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Schools For Thought: A Science of Learning in the Classroom

John T. Bruer

1. Applying What We Know in Our Schools: A New Theory of Learning

In 1956, a group of psychologists, linguists, and computer scientists met at the Massachusetts Institute of Technology for a symposium on information science (Gardner 1985). This three-day meeting was the beginning of the cognitive revolution in psychology.

That scientific revolution became a movement, and eventually a discipline, called cognitive science. Cognitive scientists study how our minds work - how we think, remember, and learn.

What Most Students Can Do Is No Longer Enough (2)
New Expectations (6)
A Science of Mind: The MIT Meeting (7)

George Miller, a psychologist, presented a version of his paper "The Magical Number Seven, Plust or Minus Two" (Miller 1956). Miller observed that the number seven appears widely in the psychological literature as a limit on the capacity of the human nervous system.

Miller concluded that short-term memory capacity must be measured in chunks, a term that was to gain wide currency in psychology.
The Revolution Becomes a Discipline (11)
Cognitive Bottlenecks and High-Order Skills (15)
Knowing Why (17)

2. The Science of Mind: Analyzing Tasks, Behavior, and Representations

A Balance-Scale Problem (19)

Research on how children learn to solve balance-scale problems illustrates the main ideas, methods, and instructional applications of cognitive science.

   RULE IV

      P1 IF weight is the same
	 THEN say "balance"

      P2 IF side X has more weight
	 THEN say "X down"

      P3 IF weight is the same AND side X has more distance
	 THEN say "X down"

      P4 IF side X has more weight AND side X has less distance
	 THEN computer torques: t1 = w1 * d1; t2 = w2 * d2

      P5 IF side X has more weight AND side X has more distance
	 THEN say "X down"

      P6 IF the torques are equal
	 THEN say "balance"

      P7 IF side X has more torque
	 THEN say "X down"


			       Figure 2.2

   The set of rules an expert might use to solve the balance-scale problem

The Human Computer and How It Works (21)

At the heart of cognitive revolution was the realization that an adequate human psychology had to include the study of how the mind processes symbols.

A symbol is an object that stands for or represents another object.
Processing and Storing Symbols: Human Memory Structures (23)

Minds differ from digital computers in some obvious ways.

Borrrowing from computing, cognitive scientists speak of our cognitive architecture, the built-in mental features that allow our minds to build and execute programs. Figure;nbsp;2.3 gives the standard picture of the human cognitive architecture.

Long-term memory has what psychologists call an associative structure. Symbol structures represent items or chunks of information in memory, and associative links tie the items together into networks of related information. We create associative links between chunks if we use the chunks together repeatedly, learn them together, or experience them together.

Cognitive psychologists have discovered that long-term memory is not a single entity; it comes in a variety of forms. At the most general level, they distinguish declarative from nondeclarative memory. Declarative memory contains a system for remembering specific events (what psychologists call episodic memory) and a system for remembering general facts and word meanings ( semantic memory). We consiously recall items from declarative memory, and we can express or describe the items we retrieve. This is not so for the contents of nondeclarative memory. Among other things, nondeclarative memory contains our memory for motor, perceptual, and cognitive skills - our memory for procedures. The contents of nondeclarative memory are not always open to conscious recall, nor can they always be expressed or accurately described.

To understand problem solving and high-order cognition, we can focus on semantic and procedural memory - our memories for facts and skills. Although semantic and procedural memory both have associative structures, their structures are slightly different. ... The associations in procedural memory form rules.

Psychologists call the associative structures in declarative memory schemas. Schemas are network structures that store our general knowledge about objects, events, or situations.

Our associative memory structures are like little theories we apply to negotiate and understand the world. The associative structures help us make predictions - as with the balance-scale - and help us make inferences that go beyond what we literally experience.

These associative structures do not simply provide a way to store information; they also influence what we notice, how we interpret it, and how we remember it.

Associative memory structures are powerful devices for organizing and deploying our skills and knowledge. Like other theories, they also actively influence what we perceive.

If long-term memory is the storehouse, then working memory is the clearinghouse. Working memory is the term psychologists use to refer to the cognitive resources we use to execute mental operations and to remember the results of those operations for short periods of time (Baddeley 1992).

Working memory's most significant characteristic is its limited capacity.

