Skip to main content

Questions tagged [intuition]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
5 votes
2 answers
443 views

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 ...
antho's user avatar
  • 151
6 votes
4 answers
2k views

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,...
Zuriel's user avatar
  • 4,275
2 votes
2 answers
197 views

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 ...
Kasimir Vilodnov's user avatar
6 votes
1 answer
312 views

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 ...
Mike's user avatar
  • 181
3 votes
1 answer
208 views

"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 ...
Adam Rubinson's user avatar
2 votes
2 answers
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 ...
Alecos Papadopoulos's user avatar
14 votes
8 answers
3k views

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 ...
Misha Lavrov's user avatar
1 vote
1 answer
277 views

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. ...
flawr's user avatar
  • 409
5 votes
2 answers
802 views

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 ...
user avatar
3 votes
4 answers
246 views

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 ...
D.R's user avatar
  • 287
5 votes
5 answers
899 views

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/...
user523384's user avatar
1 vote
2 answers
154 views

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 ...
glimpser's user avatar
  • 119
18 votes
7 answers
6k views

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 ...
katana_0's user avatar
  • 349
-2 votes
2 answers
136 views

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 ...
ELM's user avatar
  • 352
8 votes
4 answers
359 views

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 ...
Jorssen's user avatar
  • 569
5 votes
2 answers
284 views

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 ...
Zuriel's user avatar
  • 4,275
10 votes
3 answers
356 views

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 ...
Photon's user avatar
  • 602
3 votes
2 answers
620 views

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 ...
Ooker's user avatar
  • 183
8 votes
6 answers
4k views

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 $1/2$ ...
user1605665's user avatar
5 votes
0 answers
172 views

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 ...
Joseph O'Rourke's user avatar
3 votes
1 answer
613 views

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, ...
Ooker's user avatar
  • 183
5 votes
3 answers
1k views

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 ...
leeand00's user avatar
  • 173
5 votes
1 answer
259 views

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: ...
user2057368's user avatar
3 votes
1 answer
103 views

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 ...
kotoole's user avatar
  • 131
10 votes
5 answers
331 views

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 ...
mweiss's user avatar
  • 17.4k
13 votes
6 answers
5k views

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?...
Stephan Kulla's user avatar
15 votes
2 answers
2k views

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 ...
user avatar
4 votes
2 answers
1k views

Why is continuity only defined on its domain?

As mentioned in this question students sometimes struggle with the fact that continuity is only defined at points of the function's domain. For example the function $f:\mathbb R\setminus\{0\} \to \...
Stephan Kulla's user avatar
4 votes
1 answer
1k views

Why is continuity defined as a local property?

The formal definition of continuity is a local property (the definition of continuity at a point is a property of the germ of the function at this point). Why is it a good decision to make the ...
Stephan Kulla's user avatar
20 votes
7 answers
2k views

How can I motivate the formal definition of continuity?

In order to teach continuity of real valued functions $f:D\to\mathbb R$ one may start with the (in some sense wrong) intuition $f$ is continuous when its graph can be drawn without lifting the pen. ...
Stephan Kulla's user avatar
7 votes
1 answer
228 views

The role of visualization and intuition in graduate and postgraduate math and developing it

[I am not an mathematics educator; but because the process of learning is educating yourself, I'm posting it here] In Visual Complex Analysis's preface, the author gives an analogy with pseudo-deaf ...
user avatar
7 votes
1 answer
284 views

How can I improve my concept map?

I've decided to create a concept map of a chapter I covered in a textbook, it's about basic set notation. What I want is suggestions on how to improve the presentation of the map. It seems quite ...
seeker's user avatar
  • 915
5 votes
0 answers
213 views

Teaching an abstract algebra class involving modules, best way to introduce operations on modules?

At the behest of a comment on Mathoverflow, I am posting this here. I am about to teach a few classes on modules and their operations, namely the following: direct product, direct sum, and finitely ...
user5613's user avatar
12 votes
6 answers
882 views

Graphing functions from a finite field to itself

I have been teaching a ring theory course this semester, focusing on modular arithmetic and quotient rings of polynomials over fields. Several students have asked me how one could graph functions ...
Brian Rushton's user avatar
8 votes
3 answers
238 views

How important is building up intuition for a theorem before trying to prove it?

