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.

learn more… | top users | synonyms (1)

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
442 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
votes
1answer
478 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
231 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
201 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
905 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
vote
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
votes
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
votes
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
821 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
186 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 ...
3
votes
2answers
956 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
votes
2answers
436 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
314 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
vote
0answers
111 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
448 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
199 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
167 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
votes
2answers
338 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
votes
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
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 ...
1
vote
2answers
209 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
454 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
596 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 ...
13
votes
7answers
870 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
votes
3answers
210 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
votes
2answers
375 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
595 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
155 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
393 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
votes
1answer
110 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
votes
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
votes
6answers
452 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
votes
2answers
146 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 ...
54
votes
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
votes
3answers
574 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
votes
1answer
465 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
votes
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
votes
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
votes
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
votes
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 ...
1
vote
6answers
729 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
votes
5answers
1k 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 ...
1
vote
0answers
97 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
votes
1answer
423 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- ...
1
vote
4answers
201 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
242 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
vote
1answer
89 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 ...