I'm going to teach calculus for the first time to undergraduate students. I would like to know if there is some book about how to teach the concepts of calculus (e.g. limits, derivatives, etc.).

  • $\begingroup$ I don't have a book to recommend, but I would be happy to chat with you about it, and share resources. $\endgroup$
    – Sue VanHattum
    Jul 31 '15 at 21:30
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    $\begingroup$ Look at all the calculus textbooks you can get your hands on. Some of them will have ways of explaining things that you've never seen before that will be instructive. $\endgroup$ Jul 31 '15 at 23:35
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    $\begingroup$ I didn't mean buying any books. Many of your more experienced colleagues wherever you are will have a half dozen different calculus textbooks on their shelves and be quite happy to lend them to you. $\endgroup$ Aug 1 '15 at 1:16
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    $\begingroup$ I'm going to check out some of these suggestions myself. Thanks, all. user26832, you might want to make a profile with a name, so people will know who you are. You can email me at mathanthologyeditor @ gmail. $\endgroup$
    – Sue VanHattum
    Aug 1 '15 at 21:33
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    $\begingroup$ Joseph, what do you admire about his teaching? (What makes him a master teacher?) $\endgroup$
    – Sue VanHattum
    Aug 1 '15 at 23:55

There are so many Calc 1, 2, and Multi video lectures online now. Watch a few... take notes on what you think are some good traits of the person teaching. MIT's Calc course (super fast) might be helpful if you're teaching that style and speed.

Some resources

  1. This text, How to Teach Mathematics - S. Krantz, has many useful guides for the beginning person. I don't agree with all, but it's a great resource.
  2. Suzanne Kelton wrote a guide for the AMS. Its pdf is here: teaching Mathematics at the college level - pg. 31 goes into Calc stuff!
  3. Also, do not be ashamed to search for AP Calculus teachers' resources. It may be a "high school" course, but some of the teachers out there are amazing and offer a lot of guidance. Specific topic choice may vary but good teachers are that no matter what. Lin McCullin's website is an prime example. Watch those videos yourself from his site :) Ignore test-specific things. Pedagogy!
  4. If you are in the US, go to your MAA sectional meeting in the Fall!
  5. Please, please, please incorporate technology as a reasonable tool that you actually teach how to use in the right situations. This is vital and super valuable.

Mathematics should essentially be fun to do and to read and I think that generally students only love to do those things which interests them. So, one basic (and I think mandatory) component of your teaching should be about making the subject interesting for the students.

Presuming that by Calculus you mean Real Analysis I have the following suggestions for you.

  • First, let the students feel themselves the importance of a rigorous study of calculus. For this, I think that the introductory chapters of Tom M. Apostol's Calculus I and Terence Tao's Analysis I are two excellent references. Also I would like to recommend reading the last three sections of Part II of the book The Princeton Companion to Mathematics.

  • Second, when you will be giving assignments to your students try to be scientific in choosing the problems. To be precise, I don't recommend giving difficult (it's a relative term, though) problems to the beginners, nor do I recommend giving a set of problems which doesn't have any relation(s) among them. Though, it's difficult to express precisely what I mean by the above statements, I will nevertheless try to give an example which is intended to make the matter a bit more clear.

    Consider the following problems,

    1. Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $f(x)=0$ for all $x\in\mathbb{Q}$. Show that, $f(x)=0$ for all $x\in\mathbb{R}$.

    2. Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function. Let $T=\left\{\dfrac{m}{2^n}\mid m,n\in\mathbb{Z}\right\}$ such that we have $f(x)=0$ for all $x\in T$. Can we say that $f(x)=0$ for all $x\in\mathbb{R}$?

    3. Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function. Let $S=\left\{\dfrac{a}{2^n}\mid a, n\in\mathbb{Z}\right\}$ ($a$ is a constant) such that we have $f(x)=0$ for all $x\in S$. Can we say that $f(x)=0$ for all $x\in\mathbb{R}$?