For example, consider trying to prove that: If $A$ is a set and $F \subset P(A)$, then the relation $R := \{(a, b) \in A \times A $ such that for every $X \subset A - \{a, b\}$, if $X \cup \{a\} ...
Anton's user avatar
  • 143
29 votes
5 answers
1k views

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 ...
Joseph O'Rourke's user avatar
22 votes
8 answers
957 views

Non-Rigorous Use of Differentials

Consider the following example of working "directly" with differentials. One way to approach the problem of determining the arc length of the graph of a single-variable function is to imagine the arc ...
Austin Mohr's user avatar
7 votes
2 answers
598 views

Is non-standard notation useful when teaching new concepts?

I'm learning about groups and $a^n$ suddenly doesn't mean exponentiation anymore, but repetition of $\underbrace{a\circ a\circ \cdots\circ a}_n$. In some sense I think it would be useful to learn $a\!\...
Frank Vel's user avatar
  • 243
8 votes
6 answers
466 views

Any metaphors/intuitions for a limit of a sequence?

I'm writing (together with a colleague) a minicourse on mathematical analysis (currently we want to cover the Weierstrass theorem on functions on compact intervals, so the aim is to present only the ...
mbork's user avatar
  • 1,299
6 votes
1 answer
150 views

At what stage during an undergrad degree should a student move beyond just 'doing' the equations?

I ask "At what stage during an undergrad degree should a student move beyond just 'doing' the equations?" because I am taking some Year 1 maths papers in 2015. I am worried I won't be able to actually ...
user3754366's user avatar
4 votes
2 answers
296 views

Promoting intuition (for undergraduate students): visual thinking, geometic approaches, etc. in the classroom

Note: This question is ment to extend the scope of some related questions of mine. I would appreciate very much any suggestion to improve the way the question is posed. I would like to ask what is -- ...
Dal's user avatar
  • 1,111
6 votes
4 answers
421 views

Intuition behind $\zeta(2) = \frac{\pi^2}{6}$

The result $$\zeta(2) = \frac{\pi^2}{6},$$ tends to amaze young students because of its beauty. However, although in literature there are many proofs of this result, I find that none is suitable for ...
Dal's user avatar
  • 1,111
6 votes
1 answer
171 views

Intuitive explantion: What is a Finsler metric?

Neither of the two most evident sources, MathWorld: "Finsler Metric." Wikipedia: "Finsler Manifolds." seems to provide me with the high-level intuition that I could convey to students in ~10 minutes....
Joseph O'Rourke's user avatar
21 votes
3 answers
5k views

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

I believe it is the case that, between spaces, homeomorphism is stronger than homotopy equivalence which is stronger than having isomorphic homology groups. For example, the annulus and the circle ...
Joseph O'Rourke's user avatar
5 votes
1 answer
269 views

What is the best way to intuitively explain, understand and approach P vs NP

P vs NP is a million dollar millenium problem. Essentially it boils down to, If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence ...
David BasedMathematician Coven's user avatar
15 votes
4 answers
1k views

When should I say "nothing is as it seems"?

"Intuition" is the best friend and worse enemy of mathematicians! Sometimes using intuitive arguments could be very helpful to understand the nature of a phenomenon. Many of the deepest true ...
user avatar
33 votes
12 answers
5k views

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

I've noticed that series are one of the most difficult portions of calculus for new students to learn. I think the level of abstraction has to do with this. Limits, derivatives, and integrals, as ...
Brian Rushton's user avatar
26 votes
6 answers
1k views

How to present $\Bbb Z/n\Bbb Z$ to highschool level audience

I have the oportunity to talk to a highschool class about mathematics, the topic I got to present are the integers modulo $n$, ie, $\Bbb Z/n\Bbb Z$ , however I don't want to be very heavy and formal, ...
Ana Galois's user avatar
30 votes
5 answers
2k views

How should normal subgroups be introduced?

One standard definition of a normal subgroup is A subgroup $N \subset G$ is normal iff the set of left cosets $\{gN\}$ and right cosets $\{Ng\}$ coincide. There's a class of similar definitions (...
user avatar