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.

learn more… | top users | synonyms (3)

18
votes
6answers
5k views

No radical in the denominator — why? [duplicate]

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 $\sqrt{3}...
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 ...
18
votes
6answers
7k views

How to go from beginner to expert in mathematics?

I'm particularly interested in knowing the order of subjects one should study to become good in mathematics. In my not so great knowledge of the subject I would say that the best place to start could ...
18
votes
11answers
1k views

Why limits work

I'm currently a first year student in electrical engineering and computer science. I know how to compute limits, derivatives, integrals with respect to one variable that is things from one variable ...
18
votes
6answers
2k views

Why is the definition of “limit” difficult to understand at first?

Tomorrow I teach my students about limits of sequences. I have heard that the definition of limit is often difficult for students to understand, and I want to make it easier. But first I need to ...
18
votes
5answers
1k views

Alternative set theories

This is a (soft!) question for students of set theory and their teachers. OK: ZFC is the canonical set theory we all know and love. But what other, alternative set theories, should a serious student ...
18
votes
4answers
1k views

“Best practice” innovative teaching in mathematics

Our department is currently revamping our first-year courses in mathematics, which are huge classes (about 500+ students) that are mostly students who will continue on to Engineering. The existing ...
18
votes
5answers
1k 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 ...
18
votes
4answers
618 views

Is it necessary to prove everything and solve every problem in the books? [closed]

I am an undergraduate really passionate about the mathematics and microbiology. I have few big problems in learning which I would like to seek your advice. Whenever I study mathematical books (...
18
votes
4answers
2k views

Grasping mathematics

First, I'm not trying to make this sound like a "poor-me" story. I understand fully that every decision I've made leading to this is my fault. I am genuinely looking for advice. So, I am a high ...
18
votes
2answers
938 views

High school algebra textbooks for gifted students

Cross-posted to Math Educators Stack Exchange. (link) I am looking for high school algebra/mathematics textbooks targeted at talented students, as preparation for fully rigorous calculus à la Spivak. ...
18
votes
5answers
783 views

Proving a certain map on the closed unit disc must be the identity

Bounty expired. Will gladly re-create one if a satisfactory answer is posted in the future. Prove: Let $f$ be a continuous function on the closed unit disc with two properties: 1. $f$ is the ...
17
votes
4answers
6k views

Why do we need to learn Set Theory?

I was planning to write some article for the Mathematics magazine of our college and it occurred to me that it will be a good idea to write about the impact and importance of Set Theory. I plan ...
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, ...
17
votes
7answers
806 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 ...
17
votes
4answers
2k views

Common student mistakes/misconceptions in a first year calculus course

What are the common mistakes and misconceptions students make in a first year calculus course? More importantly: What can I do to prevent/rectify them? Context: Soon I will be doing some ...
17
votes
2answers
12k views

Relearning from the basics to Calculus and beyond.

Assume someone has very limited knowledge of math. (low level high school, 5-6 years ago) How would they learn from the basics of algebra, geometry and trigonometry to a solid foundation for calculus ...
17
votes
6answers
7k views

Best intuitive metaphors for math concepts (of any level)

Frequently, we introduce a new concept with a formal definition, then immediately say "Intuitively, what this means is..." What are the absolute best metaphors you've seen (for concepts of any level)? ...
17
votes
1answer
2k views

How do you remember theorems?

I am currently a Master's student in math. I do very well in my classes, understand the material, can do the proofs w/o having to read the text, etc, but as time passes, I find that I will forget ...
17
votes
1answer
2k views

Research in differential geometry

I am an 3rd year undergrad interested in mathematics and theoretical physics. I have been reading some classical differential geometry books and I want to pursue this subject further. I have three ...
17
votes
2answers
4k views

Learning Mathematics using only audio.

Are there any mathematics audio books or other audio sources for learning mathematics. I ask this because I make about 1 hour from my house to the school and staring at a screen on the car makes me ...
16
votes
13answers
2k views

How to explain the perpendicularity of two lines to a High School student?

Today I was teaching my friend from High School about linear functions. One of the exercises we had to do was finding equations of perpendicular and parallel lines. Explaining parallel equations was ...
16
votes
5answers
2k views

When I was teaching absolute function properties, I suddenly made this question …

I was teaching absolute function properties in a K-12 class. I made this question in my mind. Suppose $f(x)$ is a one-to-one function, and its definition is $f(x)=max\left \{ x,3x\right \}=ax+b|x|+c$...
16
votes
5answers
1k views

Is high school contest math useful after high school?

I've been prepping for a lot of high school math competitions this year, and I was just wondering if all the math I learn would actually mean something in college. There is a chance that all of it ...
16
votes
8answers
3k views

Why are epsilon-delta proofs difficult?

What conceptual difficulties do students learning epsilon-delta proofs have, or, why are the proofs difficult? Motivation I have to teach people and never found the epsilonistics particularly ...
16
votes
4answers
1k views

Why study Hopf Algebras?

