Studying Share self-studying mathematics tips

[Hellmut1956]

Dear young friend. I did share your interest in physics and I was not bad at school. i think the way the Khan academy proceeds in physics is a good option and I would be you, I would follow their path. The same applies to mathematics. So far to the rational way to proceed.

Emotionally physics was and is for me a fascinating science and dealing with it opens our sense for what our current science is finding out about it. Forget about quantum physics and relativity theories from your learning of physics. Look for good videos in youtube and find some that speak in a more general public kind about those fields of science. That puts you in touch with those topics and you can start to reflect about what you find interesting.

With mathematics the way it is taught at school and which you will need to master your tests in mathematics at school the Kahn academy gives you solid information to learn. But as with physics, where there are topics that you need at school in which give you a starting point, mathematics offers at least equally fascinating topics that equal in opening your eyes for a whole new way to see the world around you. i f you allow me, I would to tell you a bit about what I mean!

I did finish my studies at a technical university in 1979. That was then the last time I had to deal with mathematics in the way I did learn until then! So nearly 4 decades later the science about mathematics has also been developed very heavily. The mathematics as you learn to the end of high school really is not mathematics but is learning to apply formulas to solve equations. The top of this king of "mathematics" goes and includes calculus. Terms like differentiation and integration are ones that you can search via google and find information about it.

I like very much the current definition about the science of mathematics. Mathematics deals with finding "structures"! There is a course from the Standford University which can be taken for free here: "Introduction To Mathematical Thinking" This course has the purpose to help students to make the transition from the kind of mathematics they have been taught until then and to the kind of mathematics the students are required to learn and apply at the university. Do not let yourself be intimidated. Its even more! As you have not yet been spoiled to think the traditional way mathematics was done until about 200 years ago, you will probably have it easier to grasp this "Mathematical Way of Thinking! If you listen to the lecture of prof. Kevlin that you can see as videos, if you listen to the videos were he very detailed explains interesting topics as help for solving the challenges to think the mathematical way, you could have it easier than older ones to capture what is being taught.

 

[PlanetGazer8350]

Thank you very much for this advice, I will take it into account, and definitely check out the course on 'Mathematical Way of Thinking' . Currently, I am also in another Coursera course of Stanford University in 'Introduction to Logic'

 

[Greg Bernhardt]

Yes I want know How can I be an Expert in Trigonometry ?

Study and practice

 

[fisicist]

Hello, I am a 12 year-old boy, and I am very interested in physics and maths. For maths, would you recommend me to keep learning by myself following the order of topics for example, in each year in KhanAcademy? Or should I learn it in a different order? For physics, for the moment, I don't want anything that requires too many math, I first prefer to have a good math base that can be used for physics, and then gradually go applying it to my physics learning. So in physics, what order should I follow of topics, Eg: should I follow a certain order in each field, quantum mechanics, motion, etc (If it includes some maths in does not matter, but I prefer it to explain the concepts more strongly)? If you know about any books about physics, that mainly and strongly explain concepts (if it includes some math it doesn't matter), could you please tell me? It would be very helpful.
Currently, I am to self-studying about math, physics (mainly concepts), computer programming (before I was a bit confused and advancing little, as I was with many, now I am with just Python) and electronics. Do you think I should do them all at once, each one a smaller amount but at once, or 1 mainly at a time (and the rest but MUCH less time, so I keep thinking about them, and don`t forget), for a certain period of time, and then go changing which one I learn mainly and which one I revise in less time?

When I was in your age, I liked Jay Orear's Physics and the Feynman lectures a lot. In general: You cannot learn physics without math. So if you want to learn physics, you must learn math first, namely vectors+matrices, analysis (differentiation and integration) and later vector calculus (the gradient, divergence and curl operations and the theorems of Gauss and Stokes). The Feynman lectures cover all this, Jay Orear expects you to know differentiation.
Don't waste your time trying to learn physics without math or with as little math as possible. It doesn't work/will give you a pseudo-understanding.
I'd recommend you start with differentiation (by the way, I don't think the explanation that Feynman gives is very good; I don't think I'd have understood it there if I hadn't known it before). I would simply start here (I learned it first from the math formula reference book that we used at school, so not really a big difference): https://en.wikipedia.org/wiki/Derivative (sections 1.1, 1.4, 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 3.1, 3.2, 3.3)

 

[Christina Lin]

What are the most reliable (preferably free) online courses for algebra 2, pre calc, and calc A, B, and C that give students a deep understanding of the topics? Would I be better off with textbooks or a paid online program?

