The Role of Conceptual Entities and Their Symbols in Building Advanced Mathematical Concepts

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Subspace in linear algebra: investigating students’ concept images and interactions with the formal definition (2011, background): “For instance, Harel and Kaput (1991) studied student responses to the following problem: Let V be a subspace of a vector spaceU ; and let b be a vector inU but not in V : Is the setV þ b ¼ vþ b jv is a vector in Vf g a vector space?”
Advanced Mathematical-Thinking at Any Age: Its Nature and Its Development (2005, background): ““Economy of thought,” in this case, has to do with one’s ability to reify Solution 4 into a “solution method.” It has been shown that reification is one of the most complex processes in the conceptual development of mathematics—with the individual (e.g., Dubinsky, 1991; Greeno, 1983; Harel & Kaput,”