A VIEW ON THE TEACHERS’ OPORTUNITIES TO LEARN MATHEMATICS THROUGH TEACHING

1. A VIEW ON THE TEACHERS’ OPORTUNITIES TO LEARN MATHEMATICS THROUGH TEACHING Roza Leikin Rina Zazkis University of Haifa, Israel Simon Fraser University, Canada PME-31, July 2007, Seoul Research Forum: Learning through Teaching

2. BACKGROUND • Teachers learn through their teaching experiences(e.g., Cobb and McClain, 2001; Kennedy, 2002; Lampert & Ball, 1999; Lesh & Kelly, 1994; Mason, 1998; Ma, 1999; Shulman, 1986; Wilson, Shulman, & Richert, 1987). • Teachers’ expertise is a function of their experience(e.g., Wilson, Shulman, and Richert, 1987; Berliner, 1987; Leinhardt, 1993). • The main source of teachers’ learning through teaching (LTT) is their interactions with students and learning materials (Artzt, 2002; Leikin 2005, 2006; Steinbring, 1998; Simon, 1997)and reflection in and on action (Schön, 1993; Steinbring, 1998). Roza Leikin & Rina Zazkis, LTT

3. Interactionswith students andreflection as the main sources of LTT Roza Leikin & Rina Zazkis, LTT

4. Three dimensions of teachers’ knowledge Roza Leikin & Rina Zazkis, LTT

5. Open questions related to LTT • What exactly is being learned? • What changes in teachers’ knowledge occur? • How development of teachers’ knowledge of mathematics and pedagogy relate to each other? • What mechanisms support those changes? • Why teacher do not always learn through teaching? Roza Leikin & Rina Zazkis, LTT

6. Learning from a student’s mistake • Lora, an instructor in a course for pre-service elementary school teachers, taught a lesson on elementary number theory. • Teacher: Is number 7 a divisor of K, where K= 3456 ? • Student: It will be, once you divide by it • Teacher: What do you mean, once you divide? Do you have to divide? • Student: When you go this [points to K] divided by 7 you have 7 as a divisor, this one the dividend, and what you get also has a name, like a product but not a product… Roza Leikin & Rina Zazkis, LTT

7. What Lora learned from the above interaction? • She learned that the term “divisor” is ambiguous and a distinction is essential between: divisor of a number as a relationship in a number-theoretic sense divisor in a number sentence, as a role played in a division situation • She learned that the student assigned the meaning based on his prior schooling and not on his recent classroom experience in which the definition for a divisor was given and usage illustrated. • Before this teaching incident Lora used the term properly in either case, but was not alert to a possible misinterpretation by learners. • The student’s confusion helped her -- make the distinction, -- increased her awareness of the polysemy of the term divisor and definitions that can be conflicting. Roza Leikin & Rina Zazkis, LTT

8. Learning from students solutions • Problem On the two sides of a triangle two squares are constructed. Prove that the median of the given triangle is half of the segment that connects two vertices of the squares (see the figure) Roza Leikin & Rina Zazkis, LTT

9. Two initial teachers’ solutions: Roza Leikin & Rina Zazkis, LTT

10. Students’ solution 1 - improved –Formulating new problems Roza Leikin & Rina Zazkis, LTT

11. One more students’ solution Roza Leikin & Rina Zazkis, LTT

12. Why did not I see this? Roza Leikin & Rina Zazkis, LTT

13. What Rachel learned from the lesson? • New solutions • New theorems • New questions • Students’ creativity • Possibilities of discussion of solution elegance Roza Leikin & Rina Zazkis, LTT

14. Interaction with participants • Have you ever learned from teaching? • What have you learned? • Was it mathematics? Roza Leikin & Rina Zazkis, LTT

15. Learning from students’ questions • If AD is a hypotenuse of an external angle CAF in a triangle ABC then • “What happens if AD is parallel to BC?” Roza Leikin & Rina Zazkis, LTT

16. IS THIS KNOWLEDGE NEW? IS THIS MATHEMATICS OR PEDAGOGY? • Teachers learn both mathematics and pedagogy when teaching. • Teachers’ pedagogical content knowledge developed when they become aware of students’ unpredicted difficulties. • Through analysing sources of the difficulties and misconceptions teachers appreciate better the structure of mathematical thought. • Example 1 is a case of developing such awareness: in order to help students adopt the meaning of the term implied in a given situation the teacher had to first clarify the disparity in different uses of ‘divisor’ for herself. Roza Leikin & Rina Zazkis, LTT

17. In other (less often) situations teachers clearly learn new mathematics • This mathematics serves them in the consequent lessons in their pedagogy. • Teachers' mathematical understanding allow them to develop further students' ideas • Teachers' openness and pedagogical knowledge and skills as related to their awareness of the importance of students' autonomy in classroom mathematical discourse allow them to be more open and attentive to students. • Finally we found that teacher with more profound mathematical understanding feel 'safer' and more open to allowing students to present their mathematical ideas and ask questions. Roza Leikin & Rina Zazkis, LTT

18. Development of new mathematical knowledge • When planning the lesson teachers clearly express their “need to know the material well enough” and their “need to predict students’ possible difficulties, answers, and questions”. • Through interaction with students teachers become aware of-- new (for them) solutions to known problems, -- new properties (theorems) of the mathematical objects,-- new questions that may be asked about mathematical objects in this way they develop new mathematical connections. Roza Leikin & Rina Zazkis, LTT

19. (some) MECHANISMS OF LTT Teacher-students interactions Attention and noticing Internalmotives of interactions Unforeseen unplanned Studentinitiated Roza Leikin & Rina Zazkis, LTT

20. WHAT CHANGES IN TEACHERS’ KNOWLEDGE of mathematics OCCURS THROUGH TEACHING? • From intuitions to formal knowledge and beliefs New pieces of information are collected Familiar ideas are refined Roza Leikin & Rina Zazkis, LTT

21. Teachers’ awareness of their learning • Teachers are not always aware that they learned through their teaching and sometimes they are hesitant to admit their learning. • When they are aware of learning they are not convinced that they learned mathematics. • Very often they report “I knew this but never thought about it”. However, we consider this “thinking about it” as an indication of learning when an instructional situation presents such opportunity. • In this case LTT occurs not only in acquiring new knowledge but also in transferring existing knowledge from teachers’ passive repertoire to an active one. However, clear criteria that indicate teachers’ learning of mathematics in LTT need further development and refinement. Roza Leikin & Rina Zazkis, LTT