For questions related to the teaching and learning of mathematics. Note that Mathematics Educators StackExchange may be a better home for narrowly scoped questions on specific issues in mathematics education.

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4answers
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Do you prove all theorems whilst studying?

When you come across a new theorem, do you always try to prove it first before reading the proof within the text? I'm a CS undergrad with a bit of an interest in maths. I've not gone very far in my ...
48
votes
24answers
9k views

“Negative” versus “Minus”

As a math educator, do you think it is appropriate to insist that students say "negative $0.8$" and not "minus $0.8$" to denote $-0.8$? The so called "textbook answer" regarding this question reads: ...
48
votes
8answers
2k views

How to maintain enthusiasm and joy in teaching when the material grows stale

I recently finished my third semester of teaching calculus to freshman college students. This means I was drawing the same pictures, solving the same example problems, and discussing the same ...
47
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9answers
7k views

How to effectively study math?

Maybe this is too general for here, but I am having a lot of difficulty studying math. Just got out of the military and I guess I am not use to this yet but when I run into a problem I have trouble ...
47
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3answers
10k views

What is the importance of Calculus in today's Mathematics?

For engineering (e. g. electrical engineering) and physics, Calculus is important. But for a future mathematician, is the classical approach to Calculus still important? What is normally taught, as a ...
46
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13answers
16k views

Interesting math-facts that are visually attractive

To give a talk to 17-18 years old (who have a knack for mathematics) about how interesting mathematics (and more specifically pure mathematics) can be, I wanted to use nice facts accompanied by nice ...
45
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9answers
17k views

What is the meaning of the third derivative of a function at a point

(Originally asked on MO by AJAY.) What is the geometric, physical, or other meaning of the third derivative of a function at a point? If you have interesting things to say about the meaning of the ...
43
votes
13answers
4k views

Research done by high-school students

I'm giving a talk soon to a group of high-school students about open problems in mathematics that high-school students could understand. To inspire them, I would like to give them examples of ...
41
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12answers
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Examples of results failing in higher dimensions

A number of economists do not appreciate rigor in their usage of mathematics and I find it very discouraging. One of the examples of rigor-lacking approach are proofs done via graphs or pictures ...
37
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26answers
5k views

How to teach mathematical induction?

Some students are not convinced that a proof by mathematical induction is a proof. I have given the analogy of dominoes toppling but still some remain unconvinced. Is there very convincing way of ...
37
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19answers
8k views

Examples of mathematical induction

What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...
37
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21answers
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Mathematics problems for very young children?

This is a tall order, but since this site is for everything mathematical, here goes: What are some nice games/puzzles/problems/problem-like activities that would be appropriate for a three-five year ...
37
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2answers
4k views

When does L' Hopital's rule fail?

This thought jumped out of me during my calculus teaching seminar. It is well known that the classical L'Hospital rule claims that for the $\frac{0}{0}$ indeterminate case, we have: $$ ...
37
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7answers
2k views

Why is there antagonism towards extended real numbers?

In my backstory, I was introduced to the geometric concept of infinity rather young, through reading about the inversive plane. In the course of learning calculus, I'm pretty sure I formed a concept ...
36
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6answers
1k views

Need a result of Euler that is simple enough for a child to understand

Talking to my 8 yr old about "the greatest mathematician of all time", I said it was probably Gauss in my opinion, but that Gauss was not very kind to his kids (for example, forbidding them to go into ...
34
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8answers
2k views

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
34
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12answers
2k views

Resources for a curious beginner mathematician

I have a friend whose experience of math in school was horrible (each teacher, she says, either left or was fired the year after she had them). I've been teaching her about various things: cantor's ...
34
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3answers
2k views

Why the emphasis on Projective Space in Algebraic Geometry?

I have no doubt this is a basic question. However, I am working through Miranda's book on Riemann surfaces and algebraic curves, and it has yet to be addressed. Why does Miranda (and from what little ...
33
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19answers
36k views

How do I explain 2 to the power of zero equals 1 to a child

My daughter is stuck on the concept that $$2^0 = 1,$$ having the intuitive expectation that it be equal to zero. I have tried explaining it, but I guess not well enough. How would you explain the ...
33
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10answers
3k views

Becoming Better at Math

How can I become excellent at math? It really interests me but when I fail I become demotivated and begin to give up. EDIT: Could anyone suggest books for someone with a math education that just ...
33
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7answers
4k views

What are some good math specific study habits?

What are/ were some of your good mathematician's study habits that you found really worked for you? I'm a CS major at a respected school and have a solid GPA... However, I definitely lack when it ...
33
votes
1answer
757 views

What did mathematicians study as an undergraduate/graduate before modern mathematics such as modern algebra and analysis?

I am curious as to what mathematicians such as Leibnitz and Gauss and the Bernoulli's studied when they were students in university. I find it fascinating how we are taught calculus and abstract ...
32
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13answers
2k views

How To Present Algebraic Topology To Non-Mathematicians?

I am writing my master thesis in algebraic topology (fundamental groups) and as a system in my school students must write about one page about their theses explaining for non mathematicians the ...
32
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3answers
856 views

English words in written mathematics

I recently marked over $100$ assignments for a multivariable calculus course. One question which a lot of people did poorly was proving a given set was open. Aside from issues relating to rigour and ...
31
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12answers
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explaining the derivative of $x^x$

You set the following exercise to your calculus class: Q1. Differentiate $y(x) = x^x$. A student submits the following solution: Let $g(a)=a^x$ and $f(x)=x$. Then $y(x) = g(f(x))$, so by ...
31
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5answers
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How to pick a thesis advisor?

