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.