variables, etc., the course has students implement Euler's method in Python to integrate systems of nonlinear differential equations in the first semester.

Justin Hancock
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I don't have a class recommendation, but I think the historical development of calculus is a useful tool for an educator with this goal, and I can recommend Bressoud's recent book Calculus Reordered: A History of the Big Ideas for inspiration and guidance.

Marc Shelikoff
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