Questions related to the teaching and learning of mathematics.

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43
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23answers
14k 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 ...
3
votes
4answers
385 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 — ...
419
votes
132answers
26k 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 ...
79
votes
9answers
4k 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 ...
42
votes
4answers
4k 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 ...
6
votes
4answers
855 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 ...
25
votes
3answers
657 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 ...
40
votes
9answers
5k 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 ...
39
votes
13answers
7k 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 ...
38
votes
8answers
5k 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 ...
39
votes
9answers
11k 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 ...
5
votes
6answers
2k 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. ...
359
votes
42answers
165k 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 ...
66
votes
16answers
9k 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?
17
votes
7answers
5k 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 ...
14
votes
6answers
2k 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 ...
90
votes
7answers
109k 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 ...
24
votes
7answers
3k 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 ...
9
votes
5answers
6k 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 ...
13
votes
2answers
3k 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 ...
45
votes
12answers
3k 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 ...
41
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 ...
31
votes
7answers
1k 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 ...
22
votes
9answers
6k views

What is the (mathematical) point of geometric 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 ...
9
votes
2answers
471 views

Is there any way to read articles without subscription?

This is not mathematician question but I think it's related. How I can get access to some of "Software: Practice and Experience" articles without subscription? Any advice is welcome. Sorry if I'm ...
6
votes
3answers
444 views

Why $\frac{1}{\infty } \approx 0 $ and $ \frac{1}{0} = {\infty}$?

First I have checked the search option but found nothing relevant to my problem and also level of math. I just started learning the language of mathematics, on my own and I have trouble understanding ...
14
votes
6answers
2k views

No radical in the denominator — why?

Why do all school algebra texts define simplest form for expressions with radicals to not allow a radical in the denominator. For the classic example, $1/\sqrt{3}$ needs to be "simplified" to ...
10
votes
4answers
785 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
2
votes
5answers
1k views

Where can I find a review of discrete math

I'm looking for course notes and assignments and hopefully some example exams for Discrete Math, I'm taking a placement exam in the subject after having taken it 4 years ago.
80
votes
11answers
10k views

How to convince a math teacher of this simple and obvious fact?

I have in my presence a mathematics teacher, who asserts that $$ \frac{a}{b} = \frac{c}{d} $$ Implies: $$ a = c, \space b=d $$ She has been shown in multiple ways why this is not true: $$ ...
45
votes
13answers
6k views

Why do we need to learn integration techniques?

After a lifetime of approaching math the wrong way, I took two college math courses this quarter with a newfound zest for math. These classes are integral calc and multivariable calc. Integral calc ...
17
votes
5answers
752 views

Fun math outreach/social activities

What are some great math social activities for students? I'm looking for things that bring people together with a "light" mathematical touch. The goal is to create a stronger mathematical community in ...
10
votes
5answers
2k views

Advice for Self-Study

I am a senior in high school who has taught myself through Calculus BC and I got a 5 on the exam. However, I have taken all the math I can at my school. I have also taught myself multi-variable ...
36
votes
25answers
6k 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: ...
11
votes
9answers
769 views

Sources of problems for teaching/tutoring young mathematicians

I am tutoring several talented students, middle school level and early high school level, in mathematics. I am always looking for new sources from which to draw questions. Can anyone recommend books, ...
17
votes
4answers
3k views

Bridging any “gaps” between AP Calculus and College/Univ level Calculus II

I've been asked to tutor a soon-to-be college freshman who has taken AP Calculus and successfully earned college credit for first semester calculus. He has been admitted to an Engineering program, ...
16
votes
7answers
695 views

Exciting games and material to motivate children to math

We are a group of people trying to motivate children, especially living in the countryside, to science and math. We have different activities with children such as doing scientific experiments and ...
15
votes
5answers
2k views

What does Khan Academy have to offer? Depth? Rigor?

Khan Academy - http://www.khanacademy.org/ - is often cited as a great online resource for learning mathematics and other subjects. I have heard many good things about this website and was wondering ...
12
votes
4answers
280 views

Should the domain of a function be inferred?

It is a common practice to have students of elementary algebra infer the domain of a function as an exercise. I believe this is contrary to the spirit of the definition of a function as a collection ...
8
votes
3answers
576 views

Which of these courses to take if one intends to go to grad school in pure math (rank please)

could you rank these classes in terms of necessity to take if I intend to pursue a Ph.D in pure math? I don't know if I can fit everything, but I want to make sure I take the most important ones: ...
15
votes
8answers
1k views

When is something “obvious”?

I try to be a good student but I often find it hard to know when something is "obvious" and when it isn't. Obviously (excuse the pun) I understand that it is specific to the level at which the writer ...
11
votes
5answers
2k views

Why teach linear algebra before abstract algebra?

Is there a reason why most undergraduate curriculums put linear algebra before abstract algebra? I'm asking this because personally it seems to be much easier to understand the architecture behind ...
5
votes
2answers
333 views

Factoring Quadratics: Asterisk Method

I'm teaching my students about factoring quadratics. We've done GCF, difference of two squares, squared binomials, and grouping. One of my colleagues then found this asterisk method on line. It's ...
5
votes
2answers
438 views

Programs for precocious prodigies

I am the director of my university's mathematics honors program, and we just had an inquiry from the parent of a 15 year old who has already completed most of the math courses for a standard ...
4
votes
2answers
286 views

Majoring in maths

Does majoring in mathematics with economics (with emphasis on mathematics) have good career prospects? Does anyone know something about this course?
2
votes
3answers
3k views

Sum of digits and product of digits is equal (3 digit number)

My child got a question in school (grade) that is: Find biggest and smallest 3 digits number, which has sum of it's digits equal to product of those digits. Help please since I cannot explain my ...
8
votes
3answers
741 views

What is the probability that GCD of $(a,b)$ is $b$?

My question is quite simple. I have been googling a lot lately trying to find a solution to this: Given a sequence of n integers $[1,2,...,n]$. If we pick two numbers randomly from the set say, a and ...
7
votes
6answers
2k 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 ...
7
votes
0answers
336 views

The difference between 10 and 9.99999 … (recurring) [duplicate]

Possible Duplicate: Does .99999… = 1? At supper today my daughter was discussing her maths (she's 13) - she had been studying putting decimal numbers into what she called standard ...
4
votes
3answers
1k views

Do you need real analysis to understand complex analysis?

I'm debating whether I should take a course, in complex analysis (using Bak as a text). I've already taken Munkres level topology and "very light" real analysis (proving the basic theorems about ...