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This has certainly been tried before. See for example, H. Jerome Keisler. Elementary Calculus: An Infinitesimal Approach. On-line Edition. This has also been published in print by Dover. Kathleen Sullivan. The Teaching of Elementary Calculus Using the Nonstandard Analysis Approach. The American Mathematical Monthly Vol. 83, No. 5 (May, 1976), pp. 370-...


6

Another free option is Lehman, Leighton, and Meyer's Mathematics for Computer Science. It's written for an MIT introductory discrete math course that emphasizes training students in proof-writing.


6

Some things we're currently considering for a similar course at a large urban community college: Epp, Discrete Mathematics with Applications Artin, Algebra Gilbert, Elements of Modern Algebra Lay, Analysis with an Introduction to Proof Wade, An Introduction to Analysis And also some OER (open educational resources) options: Sundstrom, Mathematical ...


6

Daniel Solow, How To Read and Do Proofs Intended for abject beginners, unlike some of those other answers.


6

I know no better book for reasoning than Thinking Mathematically, by John Mason, Leone Burton and Kaye Stacey. It is superb in inspiring action and instilling methods of reasoning. Quoting from the introduction: Experience in working with students of all ages has convinced us that mathematical thinking can be improved by ● tackling questions conscientiously;...


5

Speaking as a former student, though an engineering one . . . It was hard enough learning to integrate tricky expressions and solve differential equations, without having to learn a new number system as well. And I don't remember ever needing the rigorous definition of a limit, standard or nonstandard. The most I needed even in complex calculus was an ...


5

Only one thing came to my mind reading your post: Concepts of Modern Mathematics written by Ian Stewart. It was an eyeopener for undergraduate me, and it is now for many of undergraduate students I teach. It is about mathematics as a whole, including a section on axioms and axiomatisation (or, as it is "Axiomatics"). To make you curious, here is how that ...


4

It is at once a philosophical, mathematical, foundational, and pedagogical issue. Rather than expand on the first sentence, I recommend Gregory Moore's book which discusses some of the first three parts (and perhaps the fourth also). As a practical matter, the Axiom of Choice is a challenge to formulate precisely and formally in a first-order language, ...


3

Much of mathematics works perfectly well with so-called "naive" set theory. That is, working with sets without worrying too much exactly what sets are. This isn't philosophically that different from calculus courses where you work with real numbers without worrying so much what they are (equivalence classes of Cauchy sequences of rational numbers), or for a ...


3

I am not completely sure I understand the distinction you are making between "axioms" and "assumptions", so this response may be missing the mark, but here is my interpretation of the question. Consider the following example of a theorem in, say, ring theory: Let $u$ be an invertible element in a ring $R$. Then... (something something something) Now ...


3

(1) The American Mathematical Society (AMS) maintains a Mathematics Subject Classification (MSC) for the mathematics literature. Here is a graph of the topics built by Andrius Kulikauskas: Image by AndriusKulikauskas. And here is a more readable detail (from roughly the center of the chart): (2) And here is another chart, more idiosyncratic, emphasizing ...


3

It's not really that relevant since the bulk of a normal calculus course (e.g. AP BC, Thomas Finney, Stewart) just does a small amount of epsilon-delta (so student is exposed to it) and then moves to "x+h". The bulk of the course is about learning derivatives, antiderivatives, methods of integration, classic applied problems, a bit if polar coordinates, bit ...


3

A book we often use in my department is Reading, Writing and Proving by Gorkin and Daepp. One feature of this book that distinguishes it from others mentioned in other answers is that it has chapters on $\mathbb{R}$, its completeness, the convergence of secuences in $\mathbb{R}$, and the Cantor-Schröder-Bernstein theorem. In other words, its focus is more ...


2

Short answer: (migrated from comments by request) Hung-Hsi Wu (Berkeley) has a home page full of links to writings of this ilk. Two specific examples that may fit the bill are ones I mentioned back in MESE 1857 (April 2014); in particular, links to a text on Pre-Algebra (pdf) and an Intro to School Algebra (pdf). Further comments: Some of H.H. Wu's ...


2

I think (counter to some others) that this is indeed a useful thing to introduce and talk about. My personal opinion is that it's something of a shame/outrage for anyone to go to college and not ever hear about what a "theorem" is. Currently I have a 2-slide talk on the axiomatic method on the first day of any of my college courses, including (especially) ...


2

I suggested back in MESE 11836 the following example from RL Wilder: With regard to the question here: Does anyone have any suggestions for a simple illustration, which ideally the students could try for themselves, of how starting from different axiom sets and following the same deductive rules leads to different consequences, that lends itself to a ...


2

I think (based on some experience with math-for-broader-purposes courses at honors freshman university level, for example) that it may be suboptimal to try to introduce "the axiomatic method" to people who have little substantive mathematical or scientific (or philosophical) experience. Namely, in my observation, people very often over-interpret the ...


2

The OP is looking at the Wikipedia article on Mathematics in general. In the first line it defines mathematics as: Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. You should be aware that this specific terminology is not used in mathematics practice ...


1

The best careful textbook you can find on elementary algebra that I'm aware of is the classic by Gelfand. It is beautifully written and completely careful-it's the book you wish you'd had in high school or grade school. I think you'll find it VERY helpful for both yourself and your students.


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Of course, I don't doubt that every mathematician knows the difference, but what if you were asked to explain it to math "newbies"? There are some ways to approach this for math newbies, one more 'to the bone' than the other. Which explanation you choose depends on the expected level of math proficiency (or tolerance). Axioms are atomic, indivisible ...


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