Contexts, Goals, Beliefs, and Learning Mathematics.

The last few years have witnessed an almost exponential growth in research into students' mathematical beliefs. This research emphasis has beer* sparked, in part, by horror stories of the apparently nonsensical things that students at all grade levels do when they attempt to solve mathematical problems. These include examples of students with relatively sophisticated concepts who use low level, primitive methods in certain situations. Further evidence for the importance of beliefs comes from findings that demonstrate that students at higher grade levels often use superficial methods or accept impossible answers rejected by children at lower grade levels. At the primary level, for example, the relational approach to arithmetic word problem solving typically evidenced by first graders is replaced by the superficial analysis of verbal problems found in many older children as they attempt to decide whether to add, subtract, multiply or divide [Carpenter, Hiebert, & Moser, 1983, p. 55]. TTie older children were second and third graders. Further, failure to relate tasks such as 6+7, 6+8, 6+9, ...,6+15