Skip to main content Skip to main content

The Learning Classroom: Theory Into Practice

How We Organize Knowledge: The Structure of the Disciplines

This program covers the ways in which the organization of knowledge and understanding can influence learning. It also introduces Bruner's and Schwab's ideas about the structure of the disciplines. Featured are a fourth-grade teacher, a 10th-grade biology teacher, and a ninth- through 12th-grade teacher, with expert commentary from Lee S. Shulman, president of the Carnegie Foundation for the Advancement of Teaching.

View Transcript

Linda Darling-Hammond: Every teacher knows that there is always too much to teach. How do you decide, subject by subject, what’s most important? How can you help your students to understand the big ideas of the disciplines they are studying? How can you get them to think and act like historians, mathematicians, or scientists?

I’m Linda Darling-Hammond, and that’s our challenge for this session of The Learning Classroom.

Every discipline has a set of big ideas that are connected to each other and provide the glue that holds the subject area together. These include ideas like ratio and proportion, balance, and equilibrium in mathematics. In biology, we might think of major themes like evolution, adaptation, and the environment. Each discipline also uses particular methods of study that reflect how knowledge is built in that field; for example, scientists do experiments, and historians evaluate historical evidence from different perspectives. To really master a discipline, students need to understand what the big ideas are and how they are connected. If they learn to think like experts in that field, their learning will be much more efficient.

Lee S. Shulman, Ph.D., President, Carnegie Foundation for the Advancement of Teaching: If I’m a physician, and I’m doing an examination of you, I don’t just randomly say, “Let me look at your nose, how’s your toe?”  I have a structure; it’s the structure of the human body and its organ systems, so I will systematically check the gastro-intestinal tract. I will check the respiratory system, the cardiovascular system, the neurological, right? Because, not only does that give me a structure for moving through the examination, it also structures my memory for any signs or symptoms that I encounter, that may be relevant to diagnosing what you have. So, every time we call something a subject matter, I would say, we call it that because it has some principle of organization that connects the ideas with one another that gives them some kind of order, some sort of meaningfulness, and it’s like a code.

Linda Darling-Hammond: Julie Helber, a fourth grade teacher at Paddock Elementary School, helps her students to think and act like scientists and mathematicians as they learn about static electricity and fractions. She guides them through their inquiries by modeling the steps that experts follow in those fields. And then she gives them an opportunity to practice doing what scientists and mathematicians do.

(classroom scene)
 Rub it on his hair too?
It’s moving!  It works! It’s moving!
Do you see his hair moving back and forth?
 It’s a magnet.
You think it’s a magnet?
 And what happens when you rub the balloon on your hair? Your hair sticks out? Why do you think that happens?
Girl in yellow: 
Since we can’t like, come to the rod our hair does and it’s the lightest thing on our body.
Ok, does anybody have another experience?
Girl in blue:
 I think ’cause when I went on the slide out there you always get shocked and stuff.

Julie Helber: The students are experiencing a first hand investigation with electricity, so they’re actually working with materials and figuring out which materials are static and which materials are not static in several trials and experiments.

Later we studied a second hand observation where the students looked at a notebook test, text from a particular student. And we read through the notebook text together to make some more meaning of what we were learning about static electricity, and we really tried to utilize the text and draw on what we already knew about static electricity to make some conclusions or some hypotheses about static electricity.

(classroom scene)
: And Michelle says she gets shocked. Are there any other cases where you’ve gotten shocked by something? Any other times? Josh?
JoshWe live on a farm, and when we had cows we had electrical wires, and, lik,e I would accidentally touch it, and I wouldn’t really be able to let go. I would really have to force myself to let go.
JulieYou have a lot of knowledge right now about static electricity. What I’m going to be giving you are some materials that you will be working with a partner to use. Okay, you’re going to be trying to create charges between the interactions of the materials that are in this bag. So how could I record your results for that? Jasper?
Um, just make two columns, one for if it had electricity and the other if it didn’t, and you can just check one.

Julie Helber: The value in setting up the experiment this way, by allowing the students to investigate something that they might not have clear and concrete directions for – there’s a lot of value in it. And what I see is the most valuable is that the students are constructing meaning by actually using these materials in a way that they have chosen, so I haven’t determined how they are going to use the materials and told them exactly what to do. Because I want them to err in what they are doing. I want them to see that they’ve tested or tried something that doesn’t work, because that’s really when they learn.

If I tell them how to do it, it’s not likely they’re going to remember it later. But if they actually learned it while they were doing it, it is likely that they would remember it later.

When there are a variety of opinions regarding a particular topic really, I think that’s the meat of teaching. The students interacting with each other, or I saying “Well, I disagree with another student,” is probably the best thing that can happen in my classroom.

Lee Shulman: The notion of structure in mathematics is, is probably one of the most obvious ones. Mathematics is a field where most people who teach mathematics will readily understand the notion that there are certain kinds of structures. There’s a notion of balance. There’s a notion of equilibrium. There are notions of ratio and proportion that just keep on coming up again and again and again and whether you’re doing primary arithmetic or you’re doing algebra, those notions return, repeat themselves and you get a sense of how these organizations really make sense of the subject.

Julie Helber: I always offer lots of options for the students to demonstrate what they know. If they need to come up today – we were doing a math lesson, and somebody said, “Can I come up there and show you?”

(classroom scene)
Julie: Is this rectangle up here divided into thirds? And you need to be able to explain it to me…. Hmm. Okay, I need somebody to say, “No,” to raise your hand and explain why you’re saying “no.” Melissa?
  First there’s two rows of four, and then they have one extra and a half. It’s a half extra part.
  Can you come up and show, because I’m not sure I get it.
:  They had added one extra part to it. Then we have that extra one.
Julie:  Okay, so how many, you’re saying that it has an extra one. What do you mean by that?
  There’s just this one extra triangle sitting there… Oh, wait.
 What are you thinking about?… Melissa, can you tell me right now what you are doing to check that?
 Um, I’m checking it by counting by these squares.
Yes, Brian you want to explain?
Here’s, if you folded this they have too little. This one has too much, this one’s right.
 Yeah, this one’s right, but these two are wrong.
Yes, this one has – they should take the half off of the eight… And put it on the seven and a half to make eight.
 Okay, so if I did draw a line like this and gave that one, then would it all be divided in thirds?
Yeah, yes.
Take that line off, take that diagonal line off. So this line off and then we would have thirds.
Okay, good!

Julie Helber: It’s very helpful for students to hear another student explaining why they arrived at a particular answer, because it may not have been the way that they were thinking.

And so by hearing how somebody else is thinking, and it goes back to me modeling how I’m thinking about things, and to maybe change their of way of thinking or to expand the way that they think.

Linda Darling-Hammond: Did you notice how Julie tried to ensure that her students would understand not only the content they were studying, but also the process of experimentation that is central to science and the search for patterns that is at the heart of mathematics?

Tenth grade biology teacher Mary Edmunds helps her students understand cell biology by first connecting the issues of disease to the drama of the science fiction novel Andromeda Strain. She then engages them in the scientific method, so they can examine the features of cell membranes for themselves.

Mary Edmunds: Adolescents have to have hands-on investigations. They have to feel like they’re a part of the learning process. They have to be able to take ownership of their own learning and they have to be able to feel that they’re a part of the learning and that they can take it out into the world and utilize it. Especially urban youth. Urban youth need to understand that they can do it. They have the tools. They are scientists, and that they can go on, and they can become a big part of this world. And the way you do that is for them to see relevance. So how do, how do you get them to see relevance in cells? They have to know that they’re made of cells. They have to know that the anthrax scare was not beyond their ability to cure. They have to know that bacteria can be grown, but that they have the scientific ability and the knowledge to destroy it. But they have to know the internal structure of a cell in order for them to take that knowledge out into the real world and solve any problems.

(classroom scene)
Mary: Did Andromeda strain like or dislike a high alkaline level?
 Dis – Dislike.
 You got it. So the optimal range for the Andromeda was not alkaline, correct? That’s why it did so well in the human body. Do we like alkaline? No, that’s why it did so well in the human body. So if we get away from our normal metabolism we do not stay at homeostatic conditions, do we? Alright, so diffusion would be best within the cell, moving around within the cell at body temperature. Let’s look at it, okay. Let’s say that this warm is body temperature. And this is cold. Who can give me a hypothesis for which of these containers will have the best diffusion, or the fastest diffusion?
 I believe that um, the, the cold water will have the most diffusion because…

Mary Edmunds: Students love to be called scientists. I mean, I like to think of myself as a scientist. And if we, if I tell them that you are now a scientist, you’re working on an investigation, they’re really more interested in it. When they put a lab coat on, a whole new aura comes over them that this is really important, that this is really scientific.

