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Currently I am going through the prealgebra art of problem solving book as I’d like to start using that series of books and may as well start with the basics to get a thorough understanding (I understand algebra and basic precalc but I want to get solid foundations).

The book is laid out in such a way where is asks you to explain why something is true, why it works or how it works. You are meant to try and analyse and think about it so you can give a concise answer. Then write an answer down and then read the solution to their explanation answer to see if you had the right understanding and explanation.

However, I have found myself while going through the basic prealgebra book that when I am asked to explain something in a problem that is trivial, when it comes to consicely writing an answer/explaintion of why such and such works or how, I keep going off on tangents writing myself notes on new insights or ways to think about such a mathematical concept or technique of a trivial problem that I come up with while writing my explanation. I could spend upwards of 20 minutes writing and just coming up with new ideas. It is very distracting, but I feel as if I need to write it all down as it comes to me.

I like having complete and utter understanding of every mathematical concept I learn and is also the reason I am going back to the basic textbooks of the AOP because I want to understand each and every concept of maths including the why and how to all these new things I learn and make my way up to more advanced books.

I want to become a mathematician or maybe a physicist, but at the rate I am going through these textbook writing all these notes to each small problem, it feels like I will never get though these textbooks in time for university where I will have to pick between mathematics or physics for a degree.

Is this normal and should I stop doing this?

I don’t want to waste my time and I am unsure if this is productive or not. Any advice or help?

Thanks.