1# Overview of Spaced Repetition {#overview}
2
3The idea is mechanically very simple: as you learn things, you write
4_flashcards_, which are question/answer pairs. Then you review them: you look at
5the question, recall the answer, and turn the card over to see the answer. Then
6you grade yourself: did you recall correctly or incorrectly?
7
8If you keep recalling correctly, the review interval grows longer; if you get it
9wrong, the interval gets shorter.
10
11Some people use paper cards, most people use software such as [Anki][anki],
12because the algorithm schedules reviews efficiently, so you don't over-review
13material. Unless you have a paper fetish, just use software.
14
15# Limiting Factors {#limit}
16
17If you're so smart, why aren't you rich? Or: if spaced repetition is so
18effective, why doesn't everyone do it? Why isn't it as common as drinking
19coffee?
20
21There are two main limiting factors to effective spaced repetition.
22
23## Habit Formation {#habit}
24
25For spaced repetition to be useful, it has to be a habit. I drill flashcards
26every day as part of my morning routine. But habit formation is difficult,
27doubly so for people who have ADHD or are low in conscientiousness.
28
29The reason you have to do it every day is that the spaced repetition algorithm
30schedules the reviews for you, freeing you from having to do that manually. But
31you don't know what cards are due in a given day until you open the app. And if
32you skip a day, those cards pile up and are due the next day.
33
34A common failure mode (and I did this more than once, before I got the hang of
35it) is to use Anki for two weeks, then drop it, and pick it back up six months
36later only to find you have 600 cards due for review. This is not encouraging,
37and it defeats the point of spaced repetition, which is to review the cards on
38the intervals the algorithm chooses.
39
40I don't have much advice in this area, except that if you have persistent
41problems with conscientiousness, untreated ADHD etc. you should address that
42first.
43
44## Card-Writing Skills {#skill}
45
46Writing effective flashcards is a skill that took me a while to acquire. Many of
47the cards I wrote in the first four or six months of using spaced repetition
48consistently turned out pretty much useless, and this can be frustrating. The
49main reason to write this post was to communicate the lessons I learnt so you
50can jump in to using spaced repetition effectively from the start.
51
52One reason this can be frustrating is you'll often remember a flashcard for the
53first few weeks of it (when you're seeing it with high frequency), but after a
54couple of months, you start failing it. It didn't take root in your long term
55memory, because it was poorly written in some way. And this long feedback cycle
56means it takes time to acquire these skills through trial and error.
57
58# Words of Encouragement {#encouragement}
59
60Learning is an automatic, instinctual process. It's a fundamental feature of
61intelligence. It's a testament to how bad schooling is that people think they
62have to have a special kind of brain to learn effectively, and that the idea of
63learning triggers aversion in people. Remember the words of Feynman: "what one
64fool can do, another can".
65
66# Rules {#rules}
67
68Here are my rules for effective spaced repetition.
69
70The rules are sorted by applicability (but not necessarily importance), with the
71more general ones first, and most specific ones last.
72
73Because many of the examples involve multiple rules at the same time, I decided
74to list the examples separately from the list of rules.
75
76## Rule: Understand First {#understand}
77
78Don't try to memorize what you don't understand. The concepts should be clear in
79your head before you try to commit them to memory. "Clear" can be a fuzzy
80thing. What I tend to do is: dig, expand, and clarify the text until I'm
81comfortable I have a good grasp of this region of the concept graph, and then
82write the flashcards.
83
84Often, when reading a book, you can't write the flashcards exactly as you read
85the text, because further information can clarify or tie together important
86concepts. It can be useful to keep a scratchpad where you write tentative
87flashcard text as you read a chapter, and at the end you organize and
88re-organize your scratchpad until you can commit it to flashcards.
89
90## Rule: Be Honest {#honest}
91
92The software doesn't know whether you recalled something correctly or not. You
93are only accountable to yourself. If you recalled something wrong, or not quite
94right, err on the side of caution and mark it forgotten.
95
96## Rule: Keep It Fun {#fun}
97
98This is _crucial_ to maintaining the habit. If reviewing flashcards feels like a
99chore, you will become averse to doing it.
100
101I used to frequently have this problem. I solved it in a few ways:
102
1031. Having a diverse knowledge base you're drilling helps, so you are not bored
104 by going through the same topic for a long time. Typically, spaced repetition
105 software will shuffle the cards, so that if you're drilling all the cards
106 across all decks, you will be surprised often.
1072. A common source of frustration is cards that are too long to recall quickly,
108 and thus feel like a chore. Break big cards down into smaller cards. It feels
109 good to be able to fly through the cards quickly.
1103. Cards that are difficult to recall are very frustrating. I solved this by
111 applying the rules described in this post.
112
113## Rule: Repeat Yourself {#repeat}
114
115Memory is frequency times volume. Individual cards should be extremely brief,
116but your deck as a whole can be as repetitive as you want.
117
118## Rule: Organize by Source {#source}
119
120Organize content by source, not topic.
121
122The reason is you'll often bring in information from multiple sources: multiple
123textbooks, plus Wikipedia, plus lecture notes, etc. Each one of these sources
124likely has a different way of organizing knowledge.
125
126Don't waste time trying to find the perfect ontology.
127
128Make a deck for each source. In the case of textbooks, make a sub-deck for each
129chapter. In the case of math textbooks, possibly make a sub-sub-deck in each
130chapter to put theorem cards.
131
132This also makes it easier to keep track of how far along you got into a text.
133
134## Rule: Write Atomic Flashcards {#atomicity}
135
136Cards should be short. They should refer to as little information as
137possible. They should be like chemical bonds, linking individual _atoms_ of
138knowledge.
139
140_This is the most important thing._ By far the worst failure mode is to
141put too much in a flashcard.
142
143There's two reasons for this rule:
144
1451. Larger cards are harder to remember.
1462. It's harder to objectively grade yourself: when you reveal the answer, you
147 might have got some things right and some things wrong. If you click forget,
148 you will be over-reviewing the parts you already know. If you click
149 remembered, you will under-review the parts you forgot.
150
151There is one exception to this: you can have big cards if you also have smaller
152cards that add up to the same information. You can think of the larger card as
153testing that you can collate the information from the smaller cards.
154
155## Rule: Write Two-Way Questions {#bidirectionality}
156
157When possible, ask questions in two directions.
158
159Whenever you have a term with a definition, the obvious thing to do is to ask
160for the definition from the term, e.g.:
161
162> Q: What is the order of a group?
163>
164> A: The cardinality of its underlying set.
165
166But you can also ask for the term from the definition, e.g.:
167
168> Q: What is the term for the cardinality of a group?
169>
170> A: The group's order.
171
172When you have some notation, like $\mathbb{R}$ for the real numbers, or $\dim V$
173for the dimension of a vector space, the natural thing to ask is what the
174notation means.
175
176> Q: What does $\mathbb{R}$ stand for?
177>
178> A: The set of real numbers.
179
180You can also ask the question backwards:
181
182> Q: What is the notation for the set of real numbers?
