What is Mathematics? Faculty vs. Student Responses N = Number L = List of topics A = Applications P = Pattern/proof/logic S = Structure/abstraction/generalization O = Other

What is Mathematics? Pre/Post Future Teachers vs. Faculty N = Number L = List of topics A = Applications P = Pattern/proof/logic S = Structure/abstraction/generalization O = Other

Acknowledgements The initial data from the 55 math/cs students and 16 math faculty was collected in 2003 as part of a larger joint study on student understanding of proof with my colleague, Dr. Curtis Bennett.

What is the focus of this investigation? This study has two parts: First, it examines what students understand the discipline of mathematics to be and compares their understanding to that of mathematics faculty. Second, it investigates whether a course designed for future K-12 math teachers can enhance students' understanding of the subject area.

What results have emerged? 1. Students have a very limited view of mathematics as being about a list of topics, primarily, numbers, and applications. 2. Faculty see mathematics as encompassing pattern/proof/logic, abstraction/generalization, and applications. 3. Almost no students (initially) considered mathematics to involve abstraction or generalization. 4. A higher percentage of future teachers initially identified patterns, proof, or logic as part of mathematics than math or computer science majors did. --CONTINUED ON NEXT PAGE------- 5. A single course CAN influence students to move toward a more expert view.

Examples of Typical Responses Numbers - the study of numbers Listing of topics (could include numbers) - algebra, pictures, numbers, everything encompasses math Applications - a mental gymnastic that helps you solve real-world and theoretical problems Pattern/Proof and/or Logic - the search for and the study of patterns Structure/Abstraction/Generalization - the analysis of abstract systems Other - a language

What was the approach? We gathered written responses to the question What is mathematics? from 55 math and computer science undergraduates 7 future K-12 teachers enrolled in a "Women and Mathematics" course (before and after the course) 16 mathematics faculty. Then we categorized the written responses using the following emergent categories (1): N = Number, L = List of topics, A = Applications, P = Pattern/ proof/logic, S = Structure/abstraction/generalization, O = Other. The emergent categories correspond to Joseph Schwab's suggested ways of structuring a discipline (2): Content boundaries (N and L), Methods of inquiry (P), Habits of mind (S), and Purposes (A). Finally, we were able to make comparisons between faculty, math/cs students, and future teachers. For the future teachers, pre-and post-course comparisons could made with faculty.

. What resources / references were helpful? (1) Coding Data Resource Kit: Visible Knowledge Project, Georgetown University. (2) Schwab, J. "Structure of the Disciplines." In G.W. Ford and L. Pugno (eds.), The Structure of Knowledge and the Curriculum. Skokie, IL: Rand McNally, 1964.

Follow-up Results and Publication There is additional evidence that something happened in that semester to change the seven future teachers ideas about math. In a final reflection they were asked whether/how their view of mathematics had changed as a result of the course. Their comments indicated that they were more aware of the role of patterns in mathematics and the desirability of understanding underlying reasons. Eighteen months after the course, follow-up interviews with three of the pre-service elementary teachers indicated that two of the three students still held their richer views. These results are available in the Electonic Proceedingsof the Eleventh Conference on Research in Undergraduate Mathematics Education.