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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15answers
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What concepts were most difficult for you to understand in Calculus? [on hold]

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 ...
15
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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, ...
8
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4answers
818 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 ...
4
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1answer
20 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, ...
104
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44answers
12k 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 ...
11
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1answer
139 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 ...
17
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1answer
525 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 ...
10
votes
2answers
476 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 ...
-1
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0answers
49 views

Why do some mathematics professors teach more/less courses than others? [migrated]

Not sure if this belongs on the Academia site, but since I'm a math major and the question is based solely on my observation of mathematics professors, I figured this site would be best. At my ...
7
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3answers
149 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, ...
1
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5answers
140 views

Why do counits go that way?

Imagine you want to motivate for an audience the definition of an adjunction in terms of unit and counit. So you can say: Often two functors $\mathcal{C} \begin{array}{c} \stackrel{\large ...
56
votes
16answers
7k 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 ...
4
votes
2answers
271 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 ...
2
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0answers
71 views

Argument for the zero vector not being defined as an eigenvector

Two days ago, my lecturer of Advanced Numerical Methods gave a review on the topic about eigenvalues and eigenvectors. Just as the lecturer presented the definition of eigenvalues and eigenvectors, a ...
18
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16answers
4k 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, ...
1
vote
1answer
54 views

Understanding and teaching the concept of derivative

I need to prepare an introductory lecture about derivatives and the concept of differentiation to a class of people with a general mathematical background (who have also studied calculus a few years ...
2
votes
8answers
337 views

A pedagogical proof that 9's can be ignored when calculating digital roots

I was asked by an elementary school teacher for a proof that you can ignore all 9's when calculating the digital root of a number. For instance, when calculating the digital root of 7593329, you ...
1
vote
0answers
42 views

Ideas for math problem solving class for undergraduate students in university

In our university there is a huge gap between two group of students. a group of them came from Math Olympiad competitions and have a very strong background from high school but others, they have just ...
6
votes
1answer
120 views

Transitioning to Higher Level Mathematics

I am just finishing grade 12 pre-calculus at my school and have strong interest in math. The problem is, it seems some important elements of higher level math are not in my schools curriculum that are ...
3
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0answers
86 views

Examples of categories which appear naturally without objects

Regarding the morphisms-only-definition of a category (which is equivalent to the usual one dealing with objects and morphisms), I would like to ask: Which examples of categories in practice appear ...
55
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25answers
6k 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 ...
2
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3answers
83 views

Teaching cardinality

I would like to give a class of 60 minutes to my undergraduate students about cardinality. I would like to begin with the definition of cardinality and end with one or two good application of this ...
4
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2answers
89 views

Is there a way to prove that the order of an element in a Group divides the order of the Group, WITHOUT USING LAGRANGE'S

This is a very easy fact we use in Group Theory, But somehow, I wondered that whether there may be another way (other than Lagrange's Theorem) to prove that the order of an element divides the order ...
19
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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.
6
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1answer
4k views

What are good resources to self-teach mathematics?

I am teaching myself mathematics using textbooks and I'm currently studying the UK a-level syllabus (I think in the USA this is equivalent to pre-college algebra & calculus). Two resources I have ...
3
votes
4answers
127 views

Good way to convince a young kid that $0*0 = 0$?

My little brother (6 years old) asked me a question ("What is $0*0$?") and gave an answer to his own question which I found ridiculous so I refuted it but he still thinks he is right. He says that ...
0
votes
1answer
36 views

Find functions with ''smart'' tangents.

This is a didactic question. Given a differentiable function $y=f(x) \;, x,y \in \mathbb{R}$, I want to construct an exercise in which we have to find a straight line that passes through a point ...
3
votes
1answer
86 views

Two categories sharing the same objects and morphisms

Is there a natural example of two categories $\mathcal{C}$, $\mathcal{C}'$ which have the same class of objects and the same class of morphisms, including source and target maps, but different ...
11
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3answers
121 views

Can you recommend a book to learn to teach math to a child?

