poses of algebra teaching, it was the foundation for modeling the variety of different approaches—both correct and incorrect—that students typically take to solving algebra problems. A number of different approaches can lead to a correct solution, and the program does not favor one over another. However, some approaches lead the student astray. If the student is working effectively on a problem, there is no computer feedback. But when a student begins down an unproductive or erroneous path, the computer program recognizes this by a process called model tracing and provides hints and instruction to guide the student’s thinking.

The Algebra Cognitive Tutor also assesses mastery of elements of the curriculum by a process called knowledge tracing. When a student’s problem-solving efforts suggest that the relevant knowledge or skill is not yet consolidated, the computer selects instruction and problems appropriate to where that student is in the learning trajectory.

In studies of cognitive tutors more generally, it was found that under controlled conditions, students could complete the curriculum in about a third of the time generally required to master the same material, with about a standard deviation (approximately a letter grade) improvement in achievement (Anderson et al., 1995). In real classrooms, the impact has generally not been as large. A third-party evaluation of the tutors suggested that the scaffolding of learning that allowed students to experience success with challenging problems produced large motivational gains (Schofield et al., 1990).

In the past, only teachers of high school students were thought to need knowledge of algebra. Although their preparation to teach does not include study of the objects and processes of high school algebra, little attention has been paid to whether or not secondary school teachers do in fact have adequate algebraic knowledge for teaching (Ferrini-Mundy and Burrill, 2002). Students preparing to be secondary school teachers typically take courses in abstract algebra and analysis, under the assumption that such mathematical background will serve them well as secondary school teachers. Yet the actual knowledge developed in such courses and its application by teachers in classrooms has not been thoroughly studied. Some research suggests that the