Our capacity to remember and process information is understandably less than our capacity to remember alone.

Working memory can hold and process only a limited amount of information, and that for only short periods of time. We can quickly exceed its capacity, and when we do that any new information coming into working memory overwrites of obliterates what was previously there. Working-memory capacity is a limiting factor in our ability to process information. It is the bottleneck in our cognitive system. Skilled thinking, problem solving, and learning depend on how well we can manage this limited resource - on how efficiently we can store, process, and move information into and out of working memory.

How does the human computer work?

Using the production system illustrated in figure 2.2 to solve the balance-scale problem shown in figure 2.1 gives a simple example.

Production systems can become very complex, but the basic mode of operation remains the same. The system looks for matches between symbols active in working memory and conditions on production rules in long-term memory. When a match is found, that rule fires, modifying the contents of working memory - and the cycle begins again. When no match can be found, the program halts. That, in short, is how cognitive scientists think the human computer works.
Problems and Representations (31)

Psychology is a science of human behavior that develops theories about how we react or respond in various situations or environments.

...cognitive scientists think of the external world in terms of task environments. A task environment is a problem plus the context in which a subject encounters the problem.

Cognitive scientists use the word problem in a special way. The idea is simple, and it borrows from our everyday use of the word. As Newell and Simon wrote, "a person is confronted with a problem when he wants something and does not know immediately what series of actions he can perform to get it" (1972, p. 72). Cognitive psychologists elaborate and refine this general notion. They think of a problem as consisting of an initial state or situation and a goal state (i.e., what the person wants). To solve a problem, a person must figure out what to do to move from the initial state to the goal state. The things a person can do, the moves he or she can make in a problem situation, cognitive psycholotists call operators.

Our initial problem representations are important because they shape the course of our problem solving. Theinitial representation determines what we take to be the initial state and can influence what we take to be the goal and the legal operators. In this way, the initial representation constrains what cognitive psychologists call the lolver's problem space. The problem space is the set of all possible knowledge states the solver can construct from the initial state using the legal operators. ... A poor initial representation can make an easy problem hard or impossible.
Analyzing the Task (34)

... cognitive scientists can discover what representations and rules people use on more complex problems. Cognitive psychologists begin their research on problem solving with what they call a task analysis. They try to define what the major variables and causes are in a given type of problem. They try to figure out what knowledge and skills the problem demands, and given those demands, what ideal performance on the problem would be. Scientifically, task analysis is essential for solving the problem that cognitive scientists have set for themselves. We can think of what cognitive scientists are trying to do in terms of an equation:
```
Task demeands + Subject's psychology = Behavior.
```
Most of the time, cognitive psychologists are trying to solve this equation for "Subject's psychology, the subject's unobservable mental processing.
Analyzing the Balance-Scale Task (35)

The beauty of the balance-scale task for developmental psychology is that it is complex enough to be interesting but simple enough for exhaustive task analysis. Two variables are relevant: the amount of weight on each arm and the distance of the weight from the fulcrum. There are three discrete outcomes: tip left, tip right, and balance. There is a simple law of torques, that solves all balance-scale problems, though few of us discover this law on our own. If weight and distance are the only two relevant variables and if the scale either tips or balances, there are only six possible kinds of balance-scale problem:
- balance problems - equal weight on each side and the weights at equal distances from the fulcrum;
- weight problems - unequal weight on each side and the weights at equal distances from the fulcrum;
- distance problems - equal weight on each side and the weights at unequal distances from the fulcrum;
- conflict-weight - one side has more weight, the other side has its weight at a greater distance from the fulcrum, and the side with greater weight goes down;
- conflict-distance - one side has more weight, the other side has its weight at a greater distance from the fulcrum, and the side with greater distance goes down;
- conflict-balance - one side has more weight, the other side has its weight at a greater distance from the fulcrum, and the scale balances.
These six possibilities cover all possible cases for how weight and distance influence the action of the scale. The six cases provide a complete theory, or task analysis, of the balance scale.

