Flash Classes: Polaris, Seeking PCK, and Five-Second Versions

Author’s note: During CFAR’s 4.5d workshops, concepts that had been formalized as “techniques,” and which could be described as algorithms and practiced in isolation, generally received 60+ minute sessions. Important concepts which did not have direct practical application, or which had not been fully pinned down, were often instead taught as 20-minute “flash classes.” The idea was that some things are well worth planting as seeds, even if there was not room in the workshop to water and grow them. There were some 30 or 40 flash classes taught at various workshops over the years; the most important dozen or so make up the next few entries in this sequence.


Polaris

Imagine the following three dichotomies:

  • A high school student mechanically following the quadratic formula, step by step, versus a mathematician who has a deep and nuanced understanding of what the quadratic formula is doing, and uses it because it’s what obviously makes sense

  • A novice dancer working on memorizing the specific steps of a particular dance, versus a novice who lets the music flow through them and tries to capture the spirit

  • A language student working on memorizing the rules of grammar and conjugation, versus one who gesticulates abundantly and patches together lots of little idioms and bits of vocabulary to get their points across

By now, you should have a set of concepts that help you describe the common threads between these three stories. You can point at goal factoring and turbocharging, and recognize ways in which the first person in each example is sort of missing the point. Those first three people, as described, are following the rules sort of just because—they’re doing what they’re supposed to do, because they’re supposed to do it, without ever pausing to ask who’s doing the supposing, and why. The latter three, on the other hand, are moved by the essence of the thing, and to the extent that they’re following a script, it’s because they see it as a useful tool, not that they feel constrained by it.

How does this apply to a rationality workshop?

Imagine you’re tutoring someone in one of the techniques—say, TAPs—and they interrupt to ask “Wait, what was step three? I can’t remember what came next,” and you realize that you don’t remember step three, either. What do you do?

You could give up, and just leave them with an incomplete version of the technique.

You could look back through the workbook, and attempt to piece together something that makes sense from bullet points that don’t really resonate with your memory of the class.

Or you could just take a broader perspective on the situation, and try to do the sensible thing. What seems like a potentially useful next question to ask? Which potential pathways look fruitful? What step three would you invent, if you were coming up with TAPs on your own, for the first time?

The basic CFAR algorithms—like the steps of a dance or the particulars of the quadratic formula—are often helpful. But they can become a crutch or a hindrance if you stick to them too closely, or follow them blindly even where they don’t seem quite right. The goal is to develop a general ability to solve problems and think strategically—ideally, you’ll use the specific, outlined steps less and less as you gain fluency and expertise. It can be valuable to start training that mindset now, even though you may not feel confident in the techniques yet.

You can think of this process as keeping Polaris in sight. There should be some sort of guiding light, some sort of known overall objective that serves as a check for whether or not you’re still pointed in the right direction. In the case of applied rationality, Polaris is not rigid, algorithmic proficiency, but a fluid and flexible awareness of all sorts of tools and techniques that mix and match and combine in whatever way you need them to.

Or, in other words: you’re here to solve your problems and achieve your goals. Everything else in this sequence is useful only insofar as it helps with that.


Seeking PCK (Pedagogical Content Knowledge)

A lot of teacher training in the USA focuses on broad teaching techniques that apply to just about any topic. Whether the topic is math, history, biology, or literature, teachers need to know how to design lesson plans and how to gain and keep control in the classroom. These domain-general teaching skills sometimes get referred to collectively as pedagogical knowledge (PK).

This is in contrast to content knowledge (CK), which is the teacher’s particular expertise in the topic being taught (e.g. knowledge of how to solve a quadratic equation).

However, in practice it’s helpful to notice that there’s a kind of knowledge that is both PK and CK. The educational profession refers to this as pedagogical content knowledge (PCK). This is knowledge about the topic being taught that is also about how students interact with the topic (and therefore how to teach that content more effectively).


For instance: what are common misconceptions about this domain? What are bad habits that typically need to be unlearned? What kinds of prompts or stimuli will actually help people identify and unlearn those bad habits, as opposed to sounding good while failing to do the trick? What’s it like to be a beginner? What’s it like to transition from beginner status to kind-of-sort-of having your feet under you (while still having lots of gaps or deficits)?

Teachers who actively try to develop PCK tend to gain a much more refined understanding of their topic, including a keen sense of which parts matter, how those parts connect, and in what order they must be explained (which is another way of saying which concepts are more fundamental, and which concepts require others as prerequisites). We encourage you to try keeping an eye out for PCK any time you begin to learn a new skill or start exploring a new domain. It will not only enrich your experience, but also make you much more likely to be able to pass the knowledge on to others.

Case study: understanding division

Suppose you’re trying to introduce the idea of division to elementary school students. You might start with a word problem like this one:

Johnny has 12 apples. He also has 4 friends who really love apples. If he gives all his apples away to his 4 friends and each friend gets the same number of apples, how many apples does each friend get?

Given some simple hands-on learning tools, a lot of elementary school students will want to count out twelve tokens and then sort them into four piles one at a time: “One for A, one for B, one for C, one for D, one for A...” They’ll stop when they run out of apple tokens, count the number in one pile, and conclude correctly that each friend gets three apples.

The teacher might then write the following on the board:

12 ÷ 4 = 3

… and say that what they’ve just done is “division,” which means that you are dividing some quantity into equal parts and looking at how much each part gets. This definition will work just fine, until the teacher introduces a problem that looks something like this:

Johnny has 12 apples. He wants to make gift bags that each contain 4 apples. How many gift bags can he make?

