Most effective method to self-learn a new mathematical topic and or concept?

[member 624364]

I am quite sure at least some members here have their own particular way to go about learning a new topic or concept. But, does anyone have any advice for self learning something that is completely new or different without the help from classmates or a teacher about?

I am self learning math, and I have had times where I was very confused/lost learning or reading a new concept.

 

[jedishrfu]

In order to help, we need more context. What is your level of math? And what math are you trying to learn?

 

[member 624364]

I’m only at algebra, as my foundations are not entirely water tight. So I have been going through multiple algebra textbooks to ensure I understand 99% of the algebra 1 and 2 material for when I try to learn trigonometry and calculus.

 

[jedishrfu]

Checkout the videos at Mathispower4u.com they cover most topics of algebra and other maths up to first year college.

There’s also Khan Academy for self learning too.

 

[Mark44]

I’m only at algebra, as my foundations are not entirely water tight. So I have been going through multiple algebra textbooks to ensure I understand 99% of the algebra 1 and 2 material for when I try to learn trigonometry and calculus.

What does "going through" entail? You should be reading the explanatory material and the examples. After that, work the problems for that section. Most textbooks provide answers to the even-numbered or odd-numbered problems. If your answers agree with those in the back of the book, try your hand on the ones without answers. Sometimes you can check your answers to verify that they are correct. For the ones you aren't able to do this, you can post questions here (in the Homework sections).

 

[symbolipoint]

This really is the best advice:

What does "going through" entail? You should be reading the explanatory material and the examples. After that, work the problems for that section. Most textbooks provide answers to the even-numbered or odd-numbered problems. If your answers agree with those in the back of the book, try your hand on the ones without answers. Sometimes you can check your answers to verify that they are correct. For the ones you aren't able to do this, you can post questions here (in the Homework sections).

Many standard Algebra 1 and Algebra 2 books contain very good instruction and if the book is of any meaningful quality, it will give example problems with full solutions in the topic section. These example problems are very important. They help you begin to learn to solve problems. You should read and reread the topic instruction/discussion, try to understand, and then you should TRY to solve each example presented on your own, as far as you possibly can; and then look to see and compare how you are doing and how the example solution is going, and try to understand what the given solution is showing. Learn from this how to understand this example and how to solve it.

Along with that, one needs to study their Mathematics course everyday, maybe 2 hours each day. Do a variety of the section's exercises, and check whatever key solutions are available (the back of the book), and redo any exercises that you did wrong; and retry those that you thought you could not do. This might require you reread the topic section again; and maybe again.

 

[jedishrfu]

Openstax is another resource for good books used in home schooling environments which can be used for self-learning too.

https://openstax.org/

and for books:

https://openstax.org/subjects

 

[fresh_42]

I am self learning math, and I have had times where I was very confused/lost learning or reading a new concept.

There is one important question that should accompany all your steps: Why?

O.k. at least the mathematical steps. Although I think, that it is always a good advice, it's not the subject here. In order to really learn math you have to understand why something is done, allowed to do or the opposite: forbidden. Why leads you to true understanding. The rest is practice, practice and practice. One standard way to answer the why question is to ask, what happens if not? E.g. why is a division by $0$ not allowed? The true reason might be a deeper understanding than what is usually subject to courses at school, but the "what if not" question already gets you an answer: If you were allowed to divide by $0$ then $0 \cdot 0 = 1 \cdot 0$ would imply (by division by $0$) that $0 = 1$ which is nonsense. So we must not allow this operation. Now you're facing a personal decision: Are you satisfied by this answer or not? Because if not, you can go on and ask why $0$ and $1$ cannot be the same? This would lead you deeper and deeper into the inner structure of mathematics and where to stop is up to you. But the why question is always at the beginning of understanding.