183>
184> A: $\mathbb{R}$
185
186## Rule: Ask Questions in Multiple Ways {#poly}
187
188Ask questions in multiple ways. Ask for formal and informal definitions of
189terms. Ask for the formal and informal statements of a theorem. Ask questions
190forwards and backwards. Add contextual questions: "what is the intutition for
191[concept]?". Add questions that link different concepts across your knowledge
192graph.
193
194The more interlinked your knowledge graph is, the better.
195
196## Rule: Concept Graphs {#graph}
197
198It can help to visualize the concepts you're acquiring as being like a graph,
199where each node represents a discrete concept having certain properties, and the
200edges in the graphs are questions which get you from one concept to another.
201
202## Rule: Cache Your Insights {#caching}
203
204When studying, you will often infer new knowledge that is not explicitly written
205down in the text by thinking about what you've just read. After verifying that
206your inference is actually true, it can be helpful to "cache your insight"---to
207make a flashcard for that new discovered piece of knowledge.
208
209## Rule: Learning Hierarchies {#hier}
210
211A lot of knowledge is hierarchical, of the form "Foo can be either A, B, or C",
212or, dually, "A is a kind of Foo". By analogy to OOP: these concepts are joined
213by superclass and subclass relations.
214
215The idea is to ask questons in the top down direction ("What are the subclasses of
216Foo?") and the bottom-up direction ("What is Bar a subclass of?").
217
218This ties into keeping flashcards atomic. Even when some information is not
219hierarchical, intrinsically, breaking down large flashcards into smaller
220flashcards is fundamentally building a hierarchy of flashcards.
221
222## Rule: Learning Sequences {#seq}
223
224In general, to learn a sequence $(A_1, \dots, A_n)$, you want to generate the
225following flashcards for each $i \in [1,n]$:
226
227| Question | Answer |
228| -------------------------------- | ----------- |
229| What is the $i$-th element? | $A_i$ |
230| What is the position of $A_i$? | $i$ |
231| What element comes after $A_i$? | $A_\{i+1\}$ |
232| What element comes before $A_i$? | $A_\{i-1\}$ |
233
234You might also want:
235
2361. A **test card:** a flashcard asking you to recite the sequence from beginning
237 to end.
2382. A **cloze sequence:** flashcard with a cloze deletion for each element in the
239 sequence, to fill in the blank given the context.
240
241The [sequence script](#seq-script) can generate these for you.
242
243How thorough you want to be depends on the nature of the information. Most of
244the time I use a cloze card and a test card.
245
246Another type of card you might use (I use this to memorize poems) is a card that
247gives you some context (the previous one or two items in the sequence) and asks
248you to fill in the blank. For example, if you wanted to learn the sequence (A,
249B, C, D), you might have these flashcards:
250
251| Question | Answer |
252| ------------------- | ------ |
253| _Beginning_, ... | A |
254| _Beginning_, A, ... | B |
255| A, B, ... | C |
256| B, C, ... | D |
257
258The [poetry script](#poetry-script) can generate these for you.
259
260# Examples {#ex}
261
262Many of these examples are overkill: we collect a lot more flashcards than the
263subject deserves. But this is to illustrate the general rules. With experience,
264you can learn how many questions a particular topic requires, and different
265volumes of your knowledge graph will be more or less interlinked.
266
267## Example: Magma Formation {#magma}
268
269From my geology notes:
270
271<div class="border-box">
272
273Magma is liquid rock under the Earth's surface.
274
275The three magma-forming processes are:
276
2771. **Increasing Temperature:** increasing temperature can melt rock.
2781. **Decreasing Pressure:** when the pressure drops, atoms are more free to
279 move, and rock becomes liquid.
2801. **Addition of Water:** water lowers the melting point of rock, because the
281 water molecules disrupt the crystal bonds.
282
283Magma forms in three places:
284
2851. **Hot spots:** as hot rock rises, pressure decreases and it becomes magma.
2862. **Rift zones:** as tectonic plates are pulled apart, hot rock rises (because
287 it is less dense) to plug the gap and melts due to decreasing pressure.
2883. **Subduction zones:** water-rich ocean lithosphere sinks into the mantle. The
289 water heats up and rises, adding water to the overlying rock, which the
290 melts.
291
292</div>
293
294Let's break this down hierarchically. We want to memorize three things:
295
2961. What magma is.
2972. How it forms.
2983. Where it forms.
299
300First, the definition:
301
302| Question | Answer |
303| ------------------------------------------------------------ | -------------------------------------- |
304| What is magma? | Liquid rock under the Earth's surface. |
305| What is the term for liquid rock under surface of the Earth? | Magma. |
306
307Second, we want to know how magma forms. A common mistake here would be to put
308the magma-forming processes _and_ their explanations in the same
309flashcard. Rather, to keep each card as small as possible, we want to separate
310the list of processes from their definitions.
311
312So we first as for a list of mechanisms:
313
314| Question | Answer |
315| ------------------------------------- | --------------------------------------------------------------- |
316| What are the magma-forming processes? | Increasing temperature, decreasing pressure, addition of water. |
317
318And then we ask for an explanation of each. We don't really need to ask why
319adding temperature melts rock:
320
321| Question | Answer |
322| ------------------------------------------------------ | --------------------------------------------------------------- |
323| Why does decreasing pressure melt rock? | Because the atoms are more free to move. |
324| Why does adding water lower the melting point of rock? | Because water molecules disrupt the bonds in the rock minerals. |
325
326Third: where magma is found. Again, we separate the list from the details:
327
328| Question | Answer |
329| ---------------------- | ------------------------------------------------------- |
330| Where does magma form? | Over hot spots, in rift zones, and in subduction zones. |
331
332Then we ask for details. For each place where magma forms, we ask both which
333processes are involved, and what the full causal explanation is. We also ask the
334question backward: which places involve a given process.
335
336| Question | Answer |
337| ------------------------------------------------------------------------------ | --------------------------------------------------------------------------------------------------------------------------------------- |
338| What magma-forming process happens over a hot spot? | Pressure-release melting. |
339| What magma-forming process happens in a rift zone? | Pressure-release melting. |
340| What magma-forming process happens in a subduction zone? | Increasing temperature and addition of water. |
341| Where does magma form due to pressure release? | Hot spots and rift zones. |
342| Where does magma form due to increasing temperature and the addition of water? | Subduction zones. |
343| How does magma form in a hot spot? | As hot mantle rock rises, the decrease in pressure causes it to melt. |
344| How does magma form in a rift zone? | As the tectonic plates move apart, hot rock rises to fill the gap, and the decrease in pressure causes it to melt. |
345| How does magma form in a subduction zone? | Waterlogged crust dives into the mantle, the water turns to steam and rises, the addition of water to overlying rock causes it to melt. |
346
347We can visualize the resulting knowledge graph like this:
348
349<img style="margin-left: auto; margin-right: auto;" src="/assets/content/effective-spaced-repetition/magma.svg"/>
350
351## Example: Plate Tectonics {#tectonics}
352
353Here's the information:
354
355<div class="border-box">
356
357The zone where two or more tectonic plates meet is called a _plate
358boundary_. There are three kinds:
359
3601. Convergent boundaries: plates come together.
3612. Divergent boundaries: plates move apart.
3623. Transform boundaries: plates slide past each other.
363
364</div>
365
366Applying the rule that cards should be two-way, we want two flashcards for the
367term 'plate boundary'.