(classroom scene)
 So since we are scientists ourselves, right? We want to find out if it’s true that the cellular membrane is permeable. Means that it lets things in and out, right? So we’re gonna use an egg, okay, everyone? We’re going to use an egg.

Mary Edmunds: You see in the classroom when I ask the students how would you get the membrane initially, that a lot of them thought, “Well, I’ll boil the egg,” – because they took their past experiences. They boil the egg, they peel it off, the membrane’s right there. But I had to get them to understand that this is a living cell, and a living cell has cyto-cell…it has cytoplasm (she corrects herself) that surrounds the nucleus, so you can’t do that, so what are you going to do?

(classroom scene)
Group 1
 It’ll start cracking, and it’ll start peeling off. You ain’t never boiled no egg have you?
 Hard-boiled eggs, like what you do for Easter, like hard-boiled eggs.
Nichole: And after awhile it’ll start to crack.
Group 2
 It could be like, maybe lemon juice.
I say vinegar.

Mary Edmunds: They have to feel secure in making a mistake and not thinking that they’re gonna be ridiculed by me or the other students in the classroom, and they have to be able to take chances. I, I don’t know everything. And the kids can ask me something, and if I don’t know it, and I tell them, you know, I’m gonna look it up, they feel more secure with me asking them questions and them not knowing the answers. But the best thing to do is to go around and catch it. Catch them while they’re doing it and get, get them to think about it.

(classroom scene)
Mary: What do you think?
Nichole: We think to boil, to boil it.
Mary: Alright, why?
Nichole: Because when you okay, like when an egg, when you boil an egg, I don’t know if everybody in here boiled an egg before, but if you keep on letting it boil the shell is going to begin to crack. And the shell is going to start, it’s gonna start to peel off, the shell gonna start to peel off.
Alright now Nichole when you peel it off, what is the first thing that you see?
Nichole: The ah, skin of the ah…
Mary: That’s right, and it would mimic the…
Class: Cell membrane.
Mary: So in theory they’re absolutely correct, because I didn’t give enough information. Now, I should have been very, very specific and said, “This was a liquid solution.” Now, remember, we – it’s only a substance that we use from the pH lab – your logic is excellent. That would be the easiest way to truly show the membrane, the easiest way to truly show the membrane. What did your group figure out?
 It, the answer is vinegar because it’ll, like, um, it’ll like melt the shell and, like, turn the inside of the egg to, like, jelly.
Mary: Alright, she’s absolutely right, absolutely right.

Mary Edmunds: And see, I had the students decide how to dissolve it. And then that made them interested. If I would’ve just said, “Okay, today you are going to dissolve the membrane of an, of an egg in vinegar,” they would’ve said, “Okay,” today. And if I would’ve asked them two days later, “What did you dissolve? What, what acid did you use?” They would say, “I don’t know. Whatever you told us to.” But see, now if I asked them, they would say, “vinegar.” And I would say why? “Because the pH is low enough to dissolve the membrane, to dissolve the eggshell without effecting the membrane.” That’s why I did that.

A couple of the groups broke their egg, accidentally, and they thought, “Oh, the experiment is now a disaster.” But that’s, I wanted them to understand that that happens to scientists. That there’s limitations in life, but the scientific word is a limitation.

(classroom scene)
Mary: What’s the problem? What happened, what is this, you’re a scientist?
 It’s a limitation.
Mary: Why is it a limitation?
Nichole: Because it was.
Girl: All the membrane of, the shell was gone for the…
Nichole: The membrane dissolved.
Mary: Okay, exactly, what is this?
Nichole: The membrane.
Mary: Okay, let’s look at it. Can you describe the membrane?
 It’s rubbery, almost like a balloon.
 Absolutely. So if it’s rubbery, do you think it’s, it’s got pores in it?
Boy: Yes.
Mary: But we can’t see them, how can we prove there are pores in this?
Nichole: With a microscope.
 Without a microscope. How could you prove in your scientific investigation…
Because the vinegar got in…

Mary Edmunds: It was a perfect opportunity for them to see the consistency of the membrane and to see, they can’t see the pores. They cannot see the pores, but they’re microscopic.

Well, they, they read that, or they learned that in some past experience that we’ve done, because I didn’t lecture to that. So that, that was a learning experience for them.

(classroom scene)
 What happened?
Girl: It broke.
Nichole: It um, it weighed more.
Mary: It weighed more. There you go it weigh…
 So it blew up like a balloon…oh!
Mary: You got it! Hey look it, the red blood cell. A red blood cell if it’s put in a hypostatic environment…

Mary Edmunds: I think it’s important for them to take the idea of cells to cellular membrane and then how the, the different solutions effect the passage across the membrane.

Problem solving is, is so essential in life. If, if students do not know how to problem solve, they’re not gonna be able to make it in college.

But when I get them to problem solve, and I can get them to problem solve, that’s where, that’s where it all comes together.

Linda Darling-Hammond: Mary’s students were able to solve both a fictional and a real life problem by working together using the scientific process. But what if Mary were teaching history instead, where the process of “finding the truth” is not so clear cut?

Lee Shulman: Let’s just take it for a given that experiment is a great idea in science. And now we start learning history. Well, how do you do an experiment in history? We then suddenly realize that that’s not the way you do history. We don’t have, we can’t put one historical period in an experimental group and another one in a control group and see what the difference is. But we’re still doing comparisons. We’re still trying to create evidence. I mean there’s certain ideas that do cut across – notions of description, of analysis, careful, careful observation, notions of what is the evidence for your claim, notions of theory. But the fundamental process of experimentation, which is at the heart of work in science, has no real analogy in history, if what you mean by experiment is that the scientist is controlling conditions and studying what happens under conditions that she herself has controlled. So what does a historian have to do? The historian in some sense has to look for comparisons where, if you will, nature has made experiments, not scientists.

Linda Darling-Hammmond: Avram Barlowe, a 12th grade history teacher at the Urban Academy, confronts this challenge every day. He has his students read the primary source materials for the period they are studying, so that they can explore the events and opinions of the time and understand the perspectives that shaped historical decisions. He focuses on central ideas like governance and human rights, and he raises essential questions like: Who should govern? And why?

(classroom scene)
Avram: Let me try to give you just a little bit more context for a second, ah and refresh your memory. Johnson decides that he’s going to implement a lot of Lincoln’s reconstruction plan, which is pretty, pretty lenient on the South. What that allows to happen is it allows state governments to be created all across the South. They pass these Black Codes that we were about to look at. Okay, these Black Codes are laws passed by Southern State governments. Now what do these laws do?

Avram Barlowe: What I asked them to do was to look at that material and ah, to actually look at the text of these laws and decide for themselves what they thought the laws were designed to do, why they thought they were created, how they would have been defended by the people who wrote them. And you’ll see, you saw in the classroom there were multiple interpretations of that. And the discussion hinges upon multiple interpretations and meaning of these codes through the looking at the evidence that the meaning presented.

(classroom scene)
 Page three Section nine, go ahead.
 Um, <reading> “If any person shall persuade or attempt to persuade, entice or cause any freed men, freeing or belonging to the servant employment…” you know, I don’t have to read all of that. Basically what it says is, it’s just like if a guy leaves his job, anyone else who tries to hire him, or tries to give him food or clothing, or anything else, they could be fined or put in jail. So it’s beyond even, like, trying to keep them down. It’s like, alright, if you want to help them we’re not going to let you.
I think this gets to the part, you’re saying that this is like slavery. Matt’s saying it’s not great, but it’s not quite the same thing as slavery. Let’s look at, try to figure out what this means and does this support Stephanie or does this support Matt.
When you look at them individually they don’t seem that bad. You know, you’re, like, okay that makes sense, it’s job training. But when you look at them as a whole, it’s just totally restricts what you’re saying. If you want to hold, you know, freed slaves up to the same standards as white citizens, why are you going to make a completely different set of books for them?

Avram Barlowe: Um, when we respond to the arguments, we respond with them on the basis of the evidence, and when we present them, we present evidence to support them.

