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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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 ...
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0answers
110 views

Figurate Numbers Project

I am teaching a course on proof. We have learned the methods of proof: direct proof, proof by contrapositive, by contradiction, by induction, etc. We have also done cardinality, modular arithmetic, ...
0
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1answer
406 views

In which branch of mathematics does “logarithm” belong? Arithmetic or algebra?

I'm currently working on an iOS & Android application for GCE O Level students. I have to classify everything neatly such that Maths never appears to be a messy subject to study and so should I ...
4
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1answer
192 views

Reading circle in mathematics?

How to learn about interesting topics in a small group of people?It seems very useful to broaden your mathematical background and get to know topics that are away from your field of specialization. ...
2
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2answers
160 views

Numeric synaesthesia: uses of and advice for learning math.

It turns out that my adolescent son might have numeric synaesthesia-- numbers have specific colors and possibly other distinguishing characteristics for him. He has shown that he can commit long ...
6
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2answers
320 views

Extremely intuitive geometric proofs for teaching

Does anyone know where I could find a book or resource of very simple intuitive proofs of the basic results in Geometry? I tutor geometry to middle schoolers, and find that due to shoddy mathematical ...
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16answers
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An example for a calculation where imaginary numbers are used but don't occur in the question or the solution.

In a presentation I will have to give an account of Hilbert's concept of real and ideal mathematics. Hilbert wrote in his treatise "Über das Unendliche" (page 14, second paragraph. Here is an English ...
0
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1answer
82 views

Basic Probability: activities

I have to talk about conditional probability, Bayes theorem and law of large numbers. It is a talk for students who are not pure mathematicians, then it is in a simple way. I have prepared some ...
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2answers
198 views

How can I infer a result using primal feasibility, dual feasibility, and complementary slackness?

I am trying to find the minimum of $-x_1$ with restrictions $\bar g\leq\bar 0$ so that $$\bar g=\begin{pmatrix} (x_1+2)^2+(x_2-4)^2-20\\ (x_1+2)^2+x_2^2-20\\ -x_1\end{pmatrix}\leq ...
6
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5answers
439 views

How to present math as something interesting?

If you are helping someone with mathematics in high school, what do you need to do to win his attention so she/he can focus on curriculum?
9
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5answers
579 views

Am I fit for higher studies/teaching in mathematics? [closed]

Let me begin with some background: I used to enjoy mathematics immensely in school, and wanted to pursue higher studies. However, everyone around me at that time told me it was a stupid area (that I ...
12
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7answers
778 views

Defining the derivative without limits

These days, the standard way to present differential calculus is by introducing the Cauchy-Weierstrass definition of the limit. One then defines the derivative as a limit, proves results like the ...
4
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3answers
209 views

Cancel before multiplying!!

$$ \binom{12}6 = \frac{12\cdot11\cdot10\cdot9\cdot8\cdot7}{6\cdot5\cdot4\cdot3\cdot2\cdot1} = 924. $$ Sometimes it's hard to talk students out of computing both the numerator and the denominator in ...
13
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2answers
365 views

Is ripping off exercises plagiarism? [closed]

Just a quick question. I teach some undergraduate mathematics. I like to produce notes that contain exercises. Sometimes I make my own exercises, sometimes I take exercises from various sources and ...
4
votes
3answers
4k views

What is the most effective way of teaching mathematics to my 7 year old kid?

I want to help my kid excel in Math. I can see she has some trouble with additions,subtraction, multiplication and division. Will it help if I let her memorize the multiplication table? or use ...
10
votes
3answers
497 views

Statements in Euclidean geometry that appear to be true but aren't

I'm teaching a geometry course this semester, involving mainly Euclidean geometry and introducing non-Euclidean geometry. In discussing the importance of deductive proof, I'd like to present some ...
2
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3answers
152 views

Proofs for Undergraduates

I am teaching a course in proof technique to undergraduate students. One of the things they can do for their project is read an involved proof and explain it, for example Gödel's Incompleteness ...
5
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4answers
378 views

Counterexamples to “Naive Induction”

I was teaching a nine-year-old friend about prime numbers. When I asked him if he thought there were finitely or infinitely many primes, he answered confidently that there must be an infinite number. ...
6
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1answer
107 views

Integral using multiple methods

I'm teaching a Calculus II class and we are covering integration techniques. We've covered $u$-substitution, integration by parts, trig integrals, trigonometric substitutions, partial fractions and ...
15
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8answers
2k views

Why are epsilon-delta proofs difficult?

What conceptual difficulties do students learning epsilon-delta proofs have, or, why are the proofs difficult? Motivation I have to teach people and never found the epsilonistics particularly ...
8
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6answers
450 views

What are the technical reasons that we must define vectors as “arrows” and carefully distinguish them from a point?

I thought I'd bring this question to math.SE, as it could spark some interesting discussion. When I first learned vectors - a long time ago and in high school - the textbook and teachers would always ...
3
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2answers
142 views

Undergraduate approach to a problem about convex functions

I am preparing some sheets of exercises that I'll assign to my undergraduate students in biology (sophomore class, or first academic year in italian universities). This is the problem: Exercise. Let ...
52
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3answers
3k views

Getting Students to Not Fear Confusion

I'm a fifth year grad student, and I've taught several classes for freshmen and sophomores. This summer, as an "advanced" (whatever that means) grad student I got to teach an upper level class: Intro ...
10
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3answers
560 views

Why this proof $0=1$ is wrong?(breakfast joke)

We have $$e^{2\pi i n}=1$$ So we have $$e^{2\pi in+1}=e$$ which implies $$(e^{2\pi in+1})^{2\pi in+1}=e^{2\pi in+1}=e$$ Thus we have $$e^{-4\pi^{2}n^{2}+4\pi in+1}=e$$ This implies ...
2
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1answer
429 views

Compact and Locally Compact Spaces

I would like to consult with anyone who is reading this post on how do you explain the distinction between compact spaces and locally compact spaces to students who had just completed topology course ...
9
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7answers
1k views

What are the practical uses of $e$?

