features consistent with the cognitive architecture and structure of knowledge as described in Chapter 3. ACT-R theory aims to describe how people acquire and organize knowledge and produce successful performance in a wide range of simple and complex cognitive tasks, and it has been subjected to rigorous scientific testing (Anderson et al., 1990). The model of learning is written as a system of “if-then” production rules that are capable of generating the multitude of solution steps and missteps typical of students. As a simple example, below is a small portion of an ACT-R production system for algebra:

Rule:

IF the goal is to solve a(bx + c) = d

THEN rewrite this as bx + c = d/a

Rule:

IF the goal is to solve a(bx + c) = d

THEN rewrite this as abx + ac = d

Bug rule:

IF the goal is to solve a(bx + c) = d

THEN rewrite this as abx + c = d

The cognitive model consists of many rules—some correct and some flawed—and their inclusion is based on empirical studies of student performance on a wide range of algebra problems. As the student is working, the tutor uses two techniques to monitor his or her activities: model tracing and knowledge tracing. Model tracing is used to monitor the student’s progress through a problem (Anderson et al., 1990). This tracing is done in the background by matching student actions to those the cognitive model might generate; the tutor is mostly silent through this process. However, when the student asks for help, the tutor has an estimate of where he or she is and can provide hints that are tailored to that student’s particular approach to the problem. Knowledge tracing is used to monitor students’ learning from problem to problem (Corbett and Anderson, 1992). A Bayesian estimation procedure, of the type described in Chapter 4, identifies students’ strengths and weaknesses by seeking a match against a subset of the production rules in the cognitive model that best captures what a student knows at that point in time. This information is used to individualize problem selection and pace students optimally through the curriculum.

Facet-Based Instruction and Assessment

The Facets program provides an example of how student performance can be described at a medium level of detail that emphasizes the progression or development toward competence and is highly useful for classroom assessment (Hunt and Minstrell, 1994; Minstrell, 2000). Developed through collaboration between Jim Minstrell (an experienced high school science teacher) and Earl Hunt (a cognitive psychologist), the assessment approach



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