368
369| Question | Answer |
370| ---------------------------------------------------------- | ------------------------------------- |
371| What is a plate boundary? | The place where tectonic plates meet. |
372| What is the term for the place where tectonic plates meet? | Plate boundary. |
373
374For the different types of plate boundary, we only ask the question in the
375top-down direction (we don't need to ask "what kind of thing is a transform
376boundary?", since the name kind of gives it away):
377
378| Question | Answer |
379| ------------------------------------- | --------------------------------- |
380| What are the types of plate boundary? | Convergent, divergent, transform. |
381
382For each kind of plate boundary, we also ask the question in two ways:
383
384| Question | Answer |
385| -------------------------------------------------------- | -------------------------------------------- |
386| Definition: convergent boundary. | Where tectonic plates come together. |
387| Definition: divergent boundary. | Where tectonic plates move apart. |
388| Definition: transform boundary. | Where tectonic plates slide past each other. |
389| Term: place where tectonic plates come together. | Convergent boundary. |
390| Term: place where tectonic plates move apart. | Divergent boundary. |
391| Term: place where tectonic plates slide past each other. | Transform boundary. |
392
393Graphically, here's how the questions link the concepts in the knowledge graph:
394
395<img style="margin-left: auto; margin-right: auto;" src="/assets/content/effective-spaced-repetition/plates.svg"/>
396
397## Example: Neural Cells {#neural}
398
399<div class="border-box">
400
401Cells in the nervous system are divided into neurons and glia. Glial cells are
402divided into macroglia and microglia. Macroglia are divided into astrocytes,
403oligodendrocytes, and Schwann cells.
404
405</div>
406
407Visually:
408
409<img style="margin-left: auto; margin-right: auto;" src="/assets/content/effective-spaced-repetition/neuro.svg"/>
410
411We first write the top-down questions:
412
413| Question | Answer |
414| ---------------------------------------------- | ------------------------------------------------ |
415| What kinds of cell make up the nervous system? | Neurons and glia. |
416| What are the kinds of glial cell? | Microglia and macroglia. |
417| What are the kinds of macroglia? | Astrocytes, oligodendrocytes, and Schwann cells. |
418
419And the bottom-up questions. We don't ask these when the answers are obvious:
420"what are microglia/macroglia a kind of" has an obvious answer.
421
422| Question | Answer |
423| --------------------------------- | ---------- |
424| Astrocytes are a kind of... | Macroglia. |
425| Oligodendrocytes are a kind of... | Macroglia. |
426| Schwann cells are a kind of... | Macroglia. |
427
428## Example: Neuron Types {#neurons}
429
430This is a brief example about keeping cards short and using hierarchies to break
431things down.
432
433From my neuroscience notes:
434
435<div class="border-box">
436
437Neurons can be divided into three categories by their function:
438
4391. **Sensory:** feed sensory information into the brain.
4402. **Motor:** send motor commands to the muscles.
4413. **Interneurons:** connect within the CNS. These are further divided into:
442 1. **Local:** form circuits with nearby neurons.
443 2. **Relay:** have long axons and communicate across brain regions.
444
445</div>
446
447Let's start by doing this the wrong way, by loading too much information
448into one card.
449
450<table>
451 <thead>
452 <tr>
453 <th>Question</th>
454 <th>Answer</th>
455 </tr>
456 </thead>
457 <tbody>
458 <tr>
459 <td>What are the functional categories of neuron?</td>
460 <td>
461 <ul>
462 <li><b>Sensory:</b> feed sensory information into the brain.</li>
463 <li><b>Motor:</b> send motor commands to the muscles.</li>
464 <li><b>Interneurons:</b> connect within the CNS.</li>
465 </ul>
466 (😩 too long)
467 </td>
468 </tr>
469 <tr>
470 <td>What are the different types of interneuron?</td>
471 <td>
472 <ul>
473 <li><b>Local:</b> form circuits with nearby neurons.</li>
474 <li><b>Relay:</b> have long axons and communicate across brain regions.</li>
475 </ul>
476 (😩 too long)
477 </td>
478 </tr>
479 </tbody>
480</table>
481
482Let's first break this down by separating _terms_ from _definitions_:
483
484| Question | Answer |
485| --------------------------------------------- | ------------------------------------------------------------------ |
486| What are the functional categories of neuron? | Sensory, motor, interneurons. |
487| What are sensory neurons? | Neurons which feed information into the brain. |
488| What are motor neurons? | Neurons which send commands to the muscles. |
489| What are interneurons? | Neurons which connect within the CNS. |
490| What are the types of interneuron? | Local, relay. |
491| What are local interneurons? | Interneurons that form circuits with nearby neurons. |
492| What are relay interneurons? | Interneurons have long axons and communicate across brain regions. |
493
494Now we ask the questions in the backward direction: from the definition to the term:
495
496| Question | Answer |
497| ---------------------------------------------------------------------------- | ------------------ |
498| What is the term for a neuron that feeds information into the brain? | Sensory neuron. |
499| What is the term for a neuron that sends commands to the muscles? | Motor neuron. |
500| What is the term for a neuron that connects within the CNS? | Interneuron. |
501| What is the term for an interneuron that forms circuits with nearby neurons? | Local interneuron. |
502| What is the term for an interneuron that communicates across brain regions? | Relay interneuron. |
503
504## Example: Vector Spaces {#vect}
505
506Here's what we want to learn:
507
508<div class="border-box">
509
510A vector space, informally, is a set whose elements---called vectors---can be
511added or scaled.
512
513More formally: a vector space over a field $\mathbb{F}$ is a set $V$ plus two
514operations:
515
5161. Vector addition: $V \times V \to V$
5171. Scalar multiplication: $V \times \mathbb{F} \to V$
518
519Satisfying the following axioms:
520
521Commutativity of Addition
522: $u + v = v + u$
523
524Associativity of Addition
525: $u + (v + w) = (u + v) + w$
526
527Identity of Addition
528: $\exists 0 \in V : v + 0 = v$
529
530Inverse of Addition
531: $\forall v \in V, \exists -v \in V : v + (-v) = 0$
532
533Identity of Scaling
534: $1v = v$
535
536Distributivity
537: $\forall v \in V, a,b \in \mathbb{F} : (a+b)v = av + bv$
538
539</div>
540
541We have to break this down. Severely. We will do this step by step.
