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A Ph.D. Program in Mathematics Education in a Dept. of Mathematics: Arizona State University as an Example Marilyn Carlson Arizona State University marilyn.carlson@asu.edu Thursday January 8, 2004, 1:00 p.m.-2:20 p.m.

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Overview of Programs Degree Options Ph.D. in Mathematics Ph.D. in Curriculum and Instruction M.N.S. in Mathematics with concentration in Math Ed. Mathematics Education Faculty Department of Mathematics and Statistics (3) College of Education (3)

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What Makes It Work? Support from Mathematics Department administrators and faculty Active participation of interested mathematicians Strong program with high standards Leadership of mathematics education faculty Strong collaborations between the College of Education and Department of Mathematics math education faculty

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What Are the Benefits? Mathematicians and Mathematics Educators collaborate in: Supervising Ph.D. students in mathematics education Developing and implementing mathematics education courses and projects Writing proposals Mathematicians and Mathematics Educators engage in regular information exchange about issues of knowing and learning mathematics.

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Required Courses Ph.D. in Mathematics Sequence of Four 500-level Math Education Courses (12 hrs) Two 400-level Qualifier Sequences(12 hrs) Three - Six Courses in College of Education(9-18 hrs) Remaining Courses in Mathematics or Statistics(18-27 hrs) Research and Dissertation(24 hrs)

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Other Requirements Ph.D. in Mathematics 2 mathematics qualifying exams Algebra, Differential Equations, Discrete, Numerical Analysis, Real Analysis, Statistics 1 written comprehensive exam Dissertation prospective defense Dissertation defense

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Required Courses Ph.D. in C & I Core Requirements (6 hours) Inquiry and Analysis(15 hours) Quantitative and Qualitative Research Major Area of Concentration(30 hours) Mathematics and Mathematics Education Internships(6 hours) Teaching and Research Cognate Study(12 hours) Mathematics Courses Dissertation Research(24 hours)

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Other Research Experiences Participation in faculty research projects Conference presentations Participation in proposal writing Participation in paper presentation

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Faculty Dept. of Mathematics Marilyn Carlson Michelle Zandieh Michael Oehrtman Phil Leonard Dennis Young College of Education Alfinio Flores Jim Middleton Jae Baek

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Ph.D. Students in Mathematics Education 8 students, Math Dept – full time TA’s and RA’s 9 students, C & I program, but mentored by Dept. of Mathematics Faculty 10 (Approx) students, C & I, mentored by C & I

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Student Research Topics Sean Larsen Supporting the Guided Reinvention of the Concepts of Group and Group Isomorphism: An Evolving Local Instruction Theory Nicole Engelke An Investigation of Cognitive Aspects of Learning Related Rate Problem Sally Jacobs Advanced Placement BC Calculus Students’ Ways of Thinking About Variable Jessica Knapp Complexities of Learning to Prove

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Student Research Projects C & I Irene (Apple) Bloom The Development of Secondary Preservice Mathematics Teachers’ Mathematical Behaviors: An Evaluation of the Effectiveness of Extended Analyses Problems Marguerite George An Examination of questioning patterns in mathematics instructors Mark Burtch Student Conjecturing in a Differential Equations Course

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Dissertation Project: Sean Larsen Mathematics TITLE: Supporting the Guided Reinvention of the Concepts of Group and Group Isomorphism: An Evolving Local Instruction Theory ABSTRACT: In this talk I will describe a local instruction theory that emerged over the course of a sequence of three teaching experiments in elementary group theory. Each of the experimental "classrooms" consisted of two university students and the teacher/ researcher. These teaching experiments were guided by the instructional design theory of Realistic Mathematics Education (RME). The goal was to promote the guided reinvention of the concepts of group and group isomorphism. The local instruction theory consists of a sequence of instructional activities and a justification (theoretical and empirical) for the sequence. The concepts of group and group isomorphism are seen as first emerging as models of students’ informal mathematical activity and then evolving into models for more formal mathematical reasoning. This process was driven by the students’ participation in classroom mathematical practices. The individual students’ mathematical development is seen as reflexively related to their participation in these mathematical practices.

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Dissertation Project: Sally Jacobs C & I TITLE: Advanced Placement BC Calculus Students’ Ways of Thinking About Variable ABSTRACT: The dissertation was an exploratory study investigating calculus students’ notions about the concept of variable in the contexts of function, limit and derivative. The findings indicate that students think about variable in qualitatively different ways, depending on whether they have a calculational versus conceptual orientation to the mathematical task at hand. Also, in the context of limit and derivative, their conceptions of variable are somewhat less flexible. Finally, a Variable Conceptions Framework for analyzing student conceptions of variable is proposed.

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