(classroom scene)
Matt: It says free, freed men and then freed Negroes, it’s like, two separate categories.
 And white people hang out with black people.
 Yeah, so I-I- I think it’s just like back in the 10 Commandments, which nobody’s complaining about. Like “Thou shalt not commit adultery,” it’s just backing up what’s moral.
 Well, it says in it that if you associate with freed men, freed Negroes, or mulattos on terms of equality, then you’re a vagrant. How is that moral and how is that anywhere in the 10 Commandments?
You really need to look at the text when she makes a kind of argument to you based on evidence. She’s saying that all white persons assembling with freed men, freed Negroes or mulattos, or even associating with freed men, freed Negroes or mulattos on terms of equality. So a white person is hanging out with a black person and treating them as an equal, she’s saying, and being a vagrant, and that’s not morality to her.

Avram Barlowe: The evidence is always central. And what’s your evidence and how do you, what, how do we analyze a piece of evidence? So there was an exchange there where that happened. Um, where a kid had not read the material beforehand, was sort of glancing through it as we spoke, and he was making some arguments, and then some other kids would say, wait a second go back to the text. You’re making generalizations about this.

(classroom scene)
 So I come back to this, what should be done with these guys who had passed these kind of laws? Should they be allowed to be in the government? Michelle’s saying that they should be as long as they…

Avram Barlowe: I’m really trying to help frame a discussion for them so the inquiry can deepen, and we can get to deeper levels of evidence and argument. And that’s basically what’s going on, it’s an exchange of ideas there. It’s not a willy-nilly exchange of ideas; it’s an exchange of ideas that’s happening within the framework of an inquiry that’s rooted in questions that I, that I, that I’ve designed.

(classroom scene)
Jose: Okay, how are you going to say you have all the rights of being an American, when like a month and a half ago I was ready to kill you, because I don’t like America, you should not be able to reap the benefits of something you were trying to destroy a couple weeks ago, that’s not right.
Michelle:  Part of the war was to keep the Union together, you know. You can’t have a country together when you’re pushing away half of it, you have to have the Congress and senators from, you know the other states, you had to have representatives and even if they did not agree with what you wanted or like what you want for the country or whatever, you still have to have it or else there would be no Union. What’s the point of having a war over keeping them together if you’re gonna push them out after when it’s over?

Avram Barlowe: They’re beginning to answer things like, they’re, they’re looking at the question of the freed men after the war in the context of, of the Emancipation Proclamation, Lincoln’s role, the slaves self-emancipation. They’re building on what they already know, so there’s an intellectual growth I think that, that, that you can see.

Lee Shulman: What’s really important as teachers teach different disciplines to students is for the students to appreciate that there are certain kinds of ideas like description, like analysis, like careful observation, like evidence, inference and theory, if you will, that are useful across disciplines. But there are other ways in which very important methods of work in one discipline just don’t show up in another discipline. And I think history and science make a lovely contrast here.

History and literature also make a lovely contrast.

There are different canons that we use for determining whether it’s a good novel, or a bad novel and they’re not the same as the ones we use for determining whether something is good history or bad history…unless we’re gonna use the novel for teaching people history. And so here again, understanding the differences in what counts as evidence and what counts as knowing a subject between subjects becomes terribly important for teachers to understand, and in turn, for students to understand.

Linda Darling-Hammond: Julie Helber, Mary Edmunds and Avram Barlowe teach the central ideas of their subject matters while they also teach their students the skills necessary to learn on their own. By helping their students to think and act like experts in each field, they’re not just giving them fish, they’re teaching them how to catch them – a skill that will keep their students nourished intellectually for the rest of their lives.

This is The Learning Classroom. Thanks for watching.

“When we want to learn something new, we have to figure out what are the main guideposts in the terrain of a body of knowledge. And that’s what we call the structure of the disciplines – the big ideas, the big modes of inquiry that are used in history, mathematics, science and so on. And when teachers provide those structures for students – so that they have a map of the territory – they can learn more effectively and hang lots of other pieces of information onto that understanding. So that the structure of knowledge itself becomes a learning and a teaching tool.” 
Linda Darling-Hammond

Key Questions

  • How does the way knowledge is organized influence learning?
  • How can teachers use the structure of a discipline to organize their teaching and enhance student learning?

Learning Objectives

  1. Structure of the disciplines – Teachers will understand that disciplines have structures representing interrelated core ideas and particular modes of inquiry. They will think about how to use these core ideas and inquiry tools to help students understand disciplinary ideas more deeply.
  2. Pedagogical content knowledge – Teachers will consider the kinds of knowledge of content and students they need in order to represent disciplinary ideas so that they are more likely to be understood.

Video Program

This episode covers the ways in which the organization of knowledge and understanding can influence learning. It also introduces Bruner’s and Schwab’s ideas about the structure of the disciplines. Julie Helber, a fourth grade teacher at Paddock Elementary School, Milan, Michigan, tenth grade biology teacher Mary Edmunds at the Detroit High School for the Performing Arts, Detroit, Michigan, and ninth through twelfth grade teacher Avram Barlowe at The Urban Academy, New York City, New York are all featured in this episode. Lee S. Shulman, president of the Carnegie Foundation for the Advancement of Teaching, provides expert commentary.

Session Content Outline

Key Questions

  • How does the way knowledge is organized influence learning?
  • How can teachers use the structure of a discipline to organize their teaching and enhance student learning?

Learning Objectives

  • Structure of the disciplines – Teachers will understand that disciplines have structures representing interrelated core ideas and particular modes of inquiry. They will think about how to use these core ideas and inquiry tools to help students understand disciplinary ideas more deeply.
  • Pedagogical content knowledge – Teachers will consider the kinds of knowledge of content and students they need in order to represent disciplinary ideas so that they are more likely to be understood.

Session Outline

There are two ways to think about the principles of learning: One is to ask, “What are the principles that are so general and universal that in some ways they apply regardless of what you are teaching and to whom?” But there are also principles of learning that are particular to the domain under study and to the kinds of understanding that the learner brings to the table. Expert teaching in a subject matter rests on an understanding of how particular students are likely to come to this knowledge – what can be done to bridge the distance between what students understand and what counts as expert knowledge in the field. This is the essence of “pedagogical content knowledge,” which is the special expertise that teachers have about subject matter and how to teach it.

At the core of pedagogical content knowledge is a deep understanding of the structure of the discipline – how knowledge is organized and pursued in a particular subject area – connected to a deep understanding of the particular students being taught. This disciplinary structure can be reflected in the curriculum and in teachers’ pedagogical strategies. The structure of the discipline affects two things:

  • How knowledge and ideas are related and interconnected
  • How inquiries are carried out

Structure of the Disciplines

  • All subject areas have structures that reveal the ways their core ideas are connected with one another.
  • Each discipline has a different structure. Subject matters differ not only in terms of their core ideas, but also in terms of how inquiries are carried out.
  • The structures of the disciplines are the building blocks for organizing the curriculum to engage students in activities and experiences around these core ideas.

Organization of Core Ideas

  • Joseph Schwab explained the notion of the “structure” of a discipline by focusing first on the way the core ideas of a discipline are organized: What are the building blocks that reflect the central concepts in a discipline, and how do they connect with one another?
  • Jerome Bruner approaches the structure of the disciplines by considering the psychology of learning and development, memory, and transfer. Bruner describes how students’ interest and enjoyment of learning can be heightened through the “sense of excitement of discovery” they experience as the structure of a discipline becomes clear to them.
  • Both Bruner and Schwab assert that students can understand the basic concepts of subject matter at an early age when they are taught in an intellectually honest way. In fact, Bruner suggests that students be taught in a spiral curriculum that introduces central concepts in the disciplines early in a child’s education and revisits these concepts again and again in the later grades in more sophisticated ways.
  • Teachers can help students understand the structure of a topic by providing an overarching conceptualization of the big ideas and then locating specific facts or information that relate to the big ideas.

Central Modes of Inquiry

Schwab also considered the central modes of inquiry and knowledge-finding tools of the disciplines:

  • How does each discipline construct, critique, and revise knowledge?
  • How do you know something is true?
  • What counts as evidence?

Structure and Curriculum

key ideas and differing ways of posing and answering questions – should inform the overall curriculum. Making decisions about what ought to be taught involves asking –

  • “What are the properties of an activity, task, or project that will lead to the greatest teaching and learning?
  • What is the organization of ideas that would make it most coherent, understandable, learnable, and transferable by a student?
  • How does one match the activity with students’ interests and abilities?” (Shulman, 2001)

Pedagogical Content Knowledge

  • “Pedagogical content knowledge” is an understanding of what kinds of experiences, what kinds of objects, and what kind of examples can be used to help students acquire an understanding of complex ideas.
  • Pedagogical content knowledge “represents the blending of content and pedagogy into an understanding of how particular topics, problems, or issues are organized, represented, and adapted to the diverse interests and abilities of learners, and presented for instruction” (Shulman, 1987, p. 8).
  • Pedagogical content knowledge is shaped not only by how teachers think about why they are teaching their subject, but also by their understanding of what they are teaching, how they are creating curriculum, and when students are ready for learning.
  • The goal of teaching is not only to encourage particular understandings, but also to develop dispositions, values, commitments, and attitudes particular to a content area.
  • One of the misconceptions about pedagogical content knowledge is that it is a memorized set of analogies, diagrams, or tricks for each of the main concepts. Rather, this approach requires ascertaining what the students already understand, or misunderstand, and applying a set of strategies to build bridges between students and content.

