Project Summary Our project had many twists and turns and ended up with four major components. We developed a typology of mathematical knowledge. A second component of our project was to perform a first look overview of the mathematical learning (in the epistemic, strategic, and social domains) of students in the mathematics program at Loyola Marymount University. A third component of the work is an analysis of a key sequence in the curriculum MATH 190/191. The fourth component of the work is an elementary understanding of what skills mathematics students transfer to other courses and to their lives in general.

The Background Questions that drove the study: 1)What knowledge from their mathematics learning do students transfer to the rest of their lives? 2) What is the evolution of a student's perception of the role of and need for proof? 3) What is the role of the mathematics workshop sequence (MATH190/191) in students' understanding of proof and how do they see their knowledge applied more liberally? Change in student understanding of proof Implications for Teaching Typology can be used for targeting instruction and gives language for complexity of the task. (Particularly as epistemic knowledge has factual, procedural, schematic, and strategic knowledge components) Epistemic learning is enhanced by community There is a risk of learners partitioning themselves into the "cans" and "cannots" that can be alleviated by tasks with initial (solo) time insufficient for completion. Habits of mind are skills most likely to transfer - they need to be incorporated into core classes

Knowledge interaction in workshop class A Typology and Taxonomy of mathematical knowledge As a first part of the project we developed a typology/taxonomy of mathematical knowledge. The categories of the typology are adapted from other works: The work of Duschl and Shavelson on science assessment, Alexander's work on Model of Domain Learning Domain Learning Typology in math This is a grid of knowledge types versus model domains of learning that Jackie and I set up. Selected Bibliography Alexander, P. (2003) The development of expertise: the journey from acclimation to proficiency. Educational Researcher. 32(8), 10-14. Barnes, L., Christensen, C., Hansen, A., Premises and Practices of Discussion Teaching in Teaching and the Case Method, Harvard Business School Press, Boston, 1994, 23-33. Duschl, R. (2003), Assessment of Inquiry, in Everyday Assessment, edited by J. Myron Atkin and Janet E. Coffey, NSTA Press, Arlington, 41-60. Pirie, S.E.B. and Kieren, T.E. (1989), A recursive theory of mathematical understanding, For the Learning of Mathematics, 9 (3), 7-11. Recio, A.M., and Godino, J.D. (2001). Institutional and personal meanings of proof. Educational Studies in Mathematics, 48(1), 83-99.

Evidence Gathered Student interviews Surveys of math majors (open-ended and Likert scale questions). Digital audio-tapes of 7 MATH 190 class days. Interviews with faculty from other disciplines (2 Education, 1 Philosophy, and 1 English). 12 Student "proof aloud" interviews. Student work from the MATH 190/191 course. Interviews with workshop students. Student focus group   Emergent Results Student knowledge progresses through major (chart) The Mathematics Workshop plays a significant role (course) Community is critical to student learning 15 minute problems play key role Student interest in explaining ideas to others plays a key role in social learning Habits of mind and writing skills transfer Pitfalls More data collected than could be analyzed in time frame. Professors in disciplines unwilling to ascribe differences of mathematics students to their mathematical training, but education professors more willing. Rubrics we intended to use were not complex enough

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