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Knowing and Learning Mathematics for Teaching: Proceedings of a Workshop (2001)
Center for Education (CFE)

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Knowing and Learning Mathematics for Teaching

Figure 1. The Knowledge Package for Subtraction

From Ma (1999). Used by permission of Lawrence Erlbaum Associates.

  • minuends between 10 and 20, e.g., 15 − 7;

  • minuends between 19 and 100, e.g., 53 − 25;

  • minuends with three or more digits, e.g., 203 − 15.

Each of these levels concerns a new idea and a new skill:

  • decomposing a ten;

  • splitting a ten from several other tens, then decomposing it;

  • successive decomposition, e.g., decomposing 1 hundred as 10 tens, then 1 ten as 10 ones.

The different levels of regrouping problems correspond to pieces of the knowledge package shown in Figure 1. However, other necessary pieces of knowledge do not occur as separate topics in the curriculum. Instead, students' learning of curriculum topics supports and is supported by knowledge of basic principles: composing and decomposing a higher value unit, the rate for composing a higher value unit, and addition and subtraction as inverse operations.

How do teachers help students focus on these basic principles? Teacher Mao (a pseudonym), speaking from thirty years of experience, described how questions can play a role in student learning:

What is the rate for composing a higher value unit? The answer is simple: 10. Ask students how many ones there are in a 10, or ask them what the rate for composing a higher value unit is, their answers will be

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