Siegler formulated some psychological hypotheses about how people might solve balance-scale problems. Using the information from the task analysis, he could test his hypotheses by giving subjects problems and observing their performance. Siegler called his hypotheses "rules" and formulated them as four production-system programs. His rules I-III are given in figure 2.6; his rule IV is the expert's production system of figure 2.2 above.
```
   RULE I

      P1 IF weight is the same
	 THEN say "balance"

      P2 IF side X has more weight
	 THEN say "X down"

   RULE II

      P1 IF weight is the same
	 THEN say "balance"

      P2 IF side X has more weight
	 THEN say "X down"

      P3 IF weight is the same AND side X has more distance
	 THEN say "X down"

   RULE III

      P1 IF weight is the same
	 THEN say "balance"

      P2 IF side X has more weight
	 THEN say "X down"

      P3 IF weight is the same AND side X has more distance
	 THEN say "X down"

      P4 IF side X has more weight AND side X has less distance
	 THEN make an educated guess

      P5 IF side X has more weight AND side X has more distance
	 THEN say "X down"

			     Figure 2.6

	 Siegler's rules I-III for the balance-scale task
			
```
The rules make different assumptions about how and when people use weight or distance information to solve the problems.

Knowing the task and having hypotheses about the subjects' psychology gave Siegler values for two of the three variables in the cognitivist's equation that interrelates task, psychology, and behavior.
Finding Out What Children Know (38)

If children use Siegler's rules, then the pattern of a child's responses to a set of balance-scale problems that contains all six types will reveal what rule that child uses. Children's responses will tell us what they know about the balance-scale task, including how they represent the problem. Siegler tested his hypotheses and predictions by giving a battery of 30 balance-scale problems to a group of 40 children...

The children's performance confirmed Sidgler's hypotheses.

As these results confirm, Siegler's rules qualify as a cognitive and developmental theory for the balance scale. As a cognitive theory should, his rules explain behavior in terms of symbol structures that children have stored in their long-term memories. The individual rules tell us what knowledge children use. The production system tells us how they organize their knowledge. Chunks of information which children encode from the task environment or generate in working memory are the conditions that cause the rules to fire. When written in a suitable computer language, the rules can be run as programs on computers, and they simulate human performance. As a good cognitive theory should, the theory embodied in Siegler's rules performs the task it explains and explains the task in terms of reresentations and mental processes.

Taken together, Siegler's four rules constitute a develpmental theory that explains development in terms of changes in knowledge structures and problem representations.

Siegler's rules also tell us what cognitive changes underlie the transition from novice to expert. On tasks like the balance scale, children progress through a series of partial understandings that graduall approach mastery.

Siegler's four rules, viewed as a developmental theory, are also a simple example of how, as Robert Glaser claimed, cognitive science can give us develpmental theories of performace change. If we know what the developmental stages are and how they differ at the level of detail provided by a cognitive theory, we ought to be able to design instruction to help children advance from one stage to the next.
Experts and Response-Time Studies (41)

How can a cognitive scientist claim that experts use rule IV and don't simply compute torques on all problems?

If experts use rule IV, then they first try to solve the problem without computing torques, and they do the numerical computation only as a last resort. This means that experts' response times on conflict problems should be longer than their response times on balance, weight, and distance problems.

Siegler tested twelve adult experts and found that they solved balance, weight, and distance problems in 1.5 to 2 seconds. To solve conflict problems, the experts took 3 to 3.5 seconds. Using response-time data, we can conclude that experts don't compute torques on all problems. Experts use rule IV.
The Balance-Scale and Learning (42)

So far, we have seen how cognitive research can generate theories about children's knowledge and how they use it to solve problems. With theories like Siegler's that describe what goes on at discrete levels of performance, we also can begin to investigate how children make transitions between levels; that is, we can study how children learn and how they might learn most effectively.

...the children who had training on conflict problems. The 8-year-olds in this group advanced two levels in their mastery on the balance scale, from rule I to rule III. The 5-year-olds in this group either stayed at rule I or became so confused and erratic that it appeared they were no longer using a rule.

From a researcher's perspective, this is a troubling result. Even if we have detailed knowledge about children's initial understanding, we can't necessarily predict how children will respond to training. There must be more involved in learning than an interaction between the children's current rule and the training they receive.
Protocol Analysis, Encoding, and Representations (44)

How are 8-year-olds different from 5-year-olds? Why do the older children, but not the younger children, learn from training on conflict problems? To answer this question, the cognitive scientist needs finer-grained data than are provided by task analyses, response patterns, and response times. Cognitive scientists use a method called protocol analysis to collect such fine-grained data.

