1We would like to thank Lael Schooler for his comments on this paper. Over its 10 year history, the research reported in this paper has been supported by contracts MDA 903-85-K-0343 and MDA 903-89-K-0190 from the Army Research Institute, a grant from the Carnegie Corporation, grants MDR-84-70337, IST-83-18629, MDR-87-15890, MDR-89-54745, and MDR-92-53161 from the National Science Foundation, and contracts N00014-84-K-0064, N00014-87-0103 and N00014-91-J-1597 from the Office of Naval Research.

[2]There were other ideas in the ACT* theory about production rule learning but these were abandoned in the ACT-R theory which is in many ways a simplification of the earlier theory.

[3]While our theoretical interpretation of the phenomenon is different, this issue of inert knowledge is, of course, a familiar problem in education (e.g., Brown, Collins, & Doguid, 1989; Cognition and Technology Group at Vanderbilt, 1990; and Whitehead, 1929).

[4]As it turns out, there are a number of different LISP tutors that we have constructed. The original tutor is called affectionately "LISP Tutor classic" in our laboratory.

[5]Students are judged to be floundering at a step in the solution when they repeat the same type of error three times or make two mistakes that the tutor does not recognize.

[6]Undoubtedly, the use of LISP creates a barrier to communication with that fraction of the readership that is not familiar with LISP. However, the semantics of LISP are not really necessary to understanding how the tutor interacts or how these interactions depend on the underlying production-rule models.

[7]Currently, the course is now a combined LISP and Prolog course in which the students learn both languages. The tutor is now delivered on a Macintosh system.

[8]For an evaluation of the contribution of the proof graph over and above any tutoring see Scheines & Sieg (in press).

[9]It got to the point were fights were occurring among students for access to the tutor.

[10]This is a one standard deviation difference.

[11]Subsequent research has suggested that teachers take about a year to become comfortable with the tutor and second year teachers have students who show achievement gains with the tutor.

[12]Public school teachers have been unwilling to allow students to progress at their own rate as enabled by the knowledge-tracing facility. Recently we have gotten Pittsburgh Public School teachers to accept such individual progress.

[13] Subjects are taking longer in the later lessons because the exercises are longer and harder.

[14]This, of course, raises interesting issues about evaluation. Since we have not gotten positive results simply by putting computers in the classroom, this indicates our positive results with project teachers is not simply a Hawthorne effect. The control classes (no tutor classes) of our project teachers do at least as well as the control and tutored classes of non-project teachers. Thus, there is something special about the combination of the tutor and the teachers' preparation to use them. We are working with the Pittsburgh Public Schools to try to develop an appropriate teacher training program.

[15]This can be viewed as embedding opportunities within the tutors for students to discover the concepts for themselves.

[16]However, it may take much longer before they really use the tutor to its full effectiveness. There is some evidence that achievement gains are higher the second year teachers work with the tutor.

[17]This only works if our tutors are bug-free, easy to use, and allow for error recovery from things like machine crashes. Typical teachers become very frustrated when their interactions with the student must focus on the computer or the tutor, rather than the subject domain being taught.

[18]Although assenting to the reviewer's comment requires interpreting "current curriculum" to Pittsburgh's algebra and geometry curriculum that follows and in some ways goes beyond the NCTM standards.