I don’t know to what degree of mathematics study you are asking about, but perhaps my history of “studying” will help you understand what helps people to study in general, and why I believe studying mathematics is different from studying other subjects.
For other subjects, it is relatively easy to study merely by doing these things:
- listening in class and making sure that you understand
- asking questions only after you have listened, and also listening to the answers the teacher gives the questions from other students to see if you understand that answer
- do ALL your classwork and ALL your homework as instructed
if possible, do your classwork or homework as part of a group, so that part of your learning process is learning how to teach someone else what you have just learned
- do your homework in a quiet place with no distractions
- do something other than homework as well, like playing a musical instrument or riding a bike
- find out what you need to study more on by going over the “questions you got wrong”
read as much of the textbook as possible, and often.
- do other reading on the subject - and lots of it!
I did all of the above, and was a “top student” in all academic subjects in high school because of it.
By far the most important thing I did was the last one on the list.
Now here is the reason why “mathematics is different”.
First of all, even when you have a very good textbook, with lots of “worked examples” and lots of “exam type questions” with answers in the back of the book, lots of “theory”, lots of illustrations of the “relevance” of the mathematics being taught
it is not that easy to learn from a textbook “by yourself”.
And unlike other academic subjects there are not many books you can read that can substitute for the school experience.
you need to have a lot of practice answering very difficult questions in mathematics, and to have a teacher who will frequently “assess” how well you are doing, because you really can’t do that for yourself
- you need to have a teacher that really understands how to teach mathematics, and how it is significantly different from other subjects
- your teacher needs to be able to make mathematics interesting, relevant, memorable, and fun - which is very difficult to do if he also does the first two things
If you have to learn mathematics from a textbook “by yourself”, yes you will gain something by “doing it the hard way”. And you may even develop the skills needed to become a world famous mathematician.
But you will lose a lot more by not having “someone there” to speed up the process. Especially at high school level.
So let me make something very very clear to you right now.
- If you do not use a textbook that much, either in classwork or for your homework, your school or your mathematics teacher are not teaching you correctly.
Now for a little bit of my background in mathematics, and why I blame my high school for not teaching me mathematics properly.
In primary school, I was lucky enough to have teachers who understood that at this level, the most important thing is to develop “basic tools” in arithmetic.
- In other words, I am old enough to not have been “damaged” by the availability of calculators to “obtain an answer” - they were not around then. My “opinion” is that there should be no calculators in school at all in the mathematics classroom, until “functions” are studied.
- For those of you with doctorates in mathematics who disagree, please tell me how many basic skills and concepts a student can learn in mathematics, merely by learning how to do “long division with decimals” on paper. And how they would learn it another way.
- So I don’t care whether those of you with doctorates in “education” agree or not. Those who can’t teach, tell others how to teach.
- In high school I was fortunate enough to have very good textbooks and very good teachers, and because I did all the things I have already recommended I was a “top student”.
- Especially by reading a lot of books.
But I have one major complaint about mathematics teaching in “North American” high schools, compared to French or German high schools.
- In my experience, many “North American” high schools do not make a distinction between “arithmetic” and “mathematics”.
- Arithmetic is just a “collection of tools”. Mathematics is both a collection of tools and a collection of problems those tools can solve.
- Yes, the focus should be “arithmetic” in primary school. But the focus should move to “mathematics” in secondary school.
There is an additional FOCUS that you NEED in “teaching and learning” mathematics, that you do not have in other subjects.
- The ability to create a problem, solve that problem, and then utilise the answer.
I would suggest that the majority of high school students in “North America” still do not really know what “mathematics” is. Or what is so “important” about it.
Mathematics is the “language” of science.
My personal experience in “North American” high schools,
and the experience of my sons in UK high schools,
tells me that there is a problem in the “English language” teaching of the LANGUAGE of mathematics. That you don’t find in France or Germany.
Although at least the problem in UK schools is more just not using the textbooks.
As we can see from the link below, it isn’t so much that UK mathematics teachers don’t know how to teach, it is more often the schools that don’t let them teach the way they should,
because it is all about “grades” instead of “deep learning that lasts”.
National Centre for Excellence in the Teaching of Mathematics
deep and lasting procedural and conceptual mathematical understanding.
This is what should be important for you:
- don’t worry about the grade
- find out what you “got wrong” on the last test
- figure out what you had not learned yet to get that answer wrong
- practice getting the answer right
learn both the procedure and the concept, as everything you learn in mathematics has both, so practice identifying both
- work with “someone else” so you can take turns learning from each other
- if you are not “clever” enough to get the right answer all the time, neither are 99% of all the “top mathematicians”
- try again
To become “fluent” in any “language”, including mathematics, focus first on the “practicalities”.
- if you are already a “top student” in other subjects, then mathematics is only going to be different because you ALSO really need a good teacher and a good textbook
- if you don’t have a good textbook or a good teacher, PAY for them if you can afford to
if you can’t afford to, at least get a “revision guide” GCSE Maths Revision Guide (with Online Edition)
in class give the teacher a chance to teach you by listening first, but then please ask questions if you don’t understand
- it helps if you LIKE mathematics, physics, and computing
there are “computer and on-line resources” that can extend the work you do in class or at home, e.g. BBC Bitesize - GCSE Maths
Now to summarise my answer
which I have put at the end only because you need to first read what I recommended for you:
The best students in mathematics, at a high school level, very often do not “study” it that much.
This is probably because we were already “top students” in other subjects, by doing everything we were supposed to do.
What helped us with what everyone else thought was a “difficult subject” was our unwillingness to just “give up” and our willingness to “think about things for longer”.
Do what you are supposed to do.
Don’t give up just because it is “difficult”.
Think about it for longer.