B Spending lots of time writing notes for trivial explanations

member 624364

Hi, possibly a weird question, I am looking for advice or suggestions.

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.
 

Drakkith

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Learn to focus. If A is true because B and C reasons, each of which are true because of D and E and F reasons, then learn to explain A in terms of B and C only and stop there. Besides, self study is different than studying with a purpose in school. You probably have the time to delve a dozen levels deep into a topic, but you won't have that much time when you're in school.
 

fresh_42

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Is this normal ...
Yes.
... and should I stop doing this?
Yes.

Your method will certainly gets you to your goal ... when you're sixty-five. It makes sense to do such exercises for subjects you're not quite sure you understood them correctly, but not all of them. It's as with almost everything in life, the clue is to find the right way in the middle. Or as my grandma used to say: "Everything which is too, is bad." So too many stupid exercises are as bad as too few. Try to understand what an object or rule or formula is about and then go ahead.
 

member 624364

Okay thanks for the advice! So if I know why something is true, I shouldn’t waste my time trying to explain it by writing down a number of explanations and notes, but instead just write a short one and move on?
 

fresh_42

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Okay thanks for the advice! So if I know why something is true, I shouldn’t waste my time trying to explain it by writing down a number of explanations and notes, but instead just write a short one and move on?
It depends on so many circumstances, as e.g. how you learn best. Me, e.g. needs to write things down in order to memorize them. And often it is very helpful to explain things to others, because it forces you to find the clue behind things, but this doesn't include yourself. So it's pointless to convince yourself, as in case you're wrong, you won't find out that you're wrong.

It's more important to remain interested and curious. Your method is poisonous to both.

Let me give you another example. $$2\cdot x = 0$$ Now you could say that this implies ##x=0##. Well, yes and no. It is only true, if there aren't numbers ##b \neq 0 \neq a## with ##a\cdot b=0##. However, the light switch on the wall doesn't follow that rule: here we have ##\text{ ON }+\text{ ON }=\text{ OFF }## or ##1+1=0##. Now ask yourself: would you have imagined such a possibility by learning from your book? Probably not, because for usual numbers we always have ##2\cdot x = 0 \Longrightarrow x=0##. So it's far more interesting to find out the differences between those very different concepts, as it is to learn a dozen times: Thou shall not divide by zero!

This means, if you want to learn something you think you know, it's often helpful and certainly more interesting, to look at it from a more general perspective. So my usual advice is: Stay curious!
 

mathwonk

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I respectfully disagree. The more I myself write about a topic the better I understand it and the longer it sticks with me, and the better able I am to make use of it in research. Dismissing a topic as soon as I see the point mentally, without following out the implications and mulling over them, and writing them up thoroughly, means I will soon forget the insight I had. It does take a lot of time, but I recommend the time as well worth it. At this early stage in your career I would pursue everything as far as interest takes me and as deeply as I can. 20 minutes is a trivial amount of time. The more you think about something usually the less trivial it becomes. But I might suggest a non trivial book like Euler's Elements of Algebra, or Euclid's Elements, or Courant's What is Mathematics?, or Gauss's Disquisitiones Arithmeticae, or Courant's Calculus and Analytic Geometry.
 
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One trick I used, while reading some complex article, was to look up word definitions only when I had spotted the word three times. After a while, it becomes automatic as you begin to infer the meaning and you only look up words that really stand out.

A related strategy is to make a checkmark or vertical bar in the margin and underline the passage in pencil (don't use highlighters ever**) for future reference. The checkmark in the margin works well with math books.

** Students get enamored with the awesome power of highlighters and as they read more highlight more. I've seen some science books where nearly every passage has been highlighted. The problem is that later on, you will view the highlighted text as distracting and it will make reviewing for tests more troublesome. Penciling is better because you can erase pencil marks especially as you realize the obviousness of what you underlined leaving the truly remarkable key stuff still underlined.

