Abstract. This study investigated the pedagogical content knowledge that a college learner who is a prospective teacher might construct for teaching two-factor multiplication. In particular in this report, we attended to learnersŐ cognitive structures for signed number multiplication, described in terms of actions, processes, objects, and schema. In closing, we suggest problem-posing, visualization of problem solving, and identifying the isomorphic relationship in between computation and visualization as tools for improving both future research and the college mathematics preparation of teachers.
Abstract. The paper describes a project that will create a tool for the preparation of Graduate Teaching Assistants (GTAs) in mathematics similar to the Integrating Mathematics and Pedagogy (IMAP) Project materials for K-12 teachers. However, among the significant differences between IMAP and the proposed cases are: (a) the video-case tools and accompanying text will be developed for an audience with a mastery of mathematics who have little or no formal training in pedagogy; (b) the case tools will be sufficiently self-contained that they can be used as part of a distance-learning course on college teaching; (c) the materials will include classroom video-clips as well as materials about out-of-classroom interactions like office hours, email communication, advising of undergraduate and graduate students, communicating with junior and senior colleagues about teaching, and testimonials by GTAs on their learning about teaching. Along the way, in developing the case materials, the authors will conduct basic, applied, and evaluation research:
Abstract. This study examines the K-12 mathematical experiences of U.S. university students via an expressive writing assignment: a mathematical autobiography. The mathematical autobiographies of 67 college students, out of over 300 enrolled in 16 sections of a college liberal arts mathematics course, were analyzed deeply using constant-comparative methods. Four categories of experience connected to aspects of mathematical self-regulation emerged as significant to the student-authors: locus of control for mathematics knowledge and learning, self-evaluation of mathematical ability, emotionally-charged epistemological views of mathematics, and mathematical decision-making habits. Interviews of 18 of the 67 students provided support and clarification of the analysis. An argument, grounded in existing research, for increased mathematical self-regulation as a result of completing the mathematics autobiography is made. Finally, connections are drawn between learning and psychological theories to support the conjecture that the assignment may be as useful to novice university teachers as it is to their students.
Abstract. Twelve of 19 college algebra classes used WeBWorK and 7 used traditional paper and pencil homework (PPH). Given the quantitative result that no significant difference in performance between WeBWorK and PPH classes was found, a qualitative analysis of 358 student and instructor surveys revealed three primary categories of student perceptions related to WeBWorK: views about its usefulness, intentionality in engaging with mathematics, and challenges to student beliefs about mathematics. Student and instructor comments are reported within the context of self-regulated learning theory. Results support the conjecture that WeBWorK is at least as effective as traditionally graded paper and pencil homework for students learning college algebra independent of socio-cultural background.
Abstract. The paper presents the guiding ideas behind our culturally responsive approach to teacher professional development and an overview of how those tenets inform, tacitly and directly, our efforts to realize the promise of the National Council of Teachers of MathematicsŐ five Process Standards. A review of the primary obstacles teachers face in implementing these standards in their own teaching and learning is followed by a description of the design elements in a university-based professional development program. Our goal is to provide an example of the foundations upon which an evolving approach to culturally responsive professional development planning has grown. We discuss research on what constitutes effective teacher professional development while noting the paucity of programs that embrace recognized needs. We do not give a prescription for effective teacher development. Instead, we speak as teacher-educators about the necessary philosophical and self-evaluative underpinnings to effective professional development and our to approach creating an environment where it is safe to leave the isolation of forced autonomy and be reflective about community, mathematical activity, and intellectual engagement.
Abstract. In two previous studies we investigated the non-routine problem solving abilities of students just finishing their first year of a traditionally taught calculus sequence. This paper reports on a similar study, using the same non-routine first-year differential calculus problems, with students who had completed one and one-half years of traditional calculus and were in the midst of an ordinary differential equations course. More than half of these students were unable to solve even one problem and more than a third made no substantial progress toward any solution. A routine test of associated algebra and calculus skills indicated that many of the students were familiar with the key calculus concepts for solving these non-routine problems; nonetheless, students often used sophisticated algebraic methods rather than calculus in approaching the non-routine problems. We suggest a possible explanation. These students may have had too few tentative solution starts in their problem situation images to help prime recall of the associated factual knowledge. We also discuss the importance of this for teaching.