This will often befuddle students who have been taught that division is equal sharing. Given the problem above, the majority of elementary students tend to make one of two errors:

  • Some students will gather twelve tokens and start counting them out into piles: “One for A, one for B, one for C… ” But after a while, they realize that they don’t know when to stop making piles—when to go back and put another token in pile A.

  • Some students will notice that four is smaller than twelve, dutifully make four piles, and sort their twelve tokens into four piles. They note that there are three tokens per pile at the end, and they proudly (and almost correctly) say that the answer is three. But in this case, if the teacher follows up and asks “three what?” the student will often say “Three apples!”

The problem is that the process for using tokens to solve this problem looks fundamentally different: the student has to do something like gather four tokens at a time and set them aside, repeat this until there aren’t any tokens left, and then count the number of collections of four tokens that have been pulled aside.

It turns out that although both word problems are represented by the symbols 12 ÷ 4 = 3, the 4 and 3 mean different kinds of things in the two problems. In the first, the equation looks like this:

(# of items) ÷ (# of groups) = (# of items per group)

And in the second, the equation looks like this:

(# of items) ÷ (# of items per group) = (# of groups)

The first version is called partitive division (after “partition”), or “equal sharing.” The second one is quotitive division (after “quotient”), or “repeated subtraction.” And even though they’re both technically forms of division and there’s a mathematical isomorphism between the two operations, they are cognitively different. In practice, you have to teach young children about these two kinds of division separately first, before you start trying to show them that they’re both unified by an underlying concept.

In this case, the PCK is awareness of the fact that there are two different kinds of division, and that students get confused if you introduce them under the same umbrella. It’s knowledge about content (partitive and quotitive division) that is relevant to knowledge about pedagogy (successful teaching requires careful disambiguation).

The main tool for developing PCK is cultivating curiosity about the students’ experience. When teaching or tutoring, rather than asking “How can I convey this idea?” or “How can I correct this person’s mistake?”, instead ask “What is it like for this person, as they encounter this material?”

The PCK on the two types of division came in part from interviewing students who were working on division problems. Sometimes the students would make errors, and the interviewer would become curious. They’d wonder what thought processes might have caused the child to make the particular mistake they did, and then try to figure out ways of testing their guesses.

For instance, maybe a child uses an equal-sharing process with tokens to solve a repeated-subtraction word problem. Rather than trying to correct the child, the interviewer might investigate whether the child is running an algorithm, asking “So, what do these tokens in these piles mean to you?”

In education research, this is sometimes called clinical interviewing, and it’s a skill that requires attention and practice. For instance, the interviewer in the above example will be less effective if they spend part of the time trying to point out the error. Instead, the interviewer has to be simply wondering— being actually curious about the child’s thinking. If the curiosity is genuine and central in the interviewer’s mind, they’re more likely to notice interesting threads to pursue and to think of useful questions to pose.

This also tends to encourage a certain kind of reflection in the student. For instance, when a child in a clinical interview thinks the interviewer is trying to get them to do something or correct a mistake, they will often start to focus on pleasing the interviewer instead of focusing on whether or not things make sense. Sometimes they become nervous or self-conscious, and other times they sacrifice effort for appearance. In contrast, a curious and effective interviewer keeps the child engaged with the problem and pointed toward comprehension.

This isn’t always the best teaching method—sometimes, it’s helpful just to give direct and clear instruction. But in general, both you and those whom you teach will gain a lot more from the experience if you keep yourself curious about the learning experience rather than on whether information has been dutifully presented.

And this goes double when the person you’re teaching is yourself.


Five-Second Versions

“Using a CFAR technique” often doesn’t mean taking out pen and paper and spending minutes or hours going through all of the steps. Instead, it involves five seconds of thought, on the fly, when a relevant situation arises.

Examples:

  • Murphyjitsu: You agree to meet a friend for coffee, and quickly run the plan through your inner simulator before ending the conversation. You hastily add “Wait! Let me make sure I have your phone number.”

  • Internal double crux: You notice that you don’t feel an urge to work on this email that you’re supposed to send. You spend a second to visualize: if you become the sort of person who does feel such an urge, will something positive result?

  • Goal factoring: You notice that you’re feeling tension between two possible outings over the weekend. You quickly identify the best thing about each, and see whether one can incorporate the other.

  • Aversion factoring: You keep feeling bad about never getting around to reading dense nonfiction books. You consider whether System 1 may be right here; perhaps it really isn’t worth the trouble to read them?

  • TAPs: You’re two minutes late for a meeting, and think about what trigger could cause you to leave five minutes earlier in the future.

  • Systemization: You feel a vague annoyance as you’re sorting through your pantry, looking for the chips, and you decide to move the bag of rice all the way to the back, where you can still reach it over smaller, more frequently used items.

Note that these five-second versions often only use a fragment of the technique (such as checking whether an aversion is well-calibrated), rather than thoroughly applying every step.

Some advantages of using five-second versions:

You can use them more often, at the moment when they’re relevant, without having to “boot up” an effortful, time-consuming mode of thinking (many CFAR instructors use these something like twenty times per day).

You can integrate them fluidly with your thinking, rather than having to interrupt your flow and remember what thoughts or activities to return to.

You can practice them many, many times.

You can develop multiple variations, including your own independent inventions.

Most importantly of all: if a larger, more effortful version of a technique is something you simply will not do, then a five-second version you will do is infinitely better than nothing.