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91 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.
Daniel R. Collins's user avatar
84 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 (...
gnasher729's user avatar
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68 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 ...
Xander Henderson's user avatar
  • 8,225
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
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56 votes

Which cognitive psychology findings are solid, that I can use to help my students?

There's a highly upvoted answer here claiming that practically no cognitive psychology findings hold up in replication. I don't think that's true at all. Sure, many findings don't hold up, but also, ...
Justin Skycak's user avatar
54 votes
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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, ...
Henry Towsner's user avatar
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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44 votes
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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 ...
BravoMath's user avatar
  • 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 ...
Daniel Hast's user avatar
  • 4,893
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 ...
Pere's user avatar
  • 1,087
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 ...
user19242's user avatar
  • 371
35 votes

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 ...
TomKern's user avatar
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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 ...
JRN's user avatar
  • 10.8k
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 ...
Joseph O'Rourke's user avatar
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 ...
Syntax Junkie's user avatar
32 votes
Accepted

Whether to tell students how difficult (you think) a problem is

In a setting where students aren't working from a book with these labels on questions, is it worthwhile for the instructor to indicate to students where the work they are asking them to do falls on a ...
Justin Skycak's user avatar
29 votes

Students can't seem to grasp the intent of tangent lines and getting general trends of derivatives from graphs

Sadly, these students seem to think of math as a bunch of rules. You have done great work, and probably gotten them a little closer. But they are resisting the reasoning that can't easily be put into ...
Sue VanHattum's user avatar
  • 20.8k
28 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 ...
Daniel R. Collins's user avatar
28 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 ...
Misha Lavrov's user avatar
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 ...
Jens Schauder's user avatar
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 ...
Daniel R. Collins's user avatar
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 ...
Carser's user avatar
  • 798
25 votes

Which cognitive psychology findings are solid, that I can use to help my students?

Having pursued the same question as the OP with same motivation for over a decade (as an undergraduate classroom instructor, and a facilitator of a faculty interest group in STEM education strategies ...
Daniel R. Collins's user avatar
24 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 ...
Xander Henderson's user avatar
  • 8,225
23 votes
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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 ...
Chris Cunningham's user avatar
22 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? ...
Nick C's user avatar
  • 9,639
22 votes
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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 ...
user1815's user avatar
  • 5,595
21 votes
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What can I do when advanced undergraduate and/or early graduate STEM students cannot perform correct math manipulations?

I am really at a loss as to why this is happening and what should be the correct remedy, as it seems that it is not easy to address this late into their study. Also there seems to be no improvement ...
Justin Skycak's user avatar
20 votes

What books are like Knuth's Surreal Numbers?

Gödel, Escher, Bach: an Eternal Golden Braid by Douglas Hofstadter is a book about Mathematics, Art, Music, Computer Science and Artificial Intelligence. It alternates story chapters and descriptive ...
Stig Hemmer's user avatar
20 votes
Accepted

Can we avoid confusion over using "let" as a quantifier?

Many logicians that I have spoken to have concurred with my assessment that this is an issue of the misleading use of "let". Many teachers use this word in two very different and incompatible ways. ...
user21820's user avatar
  • 2,649

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