Key Terms - New In This Section

  1. Amnesia – forgetting. It occurs when information has not been learned in memorable, usable ways (Shulman, 2001, interview).
  2. Central Modes Of Inquiry – how experts in each discipline construct knowledge and critique and revise knowledge that was once thought accurate and is now questioned. The manner in which experts in the discipline determine if something counts as evidence or is held to be true (Schwab, 1978; Shulman, 2001, interview).
  3. Core Ideas – the central concepts in a discipline and how they connect with one another (Schwab, 1978; Shulman, 2001, interview).
  4. Fantasia – a misconception or a set of misconceptions about ideas taught; a distorted grasp of a concept (Shulman, 2001, interview).
  5. Inertia – the absence of transfer; a situation in which students understand the ideas, but they can’t apply them outside of the immediate context in which they learned them. (Shulman, 2001, interview).
  6. Pedagogical Content Knowledge – the special expertise that teachers have about subject matter and how to teach it (Shulman, 2001, interview).
  7. Spiral Curriculum – teaching strategies that introduce central concepts in the disciplines early in a child’s education and revisit these concepts again and again in the later grades in more sophisticated ways (Bruner, 1960).
  8. Structure Of The Discipline – how knowledge is organized and pursued in a particular subject area

Questions for Reflection Step-By-Step Instructions

Step 1. The video segments in The Learning Classroom were taped as teachers worked in their own classrooms. As you watch, jot down the questions you have about what you see the teacher do and how the students respond.

Step 2. When you’re done, click on the episode title from the list and compare your questions with the Questions for Reflection and responses that our project team has anticipated.

Step 3. Review the responses we have prepared to questions that match the ones you have asked. The expert responses are not “final answers,” they are provided to give you a starting point for your own reflection. What else might you add to the response you read?

Questions for Reflection

Question 1: In the elementary classroom, Julie Helber asks the students how to record their results for their experiment. Wouldn’t it be more efficient and save time for her to provide a table or chart for the students to record their results? Wouldn’t this provide more consistency?

Response 1: By allowing the students to develop their own method of organizing their information, Julie is encouraging her students to use higher level thought processes. The students are collecting data just as a scientist would, and they are learning the importance of creating their own methods of organization so that the information makes sense to them, in order to share it with a broader audience. One chart or table for the entire class to use might create consistency in the organization of the information, but several examples will broaden the students’ repertoire of data representation.

Question 2: Julie Helber states that she wants her students to err in their experimentation. Will students become frustrated if they continually err? Is there a certain point in an inquiry in which the teacher should intervene?

Response 2: In a true inquiry it is definitely possible for frustration to occur. When students are working to make sense of a particular phenomenon it is very likely that they will become frustrated, however, it is also very likely that this is the point at which true learning is taking place. There can be a time for teacher intervention, but in order to keep the inquiry going and maintain the level of interest of the students, the teacher might offer suggestions of what other groups have tried or use questioning techniques to prompt their thinking.

Question 3: It sounds like there is a lot of student interaction and discussion taking place in Julie Helber’s classroom. With the curriculum demands an elementary teacher incurs, how does she cover the curriculum and still take the time for these discussions?

Response 3: One of the difficulties in teaching is determining the importance of what we teach as it relates to the curriculum and at what level of depth we teach the concepts or phenomenon we are exploring. Sometimes it is a judgment call for the teacher. Certain situations may warrant further discussion in order to deepen the students’ understanding, and other times the teacher may need to cut short a discussion, depending on the content. The teacher needs to have a good idea of where she wants the inquiry to lead and make decisions on the process throughout.

Question 4: I noticed that in the 10th grade science classroom Mary Edmunds has her students working in groups of four to six students. How does she ensure the participation of all group members?

Response 4: Although the video does not show how the students are being held accountable, one method she might use is to assign students specific “jobs” within the groups. By doing so, she can quickly look at what jobs are getting done to see how individual students are progressing.

It appears that the students are interacting with one another to make sense of what they are learning. They are discussing possible answers to their problem and providing positive feedback to each other. Their ideas seem to build off each other, demonstrating the participation of the group members.

Question 5: When Mary Edmunds’ students broke the shell she created a learning opportunity for the students by exploring the membrane. How does she keep the students on track and at a similar pace as the other students in the class?

Response 5: Sometimes it is not as important to keep all the students at the same pace, but rather have the students bring additional information to the broader audience. By creating a learning opportunity for the students she not only boosts their confidence (especially after breaking the egg) but also shows them how they are able to acquire information from their “mistake.” This group may not be at the same pace as the others, but they will be able to bring a new piece of evidence or information to the whole classto further their understanding of cell membranes.

Question 6: There is a lot of discussion regarding human rights and governance in Avram Barlowe’s classroom, and the students are actively engaged. Does he record the students’ ideas for later comparisons and reflections? What other methods does he employ beyond classroom discussion?

Response 6: The discussion in Avram Barlow’s classroom could serve as a springboard to many further assignments or discussions on this topic. By layering the information and building from this discussion he creates a basis from which further learning can take place. Careful organization of information by the teacher will allow students to build on what they know and take this knowledge to a higher level.

Question 7: Why is it important for Avram Barlowe to use primary source materials in his history class?

Response 7: There is so much information available to students, that if they are not using primary source materials it is quite possible that they are acquiring information that has been “slanted” based on the beliefs of the author of the documents. Primary source documents ensure the validity of the information studied and offer the students an opportunity to make their own interpretations of the material. As they try to interpret original documents, they are dealing with core concepts of history and learning to “think like historians.”

Question 8: How explicit does the teacher have to be in communicating the structure of the discipline to students?

Response 8: Generally the better a student understands the structure of the discipline the better access she will have to the specific content it holds. As Lee Shulman says, it’s like shopping for something in a supermarket – the more you know about the layout of the store, the easier it is to find the items you want to buy.

But the “big ideas” of an academic discipline are often abstract and there is no one correct way to present them. The “top-down” approach is to provide an overarching description of the big ideas and then locating specific examples of information that relate to them. The “bottom-up” approach takes specific information the students have been working with and then helping them develop a schema or concept map for how those ideas fit together.


Linda Darling-Hammond
Charles E. Ducommon Professor of Education, Stanford University

Lee S. Shulman
President, Carnegie Foundation for the Advancement of Teaching

Julie Helber
Fourth Grade Teacher, Paddock Elementary School, Milan, Michigan

Mary Edmunds
Tenth and Twelfth Grade Biology Teacher, Detroit High School for the Fine and Performing Arts, Detroit, Michigan

Avram Barlowe
Tenth through Twelfth Grade American History Teacher, Urban Academy High School, New York, New York.

Transcript of comments by Lee S. Shulman, President, Carnegie Foundation for the Advancement of Teaching

Excerpts from an interview with Lee Shulman, President of the Carnegie Foundation for the Advancement of Teaching

Recorded July, 2002

Discussion of transfer

The problem with transfer and the problem with structure are very closely related to one another. There’s a very simple fact that we all have to learn to deal with, and that is we never have enough time to learn everything we will ever need to know in order to live our lives successfully and fruitfully. So it doesn’t matter how long you would make medical school, for example. No physician could ever learn enough in medical school to anticipate every possible patient that will ever walk in her door. And therefore, the challenge of education is always to ask, “What’s the least amount of material we can teach really well that will, in turn, make it possible for those whom we teach to use that knowledge in the widest possible range of situations – including not only situations that we can anticipate but also situations that no one can anticipate.” And that’s abstractly the problem with transfer. How can you learn less, and make much more of it?

