68
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 ...
- 10.3k
61
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 ...
- 17.3k
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. ...
- 10.8k
30
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 ...
- 5,243
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. ...
- 7,369
22
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 ...
- 17.3k
22
votes
Why do most Analysis textbooks overlook, and fail to teach delta-epsilon proofs, using the K-ε principle?
My two cents on this is that it's related to two key issues that I stress in my undergraduate analysis course:
Key #1: Existence proofs have two parts: First you exhibit/construct the object, then you ...
- 7,500
20
votes
Accepted
Do undergraduates struggle with δ-ε definitions because they lack a habit of careful use of their native language?
I wouldn't say so. By studying linguistics on a deep level, this person learned to parse complicated multi-part statements and extract precise meaning from them. This skill--which people in general ...
- 854
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 ...
- 760
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 ...
- 14.2k
19
votes
Why are hand waving arguments made in textbooks of undergraduate analysis and how should readers deal with them?
A proof is meant to convince a reader of the truth of some statement. When a mathematician is communicating an argument to another mathematician, you only include the level of rigor that you need so ...
- 24.4k
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 ...
- 14.2k
18
votes
Accepted
If I take Modern Analysis next year, will I be prepared to teach multivariable/vector calculus?
It is impossible to say whether you will be ready to teach multivariable calculus after taking this course. Certainly many of the topics you list will employ some multivariable calculus, but using ...
- 24.4k
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 ...
- 14.2k
17
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 ...
- 569
17
votes
Accepted
What is important to keep in mind in grading proof-based courses?
To grade faster and provide more targeted feedback, it could be helpful to review some lists of mistakes that students commonly make during proofs. For example:
Taxonomy of bad proofs
https://www....
- 6,262
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, ...
- 17.3k
16
votes
If I take Modern Analysis next year, will I be prepared to teach multivariable/vector calculus?
Assuming you already have a good understanding of the material in a standard course in multivariable calculus and vector calculus, I would recommend that you take a class in electricity and magnetism ...
- 493
16
votes
Why do most Analysis textbooks overlook, and fail to teach delta-epsilon proofs, using the K-ε principle?
I can't comment since I lack reputation, so I'll post an answer here. I largely agree with the other answers, and can say that every time I've taught Introductory analysis after a day or so a student ...
- 171
15
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 ...
- 151
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 ...
- 24.4k
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 ...
- 2,992
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 ...
- 24.4k
13
votes
Why do most Analysis textbooks overlook, and fail to teach delta-epsilon proofs, using the K-ε principle?
The goal of teaching $\delta$-$\epsilon$ proofs is not necessarily to make students as efficient as possible in writing $\delta$-$\epsilon$ proofs. The goal is to instill understanding, of which the ...
- 811
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 ...
- 1,961
12
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 ...
- 1,926
11
votes
Accepted
What is the intuition behind the limit superior?
I have two intuitions to offer:
A sequence $(a_n)$ may have cluster points (these are points such that every neighborhood contains infinitely many elements of the sequence, or, more precisely, for ...
- 2,992
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 ...
- 2,582
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 ...
- 7,369
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