2. Mathematical habits of mind and ways of thinking for prospective teachers Gail Burrill Michigan State University

3. Background Association of Mathematics Teacher Educators (AMTE) Preliminary Teacher Education Development Study (PTEDS) (NCES) Teacher Education Development Study (TEDS) (NCES) PROM/SE Michigan State Mathematics Science Partnership (NSF) Knowledge of Algebra for Teaching (KAT) (NSF) Teachers for a New Era (Carnegie Grant) Capstone Courses ( MSU)

4. Association of Mathematics Teacher Educators (AMTE) Goals for the Preservice Education of Prospective Secondary Teachers 1.Analyze and purposefully transform their beliefs and dispositions about what mathematics is and what it means to learn, do and teach mathematics

5. Conceptual Aspects Informing Teacher Preparation © 2006 Michigan State University, Center for the Study of CurriculumSupported by NSF Grant REC-0231886 PTEDS

6. X38: Algebra Knowledge Item © 2005 MSU P-TEDSSupported by NSF Grant REC-0231886

7. Doing mathematics In general, a mathematical approach involves defining a problem through conjecturing in an established mathematical area. Conjecturing may be supported by technology, by compelling ideas based on past work, computation, or pattern exploration. ( Teachers for a New Era, 2003)

8. Doing mathematics Mathematics involves representing a mathematical concept concretely; a single concept can have multiple representations. One of the beauties of mathematics as a whole is the interplay between various areas of the subject. A particular way of representing a problem may lead to an especially efficient or enlightening result. (Teachers for a New Era, 2003)

9. Capstone Course - MSU Senior Math Majors : Deepen understanding of the mathematics needed for teaching in secondary schools. Prepare students to 1.describe connections in mathematics; 2.figure things out on their own.

10. High school math from an advanced perspective Analyses of alternative definitions, language and approaches to mathematical ideas; Extensions and generalizations of familiar theorems; Discussions of historical contexts in which concepts arose and evolved; Applications of the mathematics in a variety of settings;

11. High school math from an advanced perspective Demonstrations of alternate ways of approaching problems, with and without technology; Discussions of relations between topics studied in this course and contemporary high school curricula.

12. Habits of mind: Mathematics should make sense Know that all mathematics is not equal Understand that mathematics can be done different ways Be willing to work hard on a challenging problem for a long time Recognize that mathematics is about how mathematical results are obtained not how they are presented

13. Mathematically confident enough to take risks Using technology Asking for explanations Doing problems with students Trying something new Celebrating mistakes

14. A habit of mind

15. Polya ’ s Ten Commandments Read faces of students Give students “ know how ”, attitudes of mind, habit of methodical work Let students guess before you tell them Suggest it; do not force it down their throats (Polya, 1965, p. 116)

16. Polya ’ s Ten Commandments Be interested in the subject Know the subject Know about ways of learning Let students learn guessing Let students learn proving Look at features of problems that suggest solution methods (Polya, 1965,p. 116)