Questions about the process of learning mathematics, both inside and outside a formal environment, including learning strategies, recommendations for learning particular subjects, and studying habits.

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2
votes
2answers
109 views

Diagnosing essential Classical Mathematical Analysis I knowledge needed for II

I need to take Classical Mathematical Analysis II (Chapters 7-10: Sequences & Series of Functions, Special Functions (Exp/Log/Fourier/Gamma), Functions of Several Variables, Integration of ...
-2
votes
0answers
47 views

Why are math textbooks that are considered good books so hard to read? Why do authors make their books difficult to read? [on hold]

I've noticed that many books that are difficult to read are considered some of the best. Why does hard to read indicate that it is rigorous? For example: Rudin, Apostol, Lang, Hungerford, Ahlfors, ...
65
votes
11answers
7k views

How do people who study intensely abstract mathematics “imagine” or understand the concepts they are studying or being taught? [closed]

This question is probably to the actual people who study such mathematics, rather than any "third-party". I haven't studied any such mathematics, but I can imagine that some (probably most of it) of ...
11
votes
4answers
675 views

How can I explain topology to my grandmother?

I was recently look at a post on tex.stackexchange about explaining $\LaTeX$ to the OP's grandmother. I was wondering, could the same thing be done for topology? Except in this case the "grandmother" ...
9
votes
6answers
10k views

Math Textbooks for High School

I'm a high school student who is trying to figure out a complete course of self study for each year of high school. Is there a way to self learn grades of math without devoting too much time? For ...
13
votes
2answers
18k views

Tips for an adult to learn math — from the beginning. [closed]

First let me start with I am an adult and I can't do simple maths. I some how got through all of my math courses in University (after several attempts) but I honestly couldn't tell you how... I ...
3
votes
1answer
56 views

What is the significance of “Homomorphism”?

Certainly Homomorphism is a prerequisite to establish an “Isomorphism”(Bijection), but what does a Homomorphism tell independently when it is established between two sets? Homomorphism relates ...
1
vote
2answers
93 views

Should I continue trying to solve Spivak or pick up a lighter book?

Some background: I have no mathematical maturity. Last year I completed my schooling and the only time I picked up a math/science book was when exams were due, needless to say I haven't actually given ...
1
vote
1answer
58 views

Scratch paper alternatives? [closed]

How do you practice complicated calculation when the problem is displayed on your computer screen? Do you always have pieces of paper on the side, or do you have a Wacom tablet connected to your ...
61
votes
19answers
42k views

Good book for self study of a First Course in Real Analysis

Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction ...
54
votes
4answers
7k views

Do you prove all theorems whilst studying?

When you come across a new theorem, do you always try to prove it first before reading the proof within the text? I'm a CS undergrad with a bit of an interest in maths. I've not gone very far in my ...
1
vote
2answers
134 views

How much group theory is required before undertaking an introductory course on Galois Theory?

How much knowledge of group theory is needed in order to begin Galois Theory? Which topics are most relevant?
19
votes
3answers
1k views

Working habits in mathematics [closed]

Short version of my question: What are good and motivating working habits for a mathematician? N.B. There are similar questions to mine: see this (how to read maths) or this (efficient study) or this ...
5
votes
3answers
202 views

How to determine if I'm talented enough to study math? [closed]

After getting Bachelor's degree from Computer Science I changed my field to Applied Mathematics. The previous degree was mostly programming-oriented, so I know quite a lot about software engineering, ...
5
votes
2answers
314 views

Numerical analysis, differential equations, complex analysis for statistics

I am a student of the Statistics Department. And now I can choose a subject from such list: numerical mathematics (analysis), differential equations, complex analysis, real analysis. Can you please ...
3
votes
1answer
1k views

Best way to learn from textbooks that have the Theorem-Proof format?

