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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 ...
user52817's user avatar
  • 11k
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. ...
JRN's user avatar
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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 ...
David E Speyer's user avatar
28 votes

A visualization for the quotient rule

Depending on how much algebra you allow, you could make the exact same rectangle picture but label the sides $g(x)$ and $q(x)$ with area $f(x)$. This geometrically enforces $g(x)q(x) = f(x)$, aka $q(...
Steven Gubkin's user avatar
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 ...
mweiss's user avatar
  • 17.4k
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 ...
Aeryk's user avatar
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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 ...
user22788's user avatar
  • 854
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 ...
paul garrett's user avatar
  • 14.7k
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 ...
Steven Gubkin's user avatar
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 ...
Steven Gubkin's user avatar
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 ...
paul garrett's user avatar
  • 14.7k
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 ...
Jorssen's user avatar
  • 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....
Justin Skycak's user avatar
17 votes

How can we motivate that Newton's method is useful?

I'm going to respond from an applied math (or maybe CS) perspective: Part of the problem is that the functions that you look at in high school and the standard calculus sequence are unrealistically ...
Adam's user avatar
  • 5,873
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, ...
mweiss's user avatar
  • 17.4k
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 ...
Timothy Chow's user avatar
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 ...
Adam Boocher's user avatar
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 ...
user19682's user avatar
  • 151
14 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 ...
Michael Bächtold's user avatar
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 ...
Steven Gubkin's user avatar
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 ...
Dirk's user avatar
  • 2,991
14 votes

How can we motivate that Newton's method is useful?

In video game animation, each second thousands of equations need to be solved with code. You can't tell a computer "look at a graph"! Your students need to think beyond the setting of ...
KCd's user avatar
  • 3,536
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 ...
Daniel R. Collins's user avatar
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 ...
Misha Lavrov's user avatar
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 ...
benblumsmith's user avatar
  • 1,936
12 votes

Motivating a definition of "gap" in a line just barely more advanced than the one used in the typical first-year calculus course

In my experience, the only way to successfully teach a more sophisticated technique is to present a problem where known simpler techniques fail. For instance, anyone who's taught algebra to kids will ...
Justin Skycak's user avatar
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 ...
Gerald Edgar's user avatar
  • 7,607
11 votes

What is important to keep in mind in grading proof-based courses?

My dealing with proofs, as a student and an undergrad TA myself, left me feeling that a lot of students have one common problem with proof-based courses. They have trouble figuring out exactly which ...
Glenn Willen's user avatar
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 ...
Wrzlprmft's user avatar
  • 2,568
10 votes

Why do we study ordinary differential equations?

Of course I agree that one motivation for studying ODEs is that they have applications. But it might be useful to also point out another fact that students do not always think about: ODEs are often ...
Gustav's user avatar
  • 201

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