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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Derivative in interesting way

I am supposed to give a 15-20 minutes math lecture, where I am expecting around 20-30 people. The lecture is about derivative. Since this would be my first "class", I would appreciate any suggestions ...
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1answer
221 views

What is the use of Euler Totient or Phi Function?

What is most motivating way of introducing this function? Does it in itself have any real life applications that have an impact. I can only think of a^phi(n)=1 (mod n) which is powerful result but ...
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1k views

Number of Lines Passing Through a Given Point in the Plane

How can one prove that infinite number of lines pass through a given point in plane, using Euclid's axioms (or Hilbert's, if necessary)?
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Why $\sqrt {-1}\cdot \sqrt{-1}=-1$ rather than $\sqrt {-1}\cdot \sqrt{-1}=1$. Pre-definition reason!

It is for years that I teach complex numbers following a historical route. I start with the famous problem of Cardano: Find two numbers whose sum is equal to 10 and whose product is equal to ...
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1answer
99 views

interactive training in mathematics olympiad competition for 8th Grade: Ages 13–14.

I'll enjoy your kindness to ask this question, despite that it seems it'snt the right destination. Please show me an url, for training in mathematics olympiad competition for 6th Grade: Ages 11–12. ...
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2answers
76 views

What are some good games for teaching maths to children?

I am due to teach maths to a ten-year-old. I'd like to try out some games such as Nim and Conway's Soldiers. I've found this list on Wikipedia but Googling for more just gives me a load of Flash ...
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0answers
176 views

Is there an elementary introduction to higher order functions?

I am teaching a pre-calculus course (using the textbook by Michael Sullivan if it helps), and I realized that higher order functions seem to show up in with some frequency in pre-calculus and ...
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4answers
289 views

How to explain Fractional and Negative Exponents

My classmates doesn't understand Fractional and Negative exponents, since I was the top of my class, so they all came to me... Is there any way to explain it clearly to them?
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2answers
81 views

An Integral With an Odd Function That Isn't Contrived

When ever I teach calculus, single or multivariable, there is always the point in the text when the author covers odd functions and then gives an example of an integral to evaluates to $0$ because the ...
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1answer
476 views

What is the use of the Chinese Remainder Theorem

What is the most tangible way to introduce the Chinese Remainder Theorem? What are the practical and really interesting examples of this theorem. I am looking for examples which have a real impact on ...
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2answers
360 views

What's an induction problem that will be hard to answer with “backwards reasoning?”

I'm currently the teaching assistant for a course that serves as an introduction to rigorous proofs, and I've noticed some of my students have a tendency to try and use a sort of "backwards reasoning" ...
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1answer
282 views

Best applications-oriented introductory calculus textbooks?

Note: I've edited this question on October 9th, after establishing a bounty on it. What are the best introductory calculus textbooks that explain why calculus is important in a broad intellectual ...
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2answers
131 views

Trying to teach supremum and infimum.

I'm helping out my former calculus teacher as a volunteer calculus advisor, and I have under my supervision 5 students. They've already had an exam and... well, they failed. I read their exams and I ...
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6answers
2k views

Cool mathematics I can show to calculus students.

I am a TA for theoretical linear algebra and calculus course this semester. This is an advanced course for strong freshmen. Every discussion section I am trying to show my students (give them as a ...
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5answers
376 views

How to explain infinty to a $3^{rd}$ grader?

In my country in $3^{rd}$ grade in math kids learn the four basic arithmetic operation (addition, subtraction, multiplication and divison) up to $10 000$. My sister this year goes to $3^{rd}$ grade ...
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1answer
252 views

Workshop on Pascal's Triangle for Middle School Students

We're going to hold a three-hour math workshop for some middle school students. It'll about the Pascal's triangle. Well, we can ask the students to find patterns in the triangle, or try to prove some ...
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6answers
379 views

“$n$ is even iff $n^2$ is even” and other simple statements to teach proof-writing

I am supposed to teach undergraduate students who do not major in mathematics and I would like to give them a short introduction to mathematical reasoning and to the concept of proof. I am looking for ...
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0answers
107 views

Variation on the Sobolev space $H^1_0$

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set, let $$ C^1_0(\overline\Omega) = \{u\in C^1(\Omega)\cap C(\overline\Omega):u|_{\partial\Omega}=0\}, $$ and let $C^1_c(\Omega)$ be the space of ...
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24answers
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Why is there no “remainder” in multiplication

With division, you can have a remainder (such as $5/2=2$ remainder $1$). Now my six year old son has asked me "Why is there no remainder with multiplication"? The obvious answer is "because it ...
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2answers
151 views

Can abstract nonsense be helpful here?

Here a question for those among you, who teach Homotopics/Algebraic Topology at university. I encountered some questions that were in my view quite easier to solve in category hTop instead of Top ...
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1answer
42 views

Proofs as games?

A long time ago (but I can't remember when), I was introduced to the (pedagogical) concept of writing a proof as giving a winning strategy for a game. Basically, given a statement $\forall x\exists y ...
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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. ...
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5answers
112 views

Motivation for Studying Combinatorics (Middle School Version!)