And basically there are two kinds of transfer that you can think about. One is the transfer that occurs between learning the parts of a task and then using those parts to do something much more complicated. So, for example, when you were in first or second grade, you learned addition, and you learned subtraction, and you learned multiplication. At some point you had to learn long division. Now, that required transfer, because you had to take what you already learned about adding, and subtracting, and multiplying, and apply all three of those processes to learning a new kind of skill called division. And you can call that vertical transfer – it’s taking small pieces and putting them together into a larger more complex skill. If you’re learning a performance of some kind – football players learn very specific small skills, and then they learn to put them together into a new play that they learn to run. And that’s a very, very important part of transfer. And so as educators, we often ask ourselves, “What are those simpler skills that, again and again turn out to be useful in more complex, later things, we want students to learn. Let’s make sure we teach those simpler skills very, very well so that when they confront the more complex skills, they can put together what they already know.” Now it turns out that very often, you already have some simpler skills or simple kinds of knowledge, but when you confront the new task, you don’t realize you already know what it is you need to do that task. And that’s why problem solving, and transfer, and metacognition are so important, because they involve being wise enough to know that you already know something, and to use it when it’s necessary. Anyway, that’s one kind of transfer, from simple to complex.

A second kind of transfer occurs when you have to take what you’ve learned in one situation and apply it to a new situation at roughly the same level of complexity. And so, for example, if I have learned about the, notion of a revolution in studying the history of the United States, and then I study the history of France, can I apply the notion of revolution, can I transfer what I know about revolution, from the American context to the French context, and subsequently, to the Russian context, if I then study the Russian revolution? These are examples of a more horizontal kind of transfer, where I can take an idea from one situation and move it to another. And, you can do that within a subject matter area, so, for example, you can do it from the concept of revolution for the United States and for France, (I’m doing that from within history), or you can do it across subject matter areas. So, for example, when the teacher was teaching bridge building to the students, he used concepts like balance, and form, and function to help remind them what they needed to keep in mind as they built their bridge. Well, where have they learned those concepts before? They might’ve learned them in looking at a piece of art, and looking at a lovely painting, and a teacher might say, “Well, notice what makes this beautiful is the way in which the painter balances various elements and the connections between the form�” But now if they’ve learned those in art, if they’ve learned them well, then the teacher can draw upon those ideas in a totally different area, such as bridge building, or frankly, biology, and so there are concepts that can be used across, and that would be an example of this second kind of transfer.

It’s important for teachers to understand the concept of transfer, because it’s absolutely necessary to protect them against what is probably the worst sin of teaching, which is trying to cover all of the material with equal attention across all the things you anticipate a kid might have to know. First of all, that’s impossible. Second, it is deadly boring. And third, it’s bad psychology. The reason teachers have to understand the notion of transfer is whenever they teach, they have to ask themselves, “What is it about what I’m teaching now that will be of value, of use, a source of understanding, or of pleasure to my students at some point in the future, when they’re in a situation that is not identical to the one they’re in now?” And that has got to be a mantra for the teacher, always asking, not just “Where am I?” and “Where are they now?” but “Where might this be going?” and if they can keep on thinking of that question as they teach, which is the question of transfer, then it can transform the kind of teaching they do.

The ways a teacher can facilitate transfer – that’s a very interesting challenge. Probably the most obvious way, and a way that teachers often think that they shouldn’t do – because they almost think it’s cheating – is explicitly to point out to students the variety of situations to which what they’re learning today might be useful or transferable in the future, and give them a chance to not only talk about, but even take the time to have them DO some of those transfer kinds of tasks themselves. So for example, when teachers want students to understand that some of the very basic skills they’re learning now are going to be really important, because, at a later date, they’re going to have to put those together in some more complex processes, one way to do that is to actually to give the students first, the complex task, or process that they’re going to need the separate skills for. That gives them a kind of vision of the more complex whole that they’re aiming toward. Then, when they learn the individual processes, they can anticipate the later transfer. We do that in all kinds of situations. We can do it in arithmetic, but we also do it in something like medicine, where we teach medical students all kinds of basic ideas, and they have no idea why they’re learning these things.

Well, what we now do is, early in medical school we give them much more complex cases to deal with, and they realize they don’t know what they need to know in order to solve those more complex problems. And then we say to them, “Okay, we’re going to move back to much more basic ideas and skills that are going to be the way in which you will eventually be able to solve this kind of problem.” We do it with engineering students as well – give them a complicated design problem. And then when they realize how complicated, how many different components there are to the design problem, they are then ready to go back to the more basic processes and understand that these have transfer value to what it is they’re going to need later. So, one of the most important strategies is to actually give students the chance to encounter the variety of transfer situations, for which what they’re learning now can be very useful. And I think that’s not done very often, and it’s probably one of the most frequently missed opportunities that we as educators don’t take advantage of.

Another thing that can facilitate transfer is – as you teach the basic ideas – to have students practice talking about them with each other, and writing about themselves what these skills entail, and what these ideas mean to them, because one of the most difficult things about transfer is that you have the knowledge you need, and you don’t know you have it, or you don’t recognize that it’s useful in this new situation. People know far more than they realize they know, and they’ll give up on a more complicated task. So teaching the initial material in ways that increase the awareness of the students of what they know, how they know it, and a variety of ways of trying to represent the idea in their own head – just multiple connections – makes it more likely that when they do encounter a situation in the future where that knowledge is useful, they’ve got it packaged or organized in ways that’ll make it more available.

Discussion of learning skills in context

Learning skills in context both can increase the transfer of skills and knowledge, but it also can decrease it, if it’s not used well, and let me explain what I mean. If you use a context, if you use a particular problem, or a simulation, or if you say, “We’re going to use this for bridge building, or we’re going to do it for another, very interesting real life task,” it increases the students’ engagement with the ideas – it makes them work with the ideas much more actively and flexibly. It’s very likely they’ll do it collaboratively and talk to one another about it, which raises awareness. It has all of those virtues which makes the ideas become more alive and salient for them, and all of those things – as I said before – make it more likely that the ideas will be transferable.

The danger one has to worry about when one puts learning in context is the students somehow learn that what it is they’re doing only applies in this situation. The context takes over, and they fail to recognize the value of what they’re learning when they’re not building bridges with toothpicks, when they’re not rubbing balloons on their heads, when they’re not writing a particular essay on a particular topic that was assigned. And so the teacher is always doing this really challenging balancing act of – on one hand taking full pedagogical advantage of the richness that a context can contribute to the learning process; at the same time, constantly reminding the students, in a variety of ways that what they’re learning has value beyond the context as well – making comparisons, making analogies. So that the kids don’t think that what this is really about is toothpicks, because it’s not.

Another important way to think about transfer is to remind ourselves that classrooms are very special artificial environments that we create in order to educate. Students live most of their lives outside the classroom, and when they live their lives outside the classroom, they are living in rich environments in which a lot of learning is taking place, and so there are two kinds of transfer we have to keep very, very much in mind. One is – how can we as teachers use the rich variety of experiences that students have outside the classroom, and bring it to bear inside the classroom, so that you ask students to think about something they already know how to do outside and apply what they know inside? And of course, we also have to get the inside out – have to keep on reminding students that what they’re learning is not only valuable inside the classroom, but to get them to think about the ways that transfer can occur outside.

Let’s try to think of a few examples of that. Carol Lee, at Northwestern University, has been working on the question of how do you help students from African-American, urban communities, use the knowledge they already have about using words, rhymes, and sentences – that come out of the way in which they already use rap and things like this in their lives outside the classroom – to the interpretation, and analysis and then creation of written material inside the classroom. And she’s done stunning studies that show that when you can begin with students becoming much more conscious of the ways they’re already using language outside the classroom and then show them how they can apply those usages in the analysis of production of literature inside the classroom, you get amazing increases in learning. And so there’s an example where culture and cultural artifacts and practices outside the classroom get transferred into the classroom and facilitate learning. So that would be an example of outside-in.

We all know that almost every little boy that says he can’t learn to do division can probably do baseball batting averages in his head when he’s outside the classroom. Here’s just an example that there are all kinds of varieties of intuitive uses of mathematics that kids engage in without thinking of them in formal terms outside the classroom. And if teachers can identify those, and help the students bring what they already know and do outside the classroom into the learning of the mathematics and doings inside the classroom, there will be much more rapid learning of the mathematics.

The notion here is for the teacher to be enormously cognizant of how smart kids are outside the classroom in a variety of ways, and to have them make that connection inside. Probably the most frequent strategy that gets used to do that – and we see that on a number of the tapes – is where the teacher begins by asking the students, in one way or the other, what they already know. And as the students begin to express things they already know, the teacher begins to identify the hooks from their outside experience that she’ll then use to facilitate the learning inside the classroom.