Protocol analysis exploits this "talking to ourselves" feature of working memory. To collect fine-grained, moment-by-moment data on a subject's cognitive processing, researchers have the subject "think aloud" while solving a problem.

Protocol analysis is a fundamental method of cognitive research.

To find out why the 8-year-olds learned and the 5-year-olds didn't, Siegler and his collaborators selected several children between 5 and 10 years old for in-depth study (Klahr and Siegler 1978).

On the basis of the protocols, the difference between 5-year-olds and 8-year-olds seemed to be that the younger children saw the problems in terms of weight only, whereas the older children could see the problems in terms of weight at a distance from the fulcrum.

Can 5-year-olds learn to encode both weight and distance, or is it beyond their level of cognitive development?

Only one intervention seemed to work. The 5-year-olds had to be told explicitly what to encode and how to encode it. The instructor had to tell them what was important and teach them a strategy for remembering it.

After this training, the 5-year-olds' performance on reconstructing distance information from memory improved.

Although they now apparently encoded the information, they, like the 8-year-old rule I users, did not spontaneously start using it. They continued to use rule I. However, when these 5-year-olds were given training on conflict problems, they too progressed from rule I to rule III. They had to be taught explicitly what representation to use in order to learn from the training experience.

Students learn by modifying long-term memory structures, here called production systems. They modify their structures when they encounter problems their current rules can't solve. Some children modify their structures spontaneously; ... Other children can't. Some children have inadequate initial representations of the problem. Children have to notice the information they need and encode it if they are to build better rules.

...long-term memory structures, such as schemas and production systems, can influence what we notice, recall, and remember. The existing rules and the initial representations affect one another. Effective instruction must break into and change the interaction. Breaking into and changing the interaction often requires detailed, explicit instruction on what the initial representation should be. Often this instruction also has to include teaching an effective strategy for encoding and remembering. Students who can't learn spontaneously from new experiences need direct instruction about the relevant facts and about the strategies to use.
Rule III to Rule IV (48)

The transition from rule III to rule IV is also of educational interest.

What kind of instruction or training sessions might help older students learn rule IV? On the basis of task analysis and how the balance scale works, Siegler conjectured that there were at least two points where students might have trouble: they might not realize that balance-scale problems have quantitative, methematical solution; and, even if they did, they might have trouble figuring out which algebraic equation to apply...

In an experiment, Siegler gave 13- and 17-year-olds training experiences that included hints on quantitative encoding, or the external memory aid for hypothesis checking, or both. ... training helped students in both age groups learn rule IV, but the 13-year-olds needed more help and learned more slowly.
Cognitive Research and Effective Instruction (49)

In a summary of their work with the balance-scale task, Siegler and Klahr (1982, p. 197) conclude that their results "show that acquisition of new knowledge depends in predictable ways upon the interaction of existing knowledge, encoding processes, and the instructional environmnent." Their summary, like their work, contains all the elements that make cognitive research applicable to educational practice. The work builds on and supports the assumption that humans, like computers, are symbol processors. Task analysis, protocol analysis, response-time studies, and training studies reveal how our cognitive architecture works in solving problems.

Siegler's work shows how cognitive science "provides an empirically based technology for determining people's existing knowledge, for specifying the form of likely future knowledge states, and for choosing the types of problems that lead from present to future knowledge" (Siegler and Klahr 1982, p. 134). The following chapters describe how researchers and teachers are applying this technology to improve classroom instruction. The tasks, representations, and production systems will become more complex - the progression from novice to expert can't be captured by four rules in every domain. However, our innate cognitive architecture remains the same no matter what domain we try to master, and the methods of cognitive science yield detailed information about how we think and learn. The lessons learned on the simple balance scale apply across the curriculum.

3. Intelligent Novices: Knowing How to Learn

Since the mid 1950s cognitive science has contributed to the formulation and evolution of theories of intelligence, and so to our understanding of what causes skilled cognitive performance and what should be taught in schools. In this chapter, we will review how our understanding of intelligence and expertise has evolved over the past two decades and see how these theories have influenced educational policy and practice.

Four theories will figure in the story.

