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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504 views

The Constant Function Theorem first of all $\,$?

I quote Thomas W.Tucker $\,$ "... By the way, I view the Constant Function Theorem as even more basic than the IFT. It would be nice to use it as our theoretical cornerstone, but I know of no way to ...
2
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0answers
310 views

These unknown uniformly differentiable functions

Let $f$ be defined on $[a,b]$ and there uniformly differentiable ($\,$the $\delta$ in the definition of derivative is independent of the point). Given $\epsilon>0$, choose a partition $P \, : \, ...
1
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1answer
148 views

Turning an ellipse into a parabola

Today I was discussing circles, ellipses, hyperbolas, and parabolas in my precalculus class. We did the usual: completing the square, finding the center and radius (radii), etc. etc. But I like to ...
10
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2answers
380 views

Explaining why we can't “find” an antiderivative of $f(t) = e^{t^2}$.

We can't find $$ \int e^{t^2} \; dt $$ using basic tools from a calculus class. That is, we can't express an antiderivative of $f(t) = e^{t^2}$ using the basic operations. We can of course just ...
28
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9answers
3k views

Is this way of teaching how to solve equations dangerous somehow?

Two years ago, I bought the book Mathematics for the Nonmathematican, by Morris Kline. There I learned a new way of solving equations, which is related to the principle that states that any ...
2
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1answer
457 views

Complex division: polar form vs complex conjugate

The original problem In an electricity course which I volunteered to help with, the students solve circuits using phasors. Using phasors requires a good knowledge of complex numbers arithmetics, ...
16
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5answers
801 views

Alternative set theories

This is a (soft!) question for students of set theory and their teachers. OK: ZFC is the canonical set theory we all know and love. But what other, alternative set theories, should a serious student ...
6
votes
1answer
511 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
755 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
245 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. ...
8
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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 ...
475
votes
141answers
30k 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
679 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
213 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
158 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
183 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
547 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
291 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
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1answer
303 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
251 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
5k 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
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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
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1answer
519 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 ...
15
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1answer
535 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
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2answers
244 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
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1answer
204 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
1k 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 ...
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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 ...
16
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4answers
2k 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
909 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
votes
2answers
192 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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3answers
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 ...
8
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2answers
473 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
319 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 ...
45
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 ...
1
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0answers
113 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
490 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
203 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
170 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
365 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 ...
20
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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
230 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
463 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
621 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 ...
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7answers
1k 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 ...