Now the inside to outside is the one we probably do more frequently, and that is where we have kids study a particular problem or task – or set of processes in the classroom – and then we assign a project or an assignment of some other kind that calls upon the students to take what they’ve learned inside and apply it outside. And again, we see this in all kinds of things, we see it in kids taking the mathematics they learn in the classroom of calculation and graphing and other kinds of representation, and going to the nearest busy intersection outside the school, and beginning to do a traffic flow study to try to decide whether they need an extra stop sign or an extra stoplight. You see it when kids take things they’re learning in biology, and they get a chance to get out and study a local body of water to check for pollution levels, or to check for what kinds of wildlife live out there. And these are just examples of teachers saying, “Hey, this is a set of ideas that doesn’t stop being useful in the classroom. Let’s go out and use it in a variety of settings and see how much transfer there is that way.” So the transfer goes both ways.

Discussion of the structure of subject matter

When we say that a subject matter has a structure, what we mean is that the ideas, the facts, the principles, the theories, the skills of a subject matter are not just some sort of long list of names that can be arrayed in any order, and you simply have to sit down and memorize them, and that’s what it means to know the subject. Anything worth knowing usually has some sort of organization. And that doesn’t mean there’s only one organization for the subject, but it means if you understand the organization, you have much more likelihood of gaining some mastery over the subject, of moving around in the subject, and of using it. Let me take a trivial example. You send me to a supermarket to buy a long, long list of groceries, and the list starts with avocadoes and ends with zwieback crackers – something like that for the baby. (Do they still have zwieback for babies? I don’t know.) And I don’t know anything about the way that supermarkets are organized – I just think they’re random – it will take me forever to buy that stuff, because I’ll go down the list, and I’ll go one at a time, and I’ll be wandering all over the place. I won’t know where to look for things, and I won’t know how to organize my actions inside the supermarket. But most of us understand the structure of a supermarket. We understand that different kinds of things you might want to buy. Groceries tend to be grouped together in certain categories, but not only that, categories tend to be organized in certain ways, and sometimes we know that structure even though we haven’t explicitly thought about it. We know, for example, where to look for produce. The likelihood is that the produce is going to be along the walls of the supermarket, and for some reason, when you walk into the supermarket they’re either along the right-hand wall, or the left-hand wall. Why? I don’t know, but I know that they’re organized that way. I know that the meat and the fish counters are also likely to be along walls. Now does it have to do with the availability of water and electricity? Maybe, but I don’t know. And the canned goods are likely to be in the middle and they’re going to be organized in certain ways, right? If I know that organization, if I know that structure, then I can move through and do what I need to do and even remember what’s on that list much better than if I don’t know the structure at all. In fact, what I tend to do when I get a list of things to go to the supermarket and buy – on those rare occasions when I’m trusted alone in the supermarket; men are not very trustworthy, we are impulse buyers – I reorganize the list. And I rewrite the list, so I put all the fruits and vegetables together, and all the milk and dairy products together, because I am trying to change the task so it fits my knowledge of the structure.

We do that in other places. If I’m a physician, and I’m doing an examination of you, I don’t just randomly say, “Let me just look at your nose, how’s your toe?” I have a structure – it’s the structure of the human body and its organ systems – so I will systematically check the gastro-intestinal track. I will check the respiratory system, the cardiovascular system, the neurological, right? Because, not only does that give me a structure for moving through the examination, it also structures my memory for any signs and symptoms that I encounter, that may be relevant in diagnosing what you have. So, every time we call something a subject matter, I would say, we call it that because it has some principle of organization that connects the ideas with one another that gives them some kind of order, some sort of meaningfulness, and it’s like a code. If you were to be a teacher and you taught students about supermarkets, and you never taught them the code, you never said, “You’ve got to understand, there’s a structure here – there is a predictable way in which these things are organized.” You wouldn’t be teaching the students well, because if you understand the structure, then you could go to a supermarket in Buenos Aires, you could go to a supermarket in Tel Aviv, and you could find your away around.

Well that’s true of teaching kids mathematics; it’s true of teaching kids history; it’s true of teaching them biology; it’s true of teaching them literature. You’ve got to look for the ways in which meaning is created through organization – through the way things are connected and ordered – and if you can help students get access to that, you can give them so much more power than if you just give them long lists of things.

When we say that subject matters have structures, what we’re saying is that they have ways in which the core ideas in the subject are connected with one another so that the students can acquire meanings that would be hard to acquire otherwise. It means students beginning to detect patterns and regularities.

Discussion of looking for the structures of subject matter

Looking for structures in subject matters is really a great deal of fun, because it’s what makes the subject matters the exciting kinds of domains they are. And the way you do that is to make students much more aware of how certain ideas keep on coming up again and again in the subject area, and to make them aware of the patterns and regularities in the subject. So, for example, as you begin teaching kids arithmetic and they learn that two plus three is exactly the same as three plus two and that two times three is exactly the same as three times two, that’s a wonderful structure. We call it commutativity. It means that the order in which you do things doesn’t matter. In this case it’s the same going backwards and forwards. And then you say to them, “Well, does that mean that three minus two is the same as two minus three, or that three divided by two is the same as two divided by three?” And as they begin to realize that isn’t the case, you begin to afford them access to one of the structures of mathematics – which is that some kinds of processes do have this commutative property and others don’t. And what are the implications of that? Where can you go from there? And this is again a question of transfer. We talked about that earlier, because as you begin to get access to those structures you’re getting ideas that you can apply again, and again, and again in new situations.

And so the notion of structure in mathematics is probably one of the most obvious ones. Mathematics is a field where most people who teach mathematics will readily understand the notion that there are certain kinds of structures. There’s a notion of balance. There’s a notion of equilibrium. There are notions of ratio and proportion that just keep on coming up again, and again, and again, and whether you’re doing primary arithmetic or you’re doing algebra, those notions return, repeat themselves – and you get a sense of how these organizations really make sense of the subject. In the case of a field like literature, the structures tend to be somewhat more elusive. They aren’t as obvious. They don’t stare you in the face the way they do in mathematics. But they’re there, nevertheless, so that you find that teachers of literature will often ask students to think about notions of theme. What’s the theme of this story? And students will say, “Well, I’m not sure, what is a theme?” And so you begin to get examples and try to show them. Well, what about character? What about plot, and how is plot like or different from theme? These are aspects of the structure of literature in some sense. And, we have centuries in which people are trying to identify these structures and to use them to help people learn these ideas and appreciate these ideas. When you’re studying Shakespeare, what’s the difference between the structure of a tragedy and the structure of a comedy, and how do you know when you’re reading or watching or experiencing each one? Those are aspects of the structures of literature. So you can go field by field and get examples of structures.

Many scientists will argue that in modern biology the theory of evolution is the central structure of all of biology. And it’s got core concepts, like adaptation, like organisms and environments, like the notion of chance, and what’s the role of chance in how well organisms adapt to environments – and how does that relate to notions of evolution? I mean, these are core ideas and they keep on coming up again, and again, and again. And to teach biology without introducing students to those core ideas would be like teaching people supermarket without giving them any sense of the organization. So in just about every field of study you get a kind of grammar, a kind of syntax, a kind of structure of the field which is the code that students have to be given access to so they don’t think all their learning is a long list.

Discussion of pedagogical content knowledge

Pedagogical content knowledge is an idea that we developed to try to explain a really interesting anomaly – and the anomaly is that many people who know a subject very, very well, find it nearly impossible to teach what they know to somebody else. It’s a really intriguing problem. The world is filled with people who know how to read, but they can’t teach reading to somebody else. The world is filled with people who can write, but then their own child comes with an essay they have to write, and they ask for help, and they don’t know where to begin. There are people who are great historians and have enormous difficulty explaining the histories that they themselves know to someone else. And we certainly know that the world is filled with mathematicians who seem to find it very difficult to teach mathematics to others. So the notion of pedagogical content knowledge grows out of the question: how is it possible for someone who already knows something to teach it to someone else who doesn’t? How do you create a bridge between what you know and what somebody else does not yet know, but needs to know? And, like building any kind of bridge, it requires a different kind of understanding, a different kind of process. It requires understanding both what you know and what is already inside the students’ heads, so that you can create powerful, and flexible, and rich connections between those two.

What do those connections look like? Well, first of all, pedagogical content knowledge often takes the form of understanding what kinds of examples and analogies, metaphors, stories, drawings that you might put up – visual representatives, experiences the students have outside the classroom already – that will create some of these meaningful connections between what the students already know and what it is we want to help them know. And that’s not a trivial task. It’s an extraordinarily difficult, complex task that takes years, and years, and years to learn. And, in fact, having taught now for nearly 40 years, I’m still learning, developing pedagogical content knowledge with respect to things I know. As I get more and more insight into how to explain them to others.

