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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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 ...
4
votes
2answers
157 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 ...
3
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1answer
45 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 ...
22
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10answers
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. ...
3
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5answers
117 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 ...
2
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1answer
38 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'> ...
7
votes
4answers
291 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 ...
4
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3answers
149 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 ...
1
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0answers
31 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 ...
4
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2answers
160 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?
0
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1answer
68 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 ...
3
votes
1answer
432 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 ...
0
votes
1answer
54 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 ...
4
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4answers
320 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 ...
0
votes
1answer
57 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 ...
3
votes
2answers
292 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 ...
0
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4answers
328 views

Ceiling and floor functions

What are some real life application of ceiling and floor functions? Googling this shows some trivial applications.
6
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2answers
157 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 ...
2
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2answers
9k 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?
7
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4answers
717 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 = ...
1
vote
1answer
113 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 ...
1
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0answers
70 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 ...
22
votes
6answers
5k 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 ...
1
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2answers
2k views

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 ...
4
votes
5answers
187 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 ...
4
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4answers
144 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 ...
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 ...
5
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4answers
2k 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 ...
6
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1answer
135 views

The Value of a series

What is the value of the following series $\sum_{n=1}^\infty\sum_{m=1}^\infty\sum_{k=1}^\infty \frac{1}{mnk(m+n+k+1)}$
3
votes
2answers
131 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 ...
5
votes
7answers
333 views

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 ...
0
votes
1answer
102 views

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, ...
1
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2answers
147 views

Software for visualizing partial derivatives?

I'm whipping up a set of notes, and I want to include a diagram or two showing some partial derivatives. Specifically, a diagram would include: a 3D surface of the form z=f(x,y), a plane of the form ...
2
votes
0answers
94 views

What's the acceptance of rational trigonometry in current mathematics courses?

I've been reading about Wildberger's rational trigonometry and I'm willing to learn it. I'm wondering if it's usage is accepted in undergraduate mathematics courses. It seems there's a redefinition on ...
84
votes
12answers
11k 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: $$ ...
11
votes
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 ...
4
votes
3answers
159 views

Operations on negative integers

I was trying to teach my younger sister some math, and it drifted on to integers, and operations on negative integers. So questions like: a) $-3+2 = ?$ b) $2- (-3)= ?$ c)$-3 -2 = ?$ had to be ...
0
votes
1answer
305 views

Examples of Metaphors in Statistics and Probability?

I have a couple of questions about teaching of Probability and Statistics for high school students: 1. Can I find metaphors for the teaching of basic concepts of Probability and Statistics? (Please, ...
0
votes
1answer
142 views

Graphical Demonstration of Linear Transformations on $\mathbb{R}^2$

I'm looking for some applet, program, software, demonstration etc. to use it in a class while teaching linear transformation in $\mathbb{R}^2$, so that the students can graphically understand the ...
4
votes
1answer
171 views

Learning Complex Analysis: Integrals vs. Power Series - ordering the development of results.

Over the last few months, I have been visiting elementary complex analysis. My exposure to complex analysis is pretty much limited to the material in three books: Ahlfors, Bak/Newman, and ...
3
votes
2answers
143 views

What is an effective means to make divisibility tests a mathematical 'habit', particularly for algebra?

Divisibility tests are a useful problem-solving technique for particularly dealing with larger numbers (thousands etc) and algebraic problems. However, I have always found that many students will just ...
0
votes
2answers
146 views

Request for reference and technical support

I am going to write my master thesis in order to become a teacher of mathematics (with second subject business management). Supported by the ERASMUS program I have the opportunity to do this in ...
4
votes
3answers
91 views

First-grader problem in arithmetic

I found this problem in a text book on arithmetic for first graders (7 y.o.) of the former USSR* . The problem comes from the section that covers single-digit addition and subtraction. Here is the ...
3
votes
4answers
331 views

What is an effective means to get senior high school students to write their complete working out as part of their answer.

In Australia and in the International Baccalaureate (2 systems I have worked in), for better or worse, mathematics is assessed by criteria. This increases the importance of students to express their ...
7
votes
2answers
498 views

How to teach a High school student that complex numbers cannot be totally ordered?

I once again need your precious knowledge! I am not sure which is the best pedagogic way to teach a High school student about why complex numbers cannot be totally ordered. When I was in High school ...
10
votes
2answers
528 views

Is “problem solving” a subject to be taught?

Note: This question has been cross-posted to MathOverflow: see here. I am witnessing a new curriculum change in my country (Iran). It includes the change of all the mathematics textbooks at all ...
3
votes
0answers
92 views

How to get interest in the mathematics of tax

In a similar vein to my previous thread, I will also be teaching about the mathematics behind taxation - to a lot of people, this is very mundane - but that is not true of everyone. The practicality ...
4
votes
2answers
95 views

What is an effective and practical means to teach about natural logarithms and log laws to high school students?

My students are quite practically minded, and I have found that teaching them concepts in a practical manner to be very helpful (maths 'experiments'; modelling on the smartboard etc). I am looking ...
12
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2answers
4k views

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 ...
36
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26answers
4k views

How to teach mathematical induction?

Some students are not convinced that a proof by mathematical induction is a proof. I have given the analogy of dominoes toppling but still some remain unconvinced. Is there very convincing way of ...