I am looking for a book which contains some ideas on introducing a child to mathematics. I am not particularly looking for a textbook to be used as part of the teaching (though feel free to mention ...
25
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13answers
685 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 ...
518
votes
151answers
32k 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 ...
1
vote
2answers
76 views

Abstract/formal interest of rings

I am about to introduce first year undergrads to the concept of rings, after spending some time looking at groups; and I would like to give them more than a practical motivation (the most usual rings ...
1
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0answers
61 views

Intuition behind the link between coding theory and group theory

I am trying to find an easy link between group theory and coding theory. The usual path that most of the texts follow is that they present introductory material on groups, fields, rings, etc., and ...
2
votes
2answers
58 views

Necessity of algebraic symbolism

We solve different problems algebraically .For example,if we add $20$ with a number and the sum is $42$.What is the value of the number.To solve we denote the number as $x$ and write like this ...
92
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 ...
0
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1answer
100 views

How to explain this question to a 6 year old

My daughter who is in 1st grade is learning to grasp he meaning of multiplication and has not yet been introduced to division. she is appearing for Kangaroo Math Competition. Following question has ...
5
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3answers
98 views

Explaining that $1 \cdot 3 \cdot 5 \dotsm (2n+1) = 1 \cdot 3 \cdot 5 \dotsm (2n-1)(2n+1)$

I have a few students that are having trouble understanding that $$1 \cdot 3 \cdot 5 \dotsm (2n+1) = 1 \cdot 3 \cdot 5 \dotsm (2n-1)(2n+1),$$ specifically that $$\frac{1 \cdot 3 \cdot 5 \dotsm ...
4
votes
2answers
373 views

An example of a great explanation or freely accessible article on a math concept [closed]

Question: Give an example of a great explanation or freely accessible article on a math concept (suitable at the undergraduate or lower level), and explain why you think it is great. Possible ...
15
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3answers
391 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 ...
12
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2answers
2k views

Is there a way of intuitively grasping the magnitude of Graham's number?

I have heard it stated before that Graham's number is so vast that it is completely beyond comprehension. It is way larger than the number of atoms in the universe, so cannot be related to real ...
23
votes
11answers
2k 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. ...
1
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4answers
78 views

What is not the second derivative of a parametric equation?

1142004    Consider the parametric equations $x=f(t)$ and $y=g(t)$. To "find" $\frac{d^2y}{dx^2}$, there are three ways to go: (1) the correct one, that is, ...
6
votes
1answer
120 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'} ...
2
votes
0answers
64 views

A good way to explain $\varepsilon$-$\delta$ for chemistry / biology students?

I feel like I have a pretty good way to talk about $\varepsilon$-$\delta$ to physics and engineering students (and possibly students in comp sci). But I am not very sure what I can do for chemistry ...
2
votes
1answer
105 views

Why do we need primitive roots?

What is the most motivating way to introduce the order of a modulo n? Apart from simplifying powers of residues is there any other use of the order? Are there any examples which have a real impact on ...
0
votes
1answer
687 views

“Real world” applications of rational functions

I need a rational function/equation beyond the contrived d=rt and work problems typically given in beginner algebra. I am teaching such a class and would like to motivate the study of techniques for ...
1
vote
1answer
63 views

Why are Fourier series important?

Are there any real life applications of Fourier series? Are there examples of Fourier series which have an impact on students learning this topic. I have found the normal suspects of examples in this ...
3
votes
1answer
33 views

Find a set of minimal natural axioms, from which we construct $\mathbb{Z}$.

I am interested in this question for teaching two very different kind of students. The first (less important to me) is students in their first year in the university. I wish to construct ...
19
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9answers
1k views

Should a high school introductory calculus class teach $\varepsilon$-$\delta$ proofs?

It seems to me that most high school students are comfortable with the intuitive notion of a limit ("as $x$ gets arbitrarily close to $c$, $f(x)$ gets arbitrarily close to $L$") and gain little ...
2
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2answers
11k views

Why are prime numbers important in real life? [duplicate]

What practical use are prime numbers? Why do we emphasise the teaching of prime numbers?