ABSTRACT
This chapter illustrates some design goals and instructional techniques devised and used during a fifteen-year software development and implementation effort for mathematics instruction. When the program model is itself a direct expression of the mathematical reality, as in Darts and Green Globs, student manipulation of the model can result in direct graphic feedback, reflecting the student's input in the context of the mathematical model. Beginning in 1973, the fractions curriculum was developed in conjunction with use by students in public school classrooms. There are some similarities between the mathematical reasoning skills developing and the strategies in Sort Equivalent Fractions. The feedback advocated for improving the hypothetical fractions tutor includes direct coaching in the use of particular strategies to improve performance on computational problems. Tutors that treat student errors simply as procedural bugs, which can be corrected by review of the algorithm, neglect valuable opportunities to develop important mathematical reasoning skills with a limiting perspective of the nature of mathematics.
