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)

A review of several studies on the role of concrete and symbolic models in facilitating the learning of mathematical ideas, and a discussion of their implications for instruction are presented. (DT)

The interactive effects between mathematical achievement and manipulative v symbolic instruction with junior high school students are examined in this report. Low-achieving students benefited more from a manipulative approach to logical connectives, whereas high achieving students found the symbolic approach to be more effective. (MP)

The problems of using formalistic methods in the instruction of mathematics to elementary level children are discussed. Precision in mathematical language is not seen as a useful instructional tool, and functional modes of communication are stressed. (MP)