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The following HTML text is provided to enhance online readability. Many aspects of typography translate only awkwardly to HTML. Please use the page image as the authoritative form to ensure accuracy. standing of mathematics. They know how to guide students from their current understandings to further learning and to prepare them for future travel. Such teaching and learning is possible because the road system of fundamental mathematics has depth, breadth, and thoroughness, allowing teachers to connect student understandings with topics to be learned. This is not the case in the United States, where knowing elementary mathematics is sometimes, perhaps often, construed as knowing how to add, subtract, multiply, and divide whole numbers and fractions. 2 In terms of content, this characterization is insufficient—at the very least, elementary mathematics concerns geometry as well. But for Chinese teachers with PUFM, it is insufficient in another way. Those in Ma's study would say, “It is not enough to know how, one must also know why.” The attitude expressed by this saying may affect a teacher's own knowledge—if knowing how is insufficient, one must find a rationale for mathematical procedures. Moreover, it may affect a teacher's goals for students—if knowing how is insufficient, students must come to understand why. In contrast, a teacher without this attitude may still know how and why, but not think it important that students know both. Or a teacher without this attitude may know how, but not why—and may not be able to answer students' questions, nor see the importance of student questions. Thus a teacher's attitude may affect not only the mathematics the teacher knows but also the mathematics the teacher teaches. Other mathematical attitudes displayed by Chinese teachers include the following: claims must be justified with mathematical arguments, it is desirable to approach the same topic in multiple ways, and it is desirable to preserve the consistency of an idea in different contexts. Such attitudes may affect a teacher's knowledge by contributing to its coherence and connectedness—and also affect a teacher's teaching. These fall in the category of what Jerome Bruner (1977) calls basic attitudes and considers as one aspect of the structure of a discipline. Another aspect of disciplinary knowledge identified by Bruner is basic principles. In the case of elementary mathematics (and perhaps all disciplines), basic attitudes have a symbiotic relationship with basic principles. For example, justifications in elementary mathematics often draw on the distributive law. Solving a fraction problem in multiple ways might draw on relationships between a fraction and a division, division as the inverse of multiplication, or relationships between fractions and decimals. In the base-10 system, noting the consistency of the relationship between 10 and 1, 100 and 10, and so on leads to the idea of the rate of 10: Each unit of higher value is composed of 10 or powers of 10 lower value units. This leads to the more general principle of the rate of composing a higher valued unit—the rate is 10 in the base-10 system, but there are other possibilities. For instance, the binary system has a rate of 2. Like basic attitudes, basic principles may play a role in teaching, as well as knowing, mathematics. They may appear as parts of what Chinese teachers call a “knowledge package” for a given topic—a network of conceptual and procedural topics that support and are supported by
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