So, for example, in the teaching of mathematics – this very abstract, powerful, symbolic system – how do you teach mathematics to members of the species that are young and concrete and very much caught up in their own personal experiences? That requires pedagogical content knowledge. It requires understanding what kinds of experiences, what kinds of objects to manipulate, what kind of examples you can use to help students acquire an understanding of very, very complex ideas. And the important insight of pedagogical content knowledge was really at two levels. One was that someone who understands a subject matter deeply still doesn’t know what he needs to know in order to teach it to someone else; that subject matter knowledge is not sufficient for teaching, although it’s increasingly clear that it is necessary. So, teaching something well to someone and only having pedagogical knowledge of it, but not content knowledge as well, is also probably impossible. So there are two kinds of understanding and a very substantial part of pedagogical content knowledge is not only the understanding of the subject so you can figure out the variety of hooks that you can help create in the subject, but it’s having profound understanding of who the students are, so you understand where the hooks are in them, as well.

And that’s where questions of culture, and of language, and of the developmental status of the students become absolutely essential parts of pedagogical content knowledge. You cannot build a bridge from one side of the Golden Gate to the other if all you know about is your side of the Golden Gate, because how are you ever going to know how to anchor it at the other end unless you understand that terrain very, very fully. That’s why pedagogical content knowledge is equally concerned with a profound understanding of the subject matter at one end and of the student’s intellectual and motivational developmental and cultural standing at the other. It’s a very complex idea, but I think it captures the essential challenge of becoming an extraordinarily good teacher.

Discussion of finding the core ideas in a subject area

How do you identify the central ideas in a subject area? That is the million dollar question, really, because it is the question that the scholars in that area wrestle with constantly. I mean it’s not one of these simple questions where the answers are in the back of the book. And so it is a process of careful analysis. For example, one of the things I’ve done in preparing teachers is I’ve asked them, “Imagine now that there’s a class you’re teaching, and you’ve got 20 weeks to teach this class. Now, think about all the things you might want to teach in those 20 weeks. What if you then learned that you only had ten weeks for teaching it. What would you leave out? Would you simply lop off the last ten weeks? Or would you try to reorganize it in some fashion to make sure that certain ideas or stories or principles or concepts or facts were in it? Well, what if you learned that there are only five weeks to teach the class?” And I keep on pushing it back. And then I’ll say, well what if you had one day? And somehow in one day you had to somehow help students get access to what really counted? Now, at that point some of the students will just throw up their hands and say it’s not fair. But the very exercise of asking that kind of question forces us to rethink, and rethink, and rethink what does it mean to understand this subject well?

So if I were to say to you, “If you were an English teacher, what if you could only teach one story, one novel, or one play and that had to be in some fashion, the window to the world of literature for your students. Which play would you choose? Why? And what would you ask students to do with the play? Would it be �King Lear’? Would it be �Romeo and Juliet’? Would it be one of Arthur Miller’s plays? Would it be a Faulkner short story? Would it be Tolstoy’s War and Peace? And why?” I mean, that’s the kind of question that pushes you to begin to wrestle with, what’s the big idea here. And what’s interesting is that there may not be a single big idea.

For example, in elementary arithmetic, Dr. Lee Ping Ma has studied what she called the profound understanding of fundamental mathematics, and she discovered that in elementary arithmetic knowledge is organized into what she called knowledge packages – just clusters of ideas that are connected internally to one another, and then those ideas in turn, those packages, are connected to one another. And what she discovered was that in China the teachers and the kids understand those packages and teach them, whereas in the United States we tend not to. We tend to teach arithmetic as if it’s just a set of computational processes. And Lee Ping Ma maintains that’s the reason why consistently in international comparisons the Chinese kids dramatically outperform American kids, because we don’t stop and ask what those basic core ideas are.

Similarly, with a field like history. What are some of the core ideas about human power, about the clash of cultures, about the way in which societies organize themselves to engage in certain kinds of activities, internally and externally? And how is it that if you understand those, then you could look at different societies, different nations, over time in different places and be able to see the patterns and the regularities? But you’ve got to be able to understand those patterns and regularities yourself as a teacher before you can do that. And then you notice that some of the ideas we’ve been talking about earlier, ideas of transfer, ideas of structure, apply here as well. We’re talking about the transfer of knowledge. If you can get access to those key, core structures of a subject matter you can transfer that learning.

I mean, think of the example of learning a foreign language. If you try to learn how to conjugate every single verb in a language, one at a time, as if each verb had its own unique character, you’d spend a lifetime learning the language and every time you encountered a new verb, you would think you have to learn its conjugation from scratch. But would we learn? We learned a new language. Isn’t there a certain limited number of kinds of conjugation? And if you say, “What’s the word for to bungee jump,” and the person says, “Oh, it’s such and such.” And you think, “Ah, that’s a such and such kind of word.” You’re thinking about kinds of conjugation. They say, “Yup, that’s right.” You immediately know how to conjugate it because you understand a core principle that transfers, that applies over and over again. And they might say, “Well, it’s almost like that. There are a couple of little exceptions.” That’s fine, too, because you’re still doing variations on a theme. And we know about this musically as well. I mean, there’s certain kinds of musical performance with certain kinds of structures, and you learn to anticipate them. So, it’s different for each area. You’ve got to know it in each area. You can’t simply know it in one and say, “Well, I understand the structures that work in history, I guess I now don’t have to worry about mathematics, biology, physics, etc.” You have to know them for each subject and then begin to elaborate and work with them in your teaching the student.

Discussion of structures in science and history

Science and history make an interesting contrast if you begin to think about how notions of understanding are the same or different in those two fields, and therefore, how their teaching might differ. Let’s take a vivid example. When we first introduce science to kids, one of the first ideas they learn is the idea of an experiment. What do scientists do? They do experiments. And, you know, you put on your white coat. And as they learn about scientific method they learn about experimental groups and control groups. And they learn the great stories of scientific experiments. They learn the store of Pasteur and they learn the stories of Jonas Salk and they learn – I mean the notion of an experiment is very much at the center of what it means to do science, to generate evidence, to make inferences and come up with theories in science. Now, let me set aside for a moment the undeniable fact that there are fields of science where it’s very hard to do experiments – like astronomy. But let’s just take it for a given that experiment is a great idea in science. And now we start learning history. Well, how do you do an experiment in history? Well, you suddenly realize that that’s not the way you do history. We don’t have, we can’t put one historical period in an experimental group and another one in a control group and see what the difference is. But we’re still doing comparisons. We’re still trying to create evidence. I mean there are certain ideas that do cut across – notions of description, of analysis, careful, careful observation, notions of what is the evidence for your claim, notions of theory. But the fundamental process of experimentation which is at the heart of work in science, has no real analogy in history – if what you mean by experiment is that the scientist is controlling conditions and studying what happens under conditions that she herself has controlled.

So what does a historian have to do? The historian in some sense has to look for comparisons where, if you will, nature has made experiments, not scientists. So you ask, well, why was it that the American Revolution had these properties and proceeded this way, but the Russian Revolution, which we also call a revolution, had different properties and worked out a totally different way? Why is it that the American Revolution yielded a democratic form of government that remained stable for 200 plus years and the Russian Revolution yielded a more autocratic kind of government that had other kinds of properties. Well, you can say that’s almost like an experiment. So you’re still trying to generate evidence through comparison and contrast, careful observation and analysis. But you don’t have what the scientist in the experimental discipline has, which is control over the variables that you’re using for your work. And so what’s really important as teachers teach different disciplines to students, is for the students to appreciate that there are certain kinds of ideas like description, like analysis, like careful observation, like evidence, inference and theory, if you will, that are useful across disciplines. But there are other ways in which very important methods of work in one discipline just don’t show up in another discipline. And I think history and science make a lovely contrast here.

History and literature also make a lovely contrast. Because when a student study history, one of the first questions they always have to learn to ask is did it really happen that way? And that, that’s, you know, if somebody says that beings from outer space came down to the North American continent in 1775 in the middle of the night, they wrote the Declaration of Independence, slipped it under Thomas Jefferson’s pillow and then went back up into outer space. Well, you know, that’s a perfectly interesting narrative and the question is, but what’s your evidence that it really happened that way? Did it really happen? Now, if I write a novel and it’s a novel about beings from outer space coming down and slipping the Declaration of Independence under Thomas Jefferson’s pillow and somebody says, “That’s a perfectly awful novel �cause it didn’t really happen that way.” My response is, “You don’t get it. This is literature, this is a novel. It has different purposes. There are different canons that we use for determining whether it’s a good novel or a bad novel, and they’re not the same as the ones we use for determining whether something is good history or bad history – unless we’re going to use the novel for teaching people history.” And so here again, understanding the differences in what counts as evidence and what counts as knowing a subject between subjects becomes terribly important for teachers to understand, and in turn for students to understand.

