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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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 ...
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14answers
12k views

Why 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) ...
45
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1answer
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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 ...
10
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3answers
1k 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 ...
446
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138answers
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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 ...
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17answers
6k 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
5answers
650 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 ...
14
votes
1answer
474 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 ...
27
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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 ...
47
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20answers
3k 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 ...
20
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15answers
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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 ...
49
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9answers
17k 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
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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 ...
11
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9answers
2k 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, ...
15
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4answers
865 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 ...
223
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33answers
31k 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} ...
45
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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 ...
23
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6answers
2k views

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 ...
7
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9answers
698 views

Motivating implications of the axiom of choice?

What are some motivating consequences of the axiom of choice (or its omission)? I know that weak forms of choice are sometimes required for interesting results like Banach-Tarski; what are some ...
6
votes
6answers
2k views

Why Doesn't This Series Converge?

I am teaching a Calc II course and came across the following series when finding the interval of convergence for the Taylor series of $f(x)=\sqrt{x}$ centered at $x=1$: $$ \sum_{n=2}^\infty ...
49
votes
16answers
6k 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 ...
50
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17answers
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What are some good ways to get children excited about math?

I'm talking in the range of 10-12 years old, but this question isn't limited to only that range. Do you have any advice on cool things to show kids that might spark their interest in spending more ...
22
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10answers
1k views

Puzzles or short exercises illustrating mathematical problem solving to freshman students

At high school, the solution method to almost all mathematical exercises is to apply some technique or algorithm you have learned before. At the university, the situation is fundamentally different. ...
82
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12answers
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: $$ ...
23
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4answers
4k 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 ...
50
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19answers
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How do I convince my students that the choice of variable of integration is irrelevant?

I will be TA this semester for the second course on Calculus, which contains the definite integral. I have thought this since the time I took this course, so how do I convince my students that for a ...
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3answers
1k 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 ...
35
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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: ...
28
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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 ...
11
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2answers
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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 ...
32
votes
12answers
2k views

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 ...
11
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9answers
838 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, ...
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15answers
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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, ...
17
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7answers
706 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 ...
11
votes
15answers
9k views

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

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 ...
5
votes
3answers
884 views

Books that develop interest & critical thinking among high school students

I heard about Yakov Perelman and his books. I just finished reading his two volumes of Physics for Entertainment. What a delightful read! What a splendid author. This is the exact book I've been ...
12
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4answers
287 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 ...
4
votes
5answers
788 views

Differential Geometry of curves and surfaces: bibliography?

Dear all, next year, I will probably teach a one-semester course of Differential Geomtry of curves and surfaces. Its content must be something along the lines of the first four chapters of Do Carmo's ...
8
votes
1answer
341 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 ...
5
votes
4answers
914 views

Self-teaching myself math from pre-calc and beyond.

Going to be starting grade 12 (pre-calculus) shortly and looking to get ahead. I would like to try some more rigorous stuff on my own and have a couple questions. Ideally I would like to be prepared ...
9
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4answers
584 views

The Power of Taylor Series

I am teaching a Calculus class and we are finishing up power/Taylor series this week. The last section of the chapter is on applications, but the only ones listed there are approximating non-rational ...
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6answers
4k views

Which Mathematical Analysis I Book or Textbook Is The Best?

I'm in search of a mathematical analysis text that covers at least the same material as Walter Rudin's Principles of ... but does so in much more detail, without relegating the important results to ...
1
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1answer
601 views

Purpose of Inverse matrix

What use is the inverse matrix? I would not use it to solve linear systems but there must be some concrete or real life applications where it is used.
11
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2answers
594 views

Etymology of the word “normal” (perpendicular)

While the word "normal" is one of the most overloaded mathematical terms, in linear algebra, it is usually associated with the notion of being perpendicular to something, as in "normal vector" or ...
7
votes
3answers
5k views

How to justify small angle approximation for cosine

Everyone knows the picture that explains instantly the small angle approximation to the sine function (as defined by the parametrisation of the unit circle): "what's the length of that arc?" "See how ...
261
votes
29answers
37k views

My sister absolutely refuses to learn math [closed]

My 13-year-old sister has a problem which, given the way math is currently taught, I doubt is anything but all too common. She has a low grade in her math course and only ever attempts to memorize ...
82
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19answers
3k views

Good Physical Demonstrations of Abstract Mathematics

I like to use physical demonstrations when teaching mathematics (putting physics in the service of mathematics, for once, instead of the other way around), and it'd be great to get some more ideas to ...
65
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31answers
7k views

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

I am having an argument with someone who thinks that it's never justified to teach something that is not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is ...
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13answers
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In calculus, which questions can the naive ask that the learned cannot answer?

Number theory is known to be a field in which many questions that can be understood by secondary-school pupils have defied the most formidable mathematicians' attempts to answer them. Calculus is not ...
63
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22answers
6k views

Is math built on assumptions?

I just came across this statement when I was lecturing a student on math and strictly speaking I used: Assuming that the value of $x$ equals <something>, ... One of my students just rose ...