STUDENTS BUILDING MATHEMATICAL CONNECTIONS THROUGH COMMUNICATION

1. STUDENTS BUILDING MATHEMATICAL CONNECTIONS THROUGH COMMUNICATION Elizabeth B. Uptegrove uptegrovee@felician.edu Carolyn A. Maher cmaher@rci.rutgers.edu Rutgers University Graduate School of Education

2. Research Question

3. Theoretical Framework • Maher (1998) • Communicating their ideas helps students to develop and consolidate mathematical thinking • Justifying their thinking helps students develop mathematical reasoning skills • Sfard (2001) • Students learn to think mathematically by participating in discourse about ideas – arguing, asking questions, and anticipating feedback

4. Background • Longitudinal study (Maher, 2005) • Public school students were followed from first grade through college • After-school problem-investigation session involving four students during the sophomore year of high school (March 1998)

5. Data Sources and Analysis • Data • Videotapes (two cameras) • Student work • Analysis • Transcripts verified and reviewed • Events selected for analysis • Student work on Pascal’s Triangle • Mathematical ideas described and analyzed

6. Problems Investigated • Pizza: How many pizzas is it possible to make when there are n toppings to choose from? • Towers: How many towers n cubes tall can be built when choosing from white and blue cubes?

7. Students’ Earlier Findings • The answer to both problems is 2n • The answers to the towers problem can be enumerated by the numbers in row n of Pascal’s Triangle • C(n,r) gives the number of towers with exactly r blue cubes when selecting from blue and white cubes • They are not sure how the pizza problem fits with Pascal’s Triangle

8. Results • First, students communicated their ideas about relationships: • Between towers and Pascal’s Triangle and the binomial expansion • Between pizza problem and Pascal’s Triangle • Students provided support for these ideas • They answered questions • They responded to arguments • Then they described how the towers and pizza problems are related to each other

9. Episode 1: Towers • Students represented (a+b)2 by towers two cubes tall • Students related those towers to row 2 of Pascal’s Triangle

10. Episode 1: Towers

11. Episode 1: Towers(continued) • Building towers can be related to expanding the binomial (a+b) • Adding a blue cube is like multiplying by a • Adding a white cube is like multiplying by b

12. Episode 2: Pizzas

13. Episode 2: Pizzas • Students try to explain how the different two-topping pizzas can be found in row 2 of Pascal’s Triangle • Row 2 is: 1 2 1 1 = plain pizza 2 = one pepperoni, one pepper 1 = both toppings • But students do not identify this relationship; instead Ankur proposes: • 1 = pepperoni • 2 = both (counted twice) • 1 = peppers • ? = plain

14. Episode 3: Connecting Towers & Pizzas

15. Episode 3: Connecting Towers & Pizzas • Michael’s insight • Blue cube = topping present • White cube = topping not present • The others built on this insight

16. Analysis UsingConversational Turns • Conversational turn units are “tied sequences of utterances that constitute speakers’ turn at talk and at holding the floor” (Powell, 2003) • Used to measure students’ participation in discourse • Coded for types of discourse • Asking questions or expressing uncertainty • Answering questions • Disagreeing • Making connections • Expressing understanding

17. Students’ Participationin Discourse

18. Types of Discourse inConversational Turns

19. Summary • Episodes 1 and 2 • Students communicated their ideas about Pascal’s Triangle and its relationship to the towers and pizza problems • Students supported their ideas through demonstrations and by offering specific examples • Episode 3 • Students built on Michael’s critical insight • Students described a new connection: the isomorphism between the towers and pizza problems

20. Conclusions • The framework for developing a new connection was provided by earlier discussions about Pascal’s Triangle • Communicating about and supporting those earlier ideas helped students make this connection between two problems of equivalent structure

21. Questions?

25. Pascal’s Triangle

26. Pascal’s Triangle

27. Pascal’s Triangle