Mathematizing and Identifying: A Study of Fourth Grade Students Learning Mathematics

1. Mathematizing and Identifying:A Study of Fourth Grade StudentsLearning Mathematics CEMELA Seminar September 15, 2008 Marcy B. Wood

2. A Central Problem of Mathematics Education Research: How to account for mathematical learning and differences in mathematical learning?

3. Progression of Lenses Acquisitionist Participationist / Sociocultural

4. Progression of Lenses Participationist / Sociocultural • Account for • But what participation to focus on?

5. Narrowing Participation 2Lenses Lens #1: Commognition (Sfard 2008) Thinking = Communication Study discourse Activity = Mathematical Discourse (Mathematizing) Outcome of learning = Change in discourse

6. Participation in theCommognitive Framework The outcome of productive learning is a change in mathematizing

7. Participation in theCommognitive Framework But who students think they are seems to affect mathematizing…

8. Narrowing Participation 2Lenses Lens # 2: Identity Ex: Boaler & Greeno 2000, Jilk 2007, Martin 2000, Nasir 2002, Sfard & Prusak 2005 Definition: Significant, endorsable, reified narrative (Sfard & Prusak 2005) Arising from interactions: Positioning Theory (van Langenhove & Harré 1999, Harré & van Langenhove 1999) Direct, indirect verbal, enacted (Sfard 2007)

9. Narrowing Participation 2Lenses Lens # 2: Identity Ex: Boaler & Greeno 2000, Jilk 2007, Martin 2000, Nasir 2002, Sfard & Prusak 2005 Definition: Significant, endorsable, reified narrative (Sfard & Prusak 2005) Arising from interactions: Positioning Theory (van Langenhove & Harré 1999, Harré & van Langenhove 1999) Direct, indirect verbal, enacted (Sfard 2007)

10. Definition of Identity Narrative = an account of life events (Ochs & Capps, 2001) Significant = any change affects storyteller’s feelings about the identified Endorsable = “faithfully reflects the state of affairs” Reified = be, have, can, always, never, usually On many occasions when Josh answered math questions, his answer was correct and appropriate. Josh is smart at math. Josh is a talented math student. (Sfard & Prusak 2005, Sfard 2007)

11. Narrowing Participation 2Lenses Lens # 2: Identity Ex: Boaler & Greeno 2000, Jilk 2007, Martin 2000, Nasir 2002, Sfard & Prusak 2005 Definition: Significant, endorsable, reified narrative (Sfard & Prusak 2005) Arising from interactions: Positioning Theory (van Langenhove & Harré 1999, Harré & van Langenhove 1999) Direct, indirect verbal, enacted (Sfard 2007)

12. Identity The activity of identifying is Turning statements about activity into statements about a person (Sfard 2007)

13. Identity The activity of identifying is Turning statements about activity into statements about a person (Sfard 2007)

14. Narrowing Participation 2Lenses Lens # 2: Identity Ex: Boaler & Greeno 2000, Jilk 2007, Martin 2000, Nasir 2002, Sfard & Prusak 2005 Definition: Significant, endorsable, reified narrative (Sfard & Prusak 2005) Arising from interactions: Positioning Theory (van Langenhove & Harré 1999, Harré & van Langenhove 1999) Direct, indirect verbal, enacted (Sfard 2007)

15. Narrowing Participation The outcome of productive learning is a change in mathematizing and identifying

16. Narrowing Participation The outcome of productive learning is a change in mathematizing and identifying

17. Narrowing Participation Noticed patterns or constellations of mathematizing and identifying activity = KINDS OF LEARNING

18. Research Question Initial Question: How to account for mathematical learning and differences in mathematical learning? Research Question How do the activities of mathematizingand identifyingconnect to the development of mathematical discourse?

19. Research Question What kinds of learning(constellation of mathematizing and identifying activity) are present and how do they connect to the development of mathematical discourse? Initial Question: How to account for mathematical learning and differences in mathematical learning? Research Question How do the activities of mathematizing and identifying connect to the development of mathematical discourse?

20. Methods • Setting • 4th grade classroom • Midwest school, midsize city • Diverse SES, race/ethnicity • 20 students total, 4 focal students • Data • Mathematics lessons (3 units, 34 lessons, 70 hours of video) • Student work • Interviews, conversations with teacher

21. Methods • Task • As a group, find which of the rugs covers more area or if they cover the same amount. • Groups: • Jakeel, Rebecca, Daren • Minerva, Jessica, Bonita • Analysis • change in discourse, mathematizing, identifying

22. Findings Focus on one learner: Jakeel • Initial discourse did not address triangular spaces • Change to more mathematically desirable discourse • Shift in kind of learning during the lesson, tied to changes in mathematical discourse

23. Initial Discourse

24. Initial Discourse

27. Final Discourse Jakeel: One. This word is elongated as Jakeel simultaneously puts his index and middle fingers on two triangles. Two, three, four, five, six, seven He counts each square, pointing with his index finger to each square. Eight He puts index and middle fingers on the last two triangles.

28. Final Discourse Jakeel: One. This word is elongated as Jakeel simultaneously puts his index and middle fingers on two triangles. Two, three, four, five, six, seven He counts each square, pointing with his index finger to each square. Eight He puts index and middle fingers on the last two triangles.

29. Kind of Learning:Engaged Learning

32. Engaged Learning

33. Engaged Learning

37. Engaged Learner • Mathematizing: • Interact • Contribute • Make own sense • Adopt others’ discourse • Produce own discourse • Substantiate ideas yourself • Use others to support work toward understanding

38. Engaged Learner • Mathematizing: • Interact • Contribute • Make own sense • Adopt others’ discourse • Produce own discourse • Substantiate ideas yourself • Use others to support work toward understanding Identifying

39. Engaged Learner • Mathematizing: • Interact • Contribute • Make own sense • Adopt others’ discourse • Produce own discourse • Substantiate ideas yourself • Use others to support work toward understanding Identifying Do own work Discourse for self Part of group

40. Engaged  Directed Jakeel’s work on H and I

41. Engaged  Directed

42. Directed Learning

43. Directed Learning

44. Directed Learning • Mathematizing • Limited • Focus on physical activity • Not explanations or mathematical concepts • No adoption, production • Substantiation entirely others Identifying Passive Reactive Others are directors Discourse for others

45. Directed  Engaged

47. Final Discourse Jakeel: One. This word is elongated as Jakeel simultaneously puts his index and middle fingers on two triangles. Two, three, four, five, six, seven He counts each square, pointing with his index finger to each square. Eight He puts index and middle fingers on the last two triangles.

48. Summary • Jakeel enacts engaged learning • Exploring and talking about mathematical ideas • Identifying as capable of learning and understanding • Transitions to directed learning • Rebecca directs physical activity • Limited mathematizing • Identifying as needing to be told what to do • Back to engaged learning • Makes own sense of counting Figure J • Change in discourse (Learning Outcome!)

49. Discussion What kinds of learning are present and how do they connect to the development of mathematical discourse? • Engaged and Directed Learning • Differences in mathematizing and identifying connect to differences in mathematically desirable learning outcomes.

50. Implications • This close focus on participation • highlights students’existing activities • demonstrates impact of interaction (student and teacher)on learning • Reform mathematics • Desirable learning can happen in groups • Need to emphasize engagement with mathematical ideas