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Expansion of mathematical knowledge does not unfold in Bourbaki progression. This is true at the level of both societal and individual knowledge. Just as the invention and significant applications of calculus predated formal definitions of the real numbers (Dedekind cuts, Cauchy sequences) and a formal definition of continuity, individuals must first make ...


43

It is possible to teach real numbers in elementary school before even rigorously defining them by using what H. Wu ("The Mis-Education of Mathematics Teachers," Notices of the AMS, vol. 58, no. 3, p. 376) calls the Fundamental Assumption of School Mathematics. In terms of the nitty-gritty of classroom instruction, real numbers are handled in K–12 by what ...


15

But as far as real numbers were concerned, they were taught very early without any clear definition was even given. Were they? I would say that usually no teacher really speaks of real numbers to students until the first calculus lesson. In almost all pre-calculus exercises, students only meet rational numbers, in the form of fractions or decimal ...


11

At German universities, one of the first lectures in mathematics is "Analysis 1" which is a kind of "rigorous calculus" and there one always proceeds more or less like this: We start with an axiomatic approach to the real numbers. In short: The real numbers are a complete, ordered, Archimedean field. In practice we first introduce the axioms of a field, ...


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You may be interested in Academically Adrift: Limited Learning on College Campuses, Arum and Roksa, 2011. Also summarized in http://www.newyorker.com/arts/critics/atlarge/2011/06/06/110606crat_atlarge_menand . They have a lot of discussion of something called the Collegiate Learning Assessment, which is a standardized test of critical thinking. They find ...


7

A few recommendations: Fernando Q. Gouvêa's $p$-adic numbers: An Introduction. This text gives a good introduction to the $p$-adic number system and the properties of the space of $p$-adic numbers, vector spaces over $\mathbb{Q}_p$, and the metrically completed algebraic closure of $\mathbb{Q}_p$. There is also some discussion of $p$-adic analysis near ...


7

Introductory mathematics is not done with formal definitions. This is because most people at the corresponding age (kids) don't know what the words "formal" or "definition" actually means. This only come to play much later. You start teaching kids mathematics by showing them how to count things up to ten using their fingers. Then, you teach addition with ...


6

The reasons for not distinguish $f$ from $f(x)$ are historical. For roughly 250 years, from the first use of "function" in mathematics (by Leibniz ~1680) until ~1930, the official (and only) definition of the word was: A quantity $y$ is called a function of $x$, if its value is uniquely determined as soon as $x$ assumes a value. Look into any book from that ...


4

I do not believe there are any good such strategies, especially if you are in a Ph.D. program. I think your question is an important one because the comprehensive exam system at some places conveys some need to desperately learn lots of mathematics quickly, and as a result, many students feel this sort of pressure. The late Ken Appel told me once that nobody ...


4

I believe that, unlike many other mathematical concepts (especially more abstract ones), even fairly young people have a basic understanding of distance, including fractions to any degree of units of measurement (e..g., inches, feet, miles, etc., in the imperial measurement system and/or centimetres, metres, kilometres, etc., in the metric measurement system)...


3

I can think of two books which might provide a roadmap, though you would have to flesh it out a lot depending on your prerequisites and the timeline you propose. Frank Morgan's Real Analysis has a minimum of prereqs and a maximum of topology and series. Now, it doesn't construct the real numbers (they are just infinite series of decimals, if I recall ...


3

Since real numbers include whole numbers, rational numbers, and irrational numbers (almost the entire landscape) I’m uncertain why you think teaching does not occur. Does it not begin very early with counting and learning numbers? It isn’t until later that these numbers are labeled as real, when needing to distinguish from imaginary as introduced in pre-...


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You are confusing definitions and models. A definition of real numbers is a set of axioms they obey. Different such sets exist, but they can be shown to be equivalent. A model of real numbers is a construction like Dedekinds, which is a set of sets of sets etc. that is carefully constructed so that they obey one such set of axioms. To work with real ...


2

I trudged through Pearson's site and found that Lay's Analysis with an Intro. to Proof has PowerPoint slides on their resources page. You'll have to get an account with them to download the files, but I think that the process is straightforward for anyone who is verifiably a faculty member. See https://www.pearson.com/us/higher-education/program/Lay-...


2

I think you should consider that for most calculus students, they will have a limited exposure to e-d and then move on to massive amounts of other calculations. And that will serve the vast amount of physicists, chemists, engineers, geologists, etc. just fine. Even from a TIME usage perspective, going deeper and more intuitive into e-d may be a waste of ...


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The idea that math-study imparts certain fundamental intellectual skills was taken for granted for most of the history of Western education, and reaches back at least to Plato, who (reportedly) refused to accept students who had not mastered geometry. "Let no one ignorant of geometry enter here" was supposedly inscribed over the entrance to his Academy. In ...


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There's a really nice book by Svetlana Katok called "p-adic analysis compared to Real."


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This may not move in the directions of your greatest interests, but it is one option. (Not my field so I cannot assess comparable books.) Eschrig, Helmut. Topology and Geometry in Physics. Vol. 822. Springer Science & Business Media, 2011. Springer link. Review by Jan Jerzy Sławianowski. Publisher summary: "Written as a set of largely self-...


1

It's certainly beneficial for students (even if thy don't like maths) to be convinced that some manipulations are only valid under some conditions. Maybe the following example is both significant and short enough for a rapid illustration: Derivation under the integral sign is ("formally"): $$ \frac{d}{dt} \int f(x,t) dx = \int \frac{\partial f(x,t)}{\...


1

Interesting integral. $$ f(w)=\frac{\cos(wx)}{1+w^2} $$ has poles at $w = \pm i$. The residue at $w=i$ is given as follows: $$ \text{Res}_{w=i}\left(\frac{\cos(wx)}{1+w^2}\right) = \text{Res}_{w=i}\left(\frac{\cos(wx)}{(w+i)(w-i)}\right) = \frac{\cos(ix)}{i+i} = \frac{\cosh(x)}{2i} $$ So, if I use the usual half-circle contour $C = [-R,R] \cup C_R^+$ where $...


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This answer may seem off-topic, since this is "Math Ed" SE. But a similar question can be asked about many scientific concepts. Based on observations of the world around us animals (including humans) have built up heuristic models of physics, biology and psychology. In order to be a useful creature, one must have some predictive models, even if they are ...


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"In mathematics, one can hardly study any mathematical concept unless it is clearly and rigorously defined." There is no reason, this has to be true. It is just asserted. Two examples have been given where significant content can be learned without strict axiomatic buildup of the subject: arithmetic and calculus. We could come up with others (set theory, ...


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The answers in the comments are probably the best (in particular @James S. Cook's advice to talk to a professor at a local university if you have such a connection) but here's at least a list to get you started: Given that you've got quite a range of interested there I would suggest going through the MAA's list of Math programs: https://www.maa.org/...


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Isn't the "functional equation road" the same thing as the "differential equations road"? You can take an axiomatic approach to defining sine and cosine. See Apostol's Calculus book, page 95. Many other authors have done a similar thing and the background isn't too steep. You simply declare that there exist two functions satisfying a few properties (which ...


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