workshop on children’s development of addition and subtraction concepts taught problem solving significantly more and number facts significantly less than did teachers who had instead taken two 2-hour workshops on nonroutine problem solving. Students in the CGI teachers’ classes performed as well as students in the comparison teachers’ classes on a standardized computation test and outperformed students in the comparison teachers’ classes on complex addition and subtraction word problems.45 After teachers have studied the development of children’s mathematical thinking, they tend to place a greater emphasis on problem solving, listen to their students more and know more about their students’ abilities, and provide greater opportunity for their students to use a variety of solution methods. Gains in student achievement generally have been in the areas of understanding and problem solving, but none of the programs has led to a decline in computational skills, despite their greater emphasis on higher levels of thinking.
Case examples are yet another way to build the connections between knowledge of mathematics, knowledge of students, and knowledge of practice. Although the cases focus on classroom episodes, the discussions the teachers engage in as they reflect on the cases emphasize mathematics content and student thinking. The cases involve instruction in specific mathematical topics, and teachers analyze the cases in terms of the mathematics content being taught and the mathematical thinking reflected in the work the children produce and the interactions they engage in. Cases can be presented in writing or using multiple media such as videotapes and transcriptions of lessons. The episode in Box 10–3 is taken from a case discussion in which the case is presented through video recordings of lessons from an entire year that were captured on computer disks, together with the teacher’s plans and reflections and with samples of student work.
Notable in this example is how the teachers’ opportunities to consider mathematical ideas—in this case, functions—are set in the context of the use of those ideas in teaching. These teachers are probing the concept of functions from several overlapping perspectives. They dig into the mathematics through close work on and analysis of the task that the teacher posed. They also explore the ideas by investigating students’ work on the problem. And they revisit the mathematical ideas by looking carefully at how the teacher deals with the mathematics during the lesson.