The Process of Learning#
The process of learning consists of three stages: information acquisition, transformation, and output.
The term "information" is used here because, in my opinion, any content that I do not yet understand or have not applied is considered as information that has not been transformed into knowledge. Regardless of how well-organized, concise, or accurate the content may be, it is still considered as "disorganized information" until it is incorporated and transformed into my own cognitive system. Therefore, studying by relying solely on other people's notes is not a good method for developing a comprehensive understanding, unless it is for cramming for an upcoming exam.
Information Acquisition#
Almost all undergraduate majors have a course called "scientific literature retrieval," but in fact, acquiring knowledge in any field requires a certain level of information retrieval skills. In this era of information technology, even specialized books can be found through search engines, providing a vast amount of content. The challenge lies in finding the most suitable information and avoiding wasting time on redundant (or even incorrect) sources.
Taking linear algebra as an example, here are three factors to consider when acquiring information:
Accuracy#
Information about linear algebra can be found in textbooks, articles or notes from experts, and even answers from fellow students. However, the latter two sources cannot guarantee accuracy, while textbooks have usually undergone multiple revisions and are less likely to contain major errors. Therefore, for students who are just starting to learn linear algebra, the first priority is to choose a suitable textbook.
Readability#
Readability refers to the organization of information sources and whether they are easy to read. Many people criticize the fact that the first chapter of linear algebra textbooks in China is about "determinants" instead of explaining what linear algebra is and the problems it solves. The key issue here is whether the arrangement of the textbook is reasonable. An unreasonable arrangement can directly reduce the speed of acquiring information. In contrast, textbooks for postgraduate entrance exams often have detailed tables of contents and mind maps before each chapter, which solve part of the readability problem for students.
Completeness#
Different versions of textbooks can vary greatly in thickness. The linear algebra textbook provided by my undergraduate institution can be described as a small "calculation manual," which explains the computational methods of the subject. On the other hand, foreign textbooks often provide detailed explanations for each topic, resulting in hundreds of pages of comprehensive works. Books like "C++ Primer Plus" and "Signals and Systems" (Oppenheim) are rarely read in their entirety, but they often provide answers to confusing questions. This sense of reassurance is not provided by a "manual."
Overly concise "manual" textbooks are often difficult to digest, while overly detailed "comprehensive" textbooks can be overwhelming. Therefore, my choice is to use both! The "manual" textbook serves as the "table of contents," improving the efficiency of learning for easily understandable or less in-depth content. For parts that require deeper understanding, I refer to the "comprehensive" textbook. Based on this, I search for versions with a more readable format, and the appropriate textbook selection is complete.
Information Transformation#
This is the process of internalizing selected information into one's own cognitive system, commonly known as the "learning" process.
The most important task in this process is reading. Overall, the table of contents in the selected textbooks provides a summary and guidance for the course. In detail, each chapter provides detailed and accurate descriptions of specific knowledge. In reality, there are rarely situations where a teacher covers content that is not in the textbook. If such a situation occurs, it is often due to a problem with the choice of textbook.
However, because the mind is not adept at processing text, the process of internalizing knowledge through reading is often referred to as "struggling with textbooks." Therefore, attending lectures has gradually become the mainstream learning method. Song Hao's calculus videos on Bilibili have accumulated billions of views, and in the field of postgraduate entrance exam preparation, people are more critical of different lecturers than textbooks. Attending lectures involves both visual and auditory stimulation, making it easier than struggling with textbooks. However, lectures are essentially explanations and interpretations of textbooks, and many people rely too heavily on lectures and neglect their active role in the learning process. If one only attends lectures without reading textbooks or downplays the importance of textbooks, the efficiency of learning will be greatly reduced.
Considering the learning process that most people have developed during compulsory education, the notes taken in high school are often condensed versions of the teacher's lectures, intended for future reference, and eventually become a "notebook" similar to a manual. In my opinion, this is highly inefficient. If it is difficult to obtain various manual-style textbooks, the value of such notes may be debatable. However, since there are already tables of contents or manual-style textbooks available, such notes are unnecessary. I do not believe that these self-generated notes, which are a form of "paraphrasing," will be more accurate than what the textbook authors have written. Moreover, copying notes in class often makes it difficult to keep up with the teacher's thought process. Most importantly, there is a misconception that "writing it down = remembering it," which weakens the understanding and memory of the course content.
Therefore, perhaps the most efficient learning method is to slowly struggle with textbooks while thinking critically.
Information Output#
There are many ways to output information, which is the final step in transforming information into knowledge and, in my opinion, the most important step. The mind tends to be inert, often staying at the surface level of information without internalizing it, which is why many people say they "read the book, attended the lecture, but can't solve problems."
Explanation#
The Feynman Technique: "To test whether you truly understand something, explain it in simple language to someone else." The core of this technique lies in the final step of information output. In reality, there are many ways to output information, but explaining it to others is the highest level of verification. It requires fully internalizing the information as part of one's cognitive system and achieving a comprehensive understanding.
Explanatory Notes#
In addition to explaining to others, the other common methods are solving problems and organizing notes. In this case, organizing notes refers to creating explanatory notes that resemble explanations and summaries, aiming to verbally recount the important parts of the knowledge in written form, achieving a similar effect to the Feynman Technique. The resulting notes may not be aesthetically pleasing or complete, but they resemble lesson plans and indicate the completion of internalizing knowledge.
Problem Solving#
Another familiar method is "practicing problems," which is essentially a way to apply and express knowledge. However, this method follows a specific path guided by the question setter, and the result often represents a partial and fragmented application of knowledge, resembling a spot check. Excessive problem-solving can lead to a phenomenon similar to "overfitting," where solving problems becomes the sole focus, neglecting the application and output of knowledge. This is counterproductive and is the source of my aversion to exam-oriented education.
Conclusion#
When learning any subject, it is necessary to go through the stages of information acquisition, transformation, and output. Optimizing the details of each stage consciously is the most fundamental method for improving learning efficiency. Avoid excessive criticism of textbooks and courses. A textbook with a score of 90 or 80 does not significantly affect learning efficiency. Spending a lot of time on choosing textbooks and lecturers is not worth it. Find weaknesses in your own learning process and optimize them one by one. Reflect on yourself. I wish everyone can quickly learn any knowledge they want to master.
This article expresses personal opinions, experiences, and reflections. Please contact me for any errors, infringement, or necessary modifications in the article.
The article is not the final draft, and I apologize for any potential content changes.