Another helpful hint is to always think of an equation as a balanced scale. As long as you keep it balanced, everything is fine. Algebra has a lot to do with the solution of equations. This means to shuffle all the annoying stuff on one side of the equation until you're left with what you're interested in on the other side alone, e.g. $x$ which is $1\cdot x + 0$. The method is easy: what ever you do, do it on both sides of the scale, keep it balanced. That's basically it. The rest is practice and learning means: as soon as you've learned that a division by $0$ isn't meaningful, you won't have to re-ensure you another time. As soon as you've learnt, that $2 \cdot x = 0$ is the same as $x = 0$, you won't have to re-ensure you another time.

But we aware of the why, because there also algebraic structures, in which this cannot be concluded. E.g. a light switch switched twice leads to zero change of darkness, the same as doing nothing. But in one case we did something in the other we did nothing and yet the result is the same, Here's where abstract algebra starts: Why is the light switch different from counting apples? However, in both cases it is the why question that guides us.

 

[member 624364]

What does "going through" entail? You should be reading the explanatory material and the examples. After that, work the problems for that section. Most textbooks provide answers to the even-numbered or odd-numbered problems.

Thank you for the reply.

Sorry, let me clarify. Maybe I have the wrong terminology. When I say "going through" I simply mean I am going through the book page by page till the end, reading and trying to understand everything as well as doing the problems.

Is there a more adequate terminology as to not confuse anyone in the future?

 

[Mark44]

Thank you for the reply.

Sorry, let me clarify. Maybe I have the wrong terminology. When I say "going through" I simply mean I am going through the book page by page till the end, reading and trying to understand everything as well as doing the problems.

Is there a more adequate terminology as to not confuse anyone in the future?

The terminology is fine, but didn't provide enough explanation as to what you were doing. The approach you described here seems fine to me.

 

[member 624364]

There is one important question that should accompany all your steps: Why?

In order to really learn math you have to understand why something is done, allowed to do or the opposite: forbidden. Why leads you to true understanding. The rest is practice, practice and practice. One standard way to answer the why question is to ask, what happens if not? E.g. why is a division by $0$ not allowed? The true reason might be a deeper understanding than what is usually subject to courses at school, but the "what if not" question already gets you an answer: If you were allowed to divide by $0$ then $0 \cdot 0 = 1 \cdot 0$ would imply (by division by $0$) that $0 = 1$ which is nonsense. So we must not allow this operation. Now you're facing a personal decision: Are you satisfied by this answer or not? Because if not, you can go on and ask why $0$ and $1$ cannot be the same? This would lead you deeper and deeper into the inner structure of mathematics and where to stop is up to you. But the why question is always at the beginning of understanding.

Another helpful hint is to always think of an equation as a balanced scale. As long as you keep it balanced, everything is fine. Algebra has a lot to do with the solution of equations. This means to shuffle all the annoying stuff on one side of the equation until you're left with what you're interested in on the other side alone, e.g. $x$ which is $1\cdot x + 0$. The method is easy: what ever you do, do it on both sides of the scale, keep it balanced. That's basically it. The rest is practice and learning means: as soon as you've learned that a division by $0$ isn't meaningful, you won't have to re-ensure you another time. As soon as you've learnt, that $2 \cdot x = 0$ is the same as $x = 0$, you won't have to re-ensure you another time.

But we aware of the why, because there also algebraic structures, in which this cannot be concluded. E.g. a light switch switched twice leads to zero change of darkness, the same as doing nothing. But in one case we did something in the other we did nothing and yet the result is the same, Here's where abstract algebra starts: Why is the light switch different from counting apples? However, in both cases it is the why question that guides us.

Thank you for this very enlightening answer! It really described many things I had been thinking of and others I had not.

So I have a few questions if you don't mind.

I like how you described that 'when you finally learn/understand something you never forget. All that is needed then is practice' this is something that I have always tried to do. for instance, even the most trivial of statements i.e negative x negative = positive, 4^0 = 1 or why in school we were always taught that for fraction division we flip then multiply.