I would like to self-study the math necessary before I dive into physics, however, I would like to not break the bank in the process. I am currently enrolled in ALEKS MAT 130 and MAT 170 for algebra 2 and pre calc by ASU, but I’m not sure if they were intended for students who are getting exposed to the material for the first time. Furthermore, I’m not getting an understanding of the topics when I take algebra 2 from MAT 130. It is simply straight-up memorization. In this case, should I go for a paid online course or textbook? I don’t know about MAT 170, but if it’s anything like MAT 130, I don’t think it would be for me. I am also enrolled in Calculus 1A: Differentiation, Calculus 1B: Integration, and Calculus 1C: Coordinate systems and infinite series by MITx on Edx. I have yet to take anything from it, but it seems promising. If anybody has experience with it or suggestions for a source that provides a deeper understanding of calculus, please let me know.

 

[IGU]

ALEKS is useful for review, to check if students have any obvious holes in their knowledge, but it's not actually useful for learning anything well. If you want the best courses for students who love math and are good at it, check out Art of Problem Solving. My kid preferred the books to the online courses because he could go at his own pace.

In any case, not free. Also not particularly expensive, especially the books. But really, really good.

 

[Christina Lin]

ALEKS is useful for review, to check if students have any obvious holes in their knowledge, but it's not actually useful for learning anything well. If you want the best courses for students who love math and are good at it, check out Art of Problem Solving. My kid preferred the books to the online courses because he could go at his own pace.

In any case, not free. Also not particularly expensive, especially the books. But really, really good.

I've heard a lot of great stuff about AoPS! I will definitely check them out, and thanks for the quick response.

 

[PlanetGazer8350]

This website (http://www.openculture.com/math_free_courses) may help, all of the courses are free. Another very good website is https://openstax.org/subjects/math.

 

[geologist]

I'm interested in learning math, partially for its own sake, but mostly because I am very interested in learning scientific computing (emphasis in environmental modeling, e.g. climate, groundwater). For that I'd like to be able to cover precalculus material (algebra, trig) and basic calculus (limits, differentiation, basics of integrals).

Currently I'm going through Precalculus demystified and the Brilliant.org Algebra practice problems, which should take me through mid-May. I think two months should be a reasonable time to review algebra and trigonometry, I don't feel that it's terribly difficult (challenging, but not bang my head against the wall).

From there I'm planning on taking couple computer science online courses through edx.org (Introduction to computer science and programming using Python and Simulation and Modeling for Engineering and Science), which don't require more than a reasonable aptitude for math, probabilty/statistics, and some programming (all of which I have). Once I'm completed with these I plan on going through Elementary Calculus (with supplemental problems, e.g. brilliant.org and Paul's online math notes).

 

[geologist]

I am currently self-studying 6 subjects at a time. But I'm a bit extreme. I think 3 should be a decent number.

I know this is a bit of an old post, but I often find myself wanting to study more subjects than I should. How many hours per week do you partake in self-study?

 

[PlanetGazer8350]

I am currently learning on math, physics (mostly conceptual), artificial intelligence using tensorflow in python, and sometimes I research and keep on learning on arduino, electrical engineering, space exploration and the geology of different planets or celestial bodies, with occasionally some other small topics. However, I sometimes feel I don't have enough time to do it all at the pace and depth that I would like. I usually try to organize myself weekly or bi-weekly, as a week may sometimes not be enough for me to cover or do what I want to do, learn, or keep researching. I also try to focus on math, as it will then let me learn even more on other topics such as physics and electrical engineering. If I try to do it the other way round, with math as something secondary, when I encounter something that isn't mainly conceptual, it is very difficult for me to understand it. In addition, I think that truly learning about physics, for example, is not about the formulas at all, but the concept itself, and why the formula is the way it is, and how it has been constructed through a series of experiments. Sometimes, previously, I didn't focus much on learning math, but physics concepts, or electrical engineering, but almost always complex math appeared when researching on something, which is why I turned it the opposite way round.

