How to Use These Notes: What Learning Science Says

Reading is not learning. The six techniques on this page are the ones with the strongest evidence from cognitive science; replicated across domains, age groups, and decades. They are ranked by effect size from Dunlosky et al. (2013), the most comprehensive review to date, and each is mapped to how the notes on this site implement it.

The single most important finding: the techniques that feel easiest are the least effective, and the techniques that feel hardest are the most effective. Re-reading feels productive and produces almost no long-term retention. Retrieval practice; actively pulling information from memory; feels harder and produces dramatically better retention. The discomfort is the learning.

Think of memory as a muscle, not a container. You do not fill it by pouring information in; you strengthen it by retrieving information out. Every time you successfully recall a fact, the neural pathway gets stronger and the fact becomes easier to recall next time. Every time you re-read a fact, you create an illusion of fluency; the text looks familiar, so you think you know it, but the retrieval pathway is unstrengthened and the fact will be gone in a week.

This is why the notes have recall prompts at the top of every page (before you read, try to recall what you already know) and self-tests at the end (after you read, retrieve what you just learned). Both are retrieval practice, and both are placed where the research says they belong: before (to activate prior knowledge) and after (to consolidate).

The finding. Actively recalling information from memory produces substantially better long-term retention than re-reading the same information. Roediger and Karpicke (2006) showed that students who tested themselves after one reading retained 50% more after one week than students who re-read four times; despite the re-readers feeling more confident.

How this site implements it. Every content page has recall prompts at the top (attempt before reading; they activate prior knowledge and set the retrieval framework) and self-tests at the bottom (attempt after reading - they consolidate). The transition pages have six-question self-tests; the graduate pages have interleaving problems; the ML course has code exercises where you predict the output before running.

How to use it. After reading a section, close the page. Write down what you remember. Open the page and check. The act of trying to recall; even when you fail; strengthens the pathway more than re-reading does. This is the hypercorrection effect: errors made during retrieval practice, when corrected, produce better retention than correct answers on the first try.

Citation. Roediger, H.L. & Karpicke, J.D. (2006). “Test-enhanced learning: Taking memory tests improves long-term retention.” Psychological Science, 17(3), 249–255.

The finding. Information reviewed at increasing intervals is retained far longer than information reviewed at fixed intervals or crammed. Ebbinghaus (1885) discovered the forgetting curve; Cepeda et al. (2006) confirmed the optimal spacing.

How this site implements it. The notes do not include a spaced-repetition system (that requires a tool like Anki), but every page is self-contained - you can return to it at any interval without needing the surrounding context. The recall prompts at the top of each page are designed for a specific spacing schedule: read today, attempt the recall prompts tomorrow, attempt them again after one week, then after one month.

How to use it. Do not re-read the same page in the same session. Instead: read page A, then page B, then return to page A’s recall prompts. The intervening material creates the spacing, and the return is harder (and more effective) than immediate re-reading.

Citation. Cepeda, N.J., Pashler, H., Vul, E., Wixted, J.T., & Rohrer, D. (2006). “Distributed practice in verbal recall tasks: A review and quantitative synthesis.” Psychological Bulletin, 132(3), 354–380.

The finding. Mixing different types of problems within a practice session produces better learning than blocking by type; even though blocked practice feels easier and produces better practice performance. Rohrer and Taylor (2007) showed this for mathematics: students who practised in blocks (all volume, then all surface area) felt more confident but scored worse than students who practised interleaved.

How this site implements it. The problem sets at the end of each track mix problem types. The self-tests on the transition pages mix logic, algebra, geometry, and calculus. The recall prompts at the top of each page ask about concepts from earlier tracks.

How to use it. Do not work through a track’s problems in order without revisiting earlier material. After finishing the measure-theory page, attempt a problem from the topology page. The mixing forces you to identify which technique to use; a skill that blocked practice never develops because the technique is always the same as the previous problem.

Citation. Rohrer, D. & Taylor, K. (2007). “The shuffling of mathematics problems improves learning.” Instructional Science, 35, 481–498.

