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1500 questions
30
votes
12 answers
How can I validate the existence of percentages above 100?
I once encountered a math educator whose personal pet peeve was the "give 110%" meme. He drilled into his students that 100% was the literal maximum. Percent came from "per cent" and 100 per 100 is literally all of them. You can't eat 110% of the…
rprospero
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30
votes
5 answers
How should normal subgroups be introduced?
One standard definition of a normal subgroup is
A subgroup $N \subset G$ is normal iff the set of left cosets $\{gN\}$ and right cosets $\{Ng\}$ coincide.
There's a class of similar definitions (every left coset is also a right coset,…
user37
30
votes
10 answers
Getting students to actually read definitions
I'm teaching a second year "Introduction to Theoretical Computer Science" course, and one of the skills/habits I've tried to instill in the students is to actually read definitions, take them seriously and go back to consult them when appropriate.…
Arno
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30
votes
7 answers
Early vs. late transcendentals
There seem to be two approaches to calculus education:
Early transcendentals: introduce polynomials, rational functions, exponentials, logarithms, and trigonometric functions at the beginning of the course and use them as examples when developing…
Paul Siegel
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30
votes
8 answers
Is there a good age/level to start learning mathematical proofs?
I know from my experience I learnt proofs myself way before I learnt them in school and I felt it gave me a far better understanding of math.
What is a good point to start learning proofs? what are the pros and cons? does the whole curriculum have…
Keith Nicholas
- 401
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- 5
30
votes
7 answers
Good definition for introducing real numbers?
In the first lectures about calculus/analysis, you should introduce real numbers. Let's assume students know that rational numbers are.
What are the advantages or disadvantages in the different "definitions" of the real numbers?
I'm particulary…
Markus Klein
- 9,438
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30
votes
6 answers
How to motivate an adolescent who has fallen behind in conceptual development?
I tutor a 16 year old girl. As far as I can tell, she has average talent and interest in math.
However, her knowledge of math is that of a 10 year old or even below. She knows the basic operations on positive integers below 100, but that's mostly…
BKE
- 1,282
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30
votes
12 answers
How to give my students a straightedge instead of a ruler
I'm having a "challenge" in my geometry classes getting students to avoid using rulers as measuring devices in constructions. As natural as that usage is, they're only supposed to use them to connect points to form line segments and extend line…
Matthew Daly
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30
votes
4 answers
Open-Source Math Textbooks
It seems to me that an open-source model could work quite well for textbooks, with issues being raised by the users of the book and different forks of the project being created for different preferences, such as early transcendentals in calculus.
A…
Chris Cunningham
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30
votes
13 answers
What do you say to students who want to apply Banach-Tarski theorem in practice?
Once when I was talking about Banach-Traski theorem (paradox) I said:
OK! This is Banach-Tarski's theorem which is against our intuition but provable from our intuitive axioms! It says you can decompose a small diamond to finite parts and then…
user230
30
votes
8 answers
Good motivation for the introduction of Lebesgue integral?
When students take a course on real analysis, they have likely learned about Riemann integrals.
What is a good motivation why they have to learn a new way to integrate?
A student don't want to hear to something like "You will now learn the new…
Markus Klein
- 9,438
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30
votes
11 answers
How should a student's inefficient calculation be pointed out?
One time I watched a student solve the equation $0 = (x-2)^2-9$ for $x$ like this.
$$\begin{align*}
0 &= (x-2)^2-9
\\0 &= (x^2-4x+4)-9
\\0 &= x^2-4x-5
\\0 &= (x+1)(x-5)
\\\\x &= -1 \text{ or } x = 5
\end{align*}$$
It would…
Mike Pierce
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30
votes
9 answers
Can mathematics be learned by ONLY solving problems?
Here is the concept:
Student is presented with a problem. He/she may not even understand what is being asked, or may attempt.
Students reads a solution to the problem. In it there may be explanations about concepts - so there is a "content"…
Amir Hardoof
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29
votes
5 answers
Wonder as motivation
Like all mathematicians, I have a deep appreciation of the
beauty of mathematics. Many theorems I find amazing even after
I fully understand their proofs. (Example: Euler's formula,
$V-E+F=2-2g$. That the genus $g$ can be determined by…
Joseph O'Rourke
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29
votes
6 answers
How to encourage women to study mathematics?
What are different ways we can get women to study mathematics? In my own experience, the higher the math class, the less women in the class. Most women tend to go on the math education track and do not have to take the more advanced classes. These…
Felix Y.
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