With the number of hours each week, it really depends on which week I am on, or what I must do, even though I wouldn't really want to that thing, and rather just focus on what I truly like and interests me. To try to more or less balance what I do, with my primary focus on math, I may be 1 or 2 weeks just covering a reduced number of topics, to focus on a smaller number of topics at a time, and then I may switch some for another time period, maintaining some topics I consider very important to keep on learning further on other subjects.

 

[Terrell]

I am currently self-studying 6 subjects at a time. But I'm a bit extreme. I think 3 should be a decent number.

Please show me your ways, master.

 

[Hellmut1956]

@PlanetGazer8350: I can very much identify myself with the kind of activities you are pursuing and why! I am now retired and to keep me busy and to have my days properly organized and to satisfy my curiosity I do exercise the muscle between my ears! I started to reactivate my hobby from my days at school, model building. As my financial situation is not anymore as it used to be i decided to switch from building planes to build a sale ship from scratch. Work with wood, lamination with epoxy, later building using aluminium the project of the sail ship model became more a path that brought me in touch with many technologies. As I used to work in the semiconductor industry and later in the telecommunication industry it became evident to me that using my own build electronics would open many new ways to realize projects related to the sailship. While over the years my workshop became more and more sophisticated, milling machine and lathe purchased based on the more mechanical aspects of my project, electronic has become my main technique. Investigating my own way to combine a stepper motor to control a pulley that controls the sheet that define how much a sail can open and wishing to do it the way it was done in early 20th century i run in conflict of this part of the project with naval modelers, experts in sailboats as they claimed this was impossible due to the friction of the sheets in the pulley and in their path through the body of the sailboat. Discussing it with the community of physics they said that friction was neglectable.. So I decided to model the system that was to control my sails. Doing a lot of research, Matlab and similar tools had no price for individuals and so where financially impossible to acquire, I decided to go with Mathematica from Wolfram and their tool "SystemModeler" that uses the language Modelica. I wanted further to compare the "quality" of my Modelica models by using the ability of the Wolfram software to collect data from physical experiments and to improve this way the quality of my models.
Somewhere along the road of this activities I realized that mathematics are the language required for Physics, Electronics and Simulation. Suddenly I found a course MOOC, from Terence Tao that build the mathematics starting with the theory of numbers. The book used was available for free from Taos personal website. I fall in lough with mathematics! I even decided that it would be worthwhile to study mathematics at the "Technical University Munich". I wen to an introductory event and had the opportunity to talk with one of the mathematics professors. he told that since recent decades mathematics is viewed from the perspective of studying "structures". I had no clue of why it happen to be this way. Soon I found fascinating courses for physics, cosmology and mathematics that demonstrated the power of approaching mathematics by studying structures. Even a professor that gave a course available for free in the internet from the University of Erlangen went through the whole theoretical field of physics up to beginning master level using more and more sophisticated models of mathematical structures. Even there is a fantastic introductory course available for free as MOOC from the Stanford University named. "Introduction to Mathematical Thinking". A very worthwhile course to take. In his first lesson he defined mathematics as the science of structures!. I do not need to mention that also electronics is a science the uses mathematics. Also artificial intelligence uses statistical methods.
I have taken the effort to present how I got into this trying to confirm your opinions and so be able to pass the message to you: science is advancing so fast in many fields that even me who is 24/7 available for this studies have come to the conclusion I need a method to combine the curiosity of the fields mentioned here with the chance to benefit from what I am learning, tools for the science disciplines that affect my project of a sail model ship build from scratch in my very own way.
So I study mathematics in depth enough so that the software Mathematica takes the job to solve equations. To have my mathematics skills well enough advanced that I can capture the concepts I meet. This results in an iterative way to advance my mathematics studies.

 

[ZHHuang]

I use Understanding Analysis, by Stephen Abbot, because Rudin's book is too difficult to read. I found Understanding Analysis is thoroughly explained. It is an introduction to analysis, so the book does not contain the concept of metric space and Euclidean N-space. Self-studying is extremely time-consuming. It costs me 7 months to learn 7 chapters.

 

[Tatsuya]

Hi guys,
I'm interested in geometry and topology. What should I study if I want to be able to study topology? and in what sequence? My highest level of math education is high school and right now I can't even remember conic sections... thanks

 

[S.G. Janssens]

Hi guys,
I'm interested in geometry and topology. What should I study if I want to be able to study topology? and in what sequence? My highest level of math education is high school and right now I can't even remember conic sections... thanks

Why are you interested in topology? In order to give advice, it helps to know.

