101
votes
Unique candidate that fails
The obvious example I immediately thought of is that, if the divergent geometric series $$1 + 2 + 4 + 8 + \dotsb = \sum_{k=0}^\infty 2^k$$ converged, it would converge to $-1$.
Proof: If the series ...
- 2,602
85
votes
Accepted
What's a replacement for "married couples" in combinatorics problems?
I've been using "pets" and "owners" (as in: possible pet-shelter adoptees) in recent years.
- 20.9k
84
votes
Accepted
Unique candidate that fails
For a positive real number $x$ consider
$$x^{x^{x^{\dots}}}$$
or formally (the limit of) the sequence $a_n= x^{a_{n-1}}$ (and $a_0= 1$).
Determine the value $x$ (if it exists) such that
$$x^{x^{x^{\...
quid♦
- 7,582
82
votes
What's a replacement for "married couples" in combinatorics problems?
In the stable marriage problem, you can introduce the problem as it is. But then you ask your students how things change if you assume there are not only heterosexual but also gay and lesbian people (...
- 1,143
81
votes
Accepted
Issues with "equals", where does this come from and how do I combat it?
A surprisingly large number of students don't know what the equals sign means. Their understanding of the symbol "=" is essentially operational, not relational — they think it means "the next step" or ...
- 4,765
66
votes
What's a replacement for "married couples" in combinatorics problems?
A few possibilities off the top of my head:
Students and chairs. How many ways are there for $n$ students to sit in $k$ chairs. The game of musical chairs might be fun to play around with. One can ...
- 6,460
63
votes
Why is it possible to teach real numbers before even rigorously defining them?
Expansion of mathematical knowledge does not unfold in Bourbaki progression. This is true at the level of both societal and individual knowledge. Just as the invention and significant applications of ...
- 7,575
58
votes
Unique candidate that fails
Here is a more elementary example.
Sometimes, extraneous solutions in algebra are of the type you describe.
For instance, to solve the equation
$$\sqrt {2x-1}=-x, \quad (x \in \mathbb R)$$
we ...
- 2,420
55
votes
How to explain Monty Hall problem when they just don't get it
For some reason, the 'extend it to 100 doors and eliminate 98' explanation doesn't make it any clearer for me.
Rather than talk about probabilities as fractions, I explain it this way:
"If you ...
- 1,214
52
votes
Accepted
How rigorous should high school calculus be?
Not very rigorously at all, but that doesn't (and shouldn't) mean just memorizing calculations. (I should add that I'm basing this on my experience teaching calculus to non-major college students, ...
- 11.3k
44
votes
Accepted
Lecturers "(intentional) mistakes" as a teaching tool
Yes, showing your students that you too can make mistakes, and that real mathematics is not a linear process, are both very good ideas.
However, this does not mean that you should come to the lecture ...
- 2,602
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. ...
- 10.2k
42
votes
Is it advisable to avoid teaching "multiplication as repeated addition"?
There's a lot in this brief question, and I would like to try to give a brief answer, so I'm going to pick and choose what I respond to (and others might choose different things). Here are the parts ...
- 7,385
42
votes
Accepted
Are there science-backed effective teaching strategies?
In terms of controlled experiments, then, yes. Note that most are opposite or orthogonal to virtually all pedagogical norms in math education.
Active recall. "Put away all your notes and ...
- 534
41
votes
Can mathematics be learned by ONLY solving problems?
Such an approach seems designed to force (or at least, strongly encourage) students to learn by pattern-matching from examples. This is one of three modes of student learning in mathematics described ...
- 4,765
38
votes
Common Core, threat or menace? Or maybe ok after all?
I'm really wondering if this is the end of the world or the beginning of an improvement in the way we teach math.
I'll answer this first, and then talk a little bit about the rest of the question. ...
- 7,385
37
votes
Unique candidate that fails
If there were a linear formula for $\int_0^1 f(x) dx$ in terms of $f(0)$ and $f(1)$, it would be $\int_0^1 f(x) dx = \frac{1}{2}(f(0) + f(1))$. Proof: Suppose $\int_0^1 f(x) dx = af(0)+bf(1)$. Taking $...
- 4,312
37
votes
What's a replacement for "married couples" in combinatorics problems?
The issue is not making problems about heterosexual married couples. The issues are:
Implicitly making the assumption that all married couples are heterosexual.
Making problems about heterosexual ...
- 707
37
votes
Why's math way more puzzling, abstruse than law and medicine?
Univariate calculus — e.g. integration (see also Reddit) — is when most students find math unfathomable and labyrinthine.
Well, not really. Actually most students never reach this level of math, and ...
- 371
34
votes
What's a replacement for "married couples" in combinatorics problems?
Try objects that often occur in pairs but are distinct from each other: forks and spoons (or forks and knives), left and right shoes, salt and pepper shakers, and so on (where each fork has an ...
- 10.2k
34
votes
How to get past the "mystique" of Maths
This is indeed a challenge, especially for adults. Three suggestions, none
of which is a panacea.
(1) Emphasize a growth mindset.
Make it clear to them that learning math is a skill accessible to ...
- 28k
33
votes
Unique candidate that fails
A very simple example (if you know calculus) is finding the local minimum of a downward-opening parabola, like $y = -x^2$. The unique candidate comes from setting the derivative equal to zero, but ...
- 499
32
votes
How to teach math to someone who is neither [really] willing nor able to understand it?
I have lots of experience tutoring students like that. The main problem is that they are convinced that they can't be good in math, so your task is more that of a psychologist than a math instructor. ...
32
votes
Accepted
How to teach someone that $-3>-4$?
Draw a number line and label all the integers.
Tell him that adding $x>0$ is moving $x$ units to the right and subtracting $x>0$ is moving $x$ units to the left.
Tell him that adding $0$ is ...
- 440
32
votes
Accepted
How can teachers warn students about common mistakes without causing the student to make the mistake?
This is a 100% subjective opinion, but it is based on teaching in various venues for close to 20 years (although none of that teaching was pure math). Also, my college calculus courses are close to ...
- 1,176
32
votes
Accepted
What should be memorized in math and what should be reference table?
The goal of memorization is to reduce cognitive load. If a student plans on using derivatives as part of a larger task, and doesn't have them memorized, they need to interrupt their thought process ...
- 1,939
29
votes
Accepted
How to explain Monty Hall problem when they just don't get it
Your explanation, by the way, is very elegant. As an experienced mathematician, I see immediately that it cuts right to the heart of the matter and admits no ambiguity. Unfortunately, this is ...
- 1,083
28
votes
How to teach someone that $-3>-4$?
Again and again he finds $-4$ greater than $-3$.
Ask him who is richer, he who has a smaller debt $($like $3$ rupees$)$, or he who has a bigger debt $($like $4$ rupees$)$, assuming both persons have ...
- 397
28
votes
Accepted
Why's math way more puzzling, abstruse than law and medicine?
The perceived difficulty of abstract math is due to two factors:
You learn math at school, but it is actually very different from what you do at university. In school you are applying rules to get ...
- 404
27
votes
Unique candidate that fails
The "Only Critical Point in Town" test
Suppose I have a nice function $f : \mathbb R^n \to \mathbb R$. Suppose it has only one critical point, and that is a local maximum. Then (of course) it is ...
- 6,211
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