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Everyone has met a teacher who clearly knew the subject but could not explain it, and another who made the same topic click in five minutes. The difference is not only how much they knew. Lee Shulman gave that difference a name, pedagogical content knowledge, and it changed how teachers are trained. The idea is just as useful from the student's side of the desk. This post explains what Shulman meant, quotes his definition, and shows how to build that kind of knowledge for your own exam topics.
Who was Lee Shulman?
According to the Stanford Daily's obituary, Shulman won a scholarship to the University of Chicago, studied philosophy and psychology, and earned his MA and PhD in educational psychology there. In 1963 he joined Michigan State University, where he founded and co-directed the Institute for Research on Teaching. He came to Stanford in 1982 as the Charles E. Ducommun Professor of Education and later served as president of the Carnegie Foundation for the Advancement of Teaching. He died on 30 December 2024, at 86.
What is pedagogical content knowledge?
Shulman set out the idea in Those Who Understand: Knowledge Growth in Teaching (1986). He argued that research on teaching had developed a "missing paradigm", a blind spot about subject content, and proposed three kinds of content knowledge a teacher needs: subject matter knowledge, curricular knowledge, and pedagogical content knowledge.
He defined the third as including, for the most regularly taught topics, "the most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations—in a word, the ways of representing and formulating the subject that make it comprehensible to others." It also includes "an understanding of what makes the learning of specific topics easy or difficult: the conceptions and preconceptions that students of different ages and backgrounds bring with them."
He also expected teachers to understand not only that something is so but why it is so. He closed the paper by rejecting George Bernard Shaw's jibe about teachers: "Those who can, do. Those who understand, teach." The sentence before it puts the case directly: "the ultimate test of understanding rests on the ability to transform one's knowledge into teaching."
In a 1987 follow-up, Knowledge and Teaching: Foundations of the New Reform, he described teaching as built on comprehension and reasoning, transformation and reflection.
Why does this matter to a student?
Shulman wrote about teachers, but the two parts of his definition map onto two study habits that strong students already use.
Representations. For any core idea, can you show it more than one way? Take Newton's third law. You can state it in words, draw a free-body diagram for each of two bodies, write the equations, and give an example where the forces act on different objects: a book pushes down on a table and the table pushes up on the book. If you only have the sentence, you do not yet have the idea.
Misconceptions. Do you know the wrong answers people give, and why they are tempting? For the third law, a classic one is "the forces cancel, so nothing moves". The forces act on different bodies, so they cannot cancel. Our profile of David Hestenes and the Force Concept Inventory covers how common such beliefs are.
The third part, explaining, has direct evidence. In experiments described by Logan Fiorella in an American Psychological Association teaching resource, students learned about the Doppler effect expecting either to teach it or to take a test, and some actually taught it by recording a video lecture for a fictitious student. In the original study, Fiorella and Mayer (2013), preparing to teach helped on an immediate test, but on a test a week later the students who had actually taught on video outperformed those who had only prepared.
How do you build it for your own topics?
Use a one-page "teaching sheet" for each chapter you are revising. Take the Class 12 topic of electric potential as the example.
The core idea in one sentence. "Potential is potential energy per unit charge, so it tells you how much work it takes to bring a unit positive charge from infinity."
Three representations. The formula V = kq/r; a sketch of equipotential circles around a point charge; an analogy, such as height on a hill, with field lines pointing downhill.
Two misconceptions with the fix. "Zero field means zero potential" (false: inside a charged conducting shell the field is zero but the potential is constant and non-zero). "Potential and potential energy are the same" (energy depends on the charge you bring; potential does not).
One question you would ask a student. "Why is no work done moving a charge along an equipotential surface?"
Teach it. Explain the sheet aloud to a classmate, or record a three-minute video as if for a younger student. Notice where you stall.
For more on the teaching step, read our guide to learning by teaching.
What can a student take from how he worked?
Name what is missing. Shulman's "missing paradigm" was a gap nobody had labelled. Labelling your own gaps is the first step to fixing them.
Understanding is the ability to explain why. He asked teachers to know why a claim holds and on what grounds. Ask the same of every formula you use.
Collect cases. He argued for learning from well-documented cases. Your error log of real mistakes is your case file.
Building representations takes good examples to draw on. The Fermi Concept Library has its own lesson for every concept in physics, chemistry and mathematics, which is a useful second explanation to compare with your textbook's.
Make one teaching sheet this week for the chapter you find hardest, and explain it to someone before Sunday.