So if I were you I would:
1) Briefly scan the chapter you're studying (fast read to get the gist of it)
2) Start at the first section and read it again in more detail
3) start doing the problems and refer back to what you just read rereading if necessary.
4) add margin checkmarks where you might want to investigate it further (even on problems you feel were tricky in some way)
5) learn how to check your answers

One of my favorite math teachers in college once said "if you throw enough mud at the wall some of it will stick" so don't expect all the mud to stick on the first time. Read some, do some problems and read again as you're not sure. Remeber each section will probably have some worked problems which will be the basis for the problems that follow.
 

fresh_42

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But I might suggest a non trivial book like Euler's Elements of Algebra, or Euclid's Elements, or Courant's What is Mathematics?, or Gauss's Disquisitiones Arithmeticae, or Courant's Calculus and Analytic Geometry.
Are we still talking about prealgebra and precalculus? You need a language course first to understand those books - and I don't mean Latin.
 

member 624364

The more I myself write about a topic the better I understand it and the longer it sticks with me, and the better able I am to make use of it in research. Dismissing a topic as soon as I see the point mentally, without following out the implications and mulling over them, and writing them up thoroughly, means I will soon forget the insight I had. It does take a lot of time, but I recommend the time as well worth it. At this early stage in your career I would pursue everything as far as interest takes me and as deeply as I can. 20 minutes is a trivial amount of time. The more you think about something usually the less trivial it becomes.
Yes this is my thought process almost exactly! I am fearful that unless I follow up on my new understanding or insight that i just gained and don’t write it down. I may forget it and one day may have to read through my notes to reignite that forgotten understanding or insight I had.

I am unsure if this is truely helpful though, I have not yet had to use it, but then again I haven’t been learning maths like this for that long.
 

member 624364

Learn to focus. If A is true because B and C reasons, each of which are true because of D and E and F reasons, then learn to explain A in terms of B and C only and stop there. Besides, self study is different than studying with a purpose in school. You probably have the time to delve a dozen levels deep into a topic, but you won't have that much time when you're in school.
I see what you mean, this is pretty much what happens to me. I go deeper and deep asking why A is true which is true because B and C whic is true because blah blah blah. I seems almost endless. From now on I will try and only make a conscise explaination in whatever simple terms I know how and understand already.
 

fresh_42

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Is this for me to read?

I am certainly not ready to read anything like that hahaha.
I meant that the books @mathwonk has mentioned are all very good books and milestones in the history of mathematics. The fact that they are written in Latin aside, their language is also quite old fashioned and incomparable to modern presentations, which can be considered post-Bourbaki. I have two very good books on group theory which are written in this style (actually two volumes of one book), and one can certainly learn a lot from them. The only difficulty is, it takes too long. They are written with words, means there are only very few formulas and technical expressions. However, the fact that you can read them like a novel doesn't mean you can read them as a novel. It takes even longer to understand. Of course, the understanding will be far deeper afterwards, than reading a modern presentation of the matter, but it comes to the prize of time.

I think where we can meet is the question of practice. One really needs a lot of practice, and a lot of wrong paths, too, in order to understand the inner connections. But practice must not equal mindlessness. Practice means to apply the learnt and also to fail and find out why, which often is more useful than to succeed, however, this brings me back to grandma: nothing with a 'too' in front. If you fail too often or succeed too often, you might lose interest. And this is the worst case.
 

member 624364

I meant that the books @mathwonk
I think where we can meet is the question of practice. One really needs a lot of practice, and a lot of wrong paths, too, in order to understand the inner connections. But practice must not equal mindlessness. Practice means to apply the learnt and also to fail and find out why, which often is more useful than to succeed, however, this brings me back to grandma: nothing with a 'too' in front. If you fail too often or succeed too often, you might lose interest. And this is the worst case.
Thank you for taking the time to give me advice and everyone else too.
 

mathwonk

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all the books i recommended are available readily in English, and I have them all on my own shelf in English.. I thought this was well known.
 
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