Author's note: This was originally included in the flash class section and has been broken out on advice from readers, to be placed at the front of the handbook. It was usually a full class at the start of the workshop, and was in the flash class section merely because no fully-fledged writeup exists.

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.