To investigate questions concerning relationships among students' cognitive processes, aptitudes, later achievement and attitudes, and direct instruction, fifth- and sixth-grade students (N=72) were randomly assigned to one of six classes using a factorial assignment of ability level crossed with attitude. The questions addressed were: (1) What cognitive processes do students use during direct instruction? (2) How are students' aptitudes, including ability and attitudes, related to their reported cognitive processes during direct instruction? and (3) How are students' cognitive processes during direct instruction related to later achievement and attitudes? Videotapes of teacher and student behavior and classroom observations were made while each class of 12 students was taught a two-day lesson on probability. The teaching followed the direct instruction model and included review, development, controlled practice, and seatwork segments. After the lesson, students were interviewed about their thought processes using a "simulated-recall" procedure, and tested to assess their achievement of the basic concepts and skills taught in the lesson as well as their attitudes toward mathematics. Results showed that, independent of student ability, students' reports of their understanding of the lesson were significantly related to achievement. Moreover, students who reported using specific cognitive strategies did better on the achievement test than students who did not report using such strategies. (Author/RH)