However, I feel like when I am spending hours pondering these trivial questions that I am somehow wasting my time and also feeling guilty for doing so, or that time could be better spent just simply reading and doing more maths instead of sitting and thinking or taking a walk to jog my thoughts more. If I had decided to do this in class, I would be considered a day dreamer wasting time, as defined by the teacher. However, Is this really true or am I in fact actually developing my mathematical intuition by asking these questions, even if I do "waste" hours or possibly days?(time depends on how easily I get the "why" or "how")

In the same fashion, how might you suggest I try confirming these queries to check whether or I am understanding the "why"/"how"? Should I simply try to prove it to myself mathematically, or should I ask on these forums or do both?
By performing these "thinking sessions", are you of the opinion that It will be beneficial for me having questioned and answered these facts vs a student in school who takes said statement as fact and does not take the time nor liberty to question these facts?

I really appreciate you taking the time to respond to my question!
Thanks.

 

[member 624364]

This really is the best advice:

Many standard Algebra 1 and Algebra 2 books contain very good instruction and if the book is of any meaningful quality, it will give example problems with full solutions in the topic section. These example problems are very important. They help you begin to learn to solve problems. You should read and reread the topic instruction/discussion, try to understand, and then you should TRY to solve each example presented on your own, as far as you possibly can; and then look to see and compare how you are doing and how the example solution is going, and try to understand what the given solution is showing. Learn from this how to understand this example and how to solve it.

Along with that, one needs to study their Mathematics course everyday, maybe 2 hours each day. Do a variety of the section's exercises, and check whatever key solutions are available (the back of the book), and redo any exercises that you did wrong; and retry those that you thought you could not do. This might require you reread the topic section again; and maybe again.

Thank you for the reply.

When you say that one must study their "mathematics course" I am presuming you are talking about the mathematical topic/textbook. Not as in a real course in a school?

If that is true, then do you suggest that I should actually only do 2 hours a day and not exceed, or at a minimum I must do 2 hours a day?
Also, Is this time frame excluding any other additional studying time in unrelated topics, say programming, English or computer science etc?

Would you say that it is bad to do a full working week of intense studying then taking 2 or even 3 days off completely from studying maths because of exhaustion from maths. Also, would it be best to study over holidays too, but only doing 2 hours of it?

I feel like I am quiet 'rusty' when I have had possibly even a month or 2 weeks off from studying maths due to holidays and such things. I find that I had forgotten things. It can also take a few days to get back into the rhythm of things and remember the things I had forgotten. So, your suggestion of always doing 2 hours sounds like quite a logical idea.

Another question I have is, Do you think it is a good or bad idea to study multiple textbooks on the same topic at once, say, I am reading 3 different algebra 1 textbooks in order to try and get the most perspectives at explaining things as possible. 1 of them might be a short refresher book and the other 2 are full textbooks. Or is this simply a waste of time?
If it is any help in answering. I am not looking to have the knowledge of an engineer(Bare minimum to simply use as a tool). I am rather looking for knowledge and understanding more closely of that to a mathematician(Completely understanding the "why" and "how" of the topic at hand).

Do you have any other suggestions too?

Thank you for any replies you make in the future.

 

[symbolipoint]

Thank you for the reply.

When you say that one must study their "mathematics course" I am presuming you are talking about the mathematical topic/textbook. Not as in a real course in a school?

If that is true, then do you suggest that I should actually only do 2 hours a day and not exceed, or at a minimum I must do 2 hours a day?
Also, Is this time frame excluding any other additional studying time in unrelated topics, say programming, English or computer science etc?

Would you say that it is bad to do a full working week of intense studying then taking 2 or even 3 days off completely from studying maths because of exhaustion from maths. Also, would it be best to study over holidays too, but only doing 2 hours of it?

I feel like I am quiet 'rusty' when I have had possibly even a month or 2 weeks off from studying maths due to holidays and such things. I find that I had forgotten things. It can also take a few days to get back into the rhythm of things and remember the things I had forgotten. So, your suggestion of always doing 2 hours sounds like quite a logical idea.

