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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496 views

How does one visualize a function with a discontinuous second derivative?

Let us assume that all functions are continuous. I was teaching my calculus students the other day. We were talking about what points of non-differentiability look like. Two ways a function can fail ...
11
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2answers
718 views

Etymology of the word “normal” (perpendicular)

While the word "normal" is one of the most overloaded mathematical terms, in linear algebra, it is usually associated with the notion of being perpendicular to something, as in "normal vector" or ...
12
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1answer
242 views

Algebraic structures associated to flexagons?

Flexagons strike me as objects that would admit investigation in a first course in modern algebra. I'm surprised to be unable to find a reference discussing flexagons using modern algebra language. ...
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4answers
1k views

Should I try to change the way Abstract Algebra is taught at my university? If so, how?

[This (soft) question should be Community Wiki.] Background: A year ago, I did a one-semester long course on Abstract Algebra at my university. When we started, I was excited, because I knew the ...
465
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141answers
29k 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 ...
2
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2answers
651 views

What are the drawbacks of multiple-choice questions? [closed]

I can easily understand the advantage of multiple-choice questions for instance in grading and so. A drawback is that real life problem don't have multiple choice questions all the time for instance ...
6
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2answers
211 views

What's the problem of using a “stand-up analogy” to demonstrate the concept of set?

I was reading this text about the new math movement, there's a line in which he says: Easy as it looked, teachers didn't always get the notion of "set" straight themselves, and could teach the ...
4
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2answers
157 views

Is it necessary to know a lot of advance math to become a good junior high/high school teacher?

By "advance math" I refer to Real Analysis, Abstract Algebra and Linear Algebra (to the level of Axler). I received mainly Bs in these courses with the exception of the intro-level Linear Algebra. ...
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3answers
141 views

Where could I learn basic math terminology?

I am an english learner and I would like to learn the etymology of Mathematics. I would like to know the most common phrases in Algebra, and Geometry as well. I want to know at a level of UK's A+. ...
9
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1answer
182 views

Difference between a Lemma and a Theorem [duplicate]

What essentially is the difference between a lemma and a theorem in mathematics? More specifically, suppose you come across a general result while solving a mathematical problem, what are the ...
4
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5answers
528 views

Motivation for the importance of topology

Starting from tomorrow, I will be tutoring some undergraduate students following a course in general topology. I am looking for examples motivating the importance of topology in mathematics which can ...
6
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1answer
290 views

How to most efficiently remedy mathematical deficiencies

My spouse and I are currently both pursuing our undergraduate degrees. I'm double majoring in Computer Science & Mathematical Sciences and my spouse is double majoring in Economics & Finance. ...
1
vote
1answer
292 views

Chi-square degrees of freedom proof

I need to prove why we have the following result: When: $Y_i=\beta_0+\epsilon_i$, then: $\sum\frac{\epsilon_i^2}{\sigma^2}\sim \chi ^2(n-1)$ Thanks :)
8
votes
1answer
243 views

Pedagogy of Teaching the Inverse Matrix Method

I am teaching a group of (ordinary rather than honours) second-year engineers and we are studying matrices. I told the class today that as far as I could see we were only studying matrices and, ...
2
votes
6answers
4k views

Which Mathematical Analysis I Book or Textbook Is The Best?

I'm in search of a mathematical analysis text that covers at least the same material as Walter Rudin's Principles of ... but does so in much more detail, without relegating the important results to ...
3
votes
0answers
67 views

Teaching Student's distribution

While it is fairly straightforward to show the basics of the normal distribution in a first year undergraduate course, how does a teacher provide good intuition when the Student distribution comes in? ...
3
votes
1answer
99 views

From (algebraic) topology to geometry

I am thinking about a "correct" didactic way of linking topology (algebraic topology) to geometry. Usually, we are taught introducing geometry first, then topology, almost as an abstraction of ...
8
votes
1answer
506 views

Is “A and B imply C” equivalent to “For all A such that B, C”?

So I mostly study PDE, harmonic analysis, image processing, and so on, but for whatever reason I decided to be a TA for an undergraduate "introduction to proofs" course this semester. I suppose I ...
14
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1answer
512 views

Alternative construction of the tensor product (or: pass this secret)

The paper Tensor products and bimorphisms by B. Banachewski and E. Nelson studies tensor products (defined by classifying bimorphisms) in concrete categories. It is quite interesting that their main ...
5
votes
2answers
241 views

How does one best balance learning from a “problem based book” with supplementary material?

We all know that when learning math, one has to do more than just simply read - one must try to solve problems and work actively with the material. Many books try to force the reader to participate ...
3
votes
1answer
203 views

How should someone release their proof to the world? [closed]

Lets say someone (a reputable or non reputable mathematician) has come up with a remarkable one page proof to a famous maths problem. Lets say the proof is likely correct but hasn't been released to ...
4
votes
1answer
982 views

Teaching probability by using a deck of cards

I plan to teach two sessions of probability to 11th grade students using a deck of cards. My classes will be next week. I have already taught them the basic notions of writing sample spaces, computing ...
1
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0answers
67 views

What is the best way of introducing singular value decomposition (SVD) in a linear algebra course?

Why is it so important? Are there any applications which have a real impact?
2
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4answers
89 views

Motivating complex structure on $\mathbb{R}^2$

I'm giving a talk to a group of bright but not all that mathematically sophisticated students on the subject of complex numbers. I'd like to introduce complex numbers via geometric considerations ...
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4answers
1k views

Common student mistakes/misconceptions in a first year calculus course

What are the common mistakes and misconceptions students make in a first year calculus course? More importantly: What can I do to prevent/rectify them? Context: Soon I will be doing ...
6
votes
2answers
887 views

Useless math that become useful

I'm writing an article on Lychrel numbers and some people pointed out that this is completely useless. My idea is to amend my article with some theories that seemed useless when they are created but ...
4
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2answers
189 views

What to give as final lecture in a differential geometry class?

During the fall semester, I had to give an exercise class to second year math students, as support for a theoretical class loosely based on the book `Differential geometry of curves and surfaces' by ...
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2answers
1k views

Teaching children to convert between number bases

Richard Feynman was critical of teaching children how to convert between number bases. I'll give you an example: They would talk about different bases of numbers -- five, six, and so on -- to ...
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2answers
463 views

Do students understand infinite series before they're informally introduced?

We introduce infinite sequences and series very thoroughly in calculus classes. We first define infinite sequences, then series, carefully discussing notions of convergence, etc., and discuss all ...
4
votes
3answers
317 views

participation in 1st year introductory pure maths classes

I have just started teaching a very elementary class for 1st year students on introductory pure mathematics. ( classes at my institution are groups up to 20 students and supplement the lectures. The ...
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votes
12answers
4k views

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
112 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
votes
1answer
477 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
votes
1answer
201 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
votes
2answers
169 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
359 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
2k views

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
votes
1answer
83 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
222 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
votes
5answers
460 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
votes
5answers
616 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 ...
14
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7answers
976 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
213 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
395 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
5k 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
659 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
votes
3answers
156 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
votes
4answers
424 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
114 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 ...
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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 ...