counters. This prediction was subsequently confirmed (Carpenter, Ansell, Franke, and Fennema, 1993).

Carpenter, Fennema, and Franke (1996) propose that teachers who understand children’s thinking about arithmetic would be in a better position to craft more effective mathematics instruction. Their approach, called cognitively guided instruction, borrows from work in cognitive science to characterize the semantic structure of word problems, as well as the strategies children typically use to solve them. Cognitively guided instruction explicitly recasts this work as a coarse-grained model of student thinking that can easily be understood and used by teachers. The model allows teachers to recognize and react to ongoing events as they unfold during the course of instruction. In a sense, the work of Carpenter and colleagues suggests that teachers use this model to support continuous assessment in the classroom so that instruction can be modified frequently as needed. More detail about how this model is used in classroom practice is provided in Chapter 6.

Debugging of Computer Programs

Klahr and Carver (1988) analyzed the kinds of knowledge and reasoning skills required for students to write and debug a basic graphics design program in LOGO (a simple computer language). Beginning students were asked to write a program for drawing a house with windows and doors. Since first attempts usually involve errors (bugs), students had to learn how to debug their programs. This process involves several steps: (1) noticing and describing the discrepancies between the actual and the intended drawing, (2) considering which commands might have bugs (“buggy commands”), (3) creating a mapping between the descriptions of discrepancies and the potentially buggy commands, and (4) examining specific commands to see whether any of them was the culprit.

The investigators formulated these steps as a series of explicit rules, or “productions,” each consisting of a condition (noting, for example, whether there was a discrepancy in the orientation of the drawing) and an action (checking the values on all of the program’s “turn” commands). They wrote a debugging program, or model, based on these rules, then ran simulations to see how well the model could simulate the performance of students at two different levels of programming knowledge. When the model was set to simulate a student who had a high level of knowledge about the structure of computer programs, it quickly converged on the buggy command; when it simulated a student who lacked this knowledge, the model painstakingly examined a much greater number of possible culprits. The simulation paths followed by these two variants of the model were consistent with the behavior of real students having different levels of programming knowledge.

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