Learning Theories

Our current understanding of the cognitive processes underlying

human learning enables cognitive psychology to offer valuable

guides to the design of curricula in school subjects. This

article summarizes some principles of curriculum design drawn out

of this literature that have been applied very successfully to

middle-school instruction in mathematics and science, using the

general plan of learning from examples and by doing. The article

does not claim unique efficacy for this specific method, but

shows how experience gained from employing the theory of adaptive

production systems provides concrete practical advice for

achieving effective learning with understanding.

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Jl. of Computers in Mathematics and Science Teaching (2003) 22(4), 285-322

Designing curricula on the basis of fundamental cognitive theory is an aspiration of long standing. Within the past half century, Henry C. Morrison (1934), from the side of education, and Robert Gagne (1970), from the side of psychology, approached the design task by analyzing the material to be learned into its unitary components. Similar analysis was undertaken by Skinner and his followers using the theory of operant conditioning.

In the past two decades, the development of computer technology and artificial intelligence has stimulated a new wave of activity in curriculum design for computer-aided instruction. Computer-based curricula take such diverse forms as intelligent tutoring systems (ITS) (Anderson, Corbett, Koedinger, & Pelletier, 1995), hypertext learning environments, including multimedia encyclopedias and textbooks (Raymond & Tompa, 1988; Jacobson, Maori, Mishra, & Kolar, 1996), technology-supported learning communities (Warren, 1997), and telematics for distance education (Oliver & Reeves, 1996).

In parallel fashion, modern cognitive psychology has spurred research on curriculum design based on a variety of cognitive theories of learning (Rabinowitz, 1993; Elmore & Tennyson, 1995; Spada, 1993; Arruarte, Fernandez-Castro, & Greer, 1996). In particular, several research and demonstration projects have shown how students can acquire knowledge effectively from examples and by problem solving, using adaptive production systems as their model of knowledge and skill acquisition, and as the basis for designing the examples and problems used to implement the curriculum. Among early studies analyzing learning within the framework of adaptive production systems are Anzai and Simon (1979), Neves (1978), and Newell and Simon (1972). Two more recent examples, drawn from a much larger number, are the continuing projects and applications by Anderson, Boyle, Farrell, and Reiser (1987) and VanLehn (1987).

The Empirical Basis for this Report

In the same vein, Zhu and Simon (1987), with their associates in a Beijing public school, developed an entire three-year secondary school curriculum in algebra and geometry that enabled students to "learn from examples and doing" (henceforth LFED), with a minimum of exposition and no direct instruction. With this method students first acquire new productions by examining worked-out examples, then use them to solve new problems and receive feedback that produces further learning (1).

This curriculum is currently being followed by over 20,000 students in China. Instructional experiments have shown it to be highly effective in comparison with traditional methods (Chen, 1999; Zhu & Lee, 1998; Zhu, Lee & Zhu, 1998). The scores of the students in the experimental classes surpassed substantially those of their peers in the control groups. The differences were not only statistically significant but of practically important magnitude. An experiment in teaching the principles of buoyancy in physics using LFED was equally effective (Zhu, Nan, & Simon, 1994; Zhu, Lee, Simon, & Zhu, 1996).

Various press articles reporting this work and results have appeared in the Chinese Educational Herald (the official Chinese education newspaper), arousing wide interest. A volume, collecting the reports of the experiences of instructors who have been involved in the experiments, has been published (LFED Research group, 1999).

Scope of This Article

In this article we describe how the theory of learning from examples can guide the design of curricula and discuss a number of principles of curriculum design derived from the theory. Perhaps we should call them "heuristics" rather than "principles," as they are intended as guidelines, and not as inexorable laws.

Many of the principles are familiar, overlapping considerably with those that have been employed in other successful projects of this kind. (2) It would be worrisome if they were not, as that would imply that they were not very essential to a curriculum's effectiveness. All the groups who work in this domain draw on the same common body of theory and experiment.

Although the evidence shows that well-designed curricula for learning from examples are effective, other methods may be equally or more effective under some conditions and, given the difficulties of assessing educational procedures, the efficacy of any method can, at best, be assigned only roughly to its specific components. So this article does not claim unique efficacy for a specific method, but seeks to show how the theory of adaptive production systems provides concrete practical advice for implementing this learning method effectively.

In instruction by LFED, a number of variant procedures have been proposed. For example, although our approach and Anderson's (Anderson et al., 1995) computer tutoring systems have common origins, we are led to somewhat different prescriptions on various dimensions. For example, we place a greater emphasis on learning the conditions of productions (cues) as the central learning objective, and less emphasis on goal-driven action; and as a consequence, we encourage forward search guided by relatively specific cues, as well as backward search guided by goals. This emphasis reflects evidence that progress from novice to expert status is associated with a gradual shift from working-backward (goal-directed) methods to working-forward (recognition) methods (Bhaskar & Simon, 1977; Simon & Simon, 1978, Langley 1985, Langley, Bradshaw, & Simon, 1987, Langley, Shrager, & Saito, in press).

