# How to improve in maths?

1. Feb 16, 2013

### VertexOperator

Hi,
At the moment I am working on improving my speed, accuracy and critical thinking skills in maths so how can I improve those more efficiently and effectively?
Any advice is very much appreciated.

2. Feb 16, 2013

### Simon Bridge

"More efficiently and effectively" than what?
What do you mean by "efficient" and "effective" - how would you measure these things in this context?
What you are trying to do is get good at asking that kind of question ;)

There is no easy way - you have to get in lots of practice.
It is faster if you have a guide (than if you go it alone) - get a tutor.
See how other people do it - answer questions here by getting the OP to explore the problem and pay attention when someone else does it too.
Join skeptics organizations.

There are disciplines that go with the mindset - like asking clear questions.

3. Feb 17, 2013

### VertexOperator

I would love to know how mathematicians study! Could you please tell me?
Do I just get a lot of questions and try to solve them?

4. Feb 17, 2013

### espen180

Try to understand why things are defined the way they are. What does the definition "accomplish"? This is not always applicable, but when it is, it really helps you figure out how you can use the definition when writing proofs.

When reading proofs, try to understand them. Not just the small logical steps, but the big picture. In general, before you start writing a proof, you sshould try to convince yourself why the theorem must be true. If you can do that, your reasoning may give you a proof strategy.

Two good books that come to mind are
https://www.amazon.com/Thinking-Mathematically-2nd-J-Mason/dp/0273728911
and
https://www.amazon.com/How-Solve-It-Mathematical-Princeton/dp/069111966X/ref=pd_bxgy_b_text_z

Last edited by a moderator: May 6, 2017
5. Feb 17, 2013

### Simon Bridge

Pretty much - yep.
Go back over what I said in post #2 - that's what they do to study.
The do problems, find out how other people have attempted the same problems, and so on.
"Proofs" are "how other people did it", but in a particularly useful form.

You wanted to get better at critical inquiry skills though.

6. Feb 17, 2013

### halo31

This is a typical question that a lot of people have when trying to self study mathematics. To begin with mathematics is an amazing subject and very beautiful and there is no magical answer to becoming good at mathematics. For all of us trying to become the best takes hard work and struggling, and that too for anything in life even outside of academics. With that being said the best way to start would be getting accustomed to the different proof techniques. Luckily there are are only about 5 different ones and all are quite easy to learn. A great book would be A transition to advanced mathematics by Chartrand but a less pricier and yet great book would be the one by Peter J Eccles( I can't remember the title). Velleman's How to prove it is great but not as good as these ones. From here you can take on Number theory, Linear Algebra, Elementary Abstract Algebra and Elementary math analysis I and II. This makes up most of the core of undergraduate mathematics. There are several great books on these subjects and this is where you really learn what real math is. Being able to prove definitions and theorems as well as applying them to solve computational problems is what math really is. That connection lacks unfortunately in many high schools. Despite that its never to late to learn these things on your own. Not to mention you have these great resources such as sites like this and math stack exchange etc where you are able to get feedback on problems you are having trouble with. With all this your dreams of becoming great with mathematics lies within you.

7. Feb 17, 2013

### fibo

To improve overall abilities, you have to understand the subject conceptually.

Get a book on the history of math, and learn where ideas/theories come from.

Also do a lot of euclidean geometry problem to improve your logical thinking skills.

At the same time do as much exercises as you can in your math textbooks, do every question if you can. This will improve your accuracy.
This book is good- https://www.amazon.com/Five-Hundred...61163594&sr=1-1&keywords=500+challenging+math .

Finally to improve your speed (in algabraic manipulation)--try to work on algebraic inequality problems (problems where you have to prove with AM-GM or Cauchy-Schwarz).
These problems will help you so much, that you'll be able to do algebra in your head, and you'll be able to skip many steps.

Last edited by a moderator: May 6, 2017