activities (e.g., count by tens beginning with any number), and those that focus on developing mental computation methods.58 Although the data do not point to a single preferred instructional approach, they do suggest that effective approaches share some key features: The multidigit procedures that students use are easily understood; students are encouraged to use algorithms that they understand; instructional supports (classroom discussions, physical materials, etc.) are available to focus students’ attention on the base-10 structure of the number system and on how that structure is used in the algorithm; and students are helped to progress to using reasonably efficient but still comprehensible algorithms.59

Sixth, research on symbolic learning argues that, to be helpful, manipulatives or other physical models used in teaching must be represented by a learner both as the objects that they are and as symbols that stand for something else.60 The physical characteristics of these materials can be initially distracting to children, and it takes time for them to develop mathematical meaning for any kind of physical model and to use it effectively. These findings suggest that sustained experience with any physical models that students are expected to use may be more effective than limited experience with a variety of different models.61

In view of the attention given to the use of concrete models in U.S. school mathematics classes, we offer a special note regarding their effective use in multidigit arithmetic. Research indicates that students’ experiences using physical models to represent hundreds, tens, and ones can be effective if the materials help them think about how to combine quantities and, eventually, how these processes connect with written procedures. The models, however, are not automatically meaningful for students; the meaning must be constructed as they work with the materials. Given time to develop meaning for a model and connect it with the written procedure, students have shown high levels of performance using the written procedure and the ability to give good explanations for how they got their answers.62 In order to support understanding, however, the physical models need to show tens to be collections of ten ones and to show hundreds to be simultaneously 10 tens and 100 ones. For example, base-10 blocks have that quality, but chips all of the same size but with different colors for hundreds, tens, and ones do not.

A seventh and final observation is that the English number words and the Hindu-Arabic base-10 place-value system for writing numbers complicate the teaching and learning of multidigit algorithms in much the same way, as discussed in Chapter 5, that they complicate the learning of early number concepts.63 Closely related to the difficulties posed by the irregu-



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