The finding. For novice learners, studying worked examples; complete solutions shown step by step; produces better learning than solving equivalent problems, because solving requires the student to hold both the problem’s structure and the solution procedure in working memory simultaneously, which overloads a novice’s capacity. Sweller and Cooper (1985) called this the worked-example effect and grounded it in cognitive load theory.

How this site implements it. Every proof page has worked examples with every step shown. The measure-theory page’s Stokes’ theorem proof shows every sign’s origin. The functional analysis page’s Hahn–Banach proof shows the one-dimensional extension, then the Zorn’s lemma step, with each sub-derivation explicit.

How to use it. For novices: read the worked example first. Cover the solution. Attempt the same problem. Uncover and compare. The key is the comparison; what did you miss? For intermediates: attempt the problem first, then read the worked example and identify the differences. The progression from worked example to faded guidance to independent solving is called scaffolding.

Citation. Sweller, J. & Cooper, G.A. (1985). “The use of worked examples as a substitute for problem solving in learning algebra.” Cognition and Instruction, 2(1), 59–89.

The finding. Connecting new information to existing knowledge; asking “why?”, generating examples, explaining in your own words; produces deeper understanding and better retention than passive reading. Dunlosky et al. (2013) rated self-explanation as “moderate utility” and elaborative interrogation as promising.

How this site implements it. Every content page has an Intuition section before the formal definitions, and every proof page has a section that names where each hypothesis is spent. The intuition section is the elaboration: it connects the formal statement to something you already understand. The “where each hypothesis is spent” section is self-explanation: it forces you to see why the condition is needed, not just that it is listed.

How to use it. After reading a theorem, close the page and try to explain why the theorem is true in your own words; without the proof. Then read the proof and compare. The gap between your explanation and the proof is where your understanding is incomplete.

The finding. Information presented in both verbal and visual formats is remembered better than either alone, because the two channels are processed by different cognitive systems and create independent memory traces (Paivio, 1971).

How this site implements it. Every major concept has both a verbal description and a visual or diagrammatic representation. The topology page’s continuity section has both the epsilon-delta definition and the intuitive description. The ML course’s code examples produce visual output (plots, printed values) alongside the mathematical statements.

Re-reading. The most common study technique produces almost no long-term retention. The illusion of fluency; “I recognise this text”; is mistaken for knowledge. Dunlosky et al. rated it “low utility.”

Highlighting. The second most common technique performs no better than reading, because it requires no active processing. Highlighting selects information but does not require retrieval or elaboration.

Cramming. Massed practice produces good short-term performance and poor long-term retention, exactly as the spacing effect predicts. The crammer’s exam score will be higher than the spacer’s the next day, but the spacer’s score one month later will be dramatically higher.

Passive video watching. Watching someone else solve problems does not build the retrieval pathways that solving them yourself does. Videos are useful for initial exposure; seeing the technique demonstrated; but the learning happens when you attempt the problem yourself.

Every technique that works requires effort during learning. The effort is the signal that the retrieval pathway is being strengthened. If studying feels easy, you are not learning; you are performing the illusion of learning. The harder the retrieval, the stronger the retention.

This is Dunlosky et al.’s most important finding: the techniques students report using most (re-reading, highlighting) are the ones with the least evidence, and the techniques with the most evidence (practice testing, distributed practice) are the ones students report using least. The reason is that effective techniques feel harder, and students mistake difficulty for ineffectiveness.

These notes are designed to make the effective techniques easier to use - recall prompts at the right moments, self-tests at the right intervals, worked examples at the right depth; but the effort must come from you.

• Before You Begin; the content baseline this page’s techniques are designed to consolidate.
• Terms of Art; the vocabulary, whose acquisition follows the same retrieval-practice principles.
• Introduction to Algebraic Topology; recall prompts at the top of every page.
• Measure Theory; worked examples on every page, interleaved with the theory.
• Roediger & Karpicke (2006); Cepeda et al. (2006); Rohrer & Taylor (2007); Sweller & Cooper (1985); Dunlosky et al. (2013); the primary sources for the six techniques described here.

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