Another question I have is, Do you think it is a good or bad idea to study multiple textbooks on the same topic at once, say, I am reading 3 different algebra 1 textbooks in order to try and get the most perspectives at explaining things as possible. 1 of them might be a short refresher book and the other 2 are full textbooks. Or is this simply a waste of time?
If it is any help in answering. I am not looking to have the knowledge of an engineer(Bare minimum to simply use as a tool). I am rather looking for knowledge and understanding more closely of that to a mathematician(Completely understanding the "why" and "how" of the topic at hand).

Do you have any other suggestions too?

Thank you for any replies you make in the future.

Study "every day, about 2 hours per day" means you should or can cover the course topics as they occur in sequence as if for the course being 3 or 4 months in length. A textbook section might take you one day; or two days; or 3 days; or 4 days; or whatever time you need or find sufficient for that section for the amount of time you want to take. The time period to study during the day does not need to be exactly 2 hours. YOU do the amount of study and time that you can do.

You can skip one day in the week; or skip two days in the week; but do NOT skip more than one day at a time.

CHOICE OF BOOKS: The only important thing is to choose any single book of very good or excellent instructive quality. You can choose two books. This is your choice. Look for books that you find most useful. I like to find Algebra 1 & 2 books with enough good graphs, drawings, and pictures, and I prefer the books to have some exercise answers in the back of it.

 

[fresh_42]

However, I feel like when I am spending hours pondering these trivial questions that I am somehow wasting my time and also feeling guilty for doing so, or that time could be better spent just simply reading and doing more maths instead of sitting and thinking or taking a walk to jog my thoughts more.

This depends on so much unknowns, that I find it hard to answer. E.g. some people are good at learning many rules and algorithms to solve problems, others are better in seeing the crucial points fast. It is certainly better for understanding to go to the heart of a problem. Things you finally figured out by yourself are much harder to forget than quickly learned formulas. Time management on the other hand is equally important as you don't want to spend too much time on a minor issue. It's like always in life: the truth lies somewhere in the middle.

In the same fashion, how might you suggest I try confirming these queries to check whether or I am understanding the "why"/"how"? Should I simply try to prove it to myself mathematically, or should I ask on these forums or do both?

Again, this depends on case, time and number. Before you spent too much time on something, come on over and ask here. However, what you've written by yourself is easier to remember than what you've read or seen on videos. There is another advantage of posting questions here, which is far too often neglected. People often use the internet to find quick answers for a certain question and don't bother the way to the answer. They only want to overcome the hurdle "homework exercise 4.c". But if you're forced to explain something to others and have to think about how to do it and what exactly your difficulty is, then you'll often find the answer on the way in doing so. Teaching (or here explaining) is a good method to learn something. E.g. if you want to know why $2x=0$ implies $x=0$ then you'll probably won't take the effort to type it in on PF. On the other hand, it would force you to think about why $a\cdot b = 0$ implies $a=0$ or $b=0$ and that this is not always the case (cp. my example with the light switch above). So areas in which this holds are already more special than others. This in return means: If you want to solve $2x=0$ with ordinary numbers, then first try to figure it out by yourself. If you want to know, what it is, that this property holds or what it's called, then come over and ask. Also if you get stuck somewhere, it's better to ask than to endlessly try. You will be told, if you're questions are that simple, that you should answer them by yourself. And again, being forced to explain a situation often already reveals a way to the answer.

By performing these "thinking sessions", are you of the opinion that It will be beneficial for me having questioned and answered these facts vs a student in school who takes said statement as fact and does not take the time nor liberty to question these facts?

Let me give you another example. Problem: Solve $2x^2-10x+12=0$.

I have learned the formula $x_{1,2}= - \dfrac{p}{2} \pm \sqrt{\dfrac{p^2}{4}-q}$ where the equation has to be brought into the form $x^2+px+q=0$ first. And as I'm better in learning what to do than learning formulas, I remember it by $x_{1,2}= - \dfrac{p}{2} \pm \sqrt{\left( -\dfrac{p}{2}\right)^2 -q}\, : \,$take $-p/2$, then $\pm \sqrt{}$ and put the first squared minus $q$ under the root.

