19 votes
Accepted

Replacement for the Pac-Man grid analogy

A possibility is to show your students Google's implementation of the game Snake. If you enter the term snake into Google's search engine, the there is a box at ...
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18 votes

Lack of intuition, retention while self studying

Perhaps you should seek texts that emphasize the high-level viewpoint that you are missing in the details of the more advanced texts. Three examples: (1) Bressoud, David M. A radical approach to ...
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15 votes
Accepted

Graphing functions from a finite field to itself

Just supplementing Benjamin Dickman's nice answer, here is $x \mapsto x^2 - x$ in $\mathbb{Z}_{18}$ in the same style: For example, the pentagon wheel reflects the fact that $$(5+3k)^2-(5+3k) = 9k^2 ...
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15 votes
Accepted

Why are proofs written in flowery language incomprehensible?

To answer the question in the title, I would say that one problem with no-symbols reasonings is that one need to use a lot of pronouns. Problem is, pronouns usually leave too much ambiguity. At some ...
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14 votes
Accepted

Pedagogical challenge: Homeomorphic vs. Homotopy equivalent vs. Homologous?

Homotopy equivalence v. Homeomorphism. I believe an accessible difference between homotopy equivalence and homeomorphism is that one preserves an intuitive (though hard-to-define) topological ...
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14 votes
Accepted

Wonder as motivation

I have given this issue a lot of thought over the years -- in fact a large portion of my dissertation is devoted to related issues. I think of this challenge in terms of a "mathematical sensibility" -...
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  • 16.3k
13 votes

Graphing functions from a finite field to itself

One of the approaches taken in some areas of mathematics (e.g., in arithmetic dynamics and considerations of preperiodic points, etc) is to create these graphs by drawing discrete points and then ...
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12 votes

Wonder as motivation

This answer is from my experience running a Maths Learning Centre. I help students learn and use maths, mostly when they are struggling, and I also hear their opinions of their lecturers and other ...
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12 votes

Lack of intuition, retention while self studying

You need to pick pedagogically appropriate texts. Not the Rudin ballbusters. Pick ones that have explanations and were written for students with occasional imperfections in their previous knowledge. ...
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  • 325
11 votes
Accepted

Non-Rigorous Use of Differentials

As @BenCrowell mentioned, the transfer principle proves that direct algebraic manipulation of infinitesimals in single variable calculus is allowed. But I stumbled upon this post in theshapeofmath....
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  • 684
11 votes
Accepted

How can I motivate the formal definition of continuity?

Have a look at the paper written by Nunez et all: EMBODIED COGNITION AS GROUNDING FOR SITUATEDNESS AND CONTEXT IN MATHEMATICS EDUCATION. In essence, they argue that it is better to be causious if ...
11 votes
Accepted

Why should we study continuity?

Most functions that are studied by physicists and other scientists are continuous. However, more and more discontinuous functions are appearing in the various sciences. This is due to: Computers and ...
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  • 2,496
10 votes

Communicating to students the meaning of extremely large numbers

My favorite video for this is powers of ten from 1977. Though we can get a little smaller today, I think it still does an excellent job with getting the scale of things starting from what we know. ...
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  • 2,584
9 votes

Any metaphors/intuitions for a limit of a sequence?

My feeling is that the $\epsilon$-$\delta$ formulation is already pretty close to what one should think about limits; that is, the language can be hard to grasp at first, but the idea is very ...
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9 votes

How to effectively internalize math?

This is an excellent question. Some good advice on this can be found in the writing of Bill Thurston, some of which I have posted in an answer to this question on Math Overflow. The opening of the ...
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  • 5,877
8 votes

Pedagogical challenge: Homeomorphic vs. Homotopy equivalent vs. Homologous?

I am not entirely sure on the best way to convey the difference between homotopy equivalence and isomorphic homology groups (or even isomorphic homotopy groups, though on CW-complexes I guess this isn'...
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  • 131
8 votes
Accepted

Why is continuity only defined on its domain?

"Is the sine function continuous at Mount Everest?" This or some modification thereof could be a playful start for a discussion about the fact, which you addressed more technically, that there must ...
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  • 7,582
8 votes

Why is continuity only defined on its domain?

A somewhat analogous point of view is the one of "continuous extensions". What looks weird in saying $\frac 1 x$ is a continuous function is, of course, what is happening around $0$. What is ...
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8 votes
Accepted

How to explain multiplying and dividing by fractions with real-world examples

We have two cookies. We divide them into pieces of 1/2 cookie each and end up with four pieces. Thus 2 divided by 1/2 equals 4. We have two cookies. We take 1/2 of the collection which is one cookie. ...
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  • 2,496
8 votes

Examples of informal explanations that cause misconceptions

A point (say, in $\mathbb{R}^n$) is a vector. Vectors and points are really no different. They are both $n$-tuples in $\mathbb{R}^n$. The difference between two points (in $\mathbb{R}^n$) is a vector,...
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8 votes

Lack of intuition, retention while self studying

Your question sounds to me like: I have a terrible memory I can't intuitively understand some of the new things i come across I've seen this plenty of times, but not in maths. I've seen this in ...
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  • 181
8 votes

Intuition or geometry for Partial Fractions

Introduction I wasn't taught the partial fractions decomposition (PFD) in calculus. We didn't cover it in high school, and when I went to college, they assumed we all knew it. Somehow it was when I ...
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  • 4,579
8 votes
Accepted

How do we explain to a little child that a date in 2020 and a date in 2021 are not necessarily a year apart?

I would use a number line. This is the most straight forward way to explain being "in between" integers while giving some intuition with a visual. It is possible that a school-age child ...
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  • 788
7 votes

How should normal subgroups be introduced?

I like Gowers' fake history of normal subgroups. This is also good. Especially if you can relate it to change-of-basis, and Weyl's famous quotation "The introduction of a coördinate system is ...
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7 votes

Wonder as motivation

This spoke to me, enough that I decided to answer a related question. I had the privilege to watch Ole Hald in action when he taught a second year service class (linear algebra/diff e.q). He had won ...
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7 votes

How can I motivate the formal definition of continuity?

The prototypical way for a function to not be continuous is that of a jump discontinuity. Imagine a jump discontinuity on the order of a few micrometers, like the width of a hair. If you are tracing ...
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  • 7,585
7 votes

How important is making definitions plausible?

(1). It's going to depend on level to a huge degree. In my experience, up into advanced undergraduate you could expect definitions to be named sensibly and an instructor to explain how a sensible name ...
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  • 1,881
7 votes

How to intuitively convince the students that a strip with two full twists is homeomorphic to the standard annulus?

First of all, let me echo all the comments -- the key point here is that these surfaces are homeomorphic, but this homeomorphism cannot be realized by an isotopy. This is an important distinction! It ...
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