Aug 2, 2026

How I study math (or anything else)

Tools

Feynman Technique

Remnote

Problems

Topological Study (aka Schemas)

• Before diving into the weeds, your brain needs a map.

• Cognitive science calls this building schemas — mental frameworks that give new information a place to land

• Building a schema in two active steps:

  1. skim the chapter reading only section headers/intros, definitions, theorem statements; but not details/examples
  2. think about the purpose of this chapter/why it is useful/necessary

• Advance organizers: information presented by an instructor or prepared by a self-directed learner before the main learning material, designed to bridge the gap between what the learner already knows and what they are about to learn

  • Two types of advnace organizers for math:
  1. Expository (when the material is entirely new): a broad, conceptual analogical framework before you touch the technical math
  2. Comparative: While an expository organizer introduces a completely brand-new conceptual framework, a comparative organizer is deployed when the new material is an extension or evolution of something you already know deeply. Its primary function is twofold: a. Integration: It explicitly activates your existing mental schemas (prior knowledge). b. Discrimination: It forces you to contrast the subtle differences between the old concept and the new concept, preventing your brain from confusing the two.

• Algorithm:

  • 10-seconds per page: map the physical geography of the text
  • extract the anchor (identify peaks): copy main key words/formal statements
  • contruct the depdendency graph/flowchart
  • conjecture!

Moore method

Desirable Difficulty

• Desirable difficulties are conditions that make learning feel harder at first. However, they lead to stronger and longer-lasting learning.
• Desirable difficulties are short-term learning challenges that strengthen long-term retention and improve transfer of knowledge. These challenges boost long-term knowledge retention and transfer.
• Bjork (1994) shows testing aids understanding and memory. Learners process information deeply when tested (Bjork, 1994). Active recall helps learners build better knowledge (Bjork, 1994; Karpicke & Roediger, 2008). Karpicke & Roediger (2008) found active recall improves understanding.

Core strategies for desirable difficulties

• Spaced Practice
• Interleaving
• Testing
• Complex problem-solving without solutions (open problems)
• Cross-Domain retrieval: applying concepts from one topic to another

Tools/rituals for desirable difficulties

• Spaced Retrieval Calendars
• Interleaved Problem Sets
• Predict before even reading/test before knowing
• Elaborative self-interrogation: “Why does this make sense?” or “How does this connect to…?” creates productive struggle that deepens understanding beyond surface memorisation.
• Varied Context Practise: contextual interference creates transfer-ready knowledge that applies beyond the original learning environment
• Productive failure tasks (very hard/impossible problems): primes oneself to learn new stuff
• Feynman technique — self-explanation
• Write summaries from memory

Structural Cards

• One strategy specific to using spaced repitition for math is the use of structural cards.
• Target 3 structural categories for Remnote cards:
  • Intuition — (non-rigorous)
  • Interleaved problem bundles

Structure of mathematics (Garrity — All the math you missed…)

• As a first pass to placing structure on mathematics, we can view an area of mathematics as consisting of certain Objects, coupled with the notion of Equivalence between these objects. We can explain equivalence by looking at the allowed Maps, or functions, between the objects. At the beginning of most chapters, we will list the Objects and the Maps between the objects that are key for that subject. The Equivalence Problem is of course the problem of determining when two objects are the same, using the allowable maps.
• If the equivalence problem is easy to solve for some class of objects, then the corresponding branch of mathematics will no longer be active. If the equivalence problem is too hard to solve, with no known ways of attacking the problem, then the corresponding branch of mathematics will again not be active, though of course for opposite reasons. The hot areas of mathematics are precisely those for which there are rich partial but not complete answers to the equivalence problem
• Invariants (example: The’ goal of topology is to find enough invariants to be able to always determine when two spaces are different or the same. This has not come close to being done)
• Functions: Different areas of mathematics study different types of functions. Thus in learning a new area of mathematics, you should always “find the function” of interest.