| The studies described in this thesis explore the development of students' knowledge of mathematical procedures. Students' tendency to develop rote knowledge of procedures has been widely commented on and is generally attributed to a lack of connection to principled knowledge. I postulate an alternative endpoint for the development of procedural knowledge, one that Ryle (1949) called an “intelligent performance” and Skemp (1976) described as “relational.” Students demonstrate this capacity when they are able to flexibly use mathematical procedures, especially when they choose to deviate from established solving patterns on particular problems for greater efficiency. The purposes of these studies were (a) to demonstrate that students could develop the ability to execute mathematical procedures “intelligently,” and (b) to explore the instructional conditions that facilitate the emergence of this outcome.; In three studies, students with no prior knowledge of formal linear equation solving techniques were taught the basic transformations of this domain. After instruction, students engaged in problem-solving sessions in two conditions. In the treatment group, students completed “alternative ordering tasks,” where they were asked to re-solve previously completed problems but using a different ordering of steps. When the treatment group engaged in such tasks, the control group instead solved structurally isomorphic equations.; In Study 1, 10 students worked individually with the experimenter for 4, 30-minute problem-solving sessions. Eight of the 10 students became very successful solvers of linear equations, discovering sub-goal knowledge and also developing an efficient and consistently used solving heuristic. In Study 2, 36 students engaged in 4 one-hour individual problem-solving sessions. The treatment group students became more innovative and more flexible solvers. Study 3 replicated Study 2 using a classroom rather than an individualized learning environment; similar results were obtained.; These results suggest that alternative ordering tasks may help to support the development of flexible knowledge of procedures. Flexibility is an advantage for acquiring more advanced knowledge and also for solving specific problems most efficiently. With training on considering alternative solutions, students can be assisted in avoiding rote learning of procedures and in developing a rich understanding of why procedures work. |
