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 ...
Xander Henderson's user avatar
  • 8,185
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 ...
Joseph O'Rourke's user avatar
16 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 ...
Benoît Kloeckner's user avatar
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. ...
guest's user avatar
  • 325
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 ...
Rory Daulton's user avatar
  • 2,582
11 votes

How can we explain intuitively the convergence and divergence of these two series?

Look at a simpler example first: $(1.000000000001)^n$ compared to $0.9999999999^n$. Do they accept that the first sequence tends to $\infty$ and the second to $0$ even though it would take quite a ...
KCd's user avatar
  • 3,446
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. ...
Chris C's user avatar
  • 2,594
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 ...
Jon Bannon's user avatar
  • 6,173
8 votes
Accepted

Determining the first digit of the Quotient using hand long division efficiently?

The awkwardness of "guessing" in the division algorithm is an artifact of the base-ten representation of numbers. If you represent in binary, then your only possible "guess" is 1. In binary, your ...
user52817's user avatar
  • 10.8k
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 ...
Nicola Ciccoli's user avatar
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 ...
quid's user avatar
  • 7,692
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. ...
Rory Daulton's user avatar
  • 2,582
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,...
Joseph O'Rourke's user avatar
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 ...
Jhal's user avatar
  • 181
8 votes

How to intuitively understand how the trig ratios are calculated

If you are teaching this at an introductory level, then the algorithm that calculators use today is going to go far over their heads. (It might go over MY head!) The story of how we developed ...
Matthew Daly's user avatar
  • 5,619
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 ...
user1815's user avatar
  • 5,545
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 ...
Carser's user avatar
  • 798
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 ...
user52817's user avatar
  • 10.8k
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 ...
Nate Bade's user avatar
  • 1,941
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 ...
David E Speyer's user avatar
7 votes

Replacement for the Pac-Man grid analogy

One thing that I like about the Pacman analogy is that, if you draw Pacman's eye (or Ms. Pacman's bow, which is my usual choice), then Pacman is not mirror symmetric. This means you can talk about ...
David E Speyer's user avatar
6 votes
Accepted

Why is continuity defined as a local property?

Firstly, I don't think it entirely makes sense to ask why a property is defined to be local rather than global. Being local or global emerges from the definition itself. Continuity asks about whether ...
Jessica B's user avatar
  • 5,822
6 votes

Communicating to students the meaning of extremely large numbers

One suggestion would be to take an approach similar to that used for describing the $52!\approx 8.063*10^{67}$ ways to arrange a standard deck of playing cards as outlined in Scott Czepiel's Blog and ...
erfink's user avatar
  • 1,129
6 votes

Communicating to students the meaning of extremely large numbers

I was looking for a link for the book "The Lore of Large Numbers" that I read years ago and I thought might give you some ideas that I came to this new book "Really Big Numbers" that seems fascinating ...
Amir Asghari's user avatar
  • 4,428
6 votes

Why should we study continuity?

In our intuition, everything is continuous. But if we think long enough about real life, we discover that actually lots of things aren't. Computers generally replace continuous things with ...
Joel's user avatar
  • 161
6 votes

Lack of intuition, retention while self studying

This is probably not a real answer, but its too long for a comment. Last year I spent some time working on Stein and Shakarchi's Volume 2 with several advanced math majors. I actually think that text ...
James S. Cook's user avatar

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