How to Read Mathematics

Why a page of mathematics stalls you and what it's made of. The procedures in the later parts make more sense after this one.

1. 1The Page That Won't Let You InThe compression problem, the rereading loop and 4 ways students read.
2. 2What a Mathematical Text Is Made Of6 kinds of object, the dependency graph and where authors hide the intuition.
3. 3Four GearsReading speeds you choose on purpose and how to spend a 2-hour block.

The 6-step unpacking drill and the example bank that keeps every definition honest.

1. 4Unpacking a DefinitionThe drill, why quantifier order matters so much and a full unpacking of "bounded."
2. 5Examples, Non-Examples, and the BoundaryBuilding an example bank, making non-examples on purpose and collecting monsters.
3. 6Notation: Say It Out LoudNotation's parts of speech, a notation table and a 5-minute daily drill.

Reading a theorem before you read its proof, so you know what the proof has to deliver before it starts.

1. 7Reading a Theorem Before Its ProofSplitting the statement, restating it, auditing the hypotheses and predicting the proof.
2. 8Testing a Statement Against the EdgesA test battery, the counterexample instinct and sharpness.
3. 9The Neighborhood of a TheoremThe converse, the contrapositive and the generality ladder.

6 chapters on proofs. The last one reads a single theorem start to finish with every move in the book.

1. 10Three Ways to Read a ProofFollowing, verifying and reconstructing, and how to choose the depth on purpose.
2. 11Finding the SkeletonStructure words, margin labels, the load-bearing step and 5 proof shapes.
3. 12The Reconstruction TestWhy recognizing a proof lies to you and the gap list that fixes it.
4. 13Stuck: A ProtocolNaming the stuck, running the protocol and escalating in order.
5. 14When the Proof Uses a Technique You Have Never MetNaming the move, translating it, building a toy problem and keeping a technique card.
6. 15One Theorem, Read Start to FinishEvery move in the book on a single fixed-point theorem, including the drop-one test.

The same procedures scaled up to a chapter, a whole book on your own and a research paper.

1. 16Reading a Chapter3 passes, where the difficulty actually is and when to skip.
2. 17Working Through a Book AloneChoosing the book, the 15-page week and 5 exercises instead of 40.
3. 18Reading a Research PaperA 20-minute triage, reading for a single theorem and a 3-pass read.
4. 19Reading Above Your LevelThe one-level lookup, reading for shape and 3 questions to leave with.

What to write while you read and how to get more out of lectures, study groups and office hours.

1. 20The ArtifactsMarking the book, the summary sheet and reading with your hands.
2. 21Lectures, Groups, and Good Questions20 minutes of preparation, study groups that work and questions worth asking.
3. 22What Changes When You Get GoodYou still get stuck as often. What changes is how you read and what you know about your own gaps.

8 appendices built to sit open on your desk while you work.

1. ASeven Reading MakeoversPassages read badly, then read properly.
2. BThe Unpacking DrillThe Chapter 4 drill on a single photocopiable page.
3. CThe Stuck ProtocolThe Chapter 13 protocol on a single photocopiable page.
4. DTime Budgets and Honest SpeedsWhat each move costs in time, so slowness stops feeling like failure.
5. ESay It Out LoudHow to read symbols aloud, including Greek letters and quantifiers.
6. FFurther Reading, Honestly AnnotatedEach book says who it's for, and the free ones are marked.
7. GTips at a GlanceEvery tip in the book, collected on 2 pages.
8. HAnswers to the Questions in the TextWorked answers to the closed questions the chapters ask.