Discussion of metacognition

Metacognition is thinking about your own thinking. Sometimes we use the phrase “going meta” instead of metacognition, and what we mean by that is being able to step back and, see yourself and what your doing, as if you were someone else observing it. It’s becoming an audience for your own performance. And in this case it’s your own, intellectual performance. We know that this is extraordinarily useful when we think about learning physical skills. So that when someone is learning to play golf, we know that seeing video tape of their own swing is enormously helpful for beginning to understand what they’re doing well, and what they’re doing poorly, because, so typically, we don’t even know what we’re doing when we do it. And therefore, it’s very hard to improve a process that you’re engaged in if you don’t have any idea of what you’re doing when you engage in it. I mean it’s no accident that ballet studios have mirrors in the walls. Because even someone as exquisitely skilled as a ballet dancer, does not really understand what she looks like and what she’s doing just from trying to experience it in her body. She has to be able to see it as others might see it before she can begin to improve it, and modulate it, and notice now that physical skills are so much easier then intellectual ones, because physical skills are at the end of the day, visible. They’re there. You can see them.

Cognitive work, intellectual work, thinking and feeling is invisible, it can’t be directly observed, so the question for us is, what’s the equivalent of the mirror on the dance studio wall, of the videotape of the golf swing? We’re saying, “How do you become thoughtful about your own thinking as you’re doing mathematics and history – as you’re doing the teaching of biology, the teaching of composition.” And, that’s a great challenge. Helping people learn how to go meta on their own thought processes, which are themselves not directly visible. And yet, if you can’t do that, it becomes very difficult to improve them – to get better at them – because how can you improve on those things you can’t see or feel, and therefore you can’t understand. And that’s the great challenge, not only of cognition, but when we talk about cognitive apprenticeships for example. It’s so much more difficult to model, and to shape, and to guide processes that aren’t directly visible. It’s not like learning to become a blacksmith, or to be a shoemaker, or even to be a midwife. When learning to be a mathematical problem solver, or someone who can creatively think up narratives in writing, you need to be able to go meta on your thinking, and not only on your observable performance. So that’s what megacognition is.

I think it’s, important for teachers to give students chances to reflect on their learning, because the students are the last ones to realize, very often, what they’re doing when they’re successful and when they fail. Here again, if I’m hitting a golf ball, and I see when I finish my stroke the ball has gone two feet to the left of the tee, I’ve got pretty good evidence that there must’ve been something I’ve done wrong with my swing, because it just didn’t get anywhere. But still, I need, some way of analyzing and looking at my swing, or I have no chance to learn to hit the ball so that it’s going to go 200 yards and straight down the middle. Well, if I’m learning to write an essay, and let’s say I write two essays, and I get an A from you on one of them and a C on the other, and the teacher says “See, you CAN write a good essay, do more of what you did on the A essay, than what you did on the C essay.” And I say, “But I don’t know what I did on the A essay that was different then the C essay, I just have no idea.” That’s where it becomes terribly important for the teacher to assist the student in reflecting on their own learning, because otherwise, how do I become better, how do I move my learning from the kind of thing I did in the C situation to the kind of thing I did in the A situation, if I don’t have access to the kind of understanding of my own performance that I need to improve it.

And I think that’s the essence of it, and it’s true in field after field. It’s so true in mathematics, where kids get confronted by things like more complex contextualized word problems, and they get some right and some wrong, and they just don’t understand what it is they did when they got it right – they just don’t have that meta understanding, “What was I doing when was doing it well as against what am I doing when I’m doing it badly.” And that may be at the end of the day, the most important thing we can teach students, and what’s tragic is that very often – because of the press of time – that’s the thing we sacrifice. So for example, think about hands-on science. We design these wonderful laboratory experiences for the students, and they go, and they roll balls down inclined planes – remember that from one of the tapes. Or we have them construct their bridges out of toothpicks, and we expend so much time doing, having the experience – doing the experiment – that we don’t take the necessary time the next day, or at the end of the hour to say, “Okay, stop doing what you’re doing. What was really happening there? What were you learning? Why was it that you concluded the ball did this – it was doing this – and when the ball did that, something else was happening? Think about your own thinking here. How were you using the evidence, how were you, on what basis were you making those inferences.” And the students should then be discussing it, arguing about it, trying to make these things clear.

Very often we do these lab experiments, and we run out of time, and so at the end of the day they’ve done the experiment, they’ve had the cognition, but they have not been able to go meta, and in the absence of going meta, the cognition is almost a waste of time, because they don’t know what they know. And therefore, the likelihood that they will be able to transfer that to another situation is dramatically reduced. Metacognition is one of those processes that helps take what we learn in one situation and transforms it into a level of understanding that is much more likely to transfer to another situation. And that’s the kind of connection there is between metacognition and learning and transfer. It’s at the heart of the process of taking what you’ve learned, and making it useable in transfer situations.

It’s important for teachers to give their students MANY opportunities to reflect on their learning, because the learning itself is rarely sufficient to create understandings of a sort that can be transferred readily to other situations, and because the absence of opportunities of reflection on one’s learning is part of why some kinds of learning are simply barren and infertile, if you will, and other kinds of learning turn out to be highly productive and useable again and again. And I think the heart of it is creating opportunities to step back and analyze, and reflect on your own practice. I mean, it’s no accident that when we prepare people to do very complex and important kinds of skills, we create opportunities for reflection.

I spent years teaching medical students how to take medical histories from patients. And we didn’t just teach them to do the histories. We had them practice it and then look at a videotape of their own performance, so they could begin to see things they were doing that were and were not productive of getting good information from their patients. Without looking at their own practice they found it very, very difficult to improve that practice. When students learn to write we often have review groups, where students review each other’s essays, and critique and give feedback. Well, that’s a way of going meta, because if you can step back and say, “Suzanne, did you notice that when you used this word, it had much more power than when you used that word? Why do you think that was so? And how can we get you to use more words of the first kind than of the second kind?” And you say, “I didn’t even realize I was using those words. I mean, it just, it just came to me naturally.” And you say, “Well then, if you’re just doing it naturally, then you’re not going to get control over using it purposefully.” And that requires going meta. Then, maybe helping you reflect that way, I get some insights into my own performance as well. So these are just several examples of how it is that going meta, becomes the lever for ratcheting learning from a low level to a much higher and transferable level.


Web-Based Readings

Bransford, J. D., Brown, & A. L., Cocking, R. R. (Eds.). (2000). Effective teaching: examples in history, mathematics, and science (Chapter 7). In How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.

Gardner, H., & Boix Mansilla, V. (1994, February). Teaching for understanding within and across the disciplines. Educational Leadership,51(5).

This article by Howard Gardner describes disciplinary differences.

Related Links

The Knowledge Media Laboratory at the Carnegie Foundation
The Knowledge Media Laboratory (KML) of the Carnegie Foundation for the Advancement of Teaching researches and develops new uses of emerging technologies and new media to enhance the quality of teaching and learning.

The National Board for Professional Teaching Standards
This Web site, which describes the history and background of the National Board for Professional Teaching Standards, includes the standards for certification in various subject areas and provides suggestions for using the National Board process as a platform for reform.

National Council of Teachers of Mathematics (NCTM)
This Web site includes resources, lessons, and activities for teaching mathematics in elementary, middle and high school. NCTM principles and standards for teaching K-12 mathematics are listed on this site.

National Science Teachers Association (NSTA)
This Web site highlights teacher resources, national satellite broadcasts, and regional events for K-12 and college science education. Science Class, a monthly electronic newsletter, is available to nonmembers. NSTA membership provides support with standards, assessment, inquiry-based learning, and integrating technology and teaching.

National Council for the Social Studies (NCSS)
This Web site includes information for social studies educators, including teaching resources, information about professional development, and the national social studies standards.

National Council of Teachers of English (NCTE)
This Web site includes resources for K-12 and college teachers in English studies, literacy, and language arts. The site includes a collection of standards-based lesson plans and Web resources, an index of subject-specific resources, funding resources, and information on national conferences.

American Council on the Teaching of Foreign Languages (ACTFL)
This Web site includes online resources, publications, workshops, and proficiency testing information for foreign language educators at all levels of instruction.