542
543First, we have to separate the informal (intuitive) and formal definitions:
544
545<table>
546 <thead>
547 <tr>
548 <th>Question</th>
549 <th>Answer</th>
550 </tr>
551 </thead>
552 <tbody>
553 <tr>
554 <td>Informally: what is a vector space?</td>
555 <td>A set whose elements can be added or scaled.</td>
556 </tr>
557 <tr>
558 <td>Formally: what is a vector space?</td>
559 <td>A vector space over a field $\mathbb{F}$ is a set $V$ plus two operations: vector addition and scalar multiplication.</td>
560 </tr>
561 </tbody>
562</table>
563
564We add one brief question about notation (you may choose to skip this one, it's
565an example):
566
567<table>
568 <thead>
569 <tr>
570 <th>Question</th>
571 <th>Answer</th>
572 </tr>
573 </thead>
574 <tbody>
575 <tr>
576 <td>What are the elements of a vector space called?</td>
577 <td>Vectors.</td>
578 </tr>
579 </tbody>
580</table>
581
582Now we ask about the signatures of the operations:
583
584<table>
585 <thead>
586 <tr>
587 <th>Question</th>
588 <th>Answer</th>
589 </tr>
590 </thead>
591 <tbody>
592 <tr>
593 <td>What is the signature of vector addition?</td>
594 <td>$V \times V \to V$</td>
595 </tr>
596 <tr>
597 <td>What is the signature of scalar multiplication?</td>
598 <td>$V \times \mathbb{F} \to V$</td>
599 </tr>
600 </tbody>
601</table>
602
603Next, we ask for the axioms:
604
605<table>
606 <thead>
607 <tr>
608 <th>Question</th>
609 <th>Answer</th>
610 </tr>
611 </thead>
612 <tbody>
613 <tr>
614 <td>What are the axioms that define a vector space?</td>
615 <td>
616 <ol>
617 <li>Commutativity of Addition</li>
618 <li>Associativity of Addition</li>
619 <li>Identity of Addition</li>
620 <li>Inverse of Addition</li>
621 <li>Identity of Scaling</li>
622 <li>Distributivity</li>
623 </ol>
624 </td>
625 </tr>
626 </tbody>
627</table>
628
629Finally, we ask what each axiom means:
630
631<table>
632 <thead>
633 <tr>
634 <th>Question</th>
635 <th>Answer</th>
636 </tr>
637 </thead>
638 <tbody>
639 <tr>
640 <td>Vector spaces: state: commutativity of addition</td>
641 <td>$u + v = v + u$</td>
642 </tr>
643 <tr>
644 <td>Vector spaces: state: associativity of addition</td>
645 <td>$u + (v + w) = (u + v) + w$</td>
646 </tr>
647 <tr>
648 <td>Vector spaces: state: identity of addition</td>
649 <td>$\exists 0 \in V: v + 0 = v$</td>
650 </tr>
651 <tr>
652 <td>Vector spaces: state: inverse of addition</td>
653 <td>$\forall v \in V, \exists -v \in V: v + (-v) = 0$</td>
654 </tr>
655 <tr>
656 <td>Vector spaces: state: identity of scaling</td>
657 <td>$1v = v$</td>
658 </tr>
659 <tr>
660 <td>Vector spaces: state: distributivity</td>
661 <td>$\forall v \in V, a,b \in \mathbb{F}: (a+b)v = av + bv$</td>
662 </tr>
663 </tbody>
664</table>
665
666Graphically, you can try visualizing the flashcards and their relationships like this:
667
668<a href="/assets/content/effective-spaced-repetition/vect.svg"><img style="margin-left: auto; margin-right: auto;" src="/assets/content/effective-spaced-repetition/vect.svg"/></a>
669
670If you want to be extra thorough, you can also write the backwards questions:
671
672| Question | Answer |
673| ------------------------------------------------------------------------- | ------------------------- |
674| What is the term for a set whose elements can be added or scaled? | A vector space. |
675| Name this axiom: $u + v = v + u$ | Commutativity of Addition |
676| Name this axiom: $u + (v + w) = (u + v) + w$ | Associativity of Addition |
677| Name this axiom: $\exists 0 \in V : v + 0 = v$ | Identity of Addition |
678| Name this axiom: $\forall v \in V, \exists -v \in V : v + (-v) = 0$ | Inverse of Addition |
679| Name this axiom: $1v = v$ | Identity of Scaling |
680| Name this axiom: $\forall v \in V, a,b \in \mathbb{F} : (a+b)v = av + bv$ | Distributivity |
681
682## Example: Parity Group {#parity}
683
684<div class="border-box">
685
686The _parity group_ is a group that represents the rules for adding even and odd
687numbers. The underlying set is $\set{\text{even}, \text{odd}}$, with
688$\text{even}$ and $\text{odd}$ representing even and odd numbers
689respectively. The composition table is:
690
691| $+$ | $\text{even}$ | $\text{odd}$ |
692| $\text{even}$ | $\text{even}$ | $\text{odd}$ |
693| $\text{odd}$ | $\text{odd}$ | $\text{even}$ |
694
695The identity element is $\text{even}$. The group is Abelian.
696
697</div>
698
699We can turn this into the following flashcards:
700
701| Question | Answer |
702| ------------------------------------------------- | -------------------------------------------------------------------- |
703| What is the parity group? | The group that represents the rules for adding even and odd numbers. |
704| What is the order of the parity group? | $2$ |
705| What is the underlying set of the parity group? | $\set{\text{even}, \text{odd}}$ |
706| What is the identity element of the parity group? | $\text{even}$ |
707| What is the operation of the parity group? | Addition of even and odd numbers. |
708| $\text{even} + \text{even} = $ | $\text{even}$ |
709| $\text{even} + \text{odd} = $ | $\text{odd}$ |
710| $\text{odd} + \text{even} = $ | $\text{odd}$ |
711| $\text{odd} + \text{odd} = $ | $\text{even}$ |
712| Is the parity group Abelian? Why or why not? | Yes, because addition commutes. |
713
714## Example: Logical Consequence {#logical-consequence}
715
716From my notes on logic:
717
718<div class="border-box">
719
720The two notions of _logical consequence_ are:
721
7221. **Semantic Consequence:** $Q$ is a semantic consequence of $P$ iff, in every
723 interpretation where $P$ is true, $Q$ is also true. This is denoted $P
724 \models Q$.
7251. **Syntactic Consequence:** $Q$ is a syntactic consequence of $P$ iff there
726 exists a proof from $P$ to $Q$. This is denoted $P \vdash Q$.
727
728Semantic consequence is about interpretations, while syntactic consequence is
729about proofs.