If you are a beginning mathematics student at a university, a typical sequence would be:

calculus - analysis - (metric) topology - (general) topology,

where the latter two may be one course. (I took them separately.)

On the face of it, general topology requires very little background beyond naive set theory. I write "on the face of it", because I actually think it makes little sense to jump to general topology right away. To see where definitions come from and to gain essential intuition, I would recommend first taking a rigorous (but not necessarily long) course in single-variable analysis, treating such topics as: the triangle inequality, convergence and continuity.

The nice thing about this approach is that a lot of analysis books also treat elements of general topology, at least superficially. If you find this too much of a detour, consider beginning with a very elementary topology book, make a start and see how far you get.

Topology is broad: subfields such as algebraic topology and differential topology (with which I am not familiar) but also topological vector spaces build upon general topology. These subfields have considerably more prerequisites than general topology proper.

 

[Tatsuya]

Why are you interested in topology? In order to give advice, it helps to know.

If you are a beginning mathematics student at a university, a typical sequence would be:

calculus - analysis - (metric) topology - (general) topology,

where the latter two may be one course. (I took them separately.)

On the face of it, general topology requires very little background beyond naive set theory. I write "on the face of it", because I actually think it makes little sense to jump to general topology right away. To see where definitions come from and to gain essential intuition, I would recommend first taking a rigorous (but not necessarily long) course in single-variable analysis, treating such topics as: the triangle inequality, convergence and continuity.

The nice thing about this approach is that a lot of analysis books also treat elements of general topology, at least superficially. If you find this too much of a detour, consider beginning with a very elementary topology book, make a start and see how far you get.

Topology is broad: subfields such as algebraic topology and differential topology (with which I am not familiar) but also topological vector spaces build upon general topology. These subfields have considerably more prerequisites than general topology proper.

Thanks S.G. Janssens. I'm into topology because I'm interested in the 'shapes' and 'forms' like klein bottle and triple torus etc., which is the same reason I like geometry. I want to know the theories behind them and how to create various forms with the knowledge. Also I used to watch a TV show about maze-solving using topology and I got hooked. The pattern and analytical approach to solve the problem seem fascinating - I'm a fine art student if that helps and sorry if I've used wrong math terms. cheers

 

[S.G. Janssens]

Your question is interesting because one could perhaps think about topology as a way of capturing the essence of shape without being "hindered" by geometry. (See the paragraph halfway this page, for example.)

If you want to make your interests mathematically precise (for example, the Klein bottle can be realized as a "quotient space", which is a precise way of "gluing" objects together) and you are really mostly curious about topology and geometry without caring so much for additional mathematical "baggage", then I would suggest:

1. A good course on sets, propositions, relations and functions. (This is typically the first course that 1st-year mathematics students take in my country.)

2. A course on elementary general topology, ideally also introducing you to some geometric topology. Croom's little book "Principles of Topology" may be a good choice. There may also be full online courses that are worthwhile.

It will require investment of time and energy. The reward will be that you can understand at a much more precise level what actually fascinates you.

 

[Tatsuya]

Your question is interesting because one could perhaps think about topology as a way of capturing the essence of shape without being "hindered" by geometry. (See the paragraph halfway this page, for example.)

If you want to make your interests mathematically precise (for example, the Klein bottle can be realized as a "quotient space", which is a precise way of "gluing" objects together) and you are really mostly curious about topology and geometry without caring so much for additional mathematical "baggage", then I would suggest:

1. A good course on sets, propositions, relations and functions. (This is typically the first course that 1st-year mathematics students take in my country.)

2. A course on elementary general topology, ideally also introducing you to some geometric topology. Croom's little book "Principles of Topology" may be a good choice. There may also be full online courses that are worthwhile.

