63 votes

Why is it possible to teach real numbers before even rigorously defining them?

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 ...
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  • 7,585
62 votes

Why would you teach Calculus before teaching Real Analysis?

You may as well ask: Why teach elementary school children how to perform whole-number arithmetic without teaching them the Peano axioms first? Why teach high-school Algebra without starting with the ...
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  • 16.3k
44 votes

Why is it possible to teach real numbers before even rigorously defining them?

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. ...
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  • 10.2k
24 votes

Fourier Animation

Wikipedia always has some great animations, often by User:LucasVB or User:Cmglee. A few Fourier transform–related ones: Approximating a square wave with a Fourier transform (more of these for other ...
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  • 341
23 votes
Accepted

Should an undergraduate math program contain a course on Lebesgue integration?

I think the existing answers understate how much a standard American math major does not see the Lebesgue integral. I'm going to poke around at a variety of college websites to see how they cover this ...
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22 votes
Accepted

When did US mathematics programs start failing to prepare incoming students for books like "Baby" Rudin?

Opinion. There never was a generation of high school students in the US who could jump right into Rudin. There would be (and still is) a small portion of the top high school graduates who could. ...
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  • 6,211
21 votes

How would you explain what a PDE is to a very educated layman with no math background?

I would say something like this: "Often in complicated systems one needs to study multiple quantities, each of which varies at rates that depend on the other quantities and on how fast they are ...
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  • 16.3k
19 votes

Why would you teach Calculus before teaching Real Analysis?

Because your core assumption is bad and most students are not aspiring mathematicians Your opening statement - "Let's assume our students are actual aspiring mathematicians" - is almost certainly a ...
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19 votes

Complex numbers in high school

Although there is a tradition in the U.S. of nominal mention of complex numbers in the high school curriculum, in my observation it is invariably superficial, and complex numbers are not mentioned ...
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  • 13.5k
18 votes

Why would you teach Calculus before teaching Real Analysis?

In addition to other good answers, I do think that what "calculus" usually refers to is what people did with calculus prior to about 1830, as opposed to the foundations of it. Namely, they solved a ...
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  • 13.5k
17 votes
Accepted

Are the following topics usually in an introductory Complex Analysis class: Julia sets, Fatou sets, Mandelbrot set, etc?

No, these topics are not usually included in courses on complex analysis, for several reasons, which I will explain below. At the same time, it is easier to understand why relatively old textbooks did ...
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  • 13.5k
16 votes

Complex numbers in high school

In the United States, complex numbers are a standard part of the typical high school Algebra 2 course. Students (normally in grades 10 or 11, corresponding approximately to ages 15-17) learn to add, ...
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  • 16.3k
16 votes

Why is it possible to teach real numbers before even rigorously defining them?

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 ...
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  • 559
14 votes

How can I convince my brightest student of Cantor's theory?

Not every property is preserved by limits. Here is a more basic situation in which the same reasoning is used: For each natural number $n$, there are only finitely many natural numbers in the ...
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14 votes

Why is it possible to teach real numbers before even rigorously defining them?

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 ...
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  • 2,932
14 votes

Should an undergraduate math program contain a course on Lebesgue integration?

Is it standard for a math undergraduate program to have a course on Lebesgue integration? No (assuming that "have a course" means "require people to take such a course in order to get ...
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  • 141
13 votes

Are the following topics usually in an introductory Complex Analysis class: Julia sets, Fatou sets, Mandelbrot set, etc?

A brief stab at an answer before someone more knowledgeable comes along: My brief experience says that: no, fractal sets are not usually a topic in complex analysis (they're not in any of the 3 ...
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12 votes

Reasons for (not) distinguishing $f$ from $f(x)$

The reasons for not distinguishing $f$ from $f(x)$ are historical. For roughly 250 years, from the first use of the word "function" in mathematics (by Leibniz ~1680) until ~1930, the ...
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12 votes

Advanced Calculus vs. Analysis for a first proof-based course

This is somewhat of a hypothesis rather than a definitive answer, but one reason why vector calculus may no longer be the first proof based course at many colleges is because vector calculus involves ...
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11 votes

Evaluating the reception of (epsilon, delta) definitions

The apparent conflict between points of view expressed in the OP is illusory. There is no real conflict. The mathematics education researcher quoted in the OP is arguing that students find the ...
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  • 1,898
11 votes
Accepted

How can I motivate the formal definition of continuity?

Have a look at the paper written by Nunez et all: EMBODIED COGNITION AS GROUNDING FOR SITUATEDNESS AND CONTEXT IN MATHEMATICS EDUCATION. In essence, they argue that it is better to be causious if ...
11 votes
Accepted

Why should we study continuity?

Most functions that are studied by physicists and other scientists are continuous. However, more and more discontinuous functions are appearing in the various sciences. This is due to: Computers and ...
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  • 2,496
11 votes

Why do we study Cantor Set?

in beginning real analysis: to counter the naive notion that a "closed set" is a union of closed intervals, plus a few single points. In beginning Lebesgue measure: the easiest example of an ...
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  • 6,211
10 votes

Why would you teach Calculus before teaching Real Analysis?

In many respects this is what is done in European universities, where the degree is often a three year program. First year students reading maths take analysis. Of course they probably learned ...
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  • 7,585
10 votes

Why do we study ordinary differential equations?

ODEs are used in many models to determine how the state of this model is changing (regarding time or another variable). […] Am I missing another application […]? This may be somewhat pedantic, but I ...
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  • 2,366
9 votes

Introducing the Lebesgue integral before Riemann's

In the US, you'd be hard-pressed to find any student seeing the Lebesgue integral before having ever seen the Riemann integral. Every calculus book I've seen defines the integral as the Riemann ...
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  • 2,916
9 votes

Evaluating the reception of (epsilon, delta) definitions

The essence of your question is why do mathematicians and math educators come to different conclusions about teaching the epsilon-delta definition. May I submit that we have different primary ...
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9 votes

What is the motivation for characterizing second order linear PDEs as hyperbolic, elliptic, or parabolic?

Benjamin Dickman already linked to my answer on Math.SE, but I didn't talk too much about exactly the motivation of the characterisation for second order equations. So let me write a little bit about ...
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