Learning Procedures from Examples and by Doing

This paper describes a program that learns procedures b\ examining worked-out examples in a textbook and bv working problems I wo kinds of production (If-then) rules are created: working forward rules that produce an action when a proceduie is executed and difference rules that suggest operators from observed transformations. Dining example learning, the program examines two states in an example, fig tires out the operator that produced the second state and creates a production with some part ot the first line in the condition with the operatortor in the action. During learning b\ working problems, the progiam generates its own example trace b\ problem solving and uses the same example learning techniques. Introduction ALEX (Algebra example learner) is a program that learned to solve simple linear algebraic equations b\ examining worked-out examples and bv working test problems. The work (Neves, 1981) also included learning simple list algorithms, such as sorting and reversing a list. That work will not be reported here. An earliee version of the program was presented in Ne\es.(1978). Two important characteristics of these textbook examples ( / r / n conditions for learning, c. l . VanLehn, 1983) are that a. one procedure is taught per example and b. there are no hidden variables (however creating programs from examples or learning the commands of an operating system often requires the learner to induce variables that are not observable Dieiterich, 1984). The sub|ect mallei for learning comes fiom a high school textbook (Stem and Crabill, 1972) on elementary Algebia. The particular section of interest occurs after students are taught (in preceding chapters) exponents, the Distributive Rule, factoring, and tractions I hev then come to the first chapter in which the notion of solving an equation by manipulating symbols is introduced. I will describe the contents of this section to give an idea of what information is presented in this textbook, and which information ALEX uses to learn. The chapter on equation (solving tries to give an understanding of what an equation is and what operations can be applied to it by using a seesaw analogy with the equal sign at the fulcrum. Only operations which preserve the balance of the seesaw can be applied to the equation. ALEX does not use this information. A problem with analogy here is that it is difficult to draw the correspondence between a weight on a seesaw, and an operation on an equation for the novice user. Which math operaiions preserve equation "weight"? The next part of the chapter section teaches the legal operators of algebra by showing examples of how they work. These examples show how to add a number to both sides of an equation, how to subtract a number from both sides, how to multiply both sides by a number, and how to divide both sides by a number. The example for adding to both sides looks something like: ALEX uses these examples to learn the legal operations in algebra. Next, the book demonstrates the futility ot solving for the unknown bv randomly plugging in numbers. ALEX does not use this information f inal ly , the textbook presents a five step example that solves for the unknown (in the equation "3x 4 = 0") using legal operations, such as adding a number to both sides of an equation. After this single example six test problems are given and this section ends. The goal ot this research is to demonstrate how learning can take place by looking at the examples and b\ working the problems. Examples especially are well suited for procedute instruction as thev contain steps that will lead to the goal text (01 teacher) explanation, on the other hand, can be misleading, incomplete, 0r false. Example learning will not generally result in a complete procedure however. The student must be prepared to use problem solving along with incomplete procedures in working the problems at the end of a section. Working problems can produce learning behavior b\ having the student (0f program) generate an example trace through problem solving behaviou. Basic ideas Here I will describe some ot the basic ideas behind the ALEX program. A more detailed explanation follows this section. While learning from an example AEEX looks at two consecutive equation lines at a time. The fust line is viewed as having some operation applied to it to yield the next line The task ot the learner is to figure out the operation that took place and why it was applied in that circumstance. Once the system has this knowledge a working forward production rule is built with some part of the first line as the condition and the operator as the action. When done with those two lines the pro gram goes to the next pair of example lines, continuing until the end of the example. At the end of the example the procedure is indexed with a difference production rule. The condition of this rule is made up of the difference between the first and last lines of the example. The action is the procedure name. Procedure knowledge Procedural knowledge in ALEX is encoded as production rules (Newell & Simon, 1972; Anderson, 1983). Rules in ALEX are coded in an older version (OPS3) of the OPS family of production languages (Forgv, 1979). I will use a less formal, more English-like presentation of productions in this paper. Productions have been a popular medium for learning since Waterman (1975) because ot their modularity. The unit of knowledge is the production. A newly created production is added to the set of productions already in production memory. There is no need to look at the other productions in the set. Contrast this with a programming language where the placement ot code has important implications for what action is produced.