Doing undergraduate math exercises mentally

In summary, the conversation discusses the effectiveness of solving math problems mentally instead of writing them down. While some believe it can stimulate intellectual activity, others argue that writing down proofs and using pen and paper is crucial for understanding and improving. It is suggested to try both methods and find a balance, as relying solely on mental calculations may lead to misunderstandings and a lack of writing skills.
  • #1
Constantinos
83
1
Hey!

I was wondering, is it a bad idea to solve a math problem mentally instead of writing it down? I recently got a copy of "Understanding Analysis" and try to read it cover to cover using this method. Should I? It resembles blindfold chess in a way.
 
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  • #2
If you can do it and if you are sure you cover all the critical details (!), well, why not.
 
  • #3
It's not a 'bad' idea, but there are other parts of your brain that activate when you write down with pen and paper. I think it was N. Tesla who could do integral calculus in his mind and his teachers thought he was cheating on his tests because he would get the answers right without showing work!
 
  • #4
This might or might not relate well to the level of mathematics you intend:

Physical exercise can stimulate some intellectual mental activity, and this can include a tendency onto whatever mathematics you are currently studying. The type of exercise depends just on how the person is. For some, maybe running; or maybe jogging; for others, maybe weights; or maybe a long walk.
 
  • #5
Constantinos said:
I was wondering, is it a bad idea to solve a math problem mentally instead of writing it down? I recently got a copy of "Understanding Analysis" and try to read it cover to cover using this method. Should I? It resembles blindfold chess in a way.
I don't like the idea to work through the whole book this way. This certainly won't help you get better at writing proofs for example. You will sometimes need to write down a proof, and then read through it to see if there's any room for improvement. Then you repeat the procedure until you're satisfied. When I started doing this, I was surprised by how many times I could go through a proof and improve something each time.

But I do like the idea to do the mental thing some of the time. I like to lie on my back with my eyes closed when I try to find a way to prove something. Sometimes I end up falling asleep, but it's still a good method for me.
 
  • #6
I loved the comment about falling asleep but still being good for you. Oh god, how true is that.
Regarding reading the book you really need to try out for yourself. I know I couldn't do that myself. I read textbooks on maths or physics in places like buses etc. pretty often. However, when dealing with lots of technical details I just need to "get dirty" with writing down formulas and algebraic manipulations. You can choose to skip those details or just skim them. However when doing that (especially if you do it more often than not) you risk that you are under impression that you understood what you read but you really didn't. It is quite miserable to read through half the book being confident that it was all easy and then realize that you can't solve problems from first chapter - and that is not unlikely, especially if you are rather new to the subject.

Oh and another possbility is that you actually can fill all those technical details in without pen and paper. I for sure wish that I had that good memory and focus.

I'd say just try it and see for yourself, but once in a while make yourself "a test" with pen and paper. If it then turns out that you can really write down and use what you've learned before then it was probably good method for you.
 
  • #7
If you can, sure. However, most problems will require deep thinking now.
 
  • #8
Fredrik said:
I don't like the idea to work through the whole book this way. This certainly won't help you get better at writing proofs for example. You will sometimes need to write down a proof ...

But I do like the idea to do the mental thing some of the time.
I second this. It is a learned skill to write down a rigorous proof and to communicate your logic to others. It is also hard to review your logic if you only did it in your head. And things will get tough enough that you will want to review. Finally, don't confuse a proof and real understanding for "in your head intuition". That being said, it is nice to be able to understand things without writing every step down. Exercising your mind that way can be a good thing.
 
  • #9
Sounds like a bad idea to me. Writing skills are extremely important, it is very good to develope them at every chance you get. There is no real reason for doing advanced math in your head only, sounds like a waste of time. You would be way better off in the long run to complete a notebook containing all solutions to the problems in the book. This could potentially be useful.
 
  • #10
Sounds like a good idea, as long as it's not all the time. The main issue I see is that if you did it too often, you wouldn't learn how to write proofs. But it could be a good short-cut, if not over-used. There isn't enough time to learn everything 100% thoroughly in math. That's something you eventually learn. You should also beware of some misunderstandings that can creep in if you aren't thorough enough. Sometimes, I think I understand something, and then it turns out, I don't quite get it when I try to write it down. So, it's a bit of a balancing act.
 

1. Can mental math exercises improve my overall math skills?

Yes, regularly practicing mental math exercises can help improve your math skills. It can enhance your ability to solve problems quickly and accurately, and also strengthen your understanding of mathematical concepts.

2. Is it important to do mental math exercises even if I use a calculator?

Yes, mental math exercises are still important even if you use a calculator. They can help you develop a better understanding of mathematical concepts and improve your problem-solving skills, which can be beneficial in real-life situations where you may not have access to a calculator.

3. How can I improve my mental math abilities?

There are several ways to improve your mental math abilities. Regular practice is key, as well as breaking down complex problems into smaller, more manageable parts. You can also try using mental math strategies, such as visualizing numbers or using estimation techniques, to help solve problems more efficiently.

4. Are there any specific mental math techniques I should know?

Yes, there are various mental math techniques you can use to solve problems more efficiently. Some common techniques include using mental math shortcuts, breaking down problems into smaller parts, and using estimation to check your answers.

5. Can mental math exercises help me in other areas besides math?

Yes, mental math exercises can help improve your overall cognitive abilities, such as focus, memory, and critical thinking. They can also be beneficial in other subjects that require problem-solving skills, such as science and economics.

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