How to Use These Notes

These notes are built around a small number of findings from education research that are large in effect and robustly replicated. Knowing them changes how you should read what follows, because several of them contradict what feels productive.

1. Study worked examples before solving problems

The most replicated finding in mathematics education research over the last forty years is the worked example effect (Sweller & Cooper, 1985; Chen, Kalyuga & Sweller, 2015). Novices who study fully worked solutions learn more, and learn faster, than novices who attempt the same problems themselves.

The reason is element interactivity. Solving a problem by means-ends analysis forces you to hold in working memory your current state, the goal state, the differences between them, and candidate moves; all at once. A worked example shows you each state and the move that gets you to the next one, so the number of interacting elements collapses. What remains is the schema you were trying to learn.

So: read the worked examples first, and read them actively. Cover the solution, predict the next step, uncover, check. A worked example you read passively is worth a fraction of one you interrogate.

2. Then solve problems: the expertise reversal

The worked example effect reverses once you are no longer a novice. Kalyuga, Chandler, Tuovinen and Sweller (2001) found that for learners who already knew the material, studying worked examples was worse than solving problems: the example adds redundancy, and redundancy costs attention.

Every page here therefore fades from worked examples to completion problems to bare problems. The point at which you should stop reading examples and start solving is the point where you can predict the next step correctly most of the time. That transition is not optional; continuing to study examples after you have the schema actively slows you down.

3. Test yourself; do not reread

Retrieval practice; attempting to recall an answer rather than rereading it; produces far better retention than restudy, an effect large enough that Dunlosky et al. (2013) rated it among the two most effective techniques available to learners.

Rereading feels effective because the text becomes familiar. Familiarity is not learning: it is recognition, and recognition does not help you produce a proof in an exam. The uncomfortable version; closing the notes and trying to reconstruct the argument; is the one that works.

Each page here opens with recall prompts: questions to attempt from memory before reading. Attempting them and failing is not wasted effort; the failure is what makes the subsequent reading stick.

4. Space your practice, and mix topics

The spacing effect is, in Bjork’s words, one of the most general and robust effects in the history of research on learning and memory. Spaced practice beats massed practice on delayed tests, often by a factor of two, at equal total study time.

Interleaving; mixing problem types rather than blocking them; is the efficient way to space. It also does something spacing alone does not: it forces you to select the method, which is the part of problem solving that blocked practice never exercises. Rohrer’s work on mathematics practice shows interleaved practice substantially outperforming blocked practice on delayed tests, while learners overwhelmingly report the opposite preference.

That mismatch matters. Blocked practice produces rapid gains that feel like learning and do not survive the week. Interleaved practice feels worse and produces knowledge that does. Do not trust your feeling of fluency as a measure of learning. Bjork separates retrieval strength; how easily something comes to mind now; from storage strength; how durably it is held. Massed practice inflates the first and does little for the second.

5. Intuition first, then rigour, then both

A definition you cannot motivate is a string of symbols. Each page here follows the same order:

Motivation. A concrete problem the concept exists to solve.
Intuition. What it means, in pictures and plain language.
Definition. Precisely, with the reason for each hypothesis.
Theorem and proof. With the idea of the proof separated from its verification, because the idea is what generalises.
Worked examples. Fully worked, then faded.
Counterexamples. Where the naive version breaks, and why the hypotheses are load-bearing.
Recall prompts and problems.

Skipping to (3) and (4) is how mathematics is usually examined and almost never how it is understood. The counterexamples in (6) are the part most notes omit and most research depends on: knowing that continuity does not imply differentiability is worth less than knowing Weierstrass’ function and being able to say what it does.

6. The rigour ladder

Sections are marked so you can see how deep a given page goes:

Mark	Meaning
F Foundation	Definitions and first examples. Assumed later.
U Undergraduate	The standard degree-level treatment, with proofs.
A Advanced	Graduate-level: the machinery needed for research.
R Research	Open problems, recent results, and where the machinery fails.

A fresh undergraduate should read every F and U on a topic, then the A sections of the topics they intend to specialise in. R sections assume comfort with everything before them and are written to be read after the rest.

7. A practical schedule

The research above translates into a routine:

First pass. Read the page. Attempt the recall prompts from memory. Read the worked examples actively, predicting each step.
Second pass, 1–2 days later. Solve the completion problems without the notes. Anything you cannot do, mark.
Third pass, 1–2 weeks later. Redo the marked problems and two from an earlier topic; that is the interleaving.
Before assessment. Reconstruct the key definitions and theorem statements from memory, then check. Reconstructing a statement is harder and more useful than recognising it.

The second and third passes are where the learning happens. The first pass only makes them possible.

References

Sweller, J. & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2, 59–89.
Kalyuga, S., Chandler, P., Tuovinen, J. & Sweller, J. (2001). When problem solving is superior to studying worked examples. Journal of Educational Psychology, 93, 579–588.
Chen, O., Kalyuga, S. & Sweller, J. (2015). The worked example effect, the generation effect, and element interactivity. Journal of Educational Psychology, 107, 689–704.
Bjork, R. A. (1994). Memory and metamemory considerations in the training of human beings. In Metacognition: Knowing about Knowing. MIT Press.
Bjork, E. L. & Bjork, R. A. (2011). Making things hard on yourself, but in a good way: Creating desirable difficulties to enhance learning.
Dunlosky, J. et al. (2013). Improving students’ learning with effective learning techniques. Psychological Science in the Public Interest, 14, 4–58.
Rohrer, D. & Taylor, K. (2007). The shuffling of mathematics problems improves learning. Instructional Science, 35, 481–498.
Lyle, K. B. et al. (2020). How the amount and spacing of retrieval practice affect short- and long-term retention of mathematics knowledge. Educational Psychology Review, 32, 277–295.

How to Use These Notes