The Best Learning Methods (Especially for Math)

“Thinking you understand something after just one read” and “fear of making mistakes” are two major obstacles on the path to learning. I recently read a book dedicated to learning strategies, A Mind for Numbers: How to Excel at Math and Science (Even If You Flunked Algebra). The author argues that the brain is not a mere repository for knowledge, but rather gradually builds understanding through repeated practice and restructuring. The brain operates through two learning mechanisms: focused mode and diffuse mode.

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1. Focused Mode

This mode is typically used when learning specific concepts, such as studying in class, self-study, solving problems, and reasoning—all fall under focused mode.

2. Diffuse Mode

This is the brain’s spontaneous way of linking different pieces of knowledge, usually occurring when the body is relaxed—such as during breaks, walks, or downtime.

The author believes the most effective learning method is not relying solely on one mode but cycling through focused → rest → diffuse → focused. This loop allows the brain to work optimally. A crucial point here is the formation of knowledge chunks: how to organize fragmented information into coherent, integrated blocks. The author suggests three practical approaches:

  1. Reorganize knowledge in your own words or formats, such as taking notes.

  2. Understand concepts from different perspectives—for example, imagine explaining it as if you were a primary school student.

  3. Teach what you’ve learned to others—relatives or friends are fine, and if nobody is around, talking to yourself works too.

The author also advises against learning like a meditating monk who spends all day on a single task. This approach is not conducive to learning—the brain isn’t well-suited for it. Instead, it’s best to adopt a cycle of short bursts of study → rest → review → rest. This model resembles the Pomodoro Technique but with a key difference. The Pomodoro Technique might involve studying for 20 minutes, then resting for 5 minutes, and repeating. However, many use those 5-minute breaks to scroll on their phones, cycling through many rounds yet still struggling to learn effectively. The author stresses that breaks should avoid engaging in highly focused activities like phone use, as the brain will interpret those 5 minutes as continued “learning,” which drastically reduces efficiency. The recommended Pomodoro-style learning cycle is:

  1. Study the concept, then rest.

  2. Write down your understanding, then rest.

  3. Recall the concept from memory, solve problems, then rest.

Additionally, the author emphasizes that learning must involve creative thinking and not just focus on textbook methods. For example, there are theoretically infinitely many ways to calculate pi; the method presented in textbooks is not necessarily the best but rather a compromise designed for maximum comprehensibility.