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

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70
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34answers
25k views

Why is negative times negative = positive?

Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc. I went ahead and gave them a proof by contradiction like so: Assume ...
352
votes
35answers
42k views

Do complex numbers really exist?

Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
57
votes
1answer
3k views

Is Lagrange's theorem the most basic result in finite group theory?

Motivated by this question, can one prove that the order of an element in a finite group divides the order of the group without using Lagrange's theorem? (Or, equivalently, that the order of the group ...
15
votes
4answers
2k views

Motivation behind the definition of GCD and LCM

According to me, I can find the GCD of two integers (say $a$ and $b$) by finding all the common factors of them, and then finding the maximum of all these common factors. This also justifies the ...
101
votes
25answers
38k views

What is a good complex analysis textbook?

I'm out of college, and trying to learn complex analysis on my own. I took out Ahlfors' text from the library, but I'm finding it difficult. Any textbook recommendations? I'm probably at an ...
107
votes
8answers
7k views

Why is Euler's Gamma function the “best” extension of the factorial function to the reals?

There are lots (an infinitude) of smooth functions that coincide with f(n)=n! on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^t dt$ is "best"...
720
votes
53answers
423k views

Visually stunning math concepts which are easy to explain

Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain, but are ...
4
votes
4answers
540 views

Factoring $ac$ to factor $ax^2+bx+c$

I was watching a first-year high-school-algebra student struggle with factoring quadratics last night. Given a quadratic $ax^2+bx+c$ (I'll give you the exact example in a moment), her method — ...
632
votes
162answers
39k views

What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)

I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament,...
32
votes
3answers
2k 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 ...
56
votes
4answers
7k views

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 ...
60
votes
22answers
7k views

What is your favorite application of the Pigeonhole Principle?

The pigeonhole principle states that if $n$ items are put into $m$ "pigeonholes" with $n > m$, then at least one pigeonhole must contain more than one item. I'd like to see your favorite ...
16
votes
1answer
747 views

Alternative construction of the tensor product (or: pass this secret)

The paper Tensor products and bimorphisms by B. Banachewski and E. Nelson studies tensor products (defined by classifying bimorphisms) in concrete categories. It is quite interesting that their main ...
275
votes
33answers
34k views

Pedagogy: How to cure students of the “law of universal linearity”?

One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”: $$ \frac{1}{a+b} \mathrel{\...
66
votes
8answers
10k views

When to learn category theory?

I'm a undergraduate who wishes to learn category theory but I only have basic knowledge of linear algebra and set theory, I've also had a short course on number theory which used some basic concepts ...
55
votes
9answers
9k 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 ...
51
votes
9answers
23k 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 ...
38
votes
19answers
9k 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 $\bullet+\cdots\cdots\cdots+\bullet$ ...
7
votes
4answers
2k views

Expanding problem solving skill

I have a great passion for Math but my lack in problem solving skill always keeps me away from the "good stuff". I always wanted to be better at Math and one of the things I figured out was to keep ...
92
votes
5answers
4k views

“Advice to young mathematicians”

I have been suggested to read the Advice to a Young Mathematician section of the Princeton Companion to Mathematics, the short paper Ten Lessons I wish I had been Taught by Gian-Carlo Rota, and the ...
59
votes
9answers
32k views

Why is $\pi $ equal to $3.14159…$?

Wait before you dismiss this as a crank question :) A friend of mine teaches school kids, and the book she uses states something to the following effect: If you divide the circumference of any ...
8
votes
7answers
5k views

Easiest and most complex proof of $\gcd (a,b) \times \operatorname{lcm} (a,b) =ab.$

I'm looking for an understandable proof of this theorem, and also a complex one involving beautiful math techniques such as analytic number theory, or something else. I hope you can help me on that. ...
13
votes
13answers
4k views

How to explain the formula for the sum of a geometric series without calculus?

How to explain to a middle-school student the notion of a geometric series without any calculus (i.e. limits)? For example I want to convince my student that $$1 + \frac{1}{4} + \frac{1}{4^2} + \...
59
votes
14answers
17k views

How to effectively and efficiently learn mathematics

How do you effectively study mathematics? How does one read a maths book instead or just staring at it for hours? (Apologies in advance if the question is ill-posed or too subjective in its current ...
20
votes
2answers
6k views

Path to Basics in Algebraic Geometry from HS Algebra and Calculus?

In this question, Why study Algebraic Geometry?, Javier Álvarez, develops a succint but encompassing description of algebraic geometry and its spread across different areas of mathematics. Indeed, it ...
82
votes
19answers
14k views

How do you explain the concept of logarithm to a five year old?