We also do not, in general, introduce an initial stage of acquiring declarative knowledge which must then be transformed into procedural knowledge, but design the curriculum for direct acquisition of knowledge, even conceptual knowledge, in the form of production rules. Some attention is given to helping students learn to express important knowledge declaratively, but we have not specifically evaluated the usefulness of this step for generalization and subsequent learning. The reasons for these particular choices will appear as we proceed, but we do not claim that there is hard evidence for choosing among alternatives at this level of detail. We first introduce adaptive production systems and the methods of constructing and elaborating conditions for productions (CEC), and then discuss the principles of instructional design that follow from these methods and that have motivated our curriculum-building activities.

ADAPTIVE PRODUCTION SYSTEMS AND CEC

Current cognitive theories postulate, with good empirical support, that the knowledge enabling skilled performance is stored largely as productions: if-then statements consisting of a set of conditions followed by a set of actions, and usually designated C -> A. Whenever the conditions of a production are satisfied, the actions are evoked and carried out. A simple example of a production is:

IF (1) The goal is to carry out arithmetic computations, and

(2) There is a sequence consisting of a number followed by a plus

sign followed by a number followed by an equals sign (e.g., 6 + 3

=);

THEN Write, to the right of the equals sign, the sum of the numbers

that lie to the left of the equals sign.

A person who has this production stored in memory and is completing arithmetic equations (condition 1), upon reading or hearing such a sequence (condition 2), will add the two numbers (say, 6 and 3) and write the sum, 9. In this way, problem solving skills can be embodied in productions. To solve a problem, one must recognize the conditions that define the problem context and then execute the actions which are selected by these conditions. A production system is a set of productions of this kind, together with some rules for choosing which production to execute when more than one set of conditions is satisfied (conflict resolution rules).

Two central hypotheses provide the foundation for designing curricula based on the study of worked-out examples: (a) that human skills can be represented by productions, and (b) that these productions can be learned efficiently and with understanding by studying appropriate examples and/or by solving problems. A production system that can learn by modifying itself, altering its productions and adding additional ones to memory, is called an adaptive production system (APS).

The Student Described as an Adaptive Production System (APS)

The idea that a student can be described as an APS provided a new approach to the processes of human problem-solving and learning, and to teaching problem-solving skills. Our task in this article is to show how to identify the processes for learning to solve problems in a specific domain by specifying the production system that is to be learned. If we can specify such a system, then we can use it to design a series of problems and worked-out examples from which students can learn to solve problems in this domain.

An early application of this approach to school subjects was an APS using the LFED method, constructed by Neves (1978), which enabled a computer to learn how to solve linear equations in algebra. Shortly thereafter, Anzai and Simon (1979) used the Tower of Hanoi problem to explore further how APS's can, by solving problems, build productions that embody domain-specific knowledge. Zhu and Simon (1987), and Zhu et al., (1994, 1996) applied these ideas successfully to practical school instruction. Analyzing in detail students' processes of LFED in such fields as algebra, geometry, and buoyancy they found that students not only learned to solve specific problems, but also acquired strategies and heuristic rules for transferring their skills to new problems.

Anderson (1983, 1985, 1987), Anderson, Boyle, Corbett, & Lewis (1990), Anderson et al. (1987), Anderson et al. (1995), also taking production systems as models of students' skills, constructed computer-aided instruction (CAI) systems that were highly effective in teaching such subjects as geometry, algebra, and LISP programming. Basing their work on what they called Adaptive Control Theory (ACT), Anderson et al. (1990) assumed that knowledge is first acquired declaratively, and then converted into procedures by compilation. Thus, the students first learn verbal propositions, and then transform them into skills in the form of productions--of perceiving cues and responding to them. According to this theory, a student would first learn the proposition: "If the three sides of Triangle A are equal to the three sides of Triangle B, Triangles A and B are equal."

The student would then convert this to the procedure: "IF sides a, b, c of Triangle A are equal to sides a', b', c' of Triangle B, respectively; THEN store assertion: 'Triangle A = Triangle B'"

Following Neves (1978), Anzai and Simon (1979), and Zhu and Simon (1987), we postulate that production rules can be acquired directly, without first learning their declarative equivalents. Our protocol analyses indicate that from the onset of learning, the processes students use for explaining examples and problem solutions by drawing upon previously acquired knowledge are also used to acquire new domain-specific productions. In the succeeding problem solving, the students generalize the productions for broader application, and specialize them to handle special problem classes efficiently (Zhu et al., 1994; Zhu et al., 1996).

Whether or not it is best to acquire declarative knowledge as an intermediate step before acquiring new productions or to acquire the productions directly is still an open research question. Our classroom experiments have shown that teaching the productions directly from examples is effective.

Extracting Productions from Examples

Solving linear equations in algebra illustrates learning a skill by extracting the requisite productions from examples. In this case, four steps are required (we assume that the student already is in the habit of simplifying algebraic expressions when possible). The student has learned, just previously to encountering this example, that if the same quantity is added to or subtracted from an equation, or both sides are multiplied or divided by the same quantity (but not dividing by zero!), the solution of the equation will remain unchanged.