Mathematics education consists in the practice of teaching and learning mathematics, along with the associated research. Research in mathematics education concerns the tools, methods and approaches that facilitate the practice of mathematics or the study of this practice.

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547
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153answers
34k 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 ...
55
votes
22answers
6k 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 ...
37
votes
19answers
7k 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 $\bullet+\cdots\cdots\cdots+\bullet$ ...
6
votes
1answer
136 views

Help with diagram chasing

Given the diagram $\require{AMScd}$ \begin{CD} 0 @>>> A @>f>> B @>g>> C @>>> 0 \\ @. @V\alpha VV \#@V\beta V V\# @VV\gamma V @. \\ 0 @>>> {A'} ...
27
votes
14answers
786 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 ...
-1
votes
0answers
57 views

How can I make math joyful to learn? [closed]

As a kid I had a strong interest in games, especially competitive-type games. Chess for example was and still is fun for me, it can be described as being competitive, addicting, beautiful and filled ...
1
vote
0answers
60 views

Best intro to Fundamentals of Mathematics? [closed]

If you like mathematics it's likely that you also want to have the most solid foundations in number theory and analysis possible. I have just finished Elliott Mendelson's book Number Systems and the ...
56
votes
25answers
7k views

Easy example why complex numbers are cool

I am looking for an example explainable to someone only knowing high school mathematics why complex numbers are necessary. The best example would be possible to explain rigourously and also be clearly ...
1
vote
0answers
35 views

A harder long division puzzle than the first; what should “Algebra I” solution look like?

Here's another problem, significantly harder than the first, but still accessible to target audience. The statement of the problem (i.e., northwest corner only) comes from a PennyDell puzzle magazine: ...
6
votes
3answers
87 views

“Long-division puzzles” can help middle-grade-level students become actual problem solvers, but what should solution look like?

This is my first post. I hope it's acceptable. EDIT Since there are people to whom such notation is foreign, I will point out that the problem represents KRRAEE / KMS, where PEI is the quotient and ...
248
votes
33answers
33k 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} ...
33
votes
19answers
33k 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 ...
7
votes
8answers
827 views

Most natural intro to Complex Numbers [closed]

This is a soft question but I'm willing to ask. There are few ways to introduce the field of complex numbers, but if You had the opportunity to write an elementary textbook, what would be the most ...
74
votes
32answers
7k views

What are some conceptualizations that work in mathematics but are not strictly true?

I'm having an argument with someone who thinks it's never justified to teach something that's not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is to ...
5
votes
1answer
758 views

Soft question : First year student and confused

I hope I won't tire the fellow mathematicians with this question but I am very, very confused... I am a first year undergraduate student of Mathematics. I can't say I am a prodigy, maybe having an ...
3
votes
1answer
116 views

Students and Real Analysis

I am currently working on a project investigating why students tend to struggle when they first encounter Real Analysis and what can be done to improve the situation. I would be very grateful if any ...
0
votes
0answers
39 views

The $\epsilon$-$\delta$ definition of a limit of a linear vs a non linear function

I am teaching elementary analysis and introducing the concept of $\epsilon$-$\delta$ definition of the limit to first time learners. For example, we take $\displaystyle\lim_{x \to 2} (2x - 1) = 3$, ...
4
votes
3answers
340 views

How can I explain my 9 years old brother that $8a\cdot4a \neq 64a$

My youngest brother had a pre-algebra test yesterday and he was asked to tell if two expressions are equal or not. We agreed on most of the things but on this one I find it hard to make him accept my ...
11
votes
2answers
514 views

How to introduce type theory to newcomer

I want to introduce (dependent) type theory to some friends having background in mathematical logic and set theory. To make this introduction easy I would like to give an informal presentation that ...
55
votes
9answers
26k 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 ...
58
votes
17answers
8k views

Interesting “real life” applications of serious theorems

As a student 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: "You have a ...
19
votes
8answers
2k views

How do authors make their problems/exercises for their math books? [closed]

I want to be a math professor one day, but I'm wondering how to make my own original problems to give them to my students. I think that it is a responsibility of the professor to create original and ...
53
votes
17answers
16k views

Why is negative times negative = positive?

Someone recently asked me why a negative * a negative is positive, and why a negative * a positive is negative, etc. I went ahead and gave them a proof by contradiction like so: Assume $(-x) * (-y) ...
0
votes
0answers
38 views

Math wheel from 1917 question [duplicate]

http://www.washingtonpost.com/news/morning-mix/wp/2015/06/06/eerie-chalkboard-drawings-frozen-in-time-for-100-years-discovered-in-oklahoma-school/ I hope this came out like it was on the webpage. It ...
94
votes
11answers
6k views

Is there a domain “larger” than (i.e., a supserset of) the complex number domain?

I've been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this. In the domain of natural numbers, addition and ...
17
votes
1answer
607 views

Which universities teach true infinitesimal calculus?