Okay I understand that it cannot be explained to a 5 year old. But, how do you explain the logarithm to primary school students?
42
votes
12answers
2k views

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 ...
10
votes
4answers
1k views

What are or where can I find style guidelines for writing math?

I am a scientist writing my first manuscript with a substantial amount of mathematical methodological documentation. I am using LaTeX, but this is not my question. I would like to find a list of ...
30
votes
10answers
6k views

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 ...
18
votes
7answers
6k views

How to show every subgroup of a cyclic group is cyclic?

I'm teaching a group theory course now, and I wanted to give my students a proof that every subgroup of a cyclic group is cyclic. The easiest way I could think to do this is to say that any cyclic ...
4
votes
2answers
2k views

Prison problem: locking or unlocking every $n$th door for $ n=1,2,3,…$

I have a problem called "The Prison Problem" that I need to explain to my 9-year-old cousin. I would think that he has just started learning about divisors and perfect squares, and as such, I have a ...
64
votes
17answers
10k views

Interesting “real life” applications of serious theorems

As student in mathematics, one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like: ...
25
votes
6answers
7k views

Complete undergraduate bundle-pack [closed]

First of all I'm sorry if this is not the right place to post this. I like math a lot. But I'm not sure if i want to do a math major in college. My question is: Can you give me a list of books that ...
129
votes
9answers
210k views

How many sides does a circle have?

My son is in 2nd grade. His math teacher gave the class a quiz, and one question was this: If a triangle has 3 sides, and a rectangle has 4 sides, how many sides does a circle have? My first ...
41
votes
19answers
59k 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 ...
17
votes
9answers
4k views

Motivating infinite series

What are some good ways to motivate the material on infinite series that appears at the end of a typical American Calculus II course? My students in this course are generally from biochemistry, ...
21
votes
17answers
6k views

Explaining Horizontal Shifting and Scaling

I always find myself wanting for a clear explanation (to a college algebra student) for the fact that horizontal transformations of graphs work in the opposite way that one might expect. For example, ...
8
votes
7answers
1k views

Can this standard calculus result be explained “intuitively”

Recently I stumbled upon someone who said he wanted to understand why $\arctan x = \int\dfrac{dx}{1+x^2}$ At first I was confused. This is an easy result in any integral calculus course. But then he ...
16
votes
4answers
1k views

Fun math for young, bored kids?

For 6 months, I'll be organizing, as part as my volunteer work in an NGO, math classes with small groups (~10 students, aged 16 or 17). These classes are not compulsory, but students willing to stay ...
12
votes
5answers
419 views

Show $\lim\limits_{h\to 0} \frac{(a^h-1)}{h}$ exists without l'Hôpital or even referencing $e$ or natural log

Taking as our definition of exponentiation repeated multiplication (extended to real exponents by continuity), can we show that the limit $$\lim_{h\to 0}\dfrac{a^h-1}{h}$$ exists, without l'Hôpital,...
13
votes
4answers
1k views

Infinite Series: Fibonacci/ $2^n$

I presented the following problem to some of my students recently (from Senior Mathematical Challenge- edited by Gardiner) In the Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, 34, 55,\ldots$ each ...
16
votes
5answers
10k views

Which calculus text should I use for self-study?

I am 36 years old, and have forgotten a lot of math from high school, of which I only took up to Algebra 2. However I am teaching myself mathematics and am now, as an adult, completely fascinated ...
9
votes
1answer
480 views

Some maps of the land of mathematics?

This question is motivated by a little anecdote. I was at home teaching some secondary school math to a relative. At some relax time, he glanced at a book I had over the table - it was some text about ...
10
votes
4answers
1k views

Is there a more efficient method of trig mastery than rote memorization?

I would like to get alot better at trig than I am. What is the best/most efficient method? Thanks much in advance Joe
9
votes
6answers
3k views

What is the best base to use?

When I typed this question in google I found this link: http://octomatics.org/ Just from the graphic point of view: this system seems to be easier (when he explains that you can overlap the line). He ...
4
votes
4answers
2k views

Linear combinations of sine and cosine

If you take a linear combination of the cosine and sine function, then the result is again a sinusoid, but with a new amplitude and phase shift. $$a \cos(\theta) + b \sin(\theta) = A \cos(\theta + \...
95
votes
20answers
20k views

Visually deceptive “proofs” which are mathematically wrong

Related: Visually stunning math concepts which are easy to explain Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually ...
35
votes
7answers
6k 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 ...
28
votes
6answers
7k 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 ...
54
votes
15answers
6k views

How can I introduce complex numbers to precalculus students?

I teach a precalculus course almost every semester, and over these semesters I've found various things that work quite well. For example, when talking about polynomials and rational functions, in ...