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3
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2answers
281 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 ...
8
votes
6answers
285 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 $1/2$ ...
4
votes
0answers
136 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 ...
3
votes
1answer
192 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, ...
3
votes
1answer
260 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 ...
2
votes
0answers
169 views

Benefits of knowing theory

I've got somewhat an issue: from time to time I have to teach some math to people who either avoided it or got trough by only knowing a few working algorithms. The only thing that unites all those ...
3
votes
1answer
94 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 ...
10
votes
5answers
261 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 ...
11
votes
6answers
1k 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?...
13
votes
2answers
701 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 ...
4
votes
2answers
438 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 \...
3
votes
1answer
364 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 ...
18
votes
7answers
764 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. ...
6
votes
1answer
134 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 ...
9
votes
1answer
195 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 ...
4
votes
0answers
71 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 ...
10
votes
6answers
535 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 ...
8
votes
3answers
185 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\} ...
25
votes
5answers
901 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 ...
18
votes
7answers
576 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 ...
7
votes
2answers
428 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\!\...
8
votes
6answers
303 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 ...
6
votes
1answer
127 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 ...
4
votes
2answers
250 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 -- ...
6
votes
4answers
347 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 ...
6
votes
1answer
115 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....
16
votes
3answers
3k 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 ...
5
votes
1answer
240 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 ...
13
votes
4answers
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 ...
27
votes
11answers
3k 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 ...
24
votes
6answers
837 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, ...
18
votes
4answers
702 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 (...