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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5
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89 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 ...
0
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0answers
9 views

Need a type of function that would allow me to create “paterns” in a series of slots.

For example, lets say I have a table with 16 slots. I need them either filled or empty, and the logic behind who is filled/empty should follow a mathematical pattern. What are some functions that ...
1
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4answers
71 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, ...
4
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0answers
34 views

How can I teach Mathematics to GENIUS students? [migrated]

I'm a Math Teacher for highly talented students in High School. It's hard for me to find out how fast can I move through topics and how deep can I dive into different math subjects for enhancing ...
0
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0answers
17 views

Pedagogical question about induction/recursion

I'm having an informal debate with some internet forum friends about how "easy" induction is. My position is that it's "so easy even college freshmen are expected to understand it". But, I am ...
6
votes
1answer
111 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
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0answers
62 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 ...
0
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0answers
10 views

A wish for co-ordinate independent conic properties and utilities of polars.

In the Indian curriculum for high school (including 11th and 12th), major time and effort is dedicated to studying the conic sections. Most of the focus on these topics is knowing to write an equation ...
1
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1answer
48 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
28 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 ...
1
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0answers
41 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 ...
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0answers
27 views

Proper usage for term, addend, factor, multiplicand, expression, formula

The definitions and usage of the following words seem to vary, depending on the source text: term addend factor multiplicand expression formula The words are being used in the context of ...
-3
votes
3answers
71 views

Why does the author prefer function names after arguments?

As you can see above, the author prefers to write functions after elements, which is contrary to the century old practice of writing arguments after the functions. I wonder why he does that? Is ...
4
votes
2answers
351 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 ...
100
votes
43answers
12k views

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

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 ...
91
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 ...
9
votes
1answer
431 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 ...
9
votes
4answers
142 views

Substitution for definite integrals

In my experience, Calculus II students dislike changing bounds in definite integrals involving substitution. When facing an integral like $$\int_0^{\sqrt{\pi }} x \sin \left(x^2\right)dx,$$ for ...
5
votes
3answers
399 views

Can we teach calculus without reals?

This question is related to another question, Do we really need reals?, and could be considered a duplicate, so I would not be surprised if it will be put on hold. But I'm especially interested in ...
3
votes
1answer
58 views

Materials for teaching the axioms of the real numbers to high school students

I suddenly felt the urge to teach the axioms of the real numbers (i.e. the complete ordered field axioms) to a bright Year 10 student that I tutor, with an emphasis on the consequences of the field ...
53
votes
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 ...
0
votes
0answers
30 views

Fractal Generation Project: How to Continue after L-Systems?

Three months ago, we defined a programming project for some gifted middle school students: Writing code to generate simple fractals. Two of them have finished writing Pascal programs already! The ...
12
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6answers
246 views

Sources for mathematics outside the mathematics world

In this question I would like to ask you about material showing the uses (or occurrences) of mathematics in the everyday world. The aim is to encourage with it a group of young undergraduate ...
2
votes
1answer
158 views

Why is the axiom of choice not taught from the start to mathematics undergraduates?

I've recently discovered that the following theorems require the axiom of choice to be proven: every surjective function has a right inverse. a real-valued function that is sequentially continuous ...
7
votes
5answers
198 views

Why do we first introduce the open set definition for continuity instead of the neighborhood definition?

After (nearly) completing my course in topology, something weird just stuck out to me which I hadn't considered before. When first discussing continuity, we often use the following definition: Let ...
2
votes
4answers
94 views

Research Projects for 7th Grade Students

I'm going to define some math research projects for 7th grade students. The projects can be both purely mathematical and interdisciplinary. By the way, the students can write simple Pascal codes. I ...
15
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0answers
449 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 ...
2
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0answers
26 views

Determine the set of points $M$ of affix of $z\in\mathbb{C}$

Determine the set of points $M$ of affix of $z\in\mathbb{C}$ such that there exists at least one real $t$ satisfying $z^2=t(t-i)$ My attempt: We look for the form $z=x+yi$ and we want there is a ...
2
votes
1answer
96 views

Why is polynomial long division being taught in schools instead of Horner's method? [closed]

The Horner´s method is by a long shot easier than the Polynomial long division and serves the same purpose. Why isnt it being taught in school (in germany at least)?
0
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3answers
77 views

Is there an elementary “proof” that the first degree equation has only one solution?

I was asked from a student why the first degree equation has only one solution (if it has a solution) Let's consider the equation $2x+5-3x=-4x+14$ for example. How can I explain to a 13 year old ...
6
votes
1answer
152 views

How was real analysis & topology taught in the 70's?