My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...
8
votes
0answers
76 views

How to explain the topic of Fourier transform interactively? [closed]

This is a soft question . In the walk-in for the lectureship, I have decided to give demo lecture on the topic of Fourier transform. The principal of the institution ask me to take lecture ...
4
votes
2answers
343 views

Teaching the Concept of Infinity to Children.

I was recently out with the family and we left it up to the children where we ate lunch (11 and 9 years old). They couldn't agree and were going back and forth calling each other names. This ...
0
votes
1answer
39 views

Prove lines being parallel within a traiangle

Here is the problem: The only condition given is $DF//BC$, is it possible to prove that $GH//BC$? Please verify it. Any help will be appreciated.
1
vote
1answer
24 views

Extract sum of coefficients in a binomial expression

I have two questions: (1) Given $(1-x+x^2)^{3n}=c_0 + c_1 x + \dots +c_{6n} x^{6n}$, find $c_0+c_1+ \dots +c_n$. I manage to find $c_0+c_1+ \dots +c_{6n}$ by putting $x=1$ but I do not know how to ...
11
votes
3answers
860 views

Why study metric spaces?

Most universities have a 3rd year undergraduate analysis course in which metric spaces are studied in depth (compactness, completeness, connectedness, etc...). However, in practice it seems that most ...
0
votes
1answer
59 views

Learning math by analyzing/proving theorems?

Hello I want to learn mathematics. In order to do this I want to get familiar with formulas/theorems by taking one and just analyze it and try to manipulate it to understand it better. I wanted to ...
20
votes
11answers
9k views

3D software like GeoGebra

Does it exist a free interactive geometry software, like GeoGebra, which works for 3D geometry? I would be able to draw spheres, great circles, and so on.
108
votes
44answers
13k views

What's your favorite proof accessible to a general audience? [closed]

What math statement with proof do you find most beautiful and elegant, where such is accessible to a general audience, meaning you could state, prove, and explain it to a general audience in ...
2
votes
0answers
27 views

How should the topic of frequency domain be taught?

This is a soft question but I think it's a real fact.The Frequency domain has made revolution in the field of Mathematics, Digital Signal and image Processing. Some of concepts which are very ...
160
votes
14answers
10k views

Identification of a quadrilateral as a trapezoid, rectangle, or square

Yesterday I was tutoring a student, and the following question arose (number 76): My student believed the answer to be J: square. I reasoned with her that the information given only allows us to ...
3
votes
8answers
510 views

Ceiling and floor functions

What are some real life application of ceiling and floor functions? Googling this shows some trivial applications.
9
votes
4answers
554 views

Non-traditional math concepts for early education

I am currently working on source material for a math-related software project with my mother, who has a PhD in Elementary Education and specializes in math education. While she has quite a strong ...
2
votes
0answers
38 views

how to teach steady state in queueing - if at all? [closed]

I am teaching an undergraduate course in Operations Research to business students (they are not: maths students). I want to check, if and how teaching the steady state makes any sense. As in the ...
12
votes
1answer
300 views

Algebraic structures associated to flexagons?

Flexagons strike me as objects that would admit investigation in a first course in modern algebra. I'm surprised to be unable to find a reference discussing flexagons using modern algebra language. ...
3
votes
2answers
618 views

What's the most effective ways of teaching kids - times tables?

I'd like to help a $6$ year old who already has a pretty good grasp of $2$, $5$, and $10$ times tables.
0
votes
1answer
20 views

Algorithm for finding Complex Eigenvectors?

I'm wondering if there's a fairly easy algorithm by which one can, by hand, find eigenvectors corresponding to complex eigenvalues for small matrices. Of course, one can always row reduce, but it can ...
6
votes
2answers
408 views

How to explain lagrange multipliers to a lay audience?

So I will be giving a seminar to a scientifically mature lay audience (think bio/social science undergrad level). I have been told that I should count on less than half the audience to have experience ...
1
vote
2answers
146 views

Why study Lowest Common Multiple - LCM

What is the most motivating way to introduce LCM of two integers on a first elementary number theory course? I am looking for real life examples of LCM which have an impact. I want to be able to ...
10
votes
15answers
12k views

What concepts were most difficult for you to understand in Calculus? [closed]

I'm developing some instructional material for a Calculus 1 class and I wanted to know from experience for yourself, tutoring others, and/or helping people on this site where is the most difficulty in ...
17
votes
9answers
3k 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, ...
8
votes
4answers
905 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 ...
5
votes
1answer
115 views

Best program for creating educational math animations?

I'm looking for recommendations on what program to use for creating mathematical animations. These animations will be used in creating educational videos for high school math -- Trigonometry first, ...
11
votes
1answer
152 views

Has the age at which we teach Mathematics changed over the last two centuries?

My experience of learning Advanced Trigonometry and Calculus is that it was done to 17 and 18 year olds (School Curriculum in Australia). I assumed that it was similar in the UK, US and Europe. In ...
7
votes
3answers
165 views

Algebraic number theory topics for undergrads

What are some interesting topics or problems in algebraic number theory which could be presented to students in a first undergraduate algebra course (which covers some elementary number theory, ...