Henry Wan, Ph.D.
The true key to mastering mathematics lies in transforming knowledge from something external — merely memorized — into an internalized skillset that becomes second nature. To achieve this, students must go beyond rote learning and actively engage in a process of deep comprehension and creative exploration.
Step 1: Recall — Testing Your Understanding
After attending a math class, immediately review the concepts covered without referring to any materials. Try to recall the key points discussed by the teacher, including formulas, theorems, and their derivations. The ability to reconstruct this information from memory is the best test of how effectively you absorbed the material during class. If you struggle to recall certain details, it signals areas that require further review.
Step 2: Derivation — Reinforcing Knowledge Through Independent Thought
Take out a notebook and write down the important formulas and theorems from memory. However, do not stop at simply writing them down — challenge yourself to re-derive the formulas and prove the theorems using your own thought process. This independent thinking exercise helps pinpoint gaps in your understanding and solidifies your grasp of mathematical principles. The process of struggling through a derivation on your own is invaluable, as it forces you to connect different pieces of knowledge logically rather than just memorizing results.
Step 3: Comparison — Evaluating and Refining Methods
After completing your derivations, open your textbook and class notes to compare your approach with those presented in the materials or by the teacher. This comparison often leads to valuable insights: perhaps your method is more intuitive, or maybe the textbook’s approach is more elegant and universally applicable. By analyzing the strengths and weaknesses of different methods, you develop a more profound and flexible understanding of mathematical concepts.
Step 4: Generalization — Expanding Mathematical Thinking
True mastery of mathematics extends far beyond simply deriving formulas, proving theorems, or solving problems. It involves questioning assumptions, identifying patterns, and generalizing principles to uncover deeper insights. Consider the Pythagorean Theorem as an example. Memorizing, deriving, proving, and applying the Pythagorean formula for right triangles is just the starting point. To truly grasp its significance, challenge yourself with these thought-provoking questions:
- Why does the theorem specifically apply to right triangles?
- How does the relationship change if the triangle is acute? (Hint: The equation transforms into an inequality.)
- How does the relationship change if the triangle is obtuse? (Hint: The equation also transforms into an inequality.)
- How does the theorem evolve when extended from two dimensions to three, four, or eve n-dimensional space?
- What happens when we move from Euclidean space to spherical geometry? (On a sphere, the classical Pythagorean Theorem no longer holds, and an entirely new geometric relationship emerges.)
By exploring these deeper questions, you shift from merely using mathematical tools to truly understanding their foundations, limitations, and broader implications. This approach nurtures creativity, critical thinking, innovation, and problem-solving skills — hallmarks of true mathematical mastery.
A Real Example: From Competitive Math to Cutting-Edge Research
One of our students, inspired by this approach, successfully extended the Pythagorean theorem and the Law of Cosines to higher-dimensional spaces and eventually to spherical geometry. Under our guidance, the student refined this work into a rigorous mathematical paper, which we helped recommend for publication in a prestigious journal of the Mathematical Association of America. This outstanding academic achievement became a key stepping stone, ultimately securing the student’s admission to MIT.
Why This Method Works: From Passive Learning to Active Discovery
This structured learning approach — Recall, Derivation, Comparison, and Generalization — transforms passive reception into active engagement. Rather than simply memorizing formulas, you develop a deep contextual understanding and an independent problem-solving mindset. These habits not only enhance academic performance but also cultivate the ability to think critically, explore new ideas, and innovate — skills that extend far beyond the classroom.
Mathematics is not just about solving problems — it’s about discovering patterns, questioning assumptions, and pushing the boundaries of what you know. By following this learning strategy, you will not only excel in math but also develop a lifelong ability to think deeply and creatively.
Ivy League Education Center 