22 votes
Accepted

Fun set theory for kids

Hilbert's Hotel is a nice thought experiment for explaining results about cardinality of infinite sets and the aleph numbers. I have also used plastic bags to explain the difference between $\...
  • 1,607
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 ...
14 votes

What is a number?

I think the level of the student is very important to this question. If the student has never had an abstract math course (like my students), then the lack of a definition of "number" is a great way ...
  • 19.1k
13 votes

Why is set theory not taught at the outset of math education?

Not a complete answer (could there be one?), but too long for a comment. why is it [Set Theory] not being taught at the very outset of math education? It has been tried, most likely still is in ...
  • 271
11 votes

Fun set theory for kids

I found the formula connecting the union and intersection of two sets useful at school. $$n(A\cup B) + n(A\cap B) = n(A) + n(B)$$ Say you wish to find how many numbers from 1- 1000 inclusive are ...
  • 333
11 votes
Accepted

Is it considered a mistake to use different correct notation for writing intervals?

The goal of notation is to clearly and precisely communicate an idea. The notation $$ (a,b) = \{ x : b < x < a \} $$ fails to clearly communicate an idea, as this notation is unusual, and ...
10 votes

Examples of cultural limitations on math education

A quintessential example of a cultural clash over math education in US schools, I would say, is the entire Math Wars, an ongoing struggle over standards and pedagogy in K-12 schools. It is a struggle ...
  • 7,385
9 votes

What caused the (relatively) recent popularity of set theory?

My very incomplete understanding: I know from Henry Pollak (first-hand, but mentioned briefly here and in an article below) that one of the driving forces behind SMSG (with whom New Math is often ...
9 votes
Accepted

What caused the (relatively) recent popularity of set theory?

Whenever "the new math" comes up, I must point out that there was no one "new math" curriculum and much of that post WWII exploration into math curriculum and pedagogy had different aims and ...
  • 7,385
9 votes

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

The problem is not one of reconciling analysis and set theory at all; it's one of understanding analysis properly (and how things go sideways when dealing with the infinite). One of the first and ...
9 votes

Fun set theory for kids

Questions about infinity are one way to go. e.g. 'Are there more natural numbers or even natural numbers?' Intuition says there are more natural numbers ($\mathbb{N}$) than even natural numbers ($2\...
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8 votes

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

Here is another attempt. Consider the function $f:\mathbb{N} \to \mathbb{R}$ defined by $f(n) = \textrm{ The number of elements in the set $\{n,n+1,n+2,...2n\}$}$. In other words $f(n) = n+1$. ...
8 votes

Cognitive traps in very early set theory

I think your specific situation with the exam question is similar to a pre-Calc class that spends a good amount of time on solving trig equations, but then asks on the exam for the student to write ...
  • 6,514
8 votes

How to motivate equivalence classes

As a simple illustration of the most basic case, $\mathbb{Z}/2\mathbb{Z}$, I sometimes push the light switch of the classroom about a dozen times in a row, too fast for them to count. Then I ask: &...
8 votes

Cartesian product set

They are isomorphic. While $A\times B \neq B \times A$ for any arbitrary distinct sets $A$ and $B$, by defining the map $\phi:A\times B \to B \times A$ by $\phi(a,b)=(b,a)$, we can show that ...
  • 2,584
7 votes

Examples of cultural limitations on math education

Perhaps this historical example fits the bill: Khovanova, T., & Radul, A. (2012). Killer problems. American Mathematical Monthly, 119(10), 815-823. The piece was published earlier on the arXiv ...
6 votes

Cognitive traps in very early set theory

I think the issue here is really more one of problem solving skill or abstraction ability. You could see the same thing in various problems in computer science or in college algebra where some ...
  • 81
6 votes

Determining sets to show sufficiency of a condition?

I would not recommend to teach this method since there are some downsides. Take $A(x) \iff x \text{ is divisible by } 2$ $B(x) \iff x \text{ is divisible by } 42$ Is $A(x) \implies B(x)$ or $B(x) \...
6 votes

Determining sets to show sufficiency of a condition?

First, although you talk a bunch about cardinality, I don't see how that makes sense, so I'm going to assume you mean that you have them determine if the set corresponding to p is a subset of the set ...
6 votes

Cartesian product set

I like Chris C's answer; I will offer another point of view. Perhaps the difficulty lies in the example that has been chosen: a deck of cards isn't naturally a cartesian product for exactly the ...
6 votes

What is a number?

Whatever teachers may think about the nature of numbers, the foundations of "arithmetic" and the nature and concept of number in particular are very subtle. For a recent and sophisticated look at the ...
6 votes
Accepted

Category mistakes regarding symbols and their impact on math (mis) understanding. ( Object symbol/ sentence symbol confusion)

I like the idea here, but I agree that it misleads students, and might have the opposite of intended effect. Why not hand out a paragraph to the students, and ask them to critique it. Say that the ...
6 votes

Mnemonics to correlate the definition of "asymmetric relation" and "antisymmetric relation" with the terms

First, let's note that the terms as used by Rosen are standard definitions, as we can see on Wikipedia (here and here), as well as other resource sites. There was some question about this in the ...
6 votes

Is it considered a mistake to use different correct notation for writing intervals?

In theory you could choose to define a notation such as $$(a, b) = \{x ; b < x < a\},$$ but why would you do it? It has exactly the same power as the usual notation, and now the reader either ...
  • 4,746
5 votes

Why is set theory not taught at the outset of math education?

Because many worthwhile topics in math don't require set theory to learn them and apply them. You can get a lot of good stuff done with arithmetic, algebra, trig, calc, etc. that is uninformed by set ...
  • 295
5 votes
Accepted

When self teaching, should I learn set theory before continuing ap calculus?

Short answer: No. If you really want to know a little bit about sets, the first section of Book of Proof or Applied Discrete Structures would more than suffice for the purposes of most collegiate ...
  • 5,750
5 votes

Fun set theory for kids

A great way to get folks engaged is to bring up a contentious question. Here's one: What's the 0th power of a whole number? In particular, what's the 0th power of 0? Like Lorenzo Najt mentioned, ...
4 votes

How to motivate equivalence classes

To be honest, I wasn't sure to write this answer or not, since it was hard to choose what to write about! Believe or not, my whole PhD thesis entitled "Equivalence" was to provide a ground for ...
  • 4,314
4 votes

What is a number?

I think this question is important. I'd love to see an actual answer to it and cannot upvote it enough. I don't have an answer, but would like to share some intuitions/speculation. I think the ...
  • 5,857

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