# Questions tagged [intuition]

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### Literature on mathematical intutiion

Is there any literature on mathematical intuition? I would like to read the following, for example. How to train intuition? How to test intuition?
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### Overcoming Dyslexia and Building Intuition

I am 25 and have been studying mathematics on my own for several years, but I am still between the middle and high school levels. My main weakness is my dyslexia. I sometimes forget words or confuse ...
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### How can we explain intuitively the convergence and divergence of these two series?

It is known that $\displaystyle\sum_1^{\infty} \frac{1}{n^{1.000001}}$ converges while $\displaystyle\sum_{n\text{ is a prime number}}\frac{1}{n}$ diverges. Though we can logically prove these results,...
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### Are these explanations of variance and covariance intuitive?

When tutoring, I try to simplify concepts. I came up with these examples to explain the intention behind variance and covariance. Could you please help me find conceptual, pedagogical or mathematical ...
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### What are some ways that one can progress from stage 2 to stage 3 of the rigor stages that Terry Tao has described?

Terry Tao describes 3 stages of one's mathematics education on his web blog. 1: Pre-rigorous 2: Rigorous 3. Post-rigorous I know how one can progress from stage 1 to stage 2 (this simply can be done ...
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### "Rough subitising / estimation" for better intuition and ability to apply arithmetic

tl;dr: Why do so many students have poor intuition of numbers, and what can be done about it?  I've always been good with numbers. As a maths tutor, one of the things I notice is how poor the ...
230 views

### 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 talked with my friend on December 29 2020. Then I talked with him again on January 03, 2021. Q: What was the year when you last talked with your friend? A: 2021. Q: And what was the year the ...
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### Replacement for the Pac-Man grid analogy

To most people, a torus is a donut-like shape. Topologists like to describe the torus differently: you start with a square, and "identify opposite sides". We can imagine gluing together one ...
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### Intuition explanation about Lebesgue measure zero of the rational numbers [closed]

This is a question about the intuition of the rational number having measure zero. Let us consider followng proof: Let $I = [0,1]$ and $Q = \mathbb Q \cap I$ and let $\lambda$ be the Lebesgue measure. ...
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### Intuition or geometry for Partial Fractions

When teaching partial fractions, there's probably no way to escape the heavy algebra necessary for partial fractions, but I'm wondering how to introduce the idea in a way that is intuitive or ...
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### How to naturally encounter the properties of identity, commutativity, associativity, and distributivity (to define rings)?

(Cross posted at MSE: https://math.stackexchange.com/questions/3742948/how-did-we-isolate-the-properties-of-identity-commutativity-associativity-and) In elementary school, I remember learning about ...
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### How to intuitively understand how the trig ratios are calculated

I've asked a question on Math Stack Exchange, but it was suggested it might be a better idea to post it on this Educators instead. Here's the question link: https://math.stackexchange.com/questions/...
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### How do I assimilate mathematical concept?

Already knowing that the famous quotation "there is no royal road to mathematics", I believe that the most efficient and best way to learn mathematics is to make it intuitive to oneself, at least to ...
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### Lack of intuition, retention while self studying

I am a first year undergraduate student, currently in second semester. So basically I learnt most of the first year stuff in high school, so I have a lot of free time in this year (currently in ...
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### Should the limits of one system of elementary set theory be the limits of a student's mathematical world? [closed]

In teaching elementary set theory, suppose we refrain from emphasizing historical decisions that were made in theory construction. Is there a danger that students may see the mathematical language ...
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### Examples of informal explanations that cause misconceptions

Every good teacher knows that giving an informal explanation (that is, an explanation not based on strict definitions and maybe making use of metaphors, intuition, or everyday language) of a ...
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### How to intuitively convince the students that a strip with two full twists is homeomorphic to the standard annulus?

Intuitively speaking, one space is homemorphic to another if one can be deformed continuously to another without tearing and gluing. It is more or less easy to convince the students that a square is ...
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### How important is making definitions plausible?

During my studies I observed that while most lecturers try to explain theorems and their proofs, only very few of them try to explain definitions. However, in my opinion, definitions are the base of ...
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### Why are proofs written in flowery language incomprehensible?

Let's take an example in Wu-Ki Tung, Group theory in physics: Theorem 3.4: Irreducible representations of any abelian group must be of dimension one. Proof: Let $U(G)$ be an irreducible ...
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### How to explain multiplying and dividing by fractions with real-world examples

I'm looking for a good way to explain how multiplication and division by fractions applies in the real-world the mechanics are receiving reasonably straight forward. How can $2$ divided by $1/2$ ...
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### Intuition: 5 regular polyhedra, 6 regular 4-polytopes, and then 3 regular d-polytopes

I have struggled to offer an intuitive explanation (to U.S. college students) why the number of regular polytopes in dimension $d$ is: $d=2$, number: $\infty$. $d=3$, number: $5$, the five Platonic ...
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### Why do the stages of rigorousness have specific timestamps?

This is a reduced quote from There’s more to mathematics than rigour and proofs of Terrence Tao (emphasis mine): The “pre-rigorous” stage, in which mathematics is taught in an informal, ...
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### Determining the first digit of the Quotient using hand long division efficiently?

For instance in the following problem: _____ 48)4368 To determine an initial 9 for the first number in the Quotient, you ...
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### Benefits of knowing theory [closed]

I've got an issue: from time to time I have to teach some math to people who either avoided it, or got through by only knowing a few working algorithms. The only thing that unites all those people: ...
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### Resources on 3D transforms, vectors, coordinate systems

Background: I'm helping engineers use software to create 3D geometry in a programmatic way (similar to OpenSCAD). The functions they need to call have inputs which are low-level geometry concepts: 3D ...
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### Communicating to students the meaning of extremely large numbers

I'm planning on showing my students a "deep zoom" video of the Mandelbrot set. The video is about 15 minutes long and, at the end, shows an image that is zoomed by a factor of $10^{220}$. I'd like ...
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### Why should we study continuity?

This question is related to How can I motivate the formal definition of continuity? Imagine a student asks the question why it is worth it to study continuity. What is a good response to this question?...
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### How to effectively internalize math?

Terrence Tao commented of internalizing [here: https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/ ] "It is true that some mathematicians can be vastly more ...
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### Wonder as motivation

Like all mathematicians, I have a deep appreciation of the beauty of mathematics. Many theorems I find amazing even after I fully understand their proofs. (Example: Euler's formula, $V-E+F=2-2g$. That ...