For questions related to the teaching and learning of mathematics. Note that Mathematics Educators Stack Exchange may be a better home for narrowly scoped questions on specific issues in mathematics education.

learn more… | top users | synonyms (3)

8
votes
1answer
695 views

What pure mathematics foundations should an applied mathematician have?

I'm studying mathematics, with some statistics also, and I've always chosen applied courses. I'm getting to the point where I'm studying 3rd year undergraduate to graduate level material. My first ...
7
votes
4answers
261 views

How do I convince someone that $\mathbb{R}^2$ and its copy inside $\mathbb{R}^3$ are different?

One of my friends is taking a first course in linear algebra now, and one of the problems on his latest homework was to explain why $\mathbb{R}^2$ and $\{(a_1,a_2,a_3) \in \mathbb{R}^3 \mid a_3 = 0\}$ ...
5
votes
4answers
9k views

Why study quadrilaterals?

My niece is in the 10th grade, and they have to do lot of theorems related to quadrilaterals. And, I was surprised to know that they have to learn by rote some theorems. This has made her feel that ...
9
votes
1answer
356 views

Darboux's Integral vs. the “High School” Integral

The definition of the integral below is what I usually call the "High School definition," because that's usually where I've seen it in use. Take a partition $\Delta = \{ x_0, x_1, x_2, \ldots, x_n\...
-2
votes
3answers
143 views

What is trigonometry? [closed]

I am going to learn trigonometry next year. I am an advanced student and I like to get a head start on things. So, How do you describe and introduce Trigonometry to the advanced secondary school ...
10
votes
1answer
464 views

How often should math students take day breaks and longer breaks or vacations? Any research?

John Edensor Littlewood wrote in page 197 of Littlewood's Miscellany "For a week without teaching duties - and here I think I am preaching to the converted - I believe in on afternoon and the ...
60
votes
19answers
19k views

How do I convince my students that the choice of variable of integration is irrelevant?

I will be TA this semester for the second course on Calculus, which contains the definite integral. I have thought this since the time I took this course, so how do I convince my students that for a ...
5
votes
4answers
2k views

Teaching irrational numbers?

I'm interested in teaching the irrational numbers to high-school students, and I need your ideas on how to do this in an 'optimal' and innovative way. And my question is: What should the teacher know ...
2
votes
0answers
146 views

Exercises or courses to improve logical rigor and reasoning skills

There is plenty of math that is beautiful without needing much explanation of theory, such as fractals, geometric patterns and the Game of Life, that may interest beginners in mathematics. However, if ...
0
votes
2answers
162 views

Question about secant and cosecant.

Ok so if we take a right triangle and consider an angle $\alpha$ we get the following: From here we can define the fundamental trigonometric functions sine and cosine where $\sin(\alpha)=\frac{\...
2
votes
0answers
69 views

What is a sound curriculum for exponent rules in freshman algebra in high school?

We all know the the rules of exponents covered in freshman algebra. The question is, what is the best way to approach these topics as most 9th graders struggle in this area? I work as an after school ...
2
votes
2answers
283 views

Explaining how to simplify a quotient with negative exponents?

I work as an after school tutor at my high school. I've had kids come up to me asking how to do these types of problems: $\left(\displaystyle \frac{5xy^{-2}}{3z^{-1}} \right)^{-2}$ My approach is ...
49
votes
8answers
2k views

How to maintain enthusiasm and joy in teaching when the material grows stale

I recently finished my third semester of teaching calculus to freshman college students. This means I was drawing the same pictures, solving the same example problems, and discussing the same ...
14
votes
6answers
2k views

Self studying math, how can I learn the most?

I am currently studying Pre-Calculus on my own. I have a few texts I am working with but feel like I could learning a lot more than I am. When people typically ask these kind of questions the common ...
0
votes
1answer
1k views

Line Drawing Using Bresenham Algorithm

Indicate which raster locations would be chosen by Bersenham’s algorithm when scan converting a line from screen co-ordinates (1,1) to (8,5). First the straight values (initial values) must be found ...
1
vote
1answer
91 views

Philosophical side of MATH. knowing the path then walk it. [closed]

Can I find a book that gives me the purpose of theorems and definitions without going deep into proofs. It's just like knowing the path then walk it. That's will me the understanding reach the next ...
2
votes
2answers
980 views

Fundamental theorem of linear algebra

When I studied linear algebra we (our books, our professors) used to call Fundamental theorem of linear algebra the theorem that says: Fundamental theorem of linear algebra: A linear ...
0
votes
1answer
41 views

Standard deviation: When to use which sum-coefficient?