730
731</div>
732
733We begin with the most basic question:
734
735| Question | Answer |
736| ------------------------------------------------ | ----------------------- |
737| What are the two notions of logical consequence? | Semantic and syntactic. |
738
739Then we ask questions specifically about semantic consequence:
740
741| Question | Answer |
742| --------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------- |
743| Define semantic consequence | $Q$ is a semantic consequence of $P$ iff in every interpretation where $P$ is true, $Q$ is also true. |
744| What's the notation for "$Q$ is a semantic consequence of $P$"? | $P \models Q$ |
745| What does $P \models Q$ mean? | $Q$ is a semantic consequence of $P$ |
746| Semantic consequence connects sentences by... | Interpretations. |
747| Which notion of logical consequence involves interprerations? | Semantic consequence. |
748
749And then about syntactic consequence:
750
751| Question | Answer |
752| ---------------------------------------------------------------- | --------------------------------------------------------------------------- |
753| Define syntactic consequence | $Q$ is a syntactic consequence of $P$ iff there is a proof from $P$ to $Q$. |
754| What's the notation for "$Q$ is a syntactic consequence of $P$"? | $P \vdash Q$ |
755| What does $P \vdash Q$ mean? | $Q$ is a syntactic consequence of $P$ |
756| Syntactic consequence connects sentences by... | Proofs. |
757| Which notion of logical consequence involves proofs? | Syntactic consequence. |
758
759## Example: Periodization {#perodization}
760
761Timelines are a great example of how breaking information down hierarchically
762can help you learn long sequences. Sometimes the breakdown is already done for
763us.
764
765<div class="border-box">
766
767The [geologic time scale][gts] (GTS) divides the geological record of the Earth
768into four nested time units:
769
770[gts]: https://en.wikipedia.org/wiki/Geologic_time_scale
771
7721. The **eon** is the largest unit. Eons last hundreds of millions of years.
7732. Eons are further divided into **eras**, which last tens to hundreds of
774 millions of years.
7753. Eras are divided into **periods**, which last millions to tens of millions of
776 years.
7774. Finally, periods are divided into **epochs**, which last hundreds of
778 thousands to millions of years.
779
780The four eons, from oldest to most recent, are:
781
7821. Hadean (4.5Gya to 4Gya)
7832. Archean (4Gya to 2.5Gya)
7843. Proterozoic (2.5Gya to 538Mya)
7854. Phanerozoic (538Mya to present)
786
787</div>
788
789We want to learn the following things:
790
7911. What the geologic time scale is.
7922. How it divides the Earth's history.
7933. The four eons.
794
795Let's begin with the simplest flashcards, the definition of the GTS:
796
797| Question | Answer |
798| ----------------------------------------------------- | -------------------------------- |
799| What is the geologic time scale? | The timeline of Earth's history. |
800| What is the term for the timeline of Earth's history? | The geologic time scale. |
801
802The subdivisions form a sequence, from oldest to most recent: eon, era, period,
803epoch. So let's feed it into the [sequence script](#seq-script). Here's the
804input:
805
806```
807Geologic Time Units
808Eon
809Era
810Period
811Epoch
812```
813
814Running `cat units.txt | ./sequence.py > units.csv` and importing `units.csv`
815into Mochi, we get these flashcards:
816
817| Question | Answer |
818| ------------------------------------------------------------ | ------------------------ |
819| **Geologic Time Units:** Recall all elements of the sequence | Eon, Era, Period, Epoch. |
820| **Geologic Time Units:** What element has position 1? | Eon. |
821| **Geologic Time Units:** What element has position 2? | Era. |
822| **Geologic Time Units:** What element has position 3? | Period. |
823| **Geologic Time Units:** What element has position 4? | Epoch. |
824| **Geologic Time Units:** What is the position of Eon? | 1. |
825| **Geologic Time Units:** What is the position of Era? | 2. |
826| **Geologic Time Units:** What is the position of Period? | 3. |
827| **Geologic Time Units:** What is the position of Epoch? | 4. |
828| **Geologic Time Units:** What comes after Eon? | Era. |
829| **Geologic Time Units:** What comes after Era? | Period. |
830| **Geologic Time Units:** What comes after Period? | Epoch. |
831| **Geologic Time Units:** What comes before Era? | Eon. |
832| **Geologic Time Units:** What comes before Period? | Era. |
833| **Geologic Time Units:** What comes before Epoch? | Period. |
834
835You probably don't need _all_ of these. You can probably get away with just
836these:
837
838| Question | Answer |
839| ------------------------------------------------------------------------ | ----------------------- |
840| What are the units of the geologic time scale, from largest to smallest? | Eon, era period, epoch. |
841| What is the largest unit in the geologic time scale? | The eon. |
842| What is the second-largest unit in the geologic time scale? | The era. |
843| What is the third-largest unit in the geologic time scale? | The period. |
844| What is the smallest unit in the geologic time scale? | The epoch. |
845
846Now, since this is a concept hierarchy, we also ask the "what is" questions.
847
848| Question | Answer |
849| ------------------ | -------------------------------------- |
850| What is an eon? | A division of the geologic time scale. |
851| What is an era? | A division of the geologic time scale. |
852| What is an period? | A division of the geologic time scale. |
853| What is an epoch? | A division of the geologic time scale. |
854
855And, since units have a duration, we ask what for the duration. We do this
856forwards and backwards:
857
858| Question | Answer |
859| --------------------------------------------------------------------- | ------------------------------------------- |
860| What is the duration of an eon? | Hundreds of millions of years. |
861| Which geologic unit lasts hundreds of millions of years? | Eons. |
862| What is the duration of an era? | Tens to hundreds of millions of years. |
863| Which geologic unit lasts tens to hundreds of millions of years? | Eras. |
864| What is the duration of a period? | Millions to tens of millions of years. |
865| Which geologic unit lasts millions to tens of millions of years? | Periods. |
866| What is the duration of an epoch? | Hundreds of thousands to millions of years. |
867| Which geologic unit lasts hundreds of thousands to millions of years? | Epochs. |
868
869Now, the four eons. These form a sequence, we don't do the whole sequence script
870thing again, since you have probably, again, just use these:
871
872| Question | Answer |
873| ------------------------------- | ------------------------------------------ |
874| List eons from oldest to newest | Hadean, Archean, Proterozoic, Phanerozoic. |
875| What is the first eon? | Hadean |
876| What is the second eon? | Archean |
877| What is the third eon? | Proterozoic |
878| What is the fourth eon? | Phanerozoic |
879
880We also ask when each eon began and ended, forwards and backwards:
881
882| Question | Answer |
883| ------------------------------- | ----------- |
884| When did the Hadean begin? | 4.5 Gya |
885| When did the Hadean end? | 4 Gya |
886| Which eon began 4.5 Gya? | Hadean |
887| Which eon ended 4 Gya? | Hadean |
888| When did the Archean begin? | 4 Gya |
889| When did the Archean end? | 2.5 Gya |
890| Which eon began 4 Gya? | Archean |
891| Which eon ended 2.5 Gya? | Archean |
892| When did the Proterozoic begin? | 2.5 Gya |
893| When did the Proterozoic end? | 538 Mya |
894| Which eon began 2.5 Gya? | Proterozoic |
895| Which eon ended 538 Mya? | Proterozoic |
896| When did the Phanerozoic begin? | 538 Mya |
897| When did the Phanerozoic end? | Present |
898| Which eon began 538 Mya? | Phanerozoic |
899| Which eon is ongoing? | Phanerozoic |
900
901## Example: Rational Numbers {#ratnums}
902
903Let's commit this to spaced repetition:
904
905<div class="border-box">
906
907The set of rational numbers, denoted $\mathbb{Q}$, is the set of fractions with
908integer numerator and denominator, where the denominator is non-zero.
