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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
  • 8,288
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

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
  • 3,536
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
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
  • 11k
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

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
  • 5,629
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
  • 5,778
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
8 votes

Literature on mathematical intutiion

How to train intuition? I'll answer your first question with a couple quotes from the literature. From K. Anders Ericsson, one of the most influential researchers in the field of human expertise and ...
Justin Skycak's user avatar
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

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

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
6 votes
Accepted

Intuition explanation about Lebesgue measure zero of the rational numbers

Let me build on the idea of Steven Gubkin in his comments. One way to visualize this scenario is to use Ford circles. The standard picture is to plot a circle tangent to the $x$-axis at $\frac{p}{q}$ ...
user52817's user avatar
  • 11k
6 votes

What are some ways that one can progress from stage 2 to stage 3 of the rigor stages that Terry Tao has described?

This is not as much to answer the original question (to which the answer is just that you develop any skill by trying to practice it and evaluating the results) but to tell what my understanding of a &...
fedja's user avatar
  • 3,939
6 votes

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

For me, the intuition just comes from the integral test (which is itself intuitive since a series is just a Riemann sum of rectangles with unit width). The $n$th prime is asymptotically $n \ln n$ (...
Justin Skycak's user avatar
6 votes

Overcoming Dyslexia and Building Intuition

This might be an unpopular answer but I'm going to try to be honest and real with you. Math gets hard and unintuitive for everyone at some point, and that point is different for everyone -- for some ...
Justin Skycak's user avatar
5 votes

For calculus students, what should be the intuition or motivation behind series?

I start on the other end of the spectrum. I start off by Noting that computers can't compute sine, cosine, or anything like it. Computers can basically only compute polynomials. Sine and cosine ...
johnnyb's user avatar
  • 1,249
5 votes

Why are proofs written in flowery language incomprehensible?

Honestly, both seem pretty bad and don't prove the statement in an obvious way. For example, there is confusion betwen irreps and the representations of individual elements. You can make the $G$-...
Adam's user avatar
  • 5,883
5 votes

Examples of informal explanations that cause misconceptions

For a much lower-level topic, consider explaining to beginning algebra students why "like terms" can be combined. On a few occasions, I have resorted to reasoning with students that adding algebraic ...
Nick C's user avatar
  • 9,729
5 votes

Examples of informal explanations that cause misconceptions

I shall post my humble and incomplete list of bad explanations I've given or heard over the years: A function is continuous if you can draw its graph without lifting your pencil As you mentioned ...
Vassilis Markos's user avatar
5 votes

How do I assimilate mathematical concept?

I agree that insight sometimes takes time, but don't dismiss the benefit of exercises. Sometimes, the insight monster will lurk in the background in your subconscious. If you do exercises, it may ...
guest's user avatar
  • 77
5 votes

How to naturally encounter the properties of identity, commutativity, associativity, and distributivity (to define rings)?

This isn't exactly what you are looking for, but it's pretty close: the book "Surreal Numbers" by Donald Knuth builds up the theory of surreal numbers by telling a fictional story of two ...
Chris Cunningham's user avatar

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