This research project will carefully examine and develop a cognitive description and analysis of certain generative aspects of mathematical thinking - posing problems and formulating conjectures in the form of problems to be solved - as these processes are used by persons at varying levels of mathematical experience and maturity, ranging from sixth-grade students to secondary school mathematics teachers. Careful attention will be paid to individual differences among and within the various subject populations and also to the effects on the frequency and/or effectiveness of use of the processes of support mechanisms, such as the use of computer software support or paired problem solving. The understanding of these processes obtained from observational studies during the first two years will be operationalized in the third year of the project in an instructional study in which a set of instructional activities, based on the descriptive analysis and devised to promote the development of these generative processes, is used with middle school students. The proposed research is directed at important extensions of extant scientific findings related to mathematical thinking and problem solving. Moreover, the work proposed here is related to current efforts to reform precollege mathematics instruction to make it more intellectually stimulating for students by increasing the amount of instructional time given over the student conjecturing and investigation. Thus, the results of the proposed research program should be significant not only for the contribution they can make to the growing scientific knowledge base concerning mathematical cognition and problem solving but also for their practical relevance for precollege mathematics education.