I'm wondering when to use $\sigma = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n}(X_{i}-X_{mean})^2}$ and when to use $\sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n}(X_{i}-X_{mean})^2}$ which I have both seen in ...
0
votes
1answer
291 views

Relative Percentage vs Percentage Change

If I have a number say "500" and I say that it spiked 4 times (400%) of the original value i.e. "2,000". Does that make sense mathematically and grammatically because I'm talking about relative ...
12
votes
1answer
1k views

Teaching engineers mathematical thinking skills

In my experience, many introductory engineering mathematics textbooks these days tend to skip proofs and discuss logic only in the context of digital electronics. On the other hand, I can imagine that ...
6
votes
3answers
444 views

Teaching algebra in a culturally relevant way while fitting Common Core standards

I've been assigning algebra textbook and worksheet problems (from the publishers and my own) that look like this: Simplify the following expressions. $x^{- 3} y^2$ $c^2 d^{-5}$ $\dfrac{x^{-2}}{y^{-...
20
votes
5answers
1k views

$\epsilon, \delta$…So what?

Over the course of my studies I often encounter phrases in reference material of the type "and this avoids the need for using $\epsilon$, $\delta$ definitions" or "by this we can omit those ...
6
votes
2answers
3k views

Examples of open ended calculus “class project” ideas

I have instructed calculus I an II, each once, at the college level and would like to emphasize that math is not just about memorizing formulas and concepts for a test and that applied math is not a ...
2
votes
2answers
455 views

Examples of groups in the real world

I'm looking for some examples of groups in the real world to show students in a liberal arts math course. For example the Rubik's cube. Keep in mind these students have only a college algebra ...
131
votes
31answers
13k views

Stopping the “Will I need this for the test” question [closed]

I am a college professor in the American education system and find that the major concern of my students is trying to determine the specific techniques or problems which I will ask on the exam. This ...
6
votes
1answer
220 views

How to combat memorization

As a student in high school, I never bothered to memorize equations or methods of solving, rather I would try to identify the logic behind the operations and apply them. However, now that I've begun ...
11
votes
3answers
1k views

Why study metric spaces?

Most universities have a 3rd year undergraduate analysis course in which metric spaces are studied in depth (compactness, completeness, connectedness, etc...). However, in practice it seems that most ...
8
votes
6answers
703 views

What we're never taught explicitly

I would like to make a complaint really. School math(s) can be the most boring way to learn: sitting down and rote learning binomial expansion or the volume of a cylinder is just not interesting. It ...
5
votes
3answers
104 views

Swapping Theorems with definitions

My question is motivated from the following passage of Gian-Carlo Rota's Indiscrete Thoughts, 'Suppose you are given two formal presentations of the same mathematical theory. The definitions of the ...
2
votes
1answer
75 views

Storytelling and Applied Narrative as a Teaching Tool

Is anyone integrating storytelling or applied narrative as a technique/methodology to help teach undergraduate mathematics-based course work? If so, how are you using it and from which sources are you ...
1
vote
2answers
237 views

Explaining the concept of $z$-scores in high school statistics

The students have so far studied the uniform probability distribution and have a working familiarity with relative frequency histograms and the 68-95-99.7 empirical rule. They still have trouble with ...
275
votes
33answers
34k views

Pedagogy: How to cure students of the “law of universal linearity”?

One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”: $$ \frac{1}{a+b} \mathrel{\...
1
vote
0answers
2k views

PERCENTAGE Problem

Q: Paulson spends 75% of his income. His income is increased by 20% and he increased his expenditure by 10%.Find the percentage increase in his savings . Sol: Let the original income=Rs....
1
vote
0answers
63 views

Studies on how the wording employed on the explanation of mathematical concepts helps students to learn?

