I'm wondering if anyone knows of calculus books or other work towards teaching calculus in a less mathematically rigorous way.

I'm thinking mostly of American-style college level calculus courses introducing the derivative and integral. For whatever reason, it seems to be general practice that these courses should have a significantly higher level of mathematical rigor than the courses that proceed them. The textbooks I've seen are concerned with proofs and about rigorously justifying facts. (They usually don't prove everything, but they're typically concerned about this - for instance, noting when a claim has to wait for a more advanced course for a proof - in a way that pre-calculus textbooks aren't.)

For an example of what I mean, there's been some discussion (including here) about whether to use the conventional epsilon-delta approach to limits, or an approach using some kind of rigorous infinitesimals. But has anyone tried going all the way to working with informal infinitesimals?

But that's just an example - I'm interested in other ways people might have tried dialing down the rigor in calculus.

  • 1
    $\begingroup$ Not formed enough to be an answer, but might look at how the military teaches enlisted men. Especially nuke sailors. I seem to remember it being mostly graphical and pretty simple. But giving a lot of key insight. In general, military has a lot of experience in mass training, with limited time. Often a simpler view gives more intuition. I think of material properties better with algebra than with calculus and better with x-y than with vector calc. And tensors...eek. $\endgroup$ – guest Apr 3 '18 at 19:42
  • 3
    $\begingroup$ Do you consider the Hughes-Hallett textbook to be too rigorous? As I understand it, one if its design principles was specifically to place intuition above proof. $\endgroup$ – mweiss Apr 3 '18 at 21:00
  • 1
    $\begingroup$ HT: Probably. I do think the current curriculum that many (not all) people take gives a bite of the apple for actual calculus within pre-calc. So there is at least multiple exposures. Same thing with chemistry, physics, bio. Most people have an easier high school year before taking a year of college version (or AP). And for those who never get to the college version, that easier version gives them a lot of utility as it is basically the same stuff without some of the bells and whistles. So I wouldn't think of college calculus in isolation. $\endgroup$ – guest Apr 3 '18 at 21:27
  • 1
    $\begingroup$ Bell's A primer of Infinitesimal Analysis is not about informal infinitesimals, nor is it suited for directly teaching a first calculus course from it, but it tries to convey an intuition for (Lawvere style) infinitesimals, with many examples and applications. So maybe it can serve as a source of inspiration. $\endgroup$ – Michael Bächtold Apr 4 '18 at 8:28
  • 1
    $\begingroup$ Regarding Hughes-Hallett, see my comments in this 12 March 2012 post in the ap-calculus discussion group. And now having looked at it, I suddenly remembered a book that absolutely should be considered, but for some reason I had completely forgotten about when I wrote my answer below. $\endgroup$ – Dave L Renfro Apr 4 '18 at 18:49

Below are some books I know about. FYI, I read approximately the first half of a public library copy of Ferrar's book back in January or February of 1974 while enrolled in a 9th grade Algebra 1 course (however, in my independent reading/study of math I had completed the Algebra 2 material by this time), and I found it to be something I could understand and follow, which was definitely NOT the case with some college calculus texts I had seen at public libraries. I mention this because the book seems to be rarely mentioned on the internet, and thus will likely be overlooked by people looking for something like this. The other books, except for Brewster's book, are either well known or (because they're old enough to be in public domain) are in the process of becoming well known. Sawyer's book is not really a textbook, but I'm including it because I suspect it introduced a lot of "mathy types" in the 1960s and 1970s to calculus, as it was in the "New Mathematical Library" series of books that high school libraries back then often had several volumes from.

Calculus Made Easy by Silvanus P. Thompson (1910, 1914)

Calculus: An Intuitive and Physical Approach by Morris Kline (1967, 1977)

Calculus for Beginners by W. L. Ferrar (1967)

What is Calculus About? by W. W. Sawyer (1962).

Differential Calculus for Beginners by Alexander Knox (1884)

Elementary Illustrations of the Differential and Integral Calculus by Augustus De Morgan (1832, 1899)

Commonsense of the Calculus by George William Brewster (1923)


I can't believe I overlooked Calculus in Context below. I don't remember how I learned about it, but I've had a copy of the 1995 hardback edition since 1995 or 1996. I've used some of the material in classes (for example, Section 12.1: Stirling's Formula on pp. 690-699 and Bessel’s Equation on pp. 564-573 in classes here during the late 1990s), and I often used to flip through this book for ideas about how to introduce certain topics in class (such as Section 3.6: The Chain Rule on pp. 138-147). There’s a bit of computer program stuff in the book that’s probably out of date now, and there's some material on discrete dynamical systems topics that was avant garde in the late 1980s to late 1990s, but I was never much interested in this material and my interest in the book was for its exposition and its inclusion of certain topics that you rarely see in standard calculus texts. I’m almost certain I’ve written about this book on the internet, but for some reason I can’t find anything nontrivial now.

Even better, this book is now (legally) freely available on the internet, although if you find you really like it, then you'll probably want to get a print copy. The book (1995 edition; I haven't seen a print copy of the 2008 edition) opens nicely and the font size is medium to large (I'd guess either 12 point or 13 point), which makes it very easy to use.

Calculus in Context. The Five College Calculus Project by James Callahan, et al. (1995, 2008)

  • $\begingroup$ The single review on Ferrar at Amazon is very well done. Shares preface and table of contents and reasoned criticism/praise. Almost Renfroian. amazon.com/dp/0198531338/… $\endgroup$ – guest Apr 3 '18 at 21:37

i would add the FLCT (funny little calculus text) by robert ghrist. lovely done, focusing on ideas and concepts and still rigorous.


I'm currently working on a Calculus book and solution guide that is much less formal, but no less deep. The working title is "Calculus from the Ground Up". In fact, in some ways, it goes deeper than other calculus books, but without straining to do so. It works using infinitesimals informally. It goes onto a more formal explanation of hyperreals and hyperreal arithmetic, but only towards the end, after the student has gotten used to the idea.

If anyone is interested, feel free to email me at jonathan@bplearning.net and I'll send you a PDF of the current version. I would love any feedback on it!


Your Answer