I've always loved the theorem-proof format in textbooks, e.g. Hardy and Wright's Introduction to the theory of numbers. However, the problem is that I can't remember anything I read in this format ...
8
votes
2answers
499 views

I like math, but can't keep up with the pace. [closed]

I'm a math major. I like math. I'm comfortable with it. I'm considering to do a PhD. The thing is I always fall behind. I can't keep up with the professor, can't turn in satisfactory homework in ...
5
votes
3answers
480 views

Map of Mathematical Logic [closed]

My undergraduate University does not offer advanced courses on logic, I know truth tables, Boolean algebra, propositional calculus. However I want to pursue Mathematical Logic on the long term as a ...
3
votes
3answers
102 views

Effective Methods of Studying in different areas of Math

I apologize if this question isn't appropriate for this site, but I am looking for advice that I think other math students might be better able to give me. I am an undergraduate math major entering ...
1
vote
2answers
114 views

How do mathematicians come up with beautiful equations [closed]

In Linear regression for example, we can find weights as following: $\hat{\beta}=(X^{T}X)^{-1}X^{T}y$ how someone invented this? I mean how do they transform a problem to such an equation. And ...
4
votes
4answers
748 views

Write on my own my first mathematical induction proof

I am trying to understand how to write mathematical induction proofs. This is my first attempt. Prove that the sum of cubic positive integers is equal to the formula $$\frac{n^2 (n+1)^2}{4}.$$ I ...
1
vote
0answers
52 views

Is propositional logic enough to study real analysis?

Is it necessary to study relational logic before starting real anylisis(from Bartle and Scherbert) or propositional logic enough? Also for topics like topology and differential geometry is ...
2
votes
2answers
177 views

Theoretical and applied math help - what and why? [closed]

(Edit: If you wish to skip the prologue, you may go straight to the questions in the last few paragraphs.) I'm not very far ahead at the moment (going to begin my undergraduate years after next ...
46
votes
5answers
11k views

(Theoretical) Multivariable Calculus Textbooks [duplicate]

(Note that I have used bold text frequently simply to highlight the key points of my question for those who do not have the time to read through it thoroughly (it is not very long, however); I hope ...
5
votes
2answers
254 views

Learning to understand proofs faster?

There are many books, written by highly decorated academics, which feature proofs that I can hardly comprehend in an acceptable amount of time. Roughly each week, it happens that I find myself having ...
-1
votes
1answer
35 views

What can I do after calculus I besides calculus II? [closed]

I'm a high school senior that's just about finished studying calculus I. I probably can't take AP calculus BC, and I don't want to learn calculus II over the summer or during my spare time. I would ...
5
votes
1answer
566 views

Most efficient way to learn mathematics [closed]

So there's a lot of advice on how to learn mathematics most efficiently, and it mostly revolves around doing problems, asking questions, and considering all possible generalizations. However, I was ...
12
votes
4answers
228 views

Transitioning to Elementary School teaching and need some advice on bolstering my math skills

I'm an English major trying to make the transition to Elementary School teaching. For as long as I can remember, I've been extremely math-anxious and I've only done the bare-minimum level of Math work ...
0
votes
2answers
51 views

Value of filling the gaps in a proof

I am studying a paper "On Finite Groups with Given Conjugate Types I" recently. The author uses many words like "obviously", "clearly", "trivially", etc. in his proof. But these "obvious" implications ...
2
votes
1answer
104 views

Learning how to prove by mathematical induction for the first time [duplicate]

So, I have to prove this by mathematical induction and I have never done it! $$\sum_{i=1}^n i^2 = n(n+1)(2n+1)/6$$ However, I have learnt better the theory behind this way to prove statements and ...
1
vote
2answers
47 views

Interpretation of a statement that I think I can prove by mathematical induction

I want to prove by mathematical induction the following statement: for any sequence (a_k) of non negative real numbers holds that: $$(1+a1)(1+a2)...(1+a_n) >= 1 + a_1 + a_2 + ... a_n$$ I think ...
8
votes
5answers
3k views

How do you rebuild your Math skills after college? [closed]

I minored in Math in college (B.S. Software Engineering), and went through advanced Calc, Differential Equations, etc. But about 6 months after I graduated I had lost all my formulas. Now, about 5 ...
3
votes
2answers
82 views

Natural proofs of theorems or exercises [closed]

Some mathematicians wants to hide their reasoning and proof process so that they appear smart. As most of the mathematics lovers I am not a genius and this is why I hate the magical proofs because ...
85
votes
10answers
10k views

Math and mental fatigue [closed]

Just a soft-question that has been bugging me for a long time: How does one deal with mental fatigue when studying math? I am interested in Mathematics, but when studying say Galois Theory and ...
1
vote
0answers
44 views

Undergrad looking for study material/advice for applied mathematics.