I remember that I had to learn division in my childhood, I could handle all the other mathematical concepts that were presented until then but division was a real pain to learn, somehow the idea of $n$...
3
votes
1answer
330 views

Math Shock in graduate program

People call it Culture shock but I call it Math Shock... let me explain my Problem... First I am graduate student in a good university in USA ( I get scholarship from my country). Before I lived in ...
63
votes
24answers
13k views

How would you explain to a 9th grader the negative exponent rule?

Let us assume that the students haven't been exposed to these two rules: $a^{x+y} = a^{x}a^{y}$ and $\frac{a^x}{a^y} = a^{x-y}$. They have just been introduced to the generalization: $a^{-x} = \frac{1}...
4
votes
3answers
362 views

How can I explain my 9 years old brother that $8a\cdot4a \neq 64a$

My youngest brother had a pre-algebra test yesterday and he was asked to tell if two expressions are equal or not. We agreed on most of the things but on this one I find it hard to make him accept my ...
3
votes
0answers
77 views

Is there a link between level of abstraction and use of numbers?

One of my friend who stopped studying maths in high school told me once You study maths, can you help me fill my tax forms? In her mind, advancing in maths studies implied manipulating an ...
2
votes
1answer
45 views

Proving that there exists $w$ such that $4x < 6w < 6x$ and $\gcd(w,\frac{x\#}{6})=1$ where $x \ge 7$ and $x\#$ is the primorial

I am trying to show that for any integer $x \ge 7$, there exists $w$ with the following properties: $4x < 6w < 6x$ $\gcd\left(6w,\frac{x\#}{6}\right)=1$ I thought that this would be pretty ...
-2
votes
1answer
56 views

Learning way and Resource for Complete math Subject. [closed]

I want to learn [self learning] Mathematics from basic.What is the order [like 1) arithmetic,2) Geometry,Etc..] to learn the maths? and what is the best resource to that particular subject?
1
vote
2answers
79 views

Why is Cauchy condition for convergence not formulated in a simpler way?

The standard definition of a Cauchy sequence (e.g. it's given in Wikipedia and most textbooks I remember; admittedly those are mostly older ones) is: for every positive real $ε > 0$ there is a ...
2
votes
2answers
112 views

Information on crucial results concealed as exercises or neglected in a textbook

First, where can students find lists, information, or resources on the crucial results, inequalities, theorems, etc... which a textbook might not explictly feature or even bring up at all? Second, ...
20
votes
3answers
1k views

A problem V.I. Arnold solved as a primary school student

According to a 1995 interview that Vladimir I. Arnold gave to the Notices of the AMS, his primary school teacher I.V. Morozkin gave in 1949 (when Arnold was 12 years old) to a Soviet classroom, most ...
12
votes
5answers
268 views

Scholarly work on the beauty of math

When reading mathematical books written for a general audience, or even searching questions on this site, the adjective beautiful is often used to describe mathematics. My question is whether there ...
4
votes
3answers
258 views

Learning math for physics

I am very interested in physics and am planning to self studying it. But for this I need to be mature in various areas of math. So I want to know what is the order in which I need to learn the math ...
3
votes
2answers
292 views

What's the right moment to learn Set Theory?

I've seen a question in which the OP asked when is the right moment to learn Category Theory, it seems this moment comes a little after a course of algebra, and indeed some books on abstract algebra ...
6
votes
1answer
182 views

Soft Question: Suggestions on mathematics resources for problem solving.

I'm doing my final year of under graduation through distance education and would be appearing for entrance tests for various graduate schools in a few weeks. I am looking for a database of algorithms/...
2
votes
3answers
117 views

On the nature of a first derivative

This is a very, very basic question. Never been very involved in math but I've been learning calculus in my free time, so here goes. I have observed some various things that happen with derivatives, ...
2
votes
2answers
128 views

Mclaurin on $\arccos(\frac{n^2-1}{n^2+1})$

I have expanded $\lim_{n\to \infty} \arccos(\frac{n^2-1}{n^2+1})$ to $\arccos(1-\frac{2}{n^2})$ and now i dont know what to do. I wrote the function on walfram alpha and he told me that the result is $...
2
votes
3answers
242 views

Difference between school mathematics and university (real) mathematics [closed]

Several people I know were good in mathematics when they were in high school and they loved it but when they joined a university (specializing in mathematics) they felt mathematics is hard and that ...