86
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.
- 21.6k
83
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 ...
- 4,823
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 (...
- 1,177
67
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 ...
- 8,550
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 ...
- 7,118
53
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, ...
- 11.4k
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. ...
- 10.6k
44
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 ...
- 564
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 ...
- 4,823
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 ...
- 737
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 ...
- 10.6k
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 ...
- 28.7k
34
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
33
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 ...
- 3,713
28
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 ...
- 7,006
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
How to convince students of the integral identity $\int_0^af(x)dx=\int_0^af(a-x)dx$?
Problem of sloppy notation
The notation is sloppy. Your students are justifiably confused. We've just gotten used to it.
In order to untangle this, we need the notion of free variables and bound ...
- 421
27
votes
What methods successfully identify and eliminate severe math anxiety?
In regards to "math anxiety", the 1990 paper by Ray Hembree helped me out a lot. It's a large meta-study of about 150 papers and a total of 25,000 students. Summary of the results, as I wrote on my ...
- 21.6k
27
votes
How should a student's inefficient calculation be pointed out?
Foremost: It depends on what the lead-in lesson/topic/direction was. If this was the essential point being exercised, then I would interrupt ASAP and refocus them on the lesson/direction that just ...
- 21.6k
26
votes
What's a replacement for "married couples" in combinatorics problems?
When I taught a class about the stable marriage problem last week, I replaced "men" and "women" with "medical students" and "hospitals": the classical instance in which the Gale-Shapley algorithm is ...
- 671
26
votes
Accepted
Finding the Balance in a Math Question (Teaching)
I think it is helpful to let students know that you are looking for their thinking while problem-solving, and not just answers. Then you can ask questions like:
Find all of the points on a circle of ...
- 798
25
votes
Justifications for: Why learn mathematics?
Most of what you learn in school isn't directly useful. When I was in primary school I was taught the difference between warm and cold-blooded animals. I've never used that information; should I not ...
23
votes
Accepted
Should high school teachers say “real numbers” before teaching complex numbers?
Short Answer
You should not avoid use of the term real numbers. This is a term-of-art in mathematics, and it is important for students to learn the correct jargon. However, this technical term ...
- 7,118
22
votes
What should be memorized in math and what should be reference table?
The reasons are manifold.
One is cognitive load (@TomKern’s answer) and the distraction of looking things up sometimes causes the solution you’ve almost constructed to fall to pieces. This is ...
- 4,699
20
votes
Accepted
How should a student's inefficient calculation be pointed out?
I like your second option the best:
...wait for them to finish the calculation, or even finish the entire exercise, before I casually tell them there was a more natural way to work out that part?
...
- 8,558
20
votes
Accepted
Inability to work with an arbitrary mathematical object
I'll focus on question 2 from a perspective of "maybe the right thing to think about is: what happens in the students' minds while they read this question?"
When you say "Suppose $A⊆R$ is nonempty ...
- 20.2k
19
votes
Unique candidate that fails
Theorem: The largest positive integer is $1$.
Proof: If $n$ is a positive integer and $n \not= 1$ then $n^2 > n$, so there is an integer larger than $n$. Thus the largest integer has to be $1$. ...
- 2,888
19
votes
Why do students only see the last term of a sum abbreviated with an ellipsis?
I suspect that the issue is not so much the ellipsis per se but a problem with notation in general, and in particular with the correct use of the equals sign. At the risk of repeating what I wrote in ...
- 17k
19
votes
What books are like Knuth's Surreal Numbers?
Flatland by Edwin Abbott Abbott
It's a story about a 2-dimensional being's encounter with a three-dimensional being. (Well, there's a class(?) allegory at the beginning, but you can skip that.)
- 3,711
Only top scored, non community-wiki answers of a minimum length are eligible