33 votes

Is it too late for me to start learning mathematics?

Certainly someone your age (or even much older) can learn Calculus, even get a degree in mathematics. And get a good job afterward. That "young man's game' quote refers to doing mathematical ...
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  • 6,211
20 votes
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Is Linear Algebra Done Right too much for a beginner?

Unguided self-study of mathematics is difficult, and harder for someone with little experience at it. It is normal to take time to advance. One should think in terms of months not hours. A typical one ...
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  • 5,083
20 votes
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Dealing with disagreeable students and not compromising

It sounds like your students are not getting what they wanted from your tutelage; since they are not getting a formal credential from their work with you, their likeliest motivation is that they think ...
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  • 539
18 votes

Lack of intuition, retention while self studying

Perhaps you should seek texts that emphasize the high-level viewpoint that you are missing in the details of the more advanced texts. Three examples: (1) Bressoud, David M. A radical approach to ...
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17 votes

Effects of early study of advanced books

I have a bit of anecdotal evidence. I was unfortunately not homeschooled, nor did I have a technical childhood; I spent my childhood painting and writing short stories. I was in gifted classes, but I ...
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16 votes

How to learn math from textbooks in the right way?

I tend not to answer the questions that I have no professional knowledge about or the questions that are too general to be answered here. But, your question reminded me of my own undergraduate years ...
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  • 4,316
16 votes

How do I learn advanced mathematics without forgetting?

For context, I have a lot of experience self-learning mathematics. I spent a summer learning additional algebra, point-set topology, linear algebra, and analysis (to extend my undergraduate degree) ...
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  • 3,477
16 votes

Dealing with disagreeable students and not compromising

I think the real issue here is that you thought you were essentially doing undergraduate tutoring, and you weren't. You were doing adult education, and that is not the same thing. When someone is in ...
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16 votes

Walter Warwick Sawyer: How has reading his works changed your learning or teaching?

No, I was never inspired by him because I had never heard of him before you mentioned it. Note: My answer is for the original version of the question. Since then, the question has been edited so my ...
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15 votes

Effects of early study of advanced books

You asked for anecdotal evidence. I was a "gifted student". The school told me to teach myself 11th grade math (Trigonometry and Algebra 2) in 9th grade. I never formally learned algebra 1, but I ...
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  • 6,795
15 votes

Why are hand waving arguments made in textbooks of undergraduate analysis and how should readers deal with them?

An proof is meant to convince a reader of the truth of some statement. When a mathematician is communicating an argument to another mathematician, you only include the level of rigor that you need so ...
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15 votes
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Why are proofs written in flowery language incomprehensible?

To answer the question in the title, I would say that one problem with no-symbols reasonings is that one need to use a lot of pronouns. Problem is, pronouns usually leave too much ambiguity. At some ...
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14 votes

Inquiry about my note-taking skill

As you have noticed, mathematical text is often quite concise and it can be difficult to squeeze it into tighter space or write in your own words in a nontrivial way. Therefore it is easy to end up ...
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13 votes

Are there technique to help with poor recall?

You may have been able to remember a lot of math for a short time, but you ma have lacked understanding. When you talk to people who excelled in school mathematics, they might be perplexed by your ...
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  • 7,385
13 votes
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What is abstraction and generalization ?

By abstraction, we mean that we step back from concrete objects to consider a number of objects with identical properties simultaneously. For instance, consider the following three objects: The set ...
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  • 1,736
12 votes

Is it natural for self-learners to forget most proofs of the theorems they learn?

You will retain something as long as you practice it. It just so happens that for many, many theorems, it's the statement of the theorem that matters more than the proof. I think a good example is ...
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  • 419
12 votes
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Is it possible to improve logical thinking and problem solving abilities?

First of all I want to laud you on your knowledge of programming. You know a lot more than I did when I was your age. I tried to learn Italian after watching The Godfather but lost interest after a ...
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  • 368
12 votes
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Why's math more complicated to understand than philosophy?

If you solve a mathematical problem the wrong way, you get a wrong result, which can be checked. If you "solve" a philosophical problem there is no way to check the result in any decent timeframe. May ...
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  • 236
12 votes

Lack of intuition, retention while self studying

You need to pick pedagogically appropriate texts. Not the Rudin ballbusters. Pick ones that have explanations and were written for students with occasional imperfections in their previous knowledge. ...
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  • 325
11 votes
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Proving theorems on one's own: how long should one persist?

I think it's very commendable to try proving things yourself first; even a failed attempt has value. However, it's also important to learn from others' proofs, so don't be afraid to sneak a peek at ...
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  • 4,318
11 votes

Effects of early study of advanced books

To supplement the other U.S.-centric answer: yes, the U.S. standard curriculum through high school is not now and has not been in recent years comparable to several western European or former-easter-...
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  • 13.5k
11 votes
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How long would it take someone to master the topics in the book "Book of Proof" by Hammack and similar?

This is a text for an "introduction to proofs" course. It might not be well-known outside mathematical circles, because mathematics educators don't like to advertise this fact, but, outside ...
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10 votes
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How do you revise?

My first suggestion is not to prepare for your exams alone. Prepare for your exams with some of your colleagues, but make sure each of you reflects on your own progress toward readiness independently; ...
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  • 1,113
10 votes

What things should one know in order to enjoy their undergraduate degree?

Don't denigrate "pre-calculus mathematics and calculus." Many of the problems of students is a lack of a solid foundation in the lower mathematics, especially algebra. Make sure you can do the algebra ...
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  • 2,496
10 votes

How does a student learn to 'dig behind the scenes' or 'feel' math like a Fields Medallist?

This question makes me think of the James Gleick quote on two kinds of genius: "There are two kinds of geniuses: the 'ordinary' and the 'magicians'. An ordinary genius is a fellow whom you and I ...
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10 votes

How does a student learn to 'dig behind the scenes' or 'feel' math like a Fields Medallist?

In addition to other insightful answers/comments/remarks, apart from the issue of "degree" and "what kind of genius", I think a large part of the problem is exactly that mathematics is mostly ...
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  • 13.5k

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