909
910Formally:
911
912$$
913\mathbb{Q} = \left\{\, \frac{p}{q} \,\, \middle| \,\, p, q \in \Z \land q \neq 0 \,\right\}
914$$
915
916The $\mathbb{Q}$ stands for _quotient_.
917
918</div>
919
920Let's visualize the concept graph as we build up the flashcards. We start with
921the central node, the concept of the rational numbers:
922
923<img style="margin-left: auto; margin-right: auto;" src="/assets/content/effective-spaced-repetition/rats1.svg"/>
924
925Then we add a notation node, linked by two forward and backwards questions:
926
927<img style="margin-left: auto; margin-right: auto;" src="/assets/content/effective-spaced-repetition/rats2.svg"/>
928
929| Question | Answer |
930| ----------------------------------------------------- | ---------------------------- |
931| What is the notation for the set of rational numbers? | $\mathbb{Q}$. |
932| What does $\mathbb{Q}$ stand for? | The set of rational numbers. |
933
934Formal as well as informal definitions:
935
936<img style="margin-left: auto; margin-right: auto;" src="/assets/content/effective-spaced-repetition/rats3.svg"/>
937
938<table>
939 <thead>
940 <tr>
941 <th>Question</th>
942 <th>Answer</th>
943 </tr>
944 </thead>
945 <tbody>
946 <tr>
947 <td>Informally, what is the set of rational numbers?</td>
948 <td>The set of fractions with integer numerator and denominator, where the denominator is non-zero.</td>
949 </tr>
950 <tr>
951 <td>Formally, what is the set of rational numbers?</td>
952 <td>$\mathbb{Q} = \left\{\, \frac{p}{q} \,\, \middle| \,\, p, q \in \Z \land q \neq 0 \,\right\}$</td>
953 </tr>
954 <tr>
955 <td>What's the term for the set of integer fractions?</td>
956 <td>The rational numbers.</td>
957 </tr>
958 <tr>
959 <td>What is the name of this set? $\left\{\, \frac{p}{q} \,\, \middle| \,\, p, q \in \Z \land q \neq 0 \,\right\}$</td>
960 <td>The rational numbers.</td>
961 </tr>
962 </tbody>
963</table>
964
965And a final note on notation: what the $\mathbb{Q}$ stands for:
966
967<img style="margin-left: auto; margin-right: auto;" src="/assets/content/effective-spaced-repetition/rats4.svg"/>
968
969| Question | Answer |
970| ------------------------------------------------------ | --------------- |
971| What are the rational numbers denoted by $\mathbb{Q}$? | Q for quotient. |
972
973## Example: Regular Expressions {#regex}
974
975This is an example about asking questions in two ways.
976
977These cards go from a concept to a regex:
978
979| Question | Answer |
980| --------------------------------------- | ------ |
981| What regex matches the start of a line? | `^` |
982| What regex matches the end of a line? | `$` |
983| What regex matches a digit? | `\d` |
984
985In addition to the above, add cards that go from the regex to the concept:
986
987| Question | Answer |
988| --------------------- | -------------------- |
989| What does `^` match? | The start of a line. |
990| What does `$` match? | The end of a line. |
991| What does `\d` match? | A digit 0-9. |
992
993## Example: Voltage {#voltage}
994
995This is an example of asking questions in different ways
996
997<div class="border-box">
998
999The _voltage_ between two points $A$ and $B$ can be defined as either:
1000Voltage can be defined as:
1001
10021. The difference in electric potential between the two points.
10031. The amount of work done by a $1C$ particle as it travels from $A$ to $B$.
1004
1005</div>
1006
1007The idea here is:
1008
10091. We first ask for the definition of voltage in terms of both electric
1010 potential and work.
10112. We also ask what is the term for each definition.
1012
1013Which gives us the following flashcards:
1014
1015| Question | Answer |
1016| ------------------------------------------------------------------------------------- | ----------------------------------------------------- |
1017| What is voltage in terms of electric potential? | Difference in electric potential between two points. |
1018| What is voltage in terms of work? | Work done by a 1C particle as it travels from A to B. |
1019| What is the term for the difference in electric potential between two points? | Voltage. |
1020| What is the term for the work done by a 1C particle as it travels between two points? | Voltage. |
1021
1022## Example: Isomers {#isomers}
1023
1024<div class="border-box">
1025
1026Two chemical compounds are said to be **isomers** of each other if they have the
1027same chemical formula (same number of atoms of each element) but their
1028three-dimensional structure differs.
1029
1030Isomers can be divided into:
1031
10321. **Structural Isomers:** the chemical formula is the same but the atoms are
1033 bonded differently.
10341. **Stereoisomers:** the chemical formula and the bonds are the same but the
1035 spatial arrangement is different. These are divided into:
1036 1. **Conformational Isomers:** can be intercoverted by rotating about a sigma
1037 bond.
1038 1. **Configurational Isomers:** cannot be interconverted without breaking a
1039 bond. These are further subdivided into:
1040 1. **Enantiomers:** non-superposable mirror images. Also called _optical
1041 isomers_ because of the way they reflect plane-polarized light.
1042 1. **Diastereomers:** not enantiomers. One important subtype is:
1043 1. **Cis/Trans Isomers:** occur when two functional groups can find
1044 themselves on the same or opposite sides of a rigid structure. When
1045 both functional groups are on the same side of the rigid structure,
1046 that is a _cis_ isomer; when they are on opposite sides, that is a
1047 _trans_ isomer.
1048
1049This is an example of a _cis_ isomer:
1050
1051<img style="margin-left: auto; margin-right: auto; width: 200px;" src="/assets/content/effective-spaced-repetition/cis.svg"/>
1052
1053This is an example of a _trans_ isomer:
1054
1055<img style="margin-left: auto; margin-right: auto; width: 200px;" src="/assets/content/effective-spaced-repetition/trans.svg"/>
1056
1057</div>
1058
1059This is fairly straightforward: we have to learn a hierarchy of
1060definitions. We'll divide this into two tasks:
1061
10621. First, definitions. Ask questions from the term to the definition and from
1063 the definition to the term.
