Principal Investigators: Carol E. Malloy, Ph.D. cmalloy@email.unc Jill V. Hamm, Ph.D.

1. Mathematical IDentity Development & LEarning l l l l l l l l l l l l l l l l MIDDLE l l l l l l l l l l l l l l l l l l l l l l l l l Principal Investigators: Carol E. Malloy, Ph.D. cmalloy@email.unc.edu Jill V. Hamm, Ph.D. jhamm@email.unc.edu Judith L. Meece, Ph.D. meece@email.unc.edu University of North Carolina-Chapel Hill NSF Grant REC 0125868 Carol E. Malloy Milwaukee Mathematics Partnership August 27, 2007

2. Purpose • to better understand how mathematics reform affects students’ development as mathematics knowers and learners • to identify the processes that explain changes in students’ mathematical learning and self-conceptions

3. Framework for Looking at Reform • Reform — Teacher's use of instructional practices and curricular materials that are aligned with NCTM’s Curriculum and Teaching Standards (1989, 1991) and the Principles and Standards for School Mathematics (2000). • Use Carpenter and Lehrer (1999) model to examine how students are given opportunities to develop conceptual understanding of mathematics.

4. Assumption Reform instructional practices in mathematics education can help all student progress in their understanding and use of mathematics in their future careers.

5. Reform Instruction • Pedagogy • Content • Tasks • Mathematical interactions • Assessment

6. Looking at Instruction Pedagogy Pedagogy is seen in how a teacher's plans for and the resulting flow of the lesson including how students are given opportunities to learn. This includes the discourse that the teacher pursues in the lessons and the tools she uses.

7. Content Content includes the objectives of lesson including where the student is being led and allowed to advance and the subject matter, both procedural and conceptual, that students will gain.

8. Tasks Tasks represent the mathematical work that students are engaged in during class and the opportunity students have to internalize the work they do. Of particular interest are characteristics of classrooms and instruction that maintain high-level cognitive demands or produce a decline of high-level cognitive demands.

9. Mathematical Interaction Mathematical interaction is the mathematical conversations or discourse that results from the instruction planned and modified by the teacher and initiated by students.

10. Assessment Assessment includes the ways that the teacher determined what students had learned, specifically, evidence of student performance, the relation of student understanding to content being taught, feedback to students, and student involvement in critique.

11. Looking at Learning A class has 28 students. The ratio of girls to boys is 4 to 3. How many girls are in the class? Explain why you think your answer is correct. Concepts Assessed • Understand and apply proportional reasoning used in scaling. • Understand that a fraction always represents a part-to-whole relationship. • Understand that a ratio can represent part-to-part or part-to-whole relationships.

12. Student Responses • There are 12 girls. I used the ratio and then added them up. (Shows columns of four 4s and three 3s adding up to 18 and 12, respectively.) • 16. I got lazy and actually counted out 4,3,4,3, etc.

13. 3. 16. I set up a ratio and porportion to find the answer. I think it is correct because there should be more than half the class girls. 4. 4/7 = ?/28, 28 x 4 = 112, 112 /7 = 16 MISSING 5. There are 16 girls. I figured this out because I knew that 16/12 was the same as 4/3 and 16 + 12 gave me 28.

14. 6. There are 16 girls. I used guess and check. Students wrote in space below not on the same line: 4/3 16/12 28/4 = 7 7 boys 12 • I guess I divide 4 into 28 and the answer is the answer to the problem. 7 girls. 8. 4 X 4 = 16 , 16 + 16 = 32 girls

15. We investigated 946 students’ conceptual understanding in 44 classrooms. Teacher instructional practice was observed using pedagogy, content, tasks, assessment, and interaction. What do you think we found?

16. Differences in instruction makes a difference in what students learn.

18. Specifically, • teachers at different reform levels have subtle and substantive differences in teaching practices, • students in classrooms with the highest level of reform practice scored significantly higher on conceptual understanding, and • conceptual understanding scores correlated with End of Grade scores.

19. Questions • What do you think we should do having this knowledge? • What are small changes that can be made? • What are major changes that can be made? • How do we begin?

20. What Do Teachers Need to Know? How to help students • connect knowledge they already have • construct coherent structure for knowledge they are learning • engaging students in inquiry and problem solving • take responsibility for validating their ideas and procedures NCISLA. (2004).

21. What This Requires Teachers to Have A coherent vision of the • structure of mathematical ideas and practice they are teaching • conceptions, misconceptions, and problem-solving strategies that bring and their probable struggles • learning trajectories students are likely to follow • tasks and tools that will provide knowledge about and support student learning • scaffolding to support students to engage in sense making • class norms and activity structures that support learning

22. What Teachers Must Learn to Do? • Acknowledge and use individual student preferences in the acquisition of knowledge • Develop activities and questioning to promote mathematical discourse among students and teacher • Value student discourse and verbal knowledge • Encourage, support, and provide feedback to students as they learn

23. References • Carpenter, T. P., & Lehrer, R. (1999). Teaching and learning mathematics with understanding. In E. Fennema & T. A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 19-32). Mahwah, NJ: LEA. • Cobb, P., Wood, T. Yackel, E., & McNeal, B. (1992). Characteristics of classroom mathematics traditions: An interactional analysis. American Education Research Journal, 29, 573-604. • NCTM (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM. • NCTM (1991). Professional standards for teaching mathematics. Reston, VA: NCTM. • NCTM (2000). Principles and standards for school mathematics. Reston, VA: NCTM. • Piburn, M. & Sawada, D (2001). Reformed teaching observation protocol (RTOP) reference manual (ACEPT Technical Report IN00-3). Tempe, AZ: Arizona Collaborative for Excellence in Preparation of Teachers.