Their focus was the influence of these programs on the prospective teachers’ teaching of mathematics. Ball and Wilson found that both groups of teacher candidates lacked understanding of the underlying relationships of mathematics. At the beginning of their teacher preparation programs, 60 percent of these prospective teachers could not generate a real-world example that would demonstrate to their students an application for the division of fractions. Moreover, they still could not generate an appropriate representation of division of fractions after they had graduated from their respective preparation programs. Ball and Wilson (1990) concluded that neither group was prepared to teach mathematics for understanding or to teach mathematics in ways that differ from “telling and drilling algorithms into students.”

What else, then, needs to take place in teacher education programs to support candidates adequately in the effective teaching of science and/or mathematics? Several possible answers were revealed in a study of teachers’ understanding of mathematics conducted recently by Ma (1999). Ma studied groups of elementary school teachers in China and the United States. Despite China’s more limited teacher preparation program, Ma found that the Chinese teachers had a more profound understanding13 of the mathematics they were teaching. This deeper understanding both of mathematics content and its application allowed Chinese teachers to promote mathematical learning and inquiry more effectively than their counterparts in the United States, especially when students raised novel ideas or claims that were outside the scope of the lesson being presented in class.

Ma’s study provides some insights that might guide an upgrading of teacher knowledge in the United States. Specifically, most of the Chinese teachers only taught mathematics, up to three or four classes per day. Much of the rest of their day was unencumbered, allowing for reflection on their teaching and, perhaps more importantly, for shared study and conversation with fellow teachers about content and how to teach it. Their teaching assignments also permitted them to gain over time a better grasp of the entire elementary

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Ma (1999) described the following characteristics as evidence for a teacher’s “profound understanding of mathematics”: 1. The ability to sequence appropriately the introduction of new concepts; 2. The ability to make careful choices about problem types to be given to students in terms of number, context, and difficulty; 3. Brief but significant opportunities for students to encounter conceptual obstacles; 4. Solicitation from and discussion by students of multiple points of view about a problem; 5. Anticipation of more complex and related structures; 6. Powerful and timely introduction of generalizations.



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