TL;DR: How to take math notes that drive exam performance. Use the DTPE structure (Definition, Theorem, Proof, Example), the convention research mathematicians follow. Handwrite during lectures (Mueller & Oppenheimer 2014, math is the strongest case for handwriting). iPad + Apple Pencil + Notability ($14.99/yr) or GoodNotes ($29.99) is the optimal stack. Rewrite key theorems in LaTeX afterward (Obsidian, Notion, Overleaf). Problem sets drive recall (Karpicke & Roediger 2008, ~80% vs ~36% one-week recall). Atlas ($20/mo) for cited cross-course Q&A.
Atlas is AI-native and privacy-first by design: every answer comes back as a cited answer that links straight to the source note, and the workspace builds compounding context as you add material instead of resetting each session. The free tier covers solo use; Pro is $20/mo. Try it at atlas.
At a glance: DTPE structure: Definition + Theorem + Proof + Example. Handwriting beats laptop on conceptual recall (Mueller & Oppenheimer 2014). Apps: Notability ($14.99/yr), GoodNotes ($29.99), Obsidian (free, math plugin), Notion ($10/mo, $$ inline), Overleaf (free LaTeX). Retrieval practice: ~80% vs ~36% one-week recall (Karpicke & Roediger 2008). Polya's "How to Solve It": 1945, still canonical. LaTeX speed: ~20 wpm vs handwriting 40-50 wpm for symbols. Problem-set cadence: 5-10 problems per topic.
Math notes have a different physics from prose notes. Symbols, theorems, and proofs require structure that history or English notes do not. The students who do well in upper-division math share a method: definition-theorem-proof structure, handwritten capture, LaTeX rewrites for key theorems, and active-recall problem sets after every topic. This guide shows the system. For a method-agnostic overview of college note-taking, see our how to take good notes in college guide.
Math note-taking tools compared
| Tool | Format | Equation entry | Price | Best for |
|---|---|---|---|---|
| Paper + pencil | Analog | Hand-write | < $5 | Problem sets, exam prep |
| iPad + GoodNotes | Digital handwriting | Stylus | $9.99/yr + iPad | Lecture capture, infinite paper |
| Notability | Digital handwriting | Stylus + math conversion | $14.99/yr + iPad | Audio-synced derivations |
| Obsidian + LaTeX | Markdown | $$...$$ LaTeX | Free | Long-term reference notes |
| Notion + KaTeX | Block editor | Inline LaTeX block | Free / $10 mo | Sharing notes with classmates |
The DTPE Structure
Definition-Theorem-Proof-Example. Mirrors how research mathematicians write and what professors and graders expect.
DEFINITION 1.1 [name]
[statement]
THEOREM 1.2 [name]
[full statement, all hypotheses, all conclusions]
PROOF
[step 1 with justification]
[step 2 with justification]
...
[QED or square]
EXAMPLE
[concrete instance demonstrating the theorem]
COMMON MISTAKES
[gotchas, edge cases, sign errors]
Boxing definitions visually separates them from theorems. Numbering theorems lets you reference them later. The "Common Mistakes" section is the highest-yield addition; most lecturers warn about specific errors that show up on exams, and these warnings are the difference between a B and an A.
Hand vs Laptop vs iPad
Handwriting wins for math. Mueller and Oppenheimer 2014 found handwriting beats laptop typing on conceptual material; math is heavily conceptual. The mechanism: writing forces compression and engagement; typing enables passive transcription.
Laptop typing fails for live capture. Keyboard layouts cannot keep up with integrals, summations, matrices, and Greek letters. By the time you've finished one theorem, the lecturer is on the next.
iPad + Apple Pencil is optimal. Handwriting speed for symbol-heavy work, palm rejection, and a searchable digital archive. The de facto choice for STEM students 2020-2026. For app-by-app comparison and Cornell templates, see our how to take notes on iPad guide.
App Stack
Live capture (in lecture). Notability ($14.99/year, per Notability pricing page May 2026) for iPad. Audio sync is the killer feature for math, tap a written symbol to hear what the professor said when introducing it. GoodNotes ($29.99 one-time, per GoodNotes pricing page May 2026) is the strong alternative. The DTPE template adapts cleanly into Cornell layout; see our how to take Cornell notes guide for the cue-and-summary scaffolding.
LaTeX rewrites (post-lecture). Obsidian (free) with Math plugins, MathJax/KaTeX rendering. Notion ($10/month) for inline LaTeX via $$ delimiters. Overleaf (free) for full document LaTeX (homework, papers).
Cross-topic synthesis (finals). Atlas (free tier, $20/month Pro) for cited AI Q&A across all your notes plus textbook PDFs. Useful when prepping for finals across calc + linear alg + probability.
The 4-Step System
Step 1: Lecture Capture (Handwritten DTPE)
iPad or paper. DTPE structure. Box definitions, number theorems, indent proofs. Don't try to LaTeX live; you will fall behind.
Step 2: Same-Day Rewrite (LaTeX, Key Theorems)
Within 24 hours, rewrite the 2-3 key theorems from the lecture in LaTeX in Obsidian or Notion. The rewriting is an active-recall exercise; you can't LaTeX what you don't understand.
Step 3: Worked-Example Pass
For each theorem, work through 1-2 examples from the textbook. Annotate which step uses which theorem. This is the bridge between abstract statements and exam problems.
Step 4: Problem Set (Closed-Book)
5-10 textbook problems per topic. Closed-book, timed, on a blank page. Check against solutions afterward. Retrieval practice produced ~80% one-week recall versus ~36% for restudy in Karpicke and Roediger's 2008 experiment, and the broader retrieval-practice literature (Roediger & Butler 2011) is unusually consistent.
Don't Memorize, Derive
Memorized formulas fade by exam day; derivations stick because you understand the structure. At the end of each topic, close your notes and re-derive 3-5 key theorems on a blank page. Repeat at 1, 7, 30 days.
Polya's "How to Solve It" (1945) is still the canonical guide to math problem-solving. Four phases: understand the problem, devise a plan, carry out the plan, look back. Most struggling students skip Phase 4 (look back), which is where pattern recognition forms.
Common Mistakes
Live LaTeX during lectures. Too slow. You fall behind by Theorem 3.
Skipping the proof. Some students copy theorem statements without proofs. The proof is where the technique lives; the technique is what you reuse on exam problems.
Memorizing formulas. Fades by exam day. Derive them.
No problem sets. Re-reading notes is one of the lowest-utility study techniques (Dunlosky 2013). Closed-book problem sets drive recall.
When AI Helps
AI-grounded apps like Atlas earn their keep at finals. Ask "what does my Linear Algebra notes plus the textbook say about eigendecomposition?" and get a cited answer pulling from both. Citations matter; you must verify the math (LLMs hallucinate proofs). Atlas plus a graphing calculator (Desmos free) plus Wolfram Alpha covers most prep needs.
Atlas free tier covers individual student use; Pro at $20/month adds higher AI usage limits.
Final Take
Math notes need structure. DTPE (Definition, Theorem, Proof, Example) is the convention. Handwrite during lectures on iPad or paper; rewrite key theorems in LaTeX afterward. Problem sets are non-negotiable; re-reading notes is not enough. Atlas at finals for cross-course Q&A with citations. The system beats the tool, and the closed-book problem set beats the system. For a method-agnostic overview, see our how to take good notes guide.
