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

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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?
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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 ...
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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, ...
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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 ...
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5answers
265 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 ...
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3answers
254 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
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2answers
286 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 ...
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1answer
179 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 ...
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3answers
116 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
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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
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3answers
227 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 ...
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0answers
155 views

Learn enumerative combinatorics? [closed]

I am interested in becoming proficient in enumerative combinatorics relatively quickly. I want to be able to look at a problem briefly and think of multiple different useful approaches to it. Any ...
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2answers
3k views

How to learn calculus for beginners? [duplicate]

As a precalculus student interested in teaching myself calculus, where should I start and how should I go about learning? This question is different than past questions as I am not solely interested ...
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3answers
451 views

Being mathematically critical: how should a student approach statements that appear to be obvious?

Very occasionally, I will read or hear a theorem, and think: isn't that obvious? Not in a contemptuous "I can immediately see how to prove this" way, but rather in a "I would have thought this was ...
2
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4answers
317 views

How would you create a math class that centers on the cultural experiences of African American and Latino students [closed]

I need to write a paper on "Ethnocentric Mathematics" and I have no idea what kind of effective teaching strategies are available. We read an article from this scholar named Tate who explained that in ...
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1answer
1k views

How do you remember theorems?

I am currently a Master's student in math. I do very well in my classes, understand the material, can do the proofs w/o having to read the text, etc, but as time passes, I find that I will forget ...
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6answers
5k views

Up-to-date advice on the best way to take notes (maths)

I have read some old discussions about this topic and would like to get some up-to-date advice if possible. I'm going to start university next year (maths), and I know how important is to have a set ...
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6answers
556 views

What is the point of quadratic residues?

What is the most motivating way to introduce quadratic residues? Are there any real life examples of quadratic residues? Why is the Law of Quadratic Reciprocity considered as one of the most ...
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2answers
163 views

Is the maximal path through a math book necessarily linear?

I'm studying with two main math books (Munkres and D&F) these couple of months. My method so far is just going through the book page by page constructing everything in it (independently if I can) ...
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3answers
177 views

Applications of functions of the form $f(x)^{g(x)}$

Early on in my calculus education, I learned how to take the derivative of $x^x$ by re-writing it in the form $e^{x\ln x}$. More generally, this technique is helpful in finding the derivative of ...
4
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2answers
169 views

Is it better to teach or to grade?

As a graduate student at my university, I have the option many times of deciding what type of work I do for support. The two basic options are to teach either a calculus or college algebra course, or ...
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1answer
154 views

Reform of math symbols for high school texts

I am looking for references to papers and resources related to reforming math symbols for introductory courses at middle or high school level. Pointers to other forums also welcome. Eidt: For ...
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1answer
292 views

How can I raise my intuition in solving mathematical problem?

I am an undergraduate student studying some elementary calculus and statistics. In my honor calculus class, my professor gave one of final exam problem: $$\lim_{n \to \infty} \int_{[0,1]^{n}} ...
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2answers
943 views

Most effective order for learning different branches of mathematics. [closed]

So, I'm just a current student with a lot of interest in mathematics. Usually I am on the site looking at the questions and most of them are about things I can't currently comprehend. As I would like ...
9
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1answer
102 views

Education: Reading Proofs

I am finishing my undergraduate degree and one thing I've noticed is how little weight has been placed upon the ability to read proofs, in basically all of my math courses. In first year calculus you ...
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2answers
127 views

Motivation for abstractness

I'm seeking examples of concepts or theorems in school mathematics that are better understood when we generalize (when we deal with a more abstract concept where the former concept is a special case ...
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2answers
140 views

Books that develop ideas through tough problems?

I want examples of books that advance by first posting a hard problem, one that would be very difficult without a given idea and then proves this idea and the power of the idea by solving the problem. ...
3
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1answer
65 views

Topic for teaching assessment

I'm in the position to have a teaching assessment with a tutoring agency next week. This assessment will include me teaching the assessor a topic of my choice in 15 minutes, demonstrating the Socratic ...
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0answers
81 views

(Actual) applications of basic differential and integral methods

If this isn't the place, I apologize: At the end of my calculus class, we asked the students (among other things) what some applications of calculus methods are. Disappointingly, many focused on the ...
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5answers
754 views

Proving a certain map on the closed unit disc must be the identity

Bounty expired. Will gladly re-create one if a satisfactory answer is posted in the future. Prove: Let $f$ be a continuous function on the closed unit disc with two properties: 1. $f$ is the ...
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5answers
190 views

$\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: Overcoming Wrong Intuition

This is a widespread intuitive argument, asserting that $\mathrm{card} ( \mathbb{Q})=\mathrm{card}( \mathbb{Q^c})$: Between any two rational numbers there's an irrational one and vice versa. So ...
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0answers
234 views

What is the best way to go about learning math?