This sort of question is probably in bad taste for math.stackexchange, but is probably in high demand. (I tried to start a site on Area 51 to house questions like this, but my request was closed due ...
31
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8answers
5k views

How can I learn to “read maths” at a University level?

When I look at math, it's like my mind goes fuzzy. The only way to describe it is in terms of what I can relate it to. You know how when you read, you see the letters and words, but your brain picks ...
30
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20answers
3k views

Good math bed-time stories for children?

What are some good references/books/articles from which to derive some good bed-time math stories to pique a child's interest in math? I am fascinated by math (used to hate it as a kid) and want my ...
30
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7answers
2k views

Quotient geometries known in popular culture, such as “flat torus = Asteroids video game”

In answering a question I mentioned the Asteroids video game as an example -- at one time, the canonical example -- of a locally flat geometry that is globally different from the Euclidean plane. It ...
29
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6answers
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How to tell $i$ from $-i$?

Suppose now we are trying to explain to students who do not know complex numbers, how do we distinguish $i$ and $-i$ to them? They will object that they both squared to $-1$ and thus they are ...
29
votes
11answers
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How do you define functions for non-mathematicians?

I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to ...
29
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8answers
3k views

Very *mathematical* general physics book

I am searching for a book to study physics. So far, I've been suggested Resnick, Halliday, Krane, Physics, but it doesn't seem to be very suited for a math major. Can you suggest some more ...
29
votes
3answers
1k views

Create a Huge Problem

I am wondering if any problems have been designed that test a wide range of mathematical skills. For example, I remember doing the integral $$\int \sqrt{\tan x}\;\mathrm{d}x$$ and being impressed at ...
29
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3answers
11k views

Self-study Real analysis Tao or Rudin?

The reference requests for analysis books have become so numerous as to blot out any usefulness they could conceivably have had. So here come another one. Recently I've began to learn real analysis ...
28
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9answers
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Is this way of teaching how to solve equations dangerous somehow?

Two years ago, I bought the book Mathematics for the Nonmathematican, by Morris Kline. There I learned a new way of solving equations, which is related to the principle that states that any ...
28
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8answers
3k views

Active learning vs Passive learning in Math

I am trying to improve how I learn in general but specifically in math and a common suggestion I keep coming across is the difference between active learning and passive learning. The problem is, most ...
28
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6answers
2k views

Cool mathematics I can show to calculus students.

I am a TA for theoretical linear algebra and calculus course this semester. This is an advanced course for strong freshmen. Every discussion section I am trying to show my students (give them as a ...
28
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4answers
1k views

Understanding mathematics imprecisely

For a long time, it has been a complete mystery to me how any of my peers understood any math at all with anything short of filling in every detail, being careful about every set theoretic detail down ...
27
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10answers
8k views

What is the (mathematical) point of straightedge and compass constructions?

The ancient discipline of construction by straightedge and compass is both fascinating and entertaining. But what is its significance in a mathematical sense? It is still taught in high school ...
27
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14answers
814 views

Examples where it is easier to prove more than less

Especially (but not only) in the case of induction proofs, it happens that a stronger claim $B$ is easier to prove than the intended claim $A$ (e.g. since the induction hypothesis gives you more ...
27
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3answers
2k views

Good ways to help people learn math.

I have a student position at my university, which involves helping people who are struggling with basic calculus, precalculus, and discrete math. In general, the students that I help are hostile ...
26
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9answers
7k views

Are there contradictions in math?

Someone told me that math has a lot of contradictions. He said that a lot of things are not well defined. He told me two things that I do not know. $1+2+3+4+...=-1/12$ what is infinity $\infty$? ...
26
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6answers
1k views

Should I understand a theorem's proof before using the theorem?

I find myself embarrassed when using results in books. For example, there are so many results in Sobolev spaces that I think I would not be able to understand all of them. Yes, I could try to ...
26
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5answers
4k views

Math GRE Subject Exam [closed]

I am studying for the GRE Mathematics subject exam [1]. I am looking for tips regarding how to more effectively study for it. Does anyone know of any good study materials or have any tips in general? ...
26
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4answers
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Mind maps of Advanced Mathematics and various branches thereof

I would like to get a list of mind maps of advanced mathematics topics. As an example, I have posted one below. I would be happy if you post such other maps. Making one and posting it here is also ...
25
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11answers
4k views

Vivid examples of vector spaces?

When teaching abstract vector spaces for the first time, it is handy to have some really weird examples at hand, or even some really weird non-examples that may illustrate the concept. For example, a ...
25
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5answers
4k views

Is a good GRE score enough for a non-math graduate to be accepted in a decent pure mathematics graduate program?

I have a computer engineering degree , and i have studied several mathematics courses like single variable and multiples variables calculus , complex variables , probability , numerical analysis ... ...
25
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4answers
6k views

Teaching Introductory Real Analysis

I am currently helping teach an introduction to real analysis course at UC Berkeley. The textbook we are using in Rudin's "Principles of Mathematical Analysis" (aka baby rudin). I am trying to find ...
24
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6answers
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Why do we need to prove $e^{u+v} = e^ue^v$?

In this book I'm using the author seems to feel a need to prove $e^{u+v} = e^ue^v$ By $\ln(e^{u+v}) = u + v = \ln(e^u) + \ln(e^v) = \ln(e^u e^v)$ Hence $e^{u+v} = e^u e^v$ But we know from basic ...
24
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6answers
1k views

Why do new rules cause students to forget and misapply older rules?

I teach at a community college. I have taught everything from arithmetic to linear algebra. I have also taught at 4-year schools, but at present, I'm devoting my energies to the problem of helping ...