    These problems have a very curious property. To solve the first problem, you need to show that $\mathbb{Q}$ is dense in $\mathbb{R}$. The second problem you need to show that a certain subset of $\mathbb{Q}$ is dense in $\mathbb{R}$. Finally, the third problem asks you to prove that a subset of the subset of Problem 2 is dense in $\mathbb{R}$. So you can see that the first problem is follows from the second one, whereas the second problem follows from the third one. Can you appreciate the order in which the exercises are arranged ? Can you intuitively grasp what I meant in the above lines ?

  • Third, make your students familiar with some basic ideas of Mathematical Logic at the very beginning. For example, the concept of conjunction, disjunction, negation etc.

Finally, I think that for teaching calculus L. V. Tarasov's book Calculus: Basic Concepts for High Schools is an invaluable guide.

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    $\begingroup$ Calculus = Real Analysis is a foolish assumption to make in America,sadly. These are all good recommendations for an honors course at a typical university,but not much else. $\endgroup$ Feb 25 '18 at 19:34
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    $\begingroup$ The question does not specify USA. $\endgroup$
    – Tommi
    Jan 25 '21 at 7:18

Numbers and Functions: Steps into Analysis, R.P. Burn, Cambridge University Press, ISBN 0 521 78836 6.

This book takes a constructivist approach to the subject. It takes the form of a sequence of problems, introducing the key ideas step by step.

In his preface Burn identifies common problems student often encounter: the apparently needlessly formal approach of topics that are familiar from school as well as the insistent use of arbitrary and contrived definitions. Accordingly the book builds on high school concepts and naturally introduces key definitions.

Personally it saved me in my first year of uni.

Below is a review from 'The Mathematical Gazette'



Concepts, Problems and Solutions in School Calculus : A Dialogue Approach by Lev Tarasov and Chandra Shekhar Kumar

Employing the extremely lively form of dialogue, almost all the subjects in the syllabus are discussed comprehensively, especially questions usually considered difficult to understand. The book presents a detailed analysis of common mistakes made by students embarking on learning and applying concepts in calculus. Students will find this to be an exceptionally clear and interesting textbook which treats of complicated problems from various viewpoints and contains a great many excellent illustrations promoting a deeper understanding of the ideas and concepts involved.

The whole book is presented as a relatively free-flowing dialogue between the AUTHOR and the READER. From one discussion to another the AUTHOR will lead the inquisitive and receptive READER to different notions, ideas, and theorems of calculus, emphasizing especially complicated or delicate aspects, stressing the inner logic of proofs, and attracting the reader's attention to special points. This form of presentation will help a reader of the book in learning new definitions such as those of derivative, antiderivative, definite integral, differential equation, etc. It will lead the reader to better understanding of such concepts as numerical sequence, limit of sequence, and function. Briefly, these discussions are intended to assist pupils entering a novel world of calculus. And if in the long run the reader of the book gets a feeling of the intrinsic beauty and integrity of higher mathematics or even is appealed to it, the author will consider his mission as successfully completed.

In this edition, the entire manuscript was typeset using the LaTeX document processing system. The whole book is revised and enlarged in order to improve the text, dialogue, concepts, problems, solutions, layout, to double check almost all of the mathematical rendering, to correct all known errors, to improve the original illustrations by redrawing them with TikZ. Thus the book now appears in a form that will remain useful for at least another generation.

Infinite Numerical Sequence
Limit of Sequence
Convergent Sequence
More on Function
Limit of Function
More on The Limit of Function
Differential Equations
More on Differential Equations
Miscellaneous Problems 
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    $\begingroup$ Can you talk a little bit about this book? Answers that are just links with no context are not very high quality because when the link inevitably goes dead there is no information remaining in the answer. $\endgroup$
    – Chris Cunningham
    Jan 22 '21 at 17:22
  • $\begingroup$ A version of this book is also available here gitlab.com/mirtitles/tarasov-calculus $\endgroup$ Jan 24 '21 at 21:06
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    $\begingroup$ Your username makes it appear that you are the publisher of this book. Please edit this answer to clarify that. $\endgroup$
    – user507
    Jan 25 '21 at 22:37

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