In the pair of experiments reported here the authors investigated the relationship between meaningfulness of problem statements and subjects' use of these statements in problem-solving tasks. Subjects (96 university students) were required to memorize meaningful formulae such as "volume = area x height" or corresponding symbolic formulae such as…

The author emphasizes the importance of concepts and mathematical form in a discussion of understanding mathematics. (MN)

Discussion about the nature and varieties of mathematical understanding is presented. Symbolic understanding is defined as a mutual assimilation between a symbol system and a conceptual structure, that is dominated by the conceptual structure. (MP)

Four kinds of understanding of mathematics are suggested: instrumental, relational, intuitive, and formal. Each type of understanding is described and illustrated. (MN)

This paper examines sixth-grade children's thinking related to solving ratio and proportion problems before these children had received any instruction in the domain. It explores unitizing and norming, one of the processes identified by both content analyses (Behr, Harel, Post, & Lesh, 1990); Harel, Behr, Post & Lesh, 1990) and research in…