ARE WE IN THE BUSINESS OF CHANGING STUDENTS' BELIEFS ABOUT MATHEMATICS? Annie Selden and John Selden Research Sampler, UME Trends, Vol. 3, No. 6, January 1992, 6. If we aren't, we ought to be. Beliefs influence mathematical behavior and lie somewhere between primarily cognitive factors (one's knowledge of mathematical facts and procedures and the effective use of strategies and techniques) and primarily affective factors (anxiety, motivation, etc.). [Alan Schoenfeld, Mathematical Problem Solving, Academic Press, 1985, Chs. 5 & 10]. Clearly, what students know about mathematics determines what they can do with it, and their attitudes can affect their performance. In addition, whether overtly expressed or not, their views about the nature of mathematics can influence how they tackle mathematical problems. The Institute on Problem Solving and Thinking, an NSF-funded project at Georgia State, changed inservice teachers' beliefs from nonproductive ("I'm good in math if I can do problems fast.") to productive ("It's okay if math takes time."). [Karen Schultz, Proceedings PME-15, Vol. III, 1991]. While studying both high school geometry and college cognitive science students, Schoenfeld observed beliefs such as formal mathematics has little or nothing to do with thinking, all mathematics problems can be solved in at most ten minutes, only geniuses can create mathematics, and mathematical procedures are passed down “from above.” Behavioral consequences were that the students did not invoke formal mathematics when constructions or discoveries were called for, using instead a naive empiricism, with students giving up quickly when they forgot a formula (“either you know it or you don't”). In a study of elementary education majors in mathematics and mathematics methods courses, Dina Tirosh and Anna Graeber found that although only 10% explicitly believed “multiplication always makes bigger,” more than 50% explicitly believed “the quotient must always be less than the dividend.” [Educational Studies in Mathematics, 20, 1989]. Many of these students still thought of division in terms of the partitive model in which an object (like a pizza) is divided into equal parts. This means the divisor must be a whole number and the quotient {\it must} be less than the dividend. It may have been the source of their belief that “division always makes smaller.” Despite these students' mechanical proficiency with the division algorithm (87% correctly divided 5 by .75), this model dominated their thinking when solving word problems. Given the problem, “The price of one bolt of silk fabric is $12,000. What is the cost of .55 of the bolt?”, one student argued, “... you want to know the price of this part, a part, of the bolt. So you are going to divide .55 into 12,000 to find out what that part is.” After executing the division algorithm and obtaining 21,818, the student continued, “I moved it [the decimal point] over here [in dividend], so I move it back two places. About 218.” According to Fischbein, et al. [JRME, 16(1985)], “models become so deeply rooted in the learner's mind that they continue to exert an unconscious control over mental behavior even after the learner has acquired formal mathematical notions that are solid and correct.” Such beliefs (misconceptions) are often robust, but can be addressed by inducing cognitive