It will require investment of time and energy. The reward will be that you can understand at a much more precise level what actually fascinates you.

thank you very much S.G. Janssens! this is really helpful and now i have a clearer idea of what to do! cheers

 

[mathwonk]

a remark, motivated by requests for free books. a friend of mine wrote a math book and decided to try to publish it so as to have some income from it. she had to revise it many times over several years to satisfy the publisher, putting her research career on hold, but the result was a much better book that was ultimately recognized as the best text in its area in the country. as examples of calculus books, the best ones by all accounts are spivak, apostol, and courant, none free. so the moral is that the best books cost money, the contrapositive being that the free books are not the best. sorry about that, but those who write the really carefully polished books do deserve something to live on from that effort. my friend mike spivak lives essentially entirely from proceeds of sales of his calculus book, and he is not a rich man. so for the best results, try to pay your way, is my suggestion.

my algebra book, notes for math 843, 844, 845, free on my website, is apparently good enough that i once received an email from Wiley Interscience publishers asking me to submit it for publication, but I never did want to take the time to make it ready for publication. there are also many other much better, and more polished, books freely available online, such as Sergei Treil's Linear Algebra Done Wrong, at Brown, but many of the best still cost, and are worth, a certain price. Not everyone can afford to donate their labor and knowledge, as people do here on PF. Please try, when possible, to make an effort to support people who make a real contribution to learning.

later edit: There is one case of counterexamples to my principle of better means costlier. In the case say of the famous calculus book of George B. Thomas, the newer ones with added names like Hass, and Weir, cost 10 to 20 times more than the original ones by Thomas himself from about 1953, and are far inferior, in my opinion; and I have taught from at least 4 versions over a teaching career spanning 40 years. One of the newer ones, Thomas and Finney, 9th edition, is also available for less than $5, and is far superior to those other even newer ones costing $60 and up, but I like the 1953 version by Thomas himself, at about $5 used from abebooks. Thomas/Finney is easier to read, but the original Thomas has expert insights on how calculus is used by engineers, that the newer books omit.

 

[symbolipoint]

mathwonk, you have the right ethics in post #200, but when publishers push websites and optical information discs onto the product(a textbook) and push the price way up, something is wrong.

 

[XGWManque]

If you're completely on your own and not in any classes relevant to pure mathematics, then send me a message. I might be able to help. This goes for everybody reading this.

You are probably not going to see this, given that you haven't been active for a while-from what I can tell-but if the offer is on the table, let me know.

 

[0kelvin]

I have decided to learn calculus, linear algebra, numerical methods, all by myself. I did fail all those at university multiple times. So many times that I've lost count.

What I am doing is writing a wiki site. The first thing that I decided to do was to follow intuition and to have as many connections with geometry as possible. If something can be related to physics, statistics, experimental physics, I do it. If some concept from linear algebra can be mentioned to explain something about calculus, I do it. Now people often criticize textbooks for training monkeys by giving rules, applying in exercises and not really teaching the core concepts. Whenever possible I include proofs. If I found the proof online, give credits.

For now I'm not following the same order of a textbook. I'm doing like this. Define a function for one and multiple variables. Write the definition of a limit for one variable and mention that the same idea can be applied to two variables. Then the following page extends it to multiple variables.

If there are mistakes that happen very often I mention them through the text. There are pages dedicated to listing all possible mistakes regarding algebra, limits, derivatives, misconceptions, so on. I even created a page dedicated to listing mistakes regarding grammar that I often make. For ex: if there is a misconception about functions and I can show it with a graph that is intentionally wrong, I show the wrong graph and explain what is wrong with it.

At first I thought that It'd be more or less like a textbook, with examples on the same pages as theory. But the with formatting style of a wiki I decided to split into separated pages. Typing words is 4x faster than typing latex.

My plan is not to cover everything, but at least the core that is shared between engineering, astronomy, computer science, etc.

 

[newbie1127]

I have been self studying for sometime now, ever since Corona hit, our university assigned video call classes were poor quality and less engaging.

One day i started reading out of the reference book and found the language (which i used to think was complex) to be simple. my strategy to self study is very plain

i study with a combination of video lectures + reference book:-
1. I start by getting the outline details of the topic either from the books or the internet, just a short summary of everything i am about to learn.

2. Then, i start watching the lecture and as the professor is going through the concepts, i write my notes that i think are important. (in the beginning i used to write everything, with time i learned how to take notes)

3. Some concepts are going to be complex, and require some contemplation before moving on. so, sometimes i take my time with some things, while othertimes, its plain.