10642. Second, hierarchy: ask about subtypes and supertypes.
1065
1066So let's begin with the definitions. First we ask the questions in the forward
1067direction:
1068
1069| Question | Answer |
1070| --------------------------------- | ----------------------------------------------------------------------------------------------- |
1071| What is an isomer? | Two compounds are isomers when they have the same chemical formula but different 3D structures. |
1072| What are structural isomers? | Compounds with the same formula but the atoms have a different bond graph.. |
1073| What are stereoisomers? | Compounds with the same formula and bond graph but different spatial arrangement. |
1074| What are conformational isomers? | Isomers that can be interconverted by rotating about a sigma bond. |
1075| What are configurational isomers? | Isomers that cannot be interconverted without breaking a bond. |
1076| What are enantiomers? | Non-superposable mirror images. |
1077| What are diastereomers? | Stereoisomers that are not enantiomers. |
1078| What are cis/trans isomers? | Isomers where two functional groups are on the same or opposite sides of a rigid structure. |
1079
1080And now the definitions, in the backward direction:
1081
1082| Question | Answer |
1083| ---------------------------------------------------------------------------------------------------------------- | ---------------------- |
1084| What is the term for compounds with the same chemical formula but different 3D structures? | Isomer |
1085| What is the term for isomers with the same formula but a different bond graph? | Structural Isomer |
1086| What is the term for isomers that have the same bond graph different spatial arrangement? | Stereoisomer |
1087| What is the term for isomers that can be interconverted by rotating about a sigma bond? | Conformational Isomer |
1088| What is the term for isomers that cannot be interconverted without breaking a bond? | Configurational Isomer |
1089| What is the term for non-superposable mirror images? | Enantiomer |
1090| What is the term for stereoisomers that are not enantiomers? | Diastereomer |
1091| What is the term for isomers where two functional groups are on the same or opposite sides of a rigid structure? | Cis/Trans Isomer |
1092
1093We left some information out, to keep the cards atomic, now we have to ask
1094questions to recall that information:
1095
1096| Question | Answer |
1097| ---------------------------------------------------------------------------------------------- | ---------------------------------------------------------------------- |
1098| What is another term for enantiomer? | Optical isomer. |
1099| Why are enantiomers also called optical isomers? | Because of the way they reflect plane-polarized light. |
1100| What is an optical isomer? | Another term for enantiomer. |
1101| What is a _cis_ isomer? | One with both functional groups on the same side of a rigid structure. |
1102| What is the term for an isomer with both functional groups on the same side a rigid structure. | A _cis_ isomer. |
1103| What is a _trans_ isomer? | One with both functional groups on the same side of a rigid structure. |
1104| What is the term for an isomer with both functional groups on the same side a rigid structure. | A _trans_ isomer. |
1105
1106Now we move on to the hierarchy, which connects these concepts. We first ask the
1107questions in the downward direction, from parent to child:
1108
1109| Question | Answer |
1110| ------------------------------------------------- | ------------------------------------------------ |
1111| What are the subtypes of isomers? | Structural isomers, stereoisomers. |
1112| What are the subtypes of stereoisomers? | Conformational isomers, configurational isomers. |
1113| What are the subtypes of configurational isomers? | Enantiomers, diastereomers. |
1114| What are the subtypes of diastereomers? | Cis/trans isomers. |
1115
1116And now in the upward direction:
1117
1118| Question | Answer |
1119| ----------------------------------------- | ---------------------- |
1120| Structural isomers are a kind of ... | Isomer |
1121| Stereoisomers are a kind of ... | Isomer |
1122| Conformational isomers are a kind of ... | Stereoisomer |
1123| Configurational isomers are a kind of ... | Stereoisomer |
1124| Enantiomers are a kind of ... | Configurational isomer |
1125| Diastereomers are a kind of ... | Configurational isomer |
1126| Cis/trans isomers are a kind of ... | Diastereomer |
1127
1128And, finally, the examples:
1129
1130| Question | Answer |
1131| ------------------------------------------------------------------------------------------------------------------------------------------------------------ | ----------------- |
1132| What kind of isomer is this? <img style="margin-left: auto; margin-right: auto; width: 200px;" src="/assets/content/effective-spaced-repetition/cis.svg"/> | A _cis_ isomer. |
1133| What kind of isomer is this? <img style="margin-left: auto; margin-right: auto; width: 200px;" src="/assets/content/effective-spaced-repetition/trans.svg"/> | A _trans_ isomer. |
1134
1135## Example: Months of the Year {#months}
1136
1137Suppose you want to memorize:
1138
1139<div class="border-box">
1140
11411. January
11422. February
11433. March
11444. April
11455. May
11466. June
11477. July
11488. August
11499. September
115010. October
115111. November
115212. December
1153
1154</div>
1155
1156The index-to-element flashcards:
1157
1158| Question | Answer |
1159| ----------------------------------- | --------- |
1160| What is the 1st month of the year? | January |
1161| What is the 2nd month of the year? | February |
1162| What is the 3rd month of the year? | March |
1163| What is the 4th month of the year? | April |
1164| What is the 5th month of the year? | May |
1165| What is the 6th month of the year? | June |
1166| What is the 7th month of the year? | July |
1167| What is the 8th month of the year? | August |
1168| What is the 9th month of the year? | September |
1169| What is the 10th month of the year? | October |
1170| What is the 11th month of the year? | November |
1171| What is the 12th month of the year? | December |
1172
1173The element-to-index flashcards:
1174
1175| Question | Answer |
1176| --------------------------------------- | ------ |
1177| January is the ... month of the year. | 1 |
1178| February is the ... month of the year. | 2 |
1179| March is the ... month of the year. | 3 |
1180| April is the ... month of the year. | 4 |
1181| May is the ... month of the year. | 5 |
1182| June is the ... month of the year. | 6 |
1183| July is the ... month of the year. | 7 |
1184| August is the ... month of the year. | 8 |
1185| September is the ... month of the year. | 9 |
1186| October is the ... month of the year. | 10 |
1187| November is the ... month of the year. | 11 |
1188| December is the ... month of the year. | 12 |
1189
1190The successor flashcards:
1191
1192| Question | Answer |
1193| --------------------------- | --------- |
1194| What comes after January? | February |
1195| What comes after February? | March |
1196| What comes after March? | April |
1197| What comes after April? | May |
1198| What comes after May? | June |
1199| What comes after June? | July |
1200| What comes after July? | August |
1201| What comes after August? | September |
1202| What comes after September? | October |
1203| What comes after October? | November |
1204| What comes after November? | December |
1205
1206And the predecessor flashcards:
1207
1208| Question | Answer |
1209| ---------------------------- | --------- |
1210| What comes before February? | January |
1211| What comes before March? | February |
1212| What comes before April? | March |
1213| What comes before May? | April |
1214| What comes before June? | May |
1215| What comes before July? | June |
1216| What comes before August? | July |
1217| What comes before September? | August |
1218| What comes before October? | September |
1219| What comes before November? | October |
1220| What comes before December? | November |
1221
1222## Example: Powers of Two {#powers}
1223
1224Let's memorize the first sixteen powers of two:
1225
1226<div class="border-box">
1227
1228$$
1229\begin{align*}
12302^{2} &= 4\\
12312^{3} &= 8\\
12322^{4} &= 16\\
12332^{5} &= 32\\
12342^{6} &= 64\\
12352^{7} &= 128\\
12362^{8} &= 256\\
12372^{9} &= 512\\
12382^{10} &= 1024\\
12392^{11} &= 2048\\
12402^{12} &= 4096\\
12412^{13} &= 8192\\
12422^{14} &= 16384\\
12432^{15} &= 32768\\
12442^{16} &= 65536
1245\end{align*}
1246$$
1247
1248</div>
1249
1250The forward cards ask for the power:
1251
1252| Question | Answer |
1253| -------- | ------- |
1254| $2^2$ | $4$ |
1255| $2^3$ | $8$ |
1256| $2^4$ | $16$ |
1257| $2^5$ | $32$ |
1258| $2^6$ | $64$ |
1259| $2^7$ | $128$ |
1260| $2^8$ | $256$ |
1261| $2^9$ | $512$ |
1262| $2^{10}$ | $1024$ |
1263| $2^{11}$ | $2048$ |
1264| $2^{12}$ | $4096$ |
1265| $2^{13}$ | $8192$ |
1266| $2^{14}$ | $16384$ |
1267| $2^{15}$ | $32768$ |
1268| $2^{16}$ | $65536$ |
1269
1270While the backwards cards ask for the exponent from the power:
1271
1272| Question | Answer |
1273| -------------- | ------ |
1274| $\log_2 4$ | $2$ |
1275| $\log_2 8$ | $3$ |
1276| $\log_2 16$ | $4$ |
1277| $\log_2 32$ | $5$ |
1278| $\log_2 64$ | $6$ |
1279| $\log_2 128$ | $7$ |
1280| $\log_2 256$ | $8$ |
1281| $\log_2 512$ | $9$ |
1282| $\log_2 1024$ | $10$ |
1283| $\log_2 2048$ | $11$ |
1284| $\log_2 4096$ | $12$ |
1285| $\log_2 8192$ | $13$ |
1286| $\log_2 16384$ | $14$ |
1287| $\log_2 32768$ | $15$ |
1288| $\log_2 65536$ | $16$ |
1289
1290Finally, I have a test card that asks me to recall the entire sequence in order.