I know this is a very subjective question, but after struggling on my own for a while I figured I might as well ask it. I did all the normal math classes in college (LinAlg, MultiVariable Calc, ...
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4answers
789 views

Is studying mathematics chronologically a good idea or not and why?

In high school nowadays most mathematics you learn is fairly 'old'. You have your geometry, all of which (taught in high school) was known to the Greeks more than 2 thousand years ago. You have ...
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0answers
141 views

How would you explain the pdf of the normal distribution to high school students (11th/12th graders)

I will be teaching the normal distribution in January and I need to know how to effectively explain the concepts that does not in any way confuse students or make them feel that the material is ...
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5answers
896 views

Reflections on math education

Why is there such a big difference in math education between The Americas and (Europe and Asia) ? except for a few privileged who have the opportunity to access to math much earlier than the ordinary ...
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1answer
65 views

Simple question about power

Please solve this question and describe it. $$a^2+b^2+ab=b-a-1\Longrightarrow a^{1001}+b^{1001}=?$$ By Newton formula.
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2answers
338 views

What is a good example to show high school students why a proof for induction is a reasonable kind of proof?

I teach average-level high school students who have not had much beyond Algebra 1. I want to show them why induction makes sense. I want the sort of problem where it is intuitive that a statement is ...
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3answers
93 views

$\sin^2$ notation and uses of the alternative.

So I was taking my calculus class and I was shocked by the following: Apparently its a convention for $\sin^2(\alpha)=(\sin(\alpha))^2$ As opposed to what I thought made more sense which was ...
2
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1answer
226 views

Explaining probability theory versus statistics

I'm not sure whether this question was asked before, but it's hard to search because of lots and lots non-descriptive titles like "statistics and probability". The context: There is an anecdote I ...
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1answer
837 views

Equation For Multiples

So this is probably a super easy question for Math Stack Exchangers. Anyways I can determine the multiples of 3 up to 10 doing this. multiples of 3 up to 10 ...
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2answers
231 views

How to calculate profit share for this example [closed]

How can we calculate a profit share for Rs. 100,000 that remained in an account for 5 days only. See this question illustration below I was confused about which tags were more appropriate for ...
5
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1answer
235 views

Explain/illustrate Goedel's theorems and possible implications to non-mathematicians

I am asked to give a talk about (a) mathematical practice, (b) axiomatization, (c) Gödel's theorems and (d) possible antimechanist arguments based on the incompleteness theorems (as mentioned in P ...
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1answer
194 views

Explain mathematical practice and axiomatization to non-mathematicians

I am asked to give a talk about (a) mathematical practice, (b) axiomatization, (c) Gödel's theorems and (d) possible antimechanist arguments based on the incompleteness theorems (as mentioned in P ...
6
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1answer
288 views

What is meant by “mathematical maturity”?

I have often heard people talk of "mathematical maturity", sometimes in the sense of the maturity required to understand an area of mathematics or in the approach to a problem or proof. However, it's ...
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4answers
315 views

Formality and mathematics

Why is it important to be formal in mathematics? Is formality beneficial for students? Or is it just to scare students away from mathematics?
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3answers
1k views

How do you validate that two math expressions are equal?

Let's say you have a few expressions like the following: $$\begin{array}((x+17)^2 \\ x^2 + 34x + 289 \end{array} \\ 288 + \frac{x^2}{2} + \frac{x^2}{2} + 34x + 1 \\ [...] $$ You get the idea: ...
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0answers
82 views

Math for kids with Cuisenaire rods

I work with kids and i am searching some cool stuff to do with Cuisenaire rods. Thinking about an application i thought that i can show to my students what will be the sum of first $N\in\mathbb{N}$ ...
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4answers
565 views

Showing How Prime Factorization Helps Solving Problems

I need some problems conceivable by middle school students, which are not easy to solve unless the prime factorization of some number is known. An example: It's not easy to know wheter $n$ can be ...
3
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2answers
114 views

Motivated and unmotivated mathematics courses [closed]

The standard calculus course does not acquaint the student with the reasons why calculus has been and continues to be important in the intellectual development of humankind. Rather, it attempts to ...
3
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1answer
276 views

What are the benefits or losses of learning real analysis through a constructivist approach instead of a standard apporach?

Recently I've found some courses on real analysis that use the constructivist approach and I got curious on some aspects: What are the benefits of learning through this approach? Is it ok to learn ...