I'm looking for reasons that motivate the study of Hopf Algebra, like its applications in other branches of mathematics or maybe with physics. The first I've got is that they're interesting by ...
16
votes
4answers
607 views

Gaining Mathematical Maturity [closed]

I was redirected here by a kind fellow from math.overflow. This is not a typical math question, so I apologize if that is discourteous. I am currently a sophomore in my undergraduate mathematics ...
16
votes
3answers
323 views

Is$\frac{\sqrt{a}}{\sqrt{b}}$ the same as $\sqrt{\frac{a}{b}}$?

My idea is that the two functions are not the same since for the first function, the domain of the function is only non negative reals for the numerator and positive reals for the denominator. ...
16
votes
12answers
11k views

Which 4 maths courses to take as an Economics PhD student?

I am doing a PhD in economics and I have the chance of taking one subject a semester in the maths department (I would like to do more, but "unfortunately" I have to work on my thesis). I want to have ...
16
votes
7answers
2k views

Films about math: a question about math education and motivation for learning math [closed]

I'm interested in movies about or related with mathematics or physics, I mean not documentaries which I also consider movies, but artistic or mainstream films about math. Now I have the following in ...
16
votes
5answers
832 views

Math department space design ideas

I'm looking for resources on design space for academic departments, particularly, We've been asked to provide ideas on designing a "dream math department". I'd like to gather some ideas for doing this....
16
votes
3answers
6k views

What makes a good mathematician?

A soft question but I believe important to help get maturity in maths. Not until I got admitted to a graduate applied math program that I started to learn math. Before that, I was a life science ...
16
votes
7answers
559 views

Common inadequate definitions

I had the good fortune of assigning the following problem on the first day of my freshman honors calculus course: Is $a$ contained in the interval $[a,a)$? Explain. The majority of my students ...
16
votes
5answers
11k 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 ...
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 ...
16
votes
8answers
2k views

How to explain brackets to young students

It's amazing how simple questions from young students can often uncover unexpected gaps in (at least my) knowledge, or at least the ability to explain. Today, a student asked me why she can "forget ...
16
votes
5answers
486 views

Visualization of surface area of a sphere

I help mentor some really young, bright kids in mathematics. We were looking at geometric properties of various shapes, and one of the kids noted that the surface area of a sphere $S = 4\pi r^2$ ...
16
votes
2answers
134 views

why don't standard analysis texts relax the injectivity requirement of the multivariable change of variables theorem? [duplicate]

In standard multivariable analysis texts, the change of variables for multivariable integration in Euclidean space is almost always stated for a $C^1$ diffeomorphism $\phi$, giving the familiar ...
16
votes
3answers
463 views

which exact integration techniques belong in a first year calculus/analysis course?

At our university we are now discussing changes to the course contents and there is some heated discussion regarding integration in the first year calculus courses. Currently, the techniques of exact ...
16
votes
5answers
918 views

Reflections on math education

Why is there such a big difference in math education between The Americas and (Europe and Asia) ? except for a few privileged who have the opportunity to access to math much earlier than the ordinary ...
16
votes
4answers
506 views

Mathematical Habit

Whenever I have to do proofs or understand a concept, I start with examples and generalize. As an aspiring mathematician, I have fears that it might hurt me, especially when we have so many ...
16
votes
2answers
152 views

Math for blind people…

What happens if some blind person want to study math? Is there some "braille alphabet" for mathematical symbols? Are there math books, at least for undergraduate students, written for blind people?
16
votes
1answer
249 views

Is it a good approach to heavily depend on visualization to learn math?

I am a third year undergraduate and I am a beginner on these "real mathematics" (no pun intended). Before contacting the "real math", my math level should be considered to be "good", although I was ...
16
votes
1answer
753 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 ...
15
votes
6answers
6k views

What is the best way to develop Mathematical intuition? [closed]

I want to develop my pure mathematics knowledge and would like to know what is the best way to develop mathematical intuition? I am going through exercises that ask for proofs and I don't have the ...
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 ...
15
votes
5answers
1k views

Do non-mathematical fields use the appropriate level of analytic/probabilistic rigor?

Talking to students in different areas and taking different classes in math, physics, electrical engineering I have been struck by the differing amounts of rigor in use. I know little about economics ...
15
votes
5answers
2k views

Areas of math that can be “gamified”?

My old high school math teacher has started up a math club recently. It's become quite popular (to my pleasant surprise), and I've been looking for ways to contribute. To that end I've been looking to ...
15
votes
2answers
1k views

How to deal with exercises with no solutions given?

Probably most people will acknowledge the importance of doing exercises when reading a mathematical textbook. Here I am talking about a textbook of similar level as those ones listed in GTM. However, ...
15
votes
1answer
339 views

Do lay persons really believe that all of mathematics is known?

I once found a professor of medicine to be under the impression that everything in mathematics is already known. It's probably commonplace to hear that the masses labor under that misconception. In ...