The oldest theory maintains that a student builds up his or her intellect by mastering formal disciplines, such as Latin, Greek, logic...
In the early years of the cognitive revolution, it appeared that general skills and reasoning abilities might be at the heart of human intelligence and skilled performance.
Researchers then began to think that the key to intelligence in a domain was extensive experience with and knowledge about that domain. Expertise was domain specific.
In the early 1980s researchers turned their attention to other apparent features of expert performance. They noticed that there were intelligent novices - people who learned new fields and solved novel problems more expertly than most, regardless of how much domain-specific knowledge they possessed. Intelligent novices controlled and monitored their thought processes and made use of general, domain-independent strategies and skills where appropriate. This suggested that there was more to expert performance than just domain-specific knowledge and skills.

Perkins and Salomon call this latest theory or view the "new synthesis," because it incorporates what was correct about the earlier views, while pointing out that none of the earlier theories alone provides an adequate bases for effective educational practice. According to the new synthesis, we should combine the learning of domain-specific subject matter with the learning of general thinking skills, while also making sure that children learn to monitor nad control their thinking and learning.

The new synthesis introduces an important new idea into discussions about educational reform. The first three theories of intelligence emphasize what we should teach in our schools - formal disciplines, general thinking and learning skills, or domain-specific knowledge and skills. The new synthesis, as we shall see, implies that we should be as concerned with how we teach as we traditionally have been concerned with what we teach. The most recent research show that if we can apply the new synthesis in the classroom, we should be able to teach school subjects as high-order cognitive skills and help children become intelligent novices and expert learners.

Transfer (53)

We generally believe that learning a certain skill or subject area can help us learn a related one. ... Knowledge from the first skill or domain should transfer to the second, so there is less to learn. ... "Transfer means applying old knowledge in a setting sufficiently novel that it also requires learning new knowledge."

If this description is correct, we should be able to tell when transfer occurs. If knowledge transfers from task A to task B, then people who have learned A should be able to learn B more rapidly than people who did not first learn A.

Transfer is central to designing and developing effective instruction. Problems of transfer pervade schooling. Teachers want to teach lessons so that students can transfer what they have learned during class instruction to solve new problems at the end of a chapter. We want that learning to transfer to the unit, semester, or standardized test. Most important, we want school learning to transfer to real-world problem solving at home and on the job.

If we want to teach so as to promote transfer of knowledge, we have to answer a prior question: What kinds of knowledge and skills, if any, transfer between tasks? ... The new synthesis suggests that transfer can occur within and across domains, but only if we teach students appropriately.
Formal Disciplines and Mental Fitness (54)

Our oldest theory of expertise and intelligence goes back to the classical Greeks, who believed that mastering formal disciplines such as arithmetic and geometry, would improve general intelligence and reasoning ability. ... The theory was that these difficult formal disciplines would build general mental strength, just as rigorous physical exercise builds physical strength.

Edward Thorndike's careful studies of learning and of what knowledge transfers from one subject to another were among scientific psychology's first contributions to education. ... At the turn of the century, when Thorndike did his work, the prevailing view, derived from the ancient Greeks, was that learning formal disciplines improved general mental functioning. Thorndike, however, noted that no one had presented scientific evidence to support this view. Thorndike reasoned that if learning Latin strengthens general mental functioning, then students who had learned Latin should be able to learn other subjects more quickly. He found no evidence of this. Having learned one formal discipline did not result in more efficient learning in other domains. Mental "strength" in one domain didn't transfer to mental strength in others.

However, in some experiments where two subject domains shared surface similarity, Thorndike did observe faster learning in the second domain. He proposed a theory of "identical elements" to explain this. Thorndike suggested that where two domains share common elements of knowledge - not formal rigor - a person who has learned one of them might be able to learn the second more quickly. But, because psychologists at the turn of the century had no precise way to describe and identify "elements," Thorndike couldn't test his theory rigorously.
Elements, Productions, and Transfer (55)

Once psychologists accepted the assumption that our minds process symbols, and once the realized they could study minds as information-processing devices, it became possible to test theories such as Thorndike's. Psychologists, using the framework of computational theory, could describe "elements" as symbol structures and devise problem-solving simulations and experiments to see which symbol structures two disciplines might share.

