# Water sharpens a stone

Once upon a time I wrote an article on Habr entitled “Intensity is more important than thoroughness” about the fact that it is better to increase the number of attempts to do something than the duration of one attempt. Now I want to complement this thought.

The main idea: divide the work into working sessions in such a way as to increase the number of attempts to solve the problem. Instead of continuous work over six hours, it is often preferable to do three sessions of two hours each. Instead of three hours, sometimes it makes sense to do three sessions per hour.

### Information blocks (chunks)

To freely use new knowledge, you need to get used to it. And habit, as you know, is a matter of time. Learning How to Learn (or a great Coursera course ) refers to blocks of information as chunks, and tells you how to create them. This takes time, which in practice cannot be significantly accelerated.

First we teach what natural numbers are, then integers, then rational, then real. Then we study limits, then derivatives, then integrals, then topology, then analysis on manifolds, and so on. For each next step, we need to understand the previous one and perceive it as an information unit. When I say “diversity”, for me it is an information unit, and for a student it is an information googol: a Hausdorff topological space in which every point has a neighborhood that is homeomorphic blah. I remember the moment when for the first time I was able to say out loud the "tensor product" without internal discomfort. It took me a long time .

Therefore, it is so important to divide the training into sessions. Give yourself time to get used to the new material. When we read a complex concept and try to immediately wield it in the head, we are faced with a lack of habit. The great John von Neumann said that we do not understand mathematics, we get used to it. (It is easy to believe, if we recall that real numbers are often defined as equivalence classes of the limits of rational numbers.)

This is one of the reasons why students are advised to attend scientific seminars at the academy. Even if they don’t understand anything, they get used to using words, to theorems and even last names. And when a student reads in a textbook what he once heard at a seminar, an internal “oh, I am familiar with this!” Arises from him, and it becomes easier to learn. In the absence of a habit, the new information looks alien, and we need even more time to assimilate it.

It is also worth noting the role of sleep in organizing information. If you have time, it is useful to divide the sessions into different days in order to take advantage of the amazing ability of the brain to put in order the knowledge gained during the day during a “short coma with hallucinations and partial paralysis of the body. The morning is wiser than the evening!

### Different people

Another reason is the opportunity to take a fresh look at the tasks. After a break, rest, distraction or sleep, we start the task in a slightly different state, with a different set of thoughts, and we can see what we have not noticed at point blank before. Surely everyone had a moment when a simple task in programming could not be solved for hours, and then for no reason the solution appeared in my head.

And this is one of the reasons why it is so inefficient to do work on the day (night?) Before the deadline. Working 8-12 hours in a row on one project, we lose the opportunity to work on it “several times”, we blur our eyes, and we skip original solutions. Moreover, due to general fatigue, we can make stupid mistakes and not notice them. In general, it’s better not to do that.

Dividing the work at the session, and starting to work several times, we give the opportunity to express ideas to different people: who we are on Monday, Tuesday, Wednesday ... Moreover, doing other projects, we learn new things that sometimes unexpectedly apply in the next session.

Say, if you write a project, then go on reading the book “Perfect Code”, and then return to the project, you can fix the style and architecture in time, before it’s too late. Conversely, reading a book will do more good because you use its advice with a specific example.

### Axiom of Archimedes

If you add a small number many times with you, the sum will exceed a large number.

Another reason for the effectiveness of dividing work into sessions is the availability of small parts. The following examples are idioticly simple, and I want to reject them immediately because of their banality, but they are extremely effective. Probably everyone can convince himself that he has 5 minutes of free time every day. After all, you can just wake up five minutes earlier!

A book of a thousand pages seems out of reach for many. However, if you read only three pages each day (10 minutes in the morning before breakfast), reading the book will take a year. I'm sorry, what? Long? Better for a long time than never.

If you learn exactly one word in English every day, and keep it in your head during the day (or stick a sticker with the word on your laptop), 365 words will be typed in a year. According to my calculations, this is 365 words more than nothing.

Solve one problem, write one paragraph of text, watch one video on Coursera, do ten repetitions on the press and so on. And then the axiom of Archimedes will do everything for you.

Suppose you say that you do not have five minutes in the morning to read a book. Take two minutes or even one. Read the book while brushing your teeth. If we lower the bar to indecently simple, psychological discomfort will force us to fulfill this small portion. It’s better to read one page every day than not to read ten every day.

### Applications

To make the article more applied, I propose the following:

If you are a student and you have homework in several subjects, do not set aside one day for analysis and another for algebra. It is better to devote half a day to analysis, and half a day to algebra. And on the second day as well. Then you will start the tasks twice, possible solutions will scroll in your head during sleep, and the next day the tasks will be easier to solve.

If you work and you have several projects, work on deadlines. Having figured out what time it is necessary to complete which project and evaluate the time for the projects, you can split the work on them into separate sessions and work on several projects in turn during the day, which will allow you to take advantage of the above advantages. If there are five projects, it is not necessary to divide it into five parts every day, but it is perfectly acceptable to devote time to two or three different things every day depending on the deadlines.

If you want to read a book, but you do not have time for this, perhaps you have time for this. Put the book on your phone or in front of the alarm clock before going to bed. When you wake up, take a book, sit away from the bed and read 3 pages before you start your day. Most likely, another day you will read more than three pages or return to the book some evening. But even in this case, do not forget to read the new three pages in the new morning. Sooner or later, the book will end.

As an exercise, it is proposed to transfer these recommendations to your activities :)

PS These tips are applicable for understanding non-trivial material, organizing new concepts in your head and solving original problems. If you need to distribute leaflets for six hours, then this can be done in one sitting.