How can $e$ be used for practical mathematics? This is for a presentation on (among other numbers) $e$, aimed at people between the ages of 10 and 15. To clarify what I want: Not wanted: ...
12
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7answers
2k views

Why we need to know how to solve a quadratic?

Five years ago I was tutoring orphans in a local hospital. One of them asked me the following question when I tried to ask him to solve a quadratic: Why do I need how to solve a quadratic? I am ...
27
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6answers
2k views

How to tell $i$ from $-i$?

Suppose now we are trying to explain to students who do not know complex numbers, how do we distinguish $i$ and $-i$ to them? They will object that they both squared to $-1$ and thus they are ...
2
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6answers
5k views

Real life applications of general vector spaces

Students familiar with Euclidean space find the introduction of general vectors spaces pretty boring and abstract particularly when describing vector spaces such as set of polynomials or set of ...
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6answers
671 views

Is the Mean Value Theorem interesting to engineers, scientists, and others?

I am trying to decide whether to include whether to include the Mean Value Theorem in a calculus course I will be teaching. I am sort of leaning away from it, in light of the interesting discussion ...
15
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5answers
983 views

Best books in the genre “______ for Mathematicians”

I once heard someone (perhaps from someone famous -- anyone have a citation?) say that there ought to be a series of books called "__ for Mathematicians," each one of which would explain a different ...
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0answers
92 views

References on the equivalence of different definitions of integrability

While writing a chapter of a book about mathematical analysis, I decided to compare some definitions of integrability that are usually taught to sophomore students, in Italy. I briefly collect four ...
3
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1answer
401 views

How did the teaching of mathematics change since the 19th century?

Background: 1) In the book Gems of Geometry, chapter 9 Relativity, the author wrote: "The General Theory concerns gravitation and the mathematics behind it is considered rather difficult ( post- ...
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4answers
199 views

Examples of logs with other bases than 10

From a teaching perspective, sometimes it can be difficult to explain how logarithms work in Mathematics. I came to the point where I tried to explain binary and hexadecimal to someone who did not ...
3
votes
1answer
222 views

What is a motivating way to introduce vectors?

What is a good way to introduce vectors on a linear algebra course so that students are motivated from the start? I need an opening which will have a real impact. Are there any motivating examples?
1
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1answer
87 views

Polynomials and exponentials: showing that $n^k \le c\cdot a^n$

If $k$ is any real and $a>1$, prove that there exists a $c>0$ such that for any integer $n\ge 1,$ $$ n^k \le c\cdot a^n $$ To forestall any complaints about the imperative nature of this ...
13
votes
7answers
2k views

Teaching abstract maths concepts to young children.

I am interested in opinions and, if possible, references for published research, about the pros and cons of teaching abstract maths concepts to young children. My younger brother (five years old) ...
1
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1answer
232 views

An elementary (?) minimization problem

This morning, in Italy, there was the national exam of mathematics for students of high schools. One of the exercises asked to solve Heron's problem: given a straight line and two points lying on the ...
5
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6answers
792 views

Purpose of Linear Algebra

How much emphasizes should be on proof on a first course in Linear Algebra? I sometimes feel that they (proofs) crowd out a coherent vision for linear algebra. However I also think a central theme of ...
8
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1answer
2k views

Tips and examples for a poster presentation in pure mathematics

I will be presenting a poster in a few weeks but have no experience with them. I've seen and given plenty of talks, read and written papers, but I have never made or even seen a poster in pure ...
36
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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: ...
0
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1answer
542 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.
6
votes
2answers
255 views

Teaching permutations, How to?

I posed this question to my niece while teaching her permutations: Given four balls of different colours, and four place holders to put those balls, in how many ways can you arrange these four ...
5
votes
1answer
178 views

Forum for students - your experiences, recommendations, suggestions

Have you ever used online discussion forum or something similar for students of your course? (Or, if you are a student, have you attended a class, where something like this was used?) If yes, I'd ...
5
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3answers
264 views

Linear Algebra Journal

Is there any journal which has significant material on the teaching of linear algebra. I am investigating the most effective way to teach a course on Linear Algebra. What are the most important things ...
-2
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1answer
321 views

Mat-1.1020 L2 course material generated by students? [closed]

Mat-1.1020 L2 course is a course usually taken by theoretical-physicist-dept students in Aalto University, here official site. It is a mass course that a massive amount of students fail every year. It ...
29
votes
17answers
5k 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 $\bullet+\cdots\cdots\cdots+\bullet$ ...
5
votes
1answer
548 views

Who came up with the arrow notation $x \rightarrow y$?

I read that the arrow notation $x \rightarrow y$ was invented in the 20th century. Who introduced it? Each map needs both an explicit domain and an explicit codomain (not just a domain, as in ...
12
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5answers
251 views

Issues with text problems

When I tutor, I often see people who kind of know the stuff they cover in school at the moment and succeed at straight problems like: Find the derivative of $f(x) = \frac 12 x^2$ But when it ...
5
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2answers
296 views

certain examples of fields of fractions

Rational numbers, rational functions, and Gaussian rationals are examples of fields of fractions. In each of those cases, one knows what the quotients are long before one hears of the idea of ...