Learning to think like a mathematician: key messages from the book by Barbara Oakley

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1. Alternate between focused and absentminded thinking
Our brain has two modes of operation - focused and diffuse. Focused thinking is full attention mode. We use it when we process sequential information, when it is necessary to think logically, or when we ponder information already known (for example, multiplying numbers).

Absentminded thinking is like autopilot mode, where the brain is not focused on a particular subject, but explores all the possibilities and captures the wider picture. Thanks to scattered thinking, we see the context and can find new non-standard solutions, as well as assimilate new knowledge. Without alternating focus and relaxation, it is impossible to understand and assimilate information.

Practical advice

• If you can't find a solution to a problem, postpone it for a while, let your brain work and look for a solution in the background.
• Avoid setting (looping) on the first idea, which immediately seemed to be the only correct one; Thus, you yourself limit the capabilities of your brain and do not allow it to seek additional solutions.
• After active mental work, move around, play sports, listen to music, draw. Or just get some sleep, that's good too.

2. Actively use working memory and develop long-term
Working memory can be compared to a juggler who can hold a maximum of four objects in the air. Long-term memory is a warehouse that can hold a large amount of material. The task is to move new concepts or methods of solving problems from working memory to the "warehouse".

Practical advice

• When you read books, instead of mechanically emphasizing new concepts, speak the information out loud. This will help to avoid the "competence illusion" (feeling that you have memorized all the important thoughts).
• After you read one chapter, try to remember what it was about. For example, share new information with a friend.
• Secure the material away from your usual place of activity (for example, while walking). Your knowledge will not depend on the atmosphere, the usual points of support. Then, even during an important test, you will not be at a loss and remember everything that you have learned.

3. Study the material in portions, and then find connections between them
In order to transfer information into long-term memory and free up space in working memory, create and expand your "library of concepts and solutions." That is, break information into chunks - fragments that are interconnected by meaning. Templates from the long-term memory library are much easier to use.

4. Study the material in context
Build context. Instead of one standard way of solving a problem in a specific situation, you will see new opportunities to apply it.

Practical advice

• Never use the same problem solving method for too long. Otherwise, after a while, you will begin to mechanically apply it to other tasks for which this method is not suitable.
• Take on a variety of types of problems, analyze the solution and your mistakes.
• Try to understand the essence of the problem before solving itically, according to a template mechanically.

5. Make friends with your laziness
The habit of procrastinating is an inevitable companion to learning. A habit itself is a programmed state of the brain or, as Barbara Oakley puts it, "zombie mode." To practice more effectively, it is important to direct the zombie mode on the right track. How to do it?

Practical advice

• Find out why you are switching to procrastination mode. Find this signal and replace it with a new, more useful one. A signal can be a place, a time, a feeling of well-being, a reaction to other people, or an event. For example, get into the habit of "eating frogs all at once." Do the most important and unpleasant tasks first, as soon as you wake up or get to work.
• Use the Pomodoro technique: do short sets, for example, 20-25 minutes (or set another period of time that suits you).
• Often thoughts like “I need to solve five problems” immediately cause laziness. During your studies, focus not on the “product” but on the process - short periods of time allotted for the task.

6. Use non-standard ways to memorize
Mathematics and science can be difficult because of the abstract nature of concepts that are not tied to life. The concept and its meaning are always remembered better when there are many neural clues around it, that is, they are associated with the senses.

Practical advice

• Revive the concepts and theorems you want to learn in your mind. When solving a problem, imagine that the numbers are alive and how they behave.
• To connect your visual memory, come up with a vivid visual image. So, in the 19th century, chemists began to visualize molecules and atoms. This helped them advance in research.
• Connect motor memory. Write down the condition of the problem with your hand before proceeding with the solution. But do not overdo it: endless rewriting will not lead to an automatic response.
• Use the memory palace method. Imagine your house or other room and mentally place in it the images and concepts that you want to remember. In the future, to remember them, just return to this image and "walk" through your palace.
• Write stories or songs with concepts you need to learn.

7. Learn as a team
Study friends are external feedback and a source of new ideas, analogous to distracted thinking. Ideally, these are people who are not afraid to express criticism and point out mistakes to you.

Practical advice

• Arrange joint brainstorming sessions: this is how the skills of speaking in front of an audience, stress resistance, the ability to logically prove your idea are developed. All this will come in handy, for example, during exams or during presentations.
• Studying with like-minded people will make it easier to find your mistakes and even mistakes in textbooks.
• And most importantly, the group will help you study and move forward, even in those moments when you are lazy to study.