I'm going to teach very elementary combinatorics (limited to basic enumeration) during two weeks to middle school students. At the beginning, I want to demonstrate the importance of counting in real ...
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1answer
37 views

Identify the derivative of a distribution

When someone wants to identify the derivative of a distribution $T\in \mathcal{D}'(\mathbb{R})$, we usually write, for $\varphi\in\mathcal{D}(\mathbb{R})$ , $$<T',\varphi> = -<T,\varphi'> ...
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4answers
252 views

A Handwaving Proof of a Specific Existence and Uniqueness Theorem

My problem is as follows: Given the second order homogeneous linear differential equation with constant coefficients $$a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+c\,y(x)=0,$$ is there a good heuristic ...
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3answers
141 views

abstract algebra example book

It's very exciting when you can use the theory to solve "lower level" problems. For example, I'm looking forward to understanding why the quintic equation is not solvable. In the undergraduate ...
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0answers
29 views

The role of a uniqueness theorem for IVPs in a lower-division ODEs class

Please tell me your thoughts about this, and if you agree or disagree. I'll describe my current viewpoint, which is subject to change. Note that I've never taught a lower-division ODEs course. It ...
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2answers
148 views

What is the most motivating way to introduce modular arithmetic?

What the best way to introduce congruences in a number theory course? I am looking for something which will have an impact. What are the really interesting applications of congruent mathematics?
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1answer
64 views

Rates of Change Question with Mixed Units

I have come across the following question in a past exam paper of a module that I will be teaching this semester. The volume $V$ of m$^3$ of earth removed from a pit after $t$ hours is given by ...
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1answer
379 views

About game theory for high school students

I am a mathematician with a background in analysis who is teaching at a local high school in his spare time. There is some room for extra curricular math subjects and I want to use it for game ...
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1answer
52 views

Where are the resources on the prime number theorem?

I am looking for resources which explain the prime number theorem to 18 year old students. I am not seeking a proof of the result but something which will have an impact and motivate a student to ...
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4answers
273 views

How to Make an Introductory Class in Set Theory and Logic Exciting

I am teaching a "proof techniques" class for sophomore math majors. We start out defining sets and what you can do with them (intersection, union, cartesian product, etc.). We then move on to ...
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1answer
55 views

Are congruences difficult for beginners?

I am teaching an elementary number theory course out of Underwood Dudley's delightful book of the same name. Chapter 4 is on congruences, and it feels like Dudley devotes a substantial amount of ...
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2answers
265 views

Did Euler have a trick? [closed]

Did Euler have a trick for discovering things? Some sort of general method he could apply to mathematical objects he came across to see if they yielded any new truths? Did he just ask the right ...
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210 views

Ceiling and floor functions

What are some real life application of ceiling and floor functions? Googling this shows some trivial applications.
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2answers
153 views

Is it a problem being unable to understand a mathematical definition without examples?

I was reading a book on coding theory, there was a definition fot the Hamming's Distance and also one example. Understanding purely from the definition was hard but the example helped to give meaning ...
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2answers
8k 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?
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4answers
668 views

How to convince a high school student that differentials don't work like fractions in general?

It all started when I tried to convince a 10th grader that if $f$ is a function defined on $\mathbb{R}^n$ the differential is defined by: $\large \displaystyle df = ...
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1answer
110 views

A way to teach Archimedean property

A student asked me how to understand the Archimedean property, I tried to re-read with him what he has already done in class (well, actually copy from the blackboard in class). However I think I'm not ...
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0answers
69 views

If most of the mathematics needs a context to be not subject of interpretation, what part of mathematics doesn't need a context at all, if any?

In the past, I have asked this question here: Is mathematics the only language that is not subject of interpretation? And one of the answer started with: First, mathematics notation is subject ...
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6answers
4k views

Teaching a 4 year old maths

Im 18 years old and getting to grips with advanced mathematics (pre-university) and I have a younger brother of 4 years old (quite an age gap). I want to get him interested in learning (and away from ...
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2answers
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What is a standard precalculus syllabus?

I'm about to start teaching a calculus I class next week and I was wondering what I can expect from my students. I'm a Brit teaching in the US so I am unfamiliar with the system. I am hoping that ...
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5answers
183 views

Collecting different answers to a simple problem

I am collecting different answers to the problem below for possible publication in a pedagogical note I am writing. Please post freely (avoiding repetitions), and let your imagination go wild with ...
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4answers
138 views

Resources to help an 8yo struggling with math

Friends of mine asked me for suggestion for one of their children (age 8) who had bad scores at the local Star test (the family is based in California). Both parents work, so they have also limited ...
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8answers
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 ...
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4answers
1k 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 ...
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1answer
128 views

The Value of a series

What is the value of the following series $\sum_{n=1}^\infty\sum_{m=1}^\infty\sum_{k=1}^\infty \frac{1}{mnk(m+n+k+1)}$
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2answers
122 views

How to construct a cube

My friend has asked me this question. I have no idea how to answer, but I think the question is interesting enough to be noted here: Consider 3 pieces of wire (not necessary of equal length). Is it ...
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Is there a name for this type of logical fallacy?

Consider a statement of the form: $A$ implies $B$, where $A$ and $B$ are true, but $B$ is not implied by $A$. Example: As $3$ is odd, $3$ is prime. In this case, it is true that $3$ is odd, and ...
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Homogeneity Versus Heterogeneity in Student Groups

There is an overwhelming amount of research regarding homogeneous and heterogeneous grouping in education. The former refers to the practice of grouping "like" students together (regarding age, ...