Production systems are among the things that allow psychologists to test modern versions of Thorndike's theory. If minds are devices that execute production systems, and if (as on the balance scale) learning occurs when we add new productions to long-term memory, then we might be able to formulate and test Thorndike's claim. We can think of each individual production rule as a piece of knowledge needed for a task; we can think of it as one of Thorndike's elements. If so, the transfer of learning from one task to another should be directly related to the number of productions the tasks share.

M. K. Singley and John R. Anderson (1985) performed an elegant study to test this hypothesis. They studied the way in which secretarial students learned to use three different text editors or word processors. Two of the editors, ED and EDT, were line editors that allowed the user to edit one line at a time. EMACS was a screen editor, more like a standard word processor, that allowed the user to edit a document one screen at a time.

... Students who learned either line editor first took as long to learn the screen editor as students who started out on the screen editor. Skill on the line editors didn't transfer to the screen editor. In the case of the two line editors, students who learned one learned the second much more quickly.

Relying on computational theory, production systems, and task analysis allowed cognitive science to make precise scientific sense of Thorndike's hypothesis. The information-processing approach can give us fine-grained representations - in this case, productions - of Thordike's common elements. Cognitive research gives us methods for stating and testing claims about the transfer of knowledge between tasks.
General Methods and Intelligent Behavior (56)
Experts' Domain-Specific Knowledge (59)
Weak Methods in the Schools (64)
Metacognition (67)
Metacognition and Intelligent Novices (70)
Metacognition and Education (72)
The Final Element: General Skills Again (74)
The New Synthesis and the Teaching of High-Order Skills (77)

4. Mathematics: Making it Meaningful

Preschool Children and Number (82)
The Mental Number Line: The Heart of Math Readiness (85)
The Readiness Module (89)
Knowing Your Places: Learning Multi-Digit Arithmetic (90)
Buggy Arithmetic (91)
Marrying Concepts to Procedures (95)
The Bug Picture (98)
Word Problems: The Black Hole of Middle School Math (99)
The Adventures ofJasper Woodbury: Invitations to Thinking (100)
Jasper's Theoretical Basis: Representations and Inert Knowledge (101)
Theory into Prototype (104)
Jasper's Debut (106)
The Jasper Implementation Network (109)
Geometry Proof Tutor: An Underground Classic (112)
The Sorry State of Geometry (115)
The Expert Model (116)
The Tutor (117)
The Interface (118)
GPTutor in the Classroom (121)
From Theory to Practice to Theory (123)

5. Science: Inside the Black Box

Inside the Black Box (131)
Misconceptions and Cognitive Development 132)
The Jumping-Off Point (138)
Expert vs. Novice Physicists (142)
Problem-Solving Behavior (142)
Schemas (146)
Schemas and Problem Solving (149)
ThinkerTools: Physics in Middle School (151)
Designing ThinkerTools (153)
Motion in One Direction (154)
Motion in Two Dimensions (155)
Continuous Forces and Gravity (156)
Analyzing Trajectories (157)
The Theory of Instruction: Using the Microworlds (157)
ThinkerTools in the Schools (161)
High School Physics from a Cognitive Perspective (162)
Facets (163)
Benchmark Lessons: What Are Your Ideas Right Now? (164)
Don't Feel Dumb! (167)
Does It Work? Why? (168)
Teaching for Understanding (170)

6. Reading: Seeing the Big Picture

Cognitive Models of Skilled Reading (174)
One Loop Through the Cycle (177)
Cognitive Models and Reading Instruction (182)
Explaining the Whys (184)
Word Recognition: Accuracy Plus Speed (185)
Automaticity and the Great Debate (187)
Improving Linguistic Knowledge: Rich Vocabulary Instruction (190)
Background Knowledge in Learning to Read (194)
Background Knowledge in Reading to Learn (197)
The Big Picture: Reciprocal Teaching, Metacognition, and Reading (205)
Into the Classroom (210)

7. Writing: Transforming Knowledge

Writing: Solving Ill-Defined Problems (217)
Studying the Process: A Cognitive Model of Writing (221)
Planning and Its Subprocesses (224)
Expert, Novice, and Student Writers (227)
From Cognitive Process to Cognitive Rhetoric (232)
Constructive Planning: Building Rhetorical Representations (234)
Knowledge Telling versus Knowledge Transforming (238)
Tanya Again (240)
Writing in the Culture of School (243)
Collaborative Planning: Making Thinking Visible (246)
CSILE on Huron Street (250)