1291
1292## Example: Rilke {#rilke}
1293
1294Let's memorize this poem:
1295
1296<div class="border-box">
1297
1298**Archaic Torso of Apollo**
1299
1300Rainer Maria Rilke
1301
1302We cannot know his legendary head<br>
1303with eyes like ripening fruit. And yet his torso<br>
1304is still suffused with brilliance from inside,<br>
1305like a lamp, in which his gaze, now turned to low,
1306
1307gleams in all its power. Otherwise<br>
1308the curved breast could not dazzle you so, nor could<br>
1309a smile run through the placid hips and thighs<br>
1310to that dark center where procreation flared.
1311
1312Otherwise this stone would seem defaced<br>
1313beneath the translucent cascade of the shoulders<br>
1314and would not glisten like a wild beast's fur:
1315
1316would not, from all the borders of itself,<br>
1317burst like a star: for here there is no place<br>
1318that does not see you. You must change your life.
1319
1320</div>
1321
1322You could run this through the [sequence script](#seq-script), but that seems a
1323bit cold and mechanical. We'll use the [poetry script](#poetry-script) instead,
1324which shows us two lines of context and asks us to complete the next line. The
1325generated flashcards are:
1326
1327<table>
1328 <thead>
1329 <tr>
1330 <th>Question</th>
1331 <th>Answer</th>
1332 </tr>
1333 </thead>
1334 <tbody>
1335 <tr>
1336 <td>
1337 <i>Beginning</i><br>
1338 ...
1339 </td>
1340 <td>
1341 We cannot know his legendary head
1342 </td>
1343 </tr>
1344 <tr>
1345 <td>
1346 <i>Beginning</i><br>
1347 We cannot know his legendary head<br>
1348 ...
1349 </td>
1350 <td>
1351 with eyes like ripening fruit. And yet his torso
1352 </td>
1353 </tr>
1354 <tr>
1355 <td>
1356 We cannot know his legendary head<br>
1357 with eyes like ripening fruit. And yet his torso<br>
1358 ...
1359 </td>
1360 <td>
1361 is still suffused with brilliance from inside,
1362 </td>
1363 </tr>
1364 <tr>
1365 <td>
1366 with eyes like ripening fruit. And yet his torso<br>
1367 is still suffused with brilliance from inside,<br>
1368 ...
1369 </td>
1370 <td>
1371 like a lamp, in which his gaze, now turned to low,
1372 </td>
1373 </tr>
1374 <tr>
1375 <td>
1376 is still suffused with brilliance from inside,<br>
1377 like a lamp, in which his gaze, now turned to low,<br>
1378 ...
1379 </td>
1380 <td>
1381 gleams in all its power. Otherwise
1382 </td>
1383 </tr>
1384 </tbody>
1385</table>
1386
1387And so on. You get the pattern.
1388
1389## Example: Pharmacology {#pharma}
1390
1391<div class="border-box">
1392
1393The **dissociation constant** ($K_d$) of a drug is the drug concentration where
1394half the binding sites in an assay are occupied.
1395
1396</div>
1397
1398This will be an example of [caching your insights](#caching). From this text, we
1399can deduce more things:
1400
14011. A _high_ value of $K_d$ means the drug has a _low_ affinity for the binding
1402 sites, because it takes a higher concentration to reach the same amount of
1403 binding.
14042. A _low_ value of $K_d$ means the drug has a _high_ affinity for the binding
1405 sites, because it takes a lower concentration to reach the same occupancy.
1406
1407And from the above two facts we can also conclude:
1408
14091. $K_d$ is inversely proportional to binding affinity.
1410
1411From this, we can start asking the questions:
1412
1413| Question | Answer |
1414| -------------------------------------------------------------------------------------- | ------------------------------------------------------- |
1415| What is the term for the drug concentration where half the binding sites are occupied? | Dissociation constant. |
1416| What is the notation for the dissociation constant? | $K_d$ |
1417| What does $K_d$ stand for? | The dissociation constant. |
1418| What does a low value of $K_d$ mean? | High binding affinity. |
1419| Why does a low value of $K_d$ imply high binding affinity? | Fewer molecules are needed to reach the same occupancy. |
1420| What does a high value of $K_d$ mean? | Low binding affinity. |
1421| Why does a high value of $K_d$ imply low binding affinity? | More molecules are needed to reach the same occupancy. |
1422| If a drug's binding affinity is high, what does that tell us about $K_d$? | $K_d$ is low. |
1423| If a drug's binding affinity is low, what does that tell us about $K_d$? | $K_d$ is high. |
1424| Describe the relationship between $K_d$ and binding affinity. | $K_d$ is inversely proportional to binding affinity. |
1425| $K_d$ is \_\_\_ proportional to binding affinity. | Inversely. |
1426
1427Visually:
1428
1429<img style="margin-left: auto; margin-right: auto;" src="/assets/content/effective-spaced-repetition/pharma.svg"/>
1430
1431## Example: Misc. Examples {#misc}
1432
1433Given:
1434
1435> US Treasury bonds are called treasuries.
1436
1437You can write the following forward-backward cards:
1438
1439| Question | Answer |
1440| ------------------------------------- | ------------------ |
1441| What are US Treasury bonds nicknamed? | Treasuries. |
1442| What is nicknamed 'treasuries'? | US Treasury bonds. |
1443
1444Given:
1445
1446> The derivative of $\sin x$ is $\cos x$.
1447
1448You can ask for both the derivative and the antiderivative:
1449
1450| Question | Answer |
1451| -------------------- | -------- |
1452| Derivative: $\sin x$ | $\cos x$ |
1453| Integral: $\cos x$ | $\sin x$ |