Having spent a fair bit of time in the academy I have had the pleasure of working with brilliant researchers and brilliant teachers. Anyone who’s taken a class knows that these two groups of people are not always the same. What do brilliant researchers lack that prevents them from becoming brilliant teachers? And do brilliant teachers posses a unique skill set beyond mastery of the material that makes them so effective?

## What Teaching Isn’t

Most people who haven’t spent much time thinking about pedagogy think teaching works like this. A teacher knows all the content. Students do not. The teacher pushes the content at the students until they get it.

Teaching is actually a much more subtle and complex interplay between something the teacher would like the students to understand and the student’s current state of comprehension. In the best of cases (with sufficiently motivated teacher and student) it is even *more* complex as the teacher’s own level of comprehension of the material comes into play. A smart question from a bright student can encourage a teacher to rethink something they had previously accepted.

## Let’s Be More Specific

Comprehension of a concept is not linear, so our model for teaching should reflect that. Rather than speaking of the student’s “state of comprehension” and the teacher’s “level of comprehension” let’s talk in terms of mental models. A mental model is composed of four things.

Known | Unknown | |
---|---|---|

Known | x_{1} |
x_{2} |

Unknown | x_{3} |
x_{4} |

Before I tell you what I mean by x_{1} etc., let me specify what the values in the table are. They are Known Knowns, things I know I know. Known Unknowns, things I know I do not know. Unknown Unknowns, things I do not know I do not know. And, perhaps the trickiest of all, Unknown Knowns, or things I do not know that I know.

While the first three categories are transparent the last bears some explanation. An Unknown Known is an item of knowledge that I hold to be true such that it affects my actions and comprehension of events, BUT, I am unaware of the fact that I know this piece of knowledge.

Unknown Knowns usually become a problem when a student is learning a new concept that depends on some prior concept and their understanding of the prior concept must change in a subtle fashion. For example, a student learning about matrix multiplication might experience difficulty in remembering that matrix multiplication is not associative. Since most students learn the simple case of multiplying integers first, they think of multiplication as something inherently associative instead of an operator whose action depends on the particular mathematical objects over which it is defined.

## What Is Good Teaching?

The naive stick figure teacher above imagines teaching to consist only (or primarily) of solving students’ Unknown Unknowns (and maybe some Known Unknowns). I think teaching is a good deal more nuanced than that.

Mathematically, x_{1}, x_{2}, x_{3}, and x_{4} are partitions of a set X, whose elements are “all pieces of knowledge relevant to the concept being taught.” What is relevant for one subject may differ from what is relevant for another. One remembers the metric prefixes indirectly, by means of a mnemonic. Another by order of magnitude jumps directly related to the Latin prefixes.

Rather than specify a certain technique for teaching (e.g., Socratic questioning! Learn by discovery! Provide examples! Start from first principles!, etc.) we can instead specify a ‘goal test’ that can function as a kind of barometer for how well the teaching interaction is proceeding.

Consider a scenario with one Teacher and one Student. The Teacher’s job is to be aware of the partitioning of X into x_{1}, x_{2}, x_{3}, and x_{4} above, and teach in such a way that the Student achieves a sufficient understanding of the concept at hand. But, the distribution of X into its partitions is actually a time dependent series. As the teaching interaction progresses, the partitioning changes! And on top of that, elements may be added or subtracted from X, and elements that were previously imagined to be distinct may be revealed to be the same (or vice versa)! On top of *all* of that, the teacher must keep in mind not only the student’s partitioning of X, say X_{student}, but also her own, X_{teacher}. And both are again time-dependent series.

## The Payoff – Become A Great Communicator

The skilled teacher is one who artfully and compassionately navigates the interaction between these two knowledge distributions (and this assumes just one student, not a whole class!). It requires not only awareness of the underlying cartography of the distribution (our table above), but also the ability to know, in some sense, more than what the student currently knows about their own mental model of the concept.

What is true of teaching in this academic sense is just as true for human interaction and communication in general. Communication and discussion are not simply opportunities to shout the loudest or repeat what you already believe, but instead a navigation of this particular interaction of knowledge bases (at least when they are honest conversations, not performance pieces like TV punditry – a social performance with a much different function fit for another blog post…).

Just as not all teachers are created equal, neither are all students. A student with a sufficiently robust understanding of this dynamic can elicit much greater learning from an otherwise mediocre interaction. And of course, at many points in our lives, we are all both teacher and student.