What was the 'gold standard' textbook before Rudin? Furthermore, if anyone has knowledge of what textbooks Princeton or Harvard used back in the 1960's or 70's, I would highly appreciate it if you ...
6
votes
5answers
164 views

Topic for a lecture intended for High School students [duplicate]

I am not sure if this is the right place to post this, but here is the situation. In about two weeks or so I will be giving a 2-3 hours lecture on some topic in mathematics to freshman and sophomore ...
10
votes
2answers
98 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 ...
0
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1answer
58 views

Basic examples of probabilistic method

I'm looking for a truly basic example of probabilistic method proof which could be presented without a board (i.e. speaking only), that is, even moderately complicated calculations are not allowed. ...
2
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1answer
65 views

Undergraduate Project Suggestions

A student of mine has expressed interest in doing an independent project next quarter with me. This would not be for credit and it is purely for her own educational stimulation. She wants to study ...
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2answers
68 views

Cartesian product sets

I'm preparing a lesson on the Cartesian product of two sets and I have run into the following confusion: I understand that the Cartesian product is not a commutative operation. Generally speaking, ...
1
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2answers
65 views

Fun results from modular arithmetic

I'm trying to go through various bits of neat, fun math with some junior-high-school students in my local area, and am thinking of doing a short unit on modular arithmetic/finite groups. I'm looking ...
1
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3answers
93 views

Applications of derivatives outside mathematics and physics

I've been teaching calculus for several years and have some doubts about whether derivatives (and integration techniques) of common functions are useful and important outside mathematics and physics. ...
0
votes
1answer
20 views

How do you teach the correctness of bijections when dealing with counting in combinatorics?

Consider the problem of awarding prizes to people in a school. Let $A$ be the set of awards and $|A| = m = 3$. Let $P$ be the set of people in the school and $|P| = n$. Then in how many ways can ...
22
votes
9answers
438 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 ...
8
votes
3answers
207 views

difficulty of accepting $i^2 = -1$ for first timers [duplicate]

While teaching complex numbers for those who are encounter for the first time (usually 10th grader and 11th grader), I get the question like "Can even squared number give negative results? How ...
9
votes
3answers
396 views

Very elementary proof of that Euler's totient function is multiplicative

Well, I know two or three proofs of this fact $$\gcd(m,n)=1\implies \varphi(mn)=\varphi(m)\varphi(n)$$ where $\varphi$ is the totient function. My problem is this: I'd like to explain this to some ...
0
votes
0answers
44 views

book/exercices for a really clever studient of 14 years old

I teach some class of Math to a 14 years old boy. This boy is quite clever (and is quite bored by his math class) so i would like to give him some interesting problems in accordance with his level of ...
1
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2answers
77 views

Why does induction procedure of Euler characteristic fail for non-convex polyhedra? What am I missing?

Euler characteristic of convex polyhedra is always $V-E+F=2$. Induction procedure reduces edges and vertices until we are down to one vertex whose $V-E+F=2$ and hence you are done. The same ...
1
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1answer
66 views

Analytic skills in applied math

I am an unexperienced undergraduate student just took few basic math classes. And I have found analysis classes really interesting, like basic analysis, measure theory and functional analysis, and ...
1
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2answers
56 views

Can I define the limit of a sequence like this?

It is well-known that a sequence has a limit if and only if it is bounded and has a unique limit point. I think this is a better definition of the limit of a sequnece, comparing with the $\epsilon-N$ ...
1
vote
1answer
62 views

Using Latex in classroom instead of blackboard?

Does anyone find it doable to use Latex in a classroom setup instead of blackboard? I would think it is tempting except for the input speed one can reach. But considering the messiness of chalks, and ...
1
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2answers
119 views

How to teach Critical Thinking

I am currently tutoring a few students in an entry level physics course and had some trouble recently when it comes to helping them with problem solving. The students I am helping don't have many ...
10
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5answers
254 views

Algebraic topology in high school?

This winter I am planning on teaching a small seminar (20 lectures 45 minutes each) for high school students. I was was given the freedom to choose the topic of the seminar, but it is supposed to be ...
0
votes
1answer
45 views

Confusion on Eigenvalues of Matrix

I'm a TA with Advanced Algebra in school and teach the Jordan Form now. There are three questions about eigenvalues in this chapter: Given matrix $A$, $B$ and polynome $f$, consider the eigenvalues' ...