8. Testing, Trying, and Teaching

Cognition and Testing: From Correlations to Causes (258)
The Traditional Theory of Testing (259)
A Cognitive Theory of Learning Assessment (262)
Intelligence and Assessment: Beyond IQ (264)
Dynamic Assessment: Helping Weak Learners (269)
Motivation: Thriving versus Withering (272)
Motivation and Classroom Practices (276)
Teacher Cognition: What Does Nancy Know? (279)
The Missing Research Program (281)
Teaching and Knowledge Transforming (286)

9. Changing Our Representations: Thinking of Education in New Ways

A Science of Learning in the Classroom (290)
Representations of Research (293)
Solving an Ill-Structured Problem (294)
Knowing Why (296)

Background

jLogo Programming

Java

Appendices

Updates

Lastly

Schools For Thought: A Science of Learning in the Classroom John T. Bruer

1. Applying What We Know in Our Schools: A New Theory of Learning

What Most Students Can Do Is No Longer Enough (2)

New Expectations (6)

A Science of Mind: The MIT Meeting (7)

The Revolution Becomes a Discipline (11)

Cognitive Bottlenecks and High-Order Skills (15)

Knowing Why (17)

2. The Science of Mind: Analyzing Tasks, Behavior, and Representations

A Balance-Scale Problem (19)

The Human Computer and How It Works (21)

Processing and Storing Symbols: Human Memory Structures (23)

Problems and Representations (31)

Analyzing the Task (34)

Analyzing the Balance-Scale Task (35)

Finding Out What Children Know (38)

Experts and Response-Time Studies (41)

The Balance-Scale and Learning (42)

Protocol Analysis, Encoding, and Representations (44)

Rule III to Rule IV (48)

Cognitive Research and Effective Instruction (49)

3. Intelligent Novices: Knowing How to Learn

Transfer (53)

Formal Disciplines and Mental Fitness (54)

Elements, Productions, and Transfer (55)

General Methods and Intelligent Behavior (56)

Experts' Domain-Specific Knowledge (59)

Weak Methods in the Schools (64)

Metacognition (67)

Metacognition and Intelligent Novices (70)

Metacognition and Education (72)

The Final Element: General Skills Again (74)

The New Synthesis and the Teaching of High-Order Skills (77)

4. Mathematics: Making it Meaningful

Preschool Children and Number (82)

The Mental Number Line: The Heart of Math Readiness (85)

The Readiness Module (89)

Knowing Your Places: Learning Multi-Digit Arithmetic (90)

Buggy Arithmetic (91)

Marrying Concepts to Procedures (95)

The Bug Picture (98)

Word Problems: The Black Hole of Middle School Math (99)

The Adventures ofJasper Woodbury: Invitations to Thinking (100)

Jasper's Theoretical Basis: Representations and Inert Knowledge (101)

Theory into Prototype (104)

Jasper's Debut (106)

The Jasper Implementation Network (109)

Geometry Proof Tutor: An Underground Classic (112)

The Sorry State of Geometry (115)

The Expert Model (116)

The Tutor (117)

The Interface (118)

GPTutor in the Classroom (121)

From Theory to Practice to Theory (123)

5. Science: Inside the Black Box

Inside the Black Box (131)

Misconceptions and Cognitive Development 132)

The Jumping-Off Point (138)

Expert vs. Novice Physicists (142)

Problem-Solving Behavior (142)

Schemas (146)

Schemas and Problem Solving (149)

ThinkerTools: Physics in Middle School (151)

Designing ThinkerTools (153)

Motion in One Direction (154)

Motion in Two Dimensions (155)

Continuous Forces and Gravity (156)

Analyzing Trajectories (157)

The Theory of Instruction: Using the Microworlds (157)

ThinkerTools in the Schools (161)

High School Physics from a Cognitive Perspective (162)

Facets (163)

Benchmark Lessons: What Are Your Ideas Right Now? (164)

Don't Feel Dumb! (167)

Schools For Thought: A Science of Learning in the Classroom

John T. Bruer