Adding apples and oranges: alignment of semantic and formal knowledge.

We show that the same mechanism that mediates analogical reasoning (i.e., structural alignment) leads to interpretive "content effects" in reasoning about arithmetic word problems. Specifically, we show that both college students and textbook writers tend to construct arithmetic word problems that maintain systematic correspondence between the semantic relations that people infer from pairs of real-world objects (e.g., apples and baskets support the semantic relation CONTAIN [content, container]) and mathematical relations between arguments of arithmetic operations (e.g., DIVIDE [dividend, divisor]). Such relational alignments, to which we refer here as semantic alignments, lead to selective and sensible application of abstract formal knowledge. For example, people usually divide apples among baskets rather than baskets among apples, and readily add apples and oranges but refrain from adding apples and baskets.