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Structural Learning Theory (Joseph Scandura)
According to structural learning theory, what is learned are rules which consist of a domain, range, and procedure. There may be alternative rule sets for any given class of tasks. Problem solving may be facilitated when higher order rules are used, i.e., rules that generate new rules. Higher order rules account for creative behavior (unanticipated outcomes) as well as the ability to solve complex problems by making it possible to generate (learn) new rules.

Unlike information processing theories which often assume more complex control mechanisms and production rules, structural learning theory postulates a single, goal-switching control mechanism with minimal assumptions about the processor and allows more complex rule structures. Structural learning theory also assumes that “working memory” holds both rules and data (i.e., rules which do not act on other rules); the memory load associated with a task depends upon the rule(s) used for the task at hand.

Structural analysis is a methodology for identifying the rules to be learned for a given topic or class of tasks and breaking them done into their atomic components. The major steps in structural analysis are: (1) select a representative sample of problems, (2) identify a solution rule for each problem, (3) convert each solution rule into a higher order problem whose solutions is that rule, (4) identify a higher order solution rule for solving the new problems, (5) eliminate redundant solution rules from the rule set (i.e., those which can be derived from other rules), and (6) notice that steps 3 and 4 are essentially the same as steps 1 and 2, and continue the process iteratively with each newly-identified set of solution rules. The result of repeatedly identifying higher order rules, and eliminating redundant rules, is a succession of rule sets, each consisting of rules which are simpler individually but collectively more powerful than the ones before.

Structural learning prescribes teaching the simplest solution path for a problem and then teaching more complex paths until the entire rule has been mastered. The theory proposes that we should teach as many higher-order rules as possible as replacements for lower order rules. The theory also suggests a strategy for individualizing instruction by analyzing which rules a student has/has not mastered and teaching only the rules, or portions thereof, that have not been mastered.

Application
Structural learning theory has been applied extensively to mathematics and also provides an interpretation of Piagetian theory (Sandura & Scandura, 1980). The primary focus of the theory is problem solving instruction (Scandura, 1977). Scandura has applied the theoretical framework to the development of authoring tools and software engineering.

Example
Here is an example of structural learning theory in the context of subtraction provided by Scandura (1977):

The first step involves selecting a representative sample of problems such as 9-5, 248-13, or 801-302.
The second step is to identify the rules for solving each of the selected problems. To achieve this step, it is necessary to determine the minimal capabilities of the students (e.g., can recognize the digits 0-9, minus sign, column and rows). Then the detailed operations involved in solving each of the representative problems must be worked out in terms of the minimum capabilities of the students. For example, one subtraction rule students might learn is the “borrowing” procedure that specifies if the top number is less than the bottom number in a column, the top number in the column to the right must be made smaller by 1.
The next step is to identify any higher order rules and eliminate any lower order rules they subsume. In the case of subtraction , we could replace a number of partial rules with a single rule for borrowing that covers all cases.
The last step is to test and refine the resulting rule(s) using new problems and extend the rule set if necessary so that it accounts for all problems in the domain. In the case of subtraction, we would use problems with varying combinations of columns and perhaps different bases.
Principles
Whenever possible, teach higher order rules that can be used to derive lower order rules.
Teach the simplest solution path first and then teach more complex paths or rule sets.
Rules must be composed of the minimum capabilities possessed by the learners.
References
Scandura, J.M. (1970). The role of rules in behavior: Toward an operational definition of what (rule) is learned. Psychological Review, 77, 516-533.
Scandura, J.M. (1973). Structural Learning I: Theory and Research. London: Gordon & Breach.
Scandura, J.M. (1976). Structural Learning II: Issues and Approaches. London: Gordon & Breach.
Scandura, J.M. (1977). Problem Solving: A Structural/Process Approach with Instructional Applications. NY: Academic Press.
Scandura, J.M. & Scandura, A. (1980). Structural Learning and Concrete Operations: An Approach to Piagetian Conservation. NY: Praeger.
Scandura, J.M. (1984). Structural (cognitive task) analysis: A method for analyzing content. Part II: Precision, objectivity, and systematization. Journal of Structural Learning, 8, 1-28.
Scandura, J.M. (2004). Structural Learning Theory: Current Status and New Perspectives.
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template = input + """
Question:
How to let student learn and understanding 128 * 32, 412 * 829, 66 * 192?
Answer:
Let us think about it step by step.
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llm.invoke(template)