4. After sometime i revisit all the topics i have learned and go over them

5. After revision, i take tests to gauge my grasp on a certain subject

In my experience :

Most of the problems occur at the beginning whether it is the mental challenge of studying alone, the exhausion, the lack of self descipline, lack of experience or even, not knowing how to make notes.
there is no shortage of problems when it comes to self studying but one thing I've learned is if you're consistent and willing, most of those problems go away as soon as you find there is a problem and you get freedom to study whatever you want whenever you want and be more efficient.

Conclusion :

if you have a choice i'd say do what works for you, no need to change whatever has gotten you this far.
or your performance might take a hit.
On the other hand if you're a control freak or like things to be efficient go for it.

good luck

 

[0kelvin]

http://appliedscience.byethost7.com/index.php/Main_Page This is what I'm doing. I'm writing a lot because that's how it works. To explain concepts with words is how I approach it. I've found that once you have the concept, calculations are the easiest part. This is the complete opposite of secondary school, where teachers do repetition of calculations without giving proofs. Some proofs are not feasible because they require number theory, analysis and such. But there is so much emphasis put on mechanical calculations that almost everyone is "spoiled", being unable to see the reason of the calculation in the first place.

For example: at school teachers just tell you that to solve a linear system you can divide everything by two if every constant is a multiple of two. Then people memorize this rule without knowing what concept is behind it. I've seen an admission exam with commented solutions and one comment was that some people take an equation of a function and solve it by dividing everything by 2 or 3 if that's the case. Ok. But when it comes to plot the graph of the quadratic equation, they plot the equation after dividing everything by 2. Which means that people are memorizing rules without knowing what they are doing.

I'm doing much more progress studying like this than attending classes. The problem with classes is that either the teacher spends a lot of time answering questions from other people or the opposite, the teacher doesn't spend a lot of time explaining the concept and you are left behind.

 

[newbie1127]

http://appliedscience.byethost7.com/index.php/Main_Page This is what I'm doing. I'm writing a lot because that's how it works. To explain concepts with words is how I approach it. I've found that once you have the concept, calculations are the easiest part. This is the complete opposite of secondary school, where teachers do repetition of calculations without giving proofs. Some proofs are not feasible because they require number theory, analysis and such. But there is so much emphasis put on mechanical calculations that almost everyone is "spoiled", being unable to see the reason of the calculation in the first place.

For example: at school teachers just tell you that to solve a linear system you can divide everything by two if every constant is a multiple of two. Then people memorize this rule without knowing what concept is behind it. I've seen an admission exam with commented solutions and one comment was that some people take an equation of a function and solve it by dividing everything by 2 or 3 if that's the case. Ok. But when it comes to plot the graph of the quadratic equation, they plot the equation after dividing everything by 2. Which means that people are memorizing rules without knowing what they are doing.

I'm doing much more progress studying like this than attending classes. The problem with classes is that either the teacher spends a lot of time answering questions from other people or the opposite, the teacher doesn't spend a lot of time explaining the concept and you are left behind.

View attachment 300575

Hey,
Could you please tell me what that website is in the picture, I've been looking for good sources to understand calculus and in the picture everything about it seems to be laid out in a nice list.

 

[0kelvin]

Hey,
Could you please tell me what that website is in the picture, I've been looking for good sources to understand calculus and in the picture everything about it seems to be laid out in a nice list.

the link is in the message, the first line. I wrote it.

 

[MidgetDwarf]

I use Understanding Analysis, by Stephen Abbot, because Rudin's book is too difficult to read. I found Understanding Analysis is thoroughly explained. It is an introduction to analysis, so the book does not contain the concept of metric space and Euclidean N-space. Self-studying is extremely time-consuming. It costs me 7 months to learn 7 chapters.

But I am sure those 7 months were time well spent. Moreover, you essentially covered about two analysis classes, give or take. Provided you did most of the problems without looking at the solutions.

Have a look at Apostol's Analysis after. Or if Apostol is too hard, have a look at Bartle: Elements of Analysis.

 

[0kelvin]

Bear in mind that courses often skip chapters and a lot of details.

 

[0kelvin]

I wouldn't recommend doing what I'm doing with my wiki. It takes so much time to write and draw graphs by hand that it's extremely inefficient if you think on good grades.

The current state of my wiki is pre-calculus up to critical points for multivariable functions. I skipped integration for now. It also has introduction to computing with the C language.