I am an undergraduate math student (junior) who is looking to get a masters degree in Applied Math. I like pure math, but I want to use my education to get a great-paying job. Here are a few questions ...
11
votes
4answers
1k views

How do you resist thinking about you may not be able to go further in mathematics? [closed]

I believe that every mathematical "apprentice" more or less once found that some stuff that had been unclear somehow became clear; for example, some books might be unaccessible to you two years ago ...
1
vote
1answer
71 views

New to Abstract Algebra, need guidance [closed]

I am new to Abstract Algebra. How should I begin learning it. On the face if it, it doesn't look easy to me.I have bought the book by Prof. Gallian. Is there any other book or any videos which I can ...
16
votes
1answer
463 views

IMO programs of different nations?

We in Albania have a good team in the IMO, and this year I will probably be part of it. Since Albania does not have a public training programme, I have to consult the training programmes of other ...
1
vote
0answers
102 views

Solutions for “What is Mathematics?” by Richard Courant

I am currently reading through Courant's "What is Mathematics?" Most of the time I am not taking the exercises too seriously, given that I am reading this for pleasure, and that often my solutions are ...
0
votes
4answers
106 views

How to study Linear Algebra when lecturer moves too fast [closed]

Our Algebra and geometry class is moving so fast the lecturer does not have enough time to cover all material in class, and he is not very pedagogical at that. Trying to fill in gaps through reading ...
4
votes
3answers
83 views

What is the equivalent of musical ear training with regards to studying mathematics

When one aspires to be a professional musician, it is made clear that ear training is a very valuable skill that must be cultivated on a daily basis. The student is advised to put in the time and ...
10
votes
8answers
14k views

Can I learn Calculus on the web, for free, using simple tutorials? [closed]

I'm a lazy type of person. I love experimenting with things before learning the theory. I like to practice, and learn from my mistakes. In 3-4 months I have a basic University exam on Calculus, and ...
3
votes
2answers
82 views

The Monty Hall problem - convincing a skeptic

I have lost count of the number of times that I have been debating the solution of the Monty Hall problem with someone. Recently I had a long conversation with a colleague, who didn't seem to buy ...
1
vote
1answer
106 views

Is the i-e principle for symmetric difference useful?

Let $E$ be a set and $A$ and $B$ two subsets. We may define the symmetric difference of $A$ and $B$ by setting : $$A\Delta B:=(A\cup B)\cap (A\cap B)^c $$ There are many interesting things about ...
23
votes
8answers
1k views

Looking to attain fluency in mathematics, not academic mastery

I'm a business/international relations person and a lot of my job is flying around. I have had a lot of downtime recently, and couldn't find a sustainable hobby to fill in that time. Until I found ...
81
votes
23answers
28k views

Complete course of self-study [closed]

I am about $16$ years old and I have just started studying some college mathematics. I may never manage to get into a proper or good university (I do not trust fate) but I want to really study ...
47
votes
3answers
5k views

Motivation and methods for self-study [closed]

First, a little background: Beginning with calculus in my first semester of college, I fell in love with mathematics. That was the point at which the concepts became interesting to me, and I started ...
3
votes
1answer
264 views

I'm searching for some books with guidance into mathematical study. [closed]

Yesterday, I've found this. It's a PDF file with this purpose, from Oxford. Some weeks ago I've also found two books tha seems to fill this purpose: Prelude to Mathematics; I Want to Be a ...
13
votes
3answers
11k views

I want to start mathematics from scratch. What should I begin with?

I've just finished high school but I don't feel my knowledge of mathematics is good enough. I'd like to start again from scratch, possibly adding a bit (a lot) of problem solving to it. What is the ...
3
votes
3answers
436 views

For Maths Major, advice for perfect book to learn Algorithms and Date Structures

Purpose: Self-Learning, NOT for course or exam Prerequisite: Done a course in basic data structures and algorithms, but too basic, not many things. Major: Bachelor, Mathematics My Opinion: Prefer ...