I think more common, given $Ax^2+Bx+C=0\,,$ is $x_{1,2} = - \dfrac{B}{2A} \pm \sqrt{\dfrac{B^2-4AC}{4A^2}}$

But in any case, the principle is the same:
$Ax^2+Bx+C=Ax^2 + 2 \cdot \sqrt{A} \cdot \dfrac{B}{2\sqrt{A}} \cdot x + \left(\dfrac{B}{2\sqrt{A}}\right)^2 - \left(\dfrac{B}{2\sqrt{A}}\right)^2 +C = \left( \sqrt{A}x + \dfrac{B}{2\sqrt{A}} \right)^2 - \left(\dfrac{B}{2\sqrt{A}}\right)^2 +C = 0$ and then solve for $x$.
Once you've understood this, you can still use (the faster) formula, but can always rollback to this principle as in case you're not sure, which I just did to derive the second formula that I haven't learnt.

This should illustrate the difference between learning and understanding. Wikipedia is better to look up the formula, we are better to understand why this formula holds.
Here's another read which might help you:
https://www.physicsforums.com/insights/10-math-tips-save-time-avoid-mistakes/

 

[IGU]

However, I feel like when I am spending hours pondering these trivial questions that I am somehow wasting my time and also feeling guilty for doing so, or that time could be better spent

No, this is spot on. This is what math is. Mathematicians spend most of their time lost, wandering in the wilderness. If you come away from hours working on a problem with no solution but perhaps understanding a bit more about the problem, that's mathematics. If you come away with a straightforward solution after a few minutes, the problem was too easy.

 

[symbolipoint]

I’m only at algebra, as my foundations are not entirely water tight. So I have been going through multiple algebra textbooks to ensure I understand 99% of the algebra 1 and 2 material for when I try to learn trigonometry and calculus.

In school or not? High School, or college, now or later?

New material or new topics if studying Algebra 1 or Algebra 2, is often confusing. You may be completely confused on first day of a new topic, and you must study for one or two or three days (not more than maybe 3 hours per day) for most of that confusion to leave. This is how learning new topics in Mathematics is. You must tolerate confusion, and you must study regularly (and also rest) before you understand the topic.

 

[clope023]

This is a decent resource:

https://www.amazon.com/dp/B000MAHBYQ/?tag=pfamazon01-20

 

[symbolipoint]

This is a decent resource:

https://www.amazon.com/dp/B000MAHBYQ/?tag=pfamazon01-20

I'm not saying that a book telling how is bad; but, really nobody needs to buy of book about how to study. One should study frequently and intently, several weeks at a time, almost everyday, reread, and work on the exercises. Study sessions must not have interruptions. Review sections already studied during the study season, so as to not forget them too much.

Here's something more to think about. You go to the class meeting. You listen and write, and try to think. Within 20 minutes after the end of the class meeting, start studying your notes and review the relevant covered material of the lecture from your textbook (even if you did review it before the class meeting).

 

[clope023]

I'm not saying that a book telling how is bad; but, really nobody needs to buy of book about how to study. One should study frequently and intently, several weeks at a time, almost everyday, reread, and work on the exercises. Study sessions must not have interruptions. Review sections already studied during the study season, so as to not forget them too much.

Here's something more to think about. You go to the class meeting. You listen and write, and try to think. Within 20 minutes after the end of the class meeting, start studying your notes and review the relevant covered material of the lecture from your textbook (even if you did review it before the class meeting).

A dedicated system of sorts is more useful than going at it randomly though.

 

[symbolipoint]

A dedicated system of sorts is more useful than going at it randomly though.

Sure. We are often (at least I was) given guidance in high school and in college about how to study. This was usually informal although presented during class time. Intro and general and intermediate levels of courses. The advice was clear enough, good enough, not too lengthy.