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Knowing What Students Know: The Science and Design of Eduacational Assessment
Models of Learning Arithmetic
Researchers have conducted a broad range of inquiries about the cognitive foundations of arithmetic, beginning with infants’ sense of number and culminating in the arithmetic basis of algebra. Particularly intriguing from the standpoint of cognitive theory are children’s skills in reasoning about arithmetic word problems. Researchers generally attempt to account for problem difficulty, transitions in children’s solution strategies, and errors. For example, children find the first of the following two problems easier than the second:
Joe has 5 marbles. Then Tom gives him 6 more marbles. How many marbles does he have now?
Melissa has 5 pieces of candy. Elaine gives her some more pieces of candy. Now Melissa has 11 pieces of candy. How many pieces of candy did Elaine give her?
Carpenter and Moser (1982) and Steffe (1970) attribute the relative difficulty of these problems to their semantic structure. Both problems involve actions, which makes them generally easier for children to solve than nonaction problems. But the second problem is more difficult than the first because it includes an unknown change quantity, and children have difficulty imagining actions that involve combinations of unspecified quantities. Unlike adults, children perceive the problems not as involving different operators (e.g., 5 + 6 = ? and 11–5 = ?), but as variants of combining sets (e.g., 5 + 6 = ? and 5 + ? = 11). Thus it is somewhat more difficult for them to invoke a simple counting strategy to solve the second problem. In contrast, the first can be solved quite easily by counting all the marbles.
Other researchers have supplemented these accounts by building explicit models of student knowledge and cognitive processes. For example, Riley, Greeno and Heller (1983) accounted for differences in problem difficulty by appealing to problem schemas that differentiate among problem types. Under this theory, certain problem schemas are activated by the semantic structure of word problems. Once activated, the schemas invoke associated strategies, such as finding differences between sets. The Riley, Greeno and Heller (1983) model was subsequently augmented by Kintsch and Greeno (1985) to include natural language processing of the problems. A somewhat different set of assumptions guided Briars and Larkin (1984), who assumed that children would use concrete objects, such as teddy bears or chips, to model the relations evident in the semantic structure of a problem. This model predicted that children could solve a wide range of problems, including those typically thought of as multiplication or division, if they could “directly model” (e.g., represent) sets and their relations with