This report attempts to discover how students think about and use mathematics in open-ended questions. Part I, "Open-ended Questions in Mathematics," describes the rationale for adding open-ended questions to the grade 12 mathematics test of the California Assessment Program (CAP). Part II describes the findings from students' responses to five selected open-ended problems from the 1987 test. Strengths and weaknesses in the students' responses are examined, including students' misconceptions. In part III the CAP advisory committee's reasons for developing the scoring procedure used for the five open-ended questions are discussed. Five scoring procedures for five problems are presented. Part IV includes 20 student responses that have been scored and two responses that are in the ten problems given in 1987 and 1988. A set of nine open-ended problems were presented. (YP)