Notes Projects

Math books demand a fundamentally different approach than novels. Without a teacher guiding you, developing an effective strategy becomes crucial. While learners vary in pace, background, and style one principle stands true: understanding requires active engagement.

As Jay Cummings emphasizes in his "Proofs: A Long-Form Mathematics Textbook":

"There is another set of goals that has to come from you. To go beyond rote learning - to really understand mathematics - requires you to struggle with the material. As you are introduced to a proof, I hope you do not just passively read it without challenging yourself to figure out portions on your own. I encourage you to work through plenty of exercises, to read extra proofs on your own, and to organize study groups to discuss the material with others. Challenge yourself and you will grow faster. These are the soft skills that only you can instill, and I hope you put in the work to do so."

The authors of "Discrete Mathematics and Its Applications" book - Donald Bindner and Martin J. Erickson also state that mathematics is an active endeavor. You cannot learn mathematics passively. Watching lectures, listening to explanations, or skimming textbooks will not suffice. To truly master the subject, you must engage actively with the material:

"How does one remember all the definitions and theorems? The way to remember something is to apply it, and one way to apply your knowledge is by solving problems. Perhaps the most important activity in learning mathematics is problem solving. When you solve a problem, you put your understanding to a test.[...] Work on your mathematics every day. When you're at rest, the ideas will continue to click and the next day you will go further and learn more."

Cal Newport's example

Honesty with yourself about your understanding is crucial, especially when no one else is there to test you. To truly gauge whether you've grasped the material, you need to actively challenge your knowledge. A powerful method for this comes from Cal Newport, a computer science professor and books author, who applied it rigorously during his student years.

In every lecture, the professor would write a statement like "If a number is a perfect square, it must be odd" and walk through its logic. As the class advanced, Newport adopted a strict rule: "I had to understand every proof so well I could rewrite it from memory with pencil and paper - no notes allowed. Not one proof could be skipped."

He started a routine: buying stacks of paper and copying each theorem from class onto its own sheet. At the top, he'd write the problem; below, he'd try to rebuild the proof from scratch, not just copy it. "I aimed to do this weekly," he admits, "but I often fell behind." More than once, he realized he didn't truly get a proof: "I'd hit a wall and realize my notes made no sense. So I'd hunt in textbooks, Google the problem, or ask the professor until it clicked. Not knowing was never an option."

Before exams, he switched to intense review. He split his proofs into two piles: "Got it" and "Need work." He'd drill the tricky ones repeatedly until he could flawlessly recreate every proof. By test day, he could recite them "without hesitation or errors," far ahead of classmates cramming last-minute.

This method took serious effort, but Newport calls it smart work: "If you added up all the hours spent rewriting proofs, hunting answers, and reviewing, it's a ton of time. Trying to do this in two days would be impossible." Yet it paid off. While peers scrambled to relearn material, he stayed calm, his skills built steadily over weeks. "It wasn’t torture," he adds. "I tackled it fresh each time, spreading the work out. That's how you master tough subjects."

The Proof Internalization Workflow

Additionally, there is a technique I rely on. When encountering a theorem and its proof in a math book, resist the urge to passively read it. Instead, challenge yourself to recreate the proof independently.

Start by writing down the theorem, then dive into scratch work:

Test examples: Generate concrete cases to see how the theorem behaves. Try simple, edge, and extreme examples to build intuition.

Seek counterexamples: Even if none exist, probing the theorem's boundaries sharpens your understanding.

If you hit a wall staring at a blank page - peek at the first sentence of the book's proof for a hint. Close the book and let that clue guide your next attempt. Repeat this cycle: struggle, hint, retry. Often, a single sentence will spark a breakthrough, letting you craft your own solution.

Sometimes, though, the proof remains unclear. In these cases, temporarily set it aside. Spaced repetition - a technique often tied to memorization - can help. Let your subconscious mull over the problem during breaks: nap, walk, or do routine tasks like washing dishes. Avoid "pseudo-breaks" like scrolling social media; these drain mental energy instead of replenishing it.

By alternating focused effort with strategic rest, you let complex ideas settle and connect. Over time, this iterative process transforms into intuitive understanding.