Supporting Productive Whole Class Discussions

1. Supporting Productive Whole Class Discussions Paul Cobb Vanderbilt University

2. Overview • Classroom social and sociomathematical norms • The norm of what counts as an acceptable mathematical argument • Calculational and conceptual discourse • Reflective shifts in classroom discourse

3. Organization of Classroom Activities • Initial discussion during which the teacher introduces instructional activities • Students’ work on the instructional activities • Concluding whole class discussion of students’ solutions and interpretations

4. Organization of Classroom Activities • Teacher’s goal: To achieve a mathematical agenda by building on students’ contributions

5. The Swing of the Pendulum • Student-centered approaches • Celebrate students’ discoveries and methods as ends in themselves • Teacher-centered approaches • Focus on conveying mathematical ideas and procedures to students

6. Transcending This Dichotomy • Keep one eye on the mathematical horizon and the other on students’ current understandings, concerns, and interests (Deborah Ball, 1993)

7. Making Sense of Classrooms • What do students have to know and do to be effective? • What obligations do they have to fulfill?

8. Making Sense of Classrooms • It is just a class. Most classes teach then they give you class work then homework. She [the teacher] goes over the homework. Then she goes over new stuff. Then we start on homework. And then it is time to go.

9. Classroom Norms • Classroom social norms -- general classroom obligations • Explain and justify solutions • Attempt to make sense of explanations given by others • Indicate understanding and non-understanding • Ask clarifying questions • Question alternatives when conflicts in interpretations have become apparent

10. Classroom Norms • Sociomathematical norms -- specifically mathematical obligations • What counts as a different mathematical solution • What counts as an efficient mathematical solution • What counts as a sophisticated mathematical solution • What counts as an acceptable mathematical explanation

11. Classroom Norms • You talk about your way, or you add something to someone else's way. You can't just say that you agree or you disagree. Mrs. M [the teacher] makes you explain it. You have to ask questions about things that you don't understand. • You have to do a good job explaining how you looked at the problem. That's important since you didn't talk with everybody else when you were doing the problem.

12. Establishing Classroom Norms • Scaffolding and holding students accountable • Indicate understanding and non-understanding • Ask clarifying questions

13. Equity In Students’ Access To Significant Mathematical ideas • All students are able to participate substantially in classroom activities • All students see reason and purpose to engage in classroom activities • Students view classroom activities as worthy of their engagement

14. Equity In Students’ Access To Significant Mathematical ideas • Differing norms of participation, language, and communication • Potential conflicts with the norms that the teacher seeks to establish in the classroom • Explicit negotiation of classroom norms is a critical aspect of equitable instructional practice

15. What Counts as an Acceptable Mathematical Explanation • Calculational explanation • Explain the process of arriving at a result or answer • Conceptual explanation • Also explain the reasons for this process process

16. Calculational and Conceptual Discourse • Chris and Juan have 12 candies altogether. Chris has 8 candies. How many candies does Juan have?

17. Calculational and Conceptual Discourse • A directory of 62 pages has 45 names per page. How many names are in the directory?

18. Calculational and Conceptual Discourse • Initial activities for linear measurement • Counting the first step

19. Calculational and Conceptual Discourse

20. Calculational and Conceptual Discourse

21. Calculational and Conceptual Discourse • Casey: And I was saying, see like there’s seven green that last longer. • Teacher: OK, the greens are the Always Ready, so let’s make sure we keep up with which set is which, OK. • Casey: OK, the Always Ready are more consistent with the seven right there, and then seven of the Tough ones are like further back, I just saying ‘cause like seven out of ten of the greens were the longest, and like ...

22. Calculational and Conceptual Discourse • Ken: Good point. • Janice: I understand. • Teacher: You understand? OK Janice, I’m not sure I do, so could you say it for me? • Janice: She’s saying that out of ten of the batteries that lasted the longest, seven of them are green, and that’s the most number, so the Always Ready batteries are better because more of those batteries lasted longer.

23. Calculational and Conceptual Discourse

24. Calculational and Conceptual Discourse • Teacher: So maybe, Casey, you can explain to us why you chose 10, that would be really helpful. • Casey: Alright, because there’s ten of the Always Ready and there’s ten of the Tough Cell, there’s 20, and half of 20 is ten. • Teacher: And why would it be helpful for us to know about the top ten, why did you choose that, why did you choose ten instead of twelve? • Casey: Because I was trying to go with the half.

25. Calculational and Conceptual Discourse

26. Calculational and Conceptual Discourse • Brad: See, there’s still green ones [Always Ready] behind 80, but all of the Tough Cell is above 80. I would rather have a consistent battery that I know will get me over 80 hours than one that you just try to guess. • Teacher: Why were you picking 80? • Brad: Because most of the Tough Cell batteries are all over 80.

27. Calculational and Conceptual Discourse • Jennifer: Even though seven of the ten longest lasting batteries are Always Ready ones, the two lowest are also Always Ready and if you were using those batteries for something important then you might end up with one of those bad batteries. • Barry: The other thing is that I think you also need to know something about that or whatever you’re using them [the batteries] for. • Teacher: You bet.

28. Calculational and Conceptual Discourse • S: I knew what they [the other students] did so I didn't want them to tell me what they were doing, but what were they thinking, yeah, what was your intention.

29. Calculational and Conceptual Discourse • S: You can't just talk about your conclusion because that doesn't let anybody know why you did things. • I: Is that important? • S: If you don't talk about what you were thinking about then we don't know if it all is okay … we can't figure out if it is a good point.

30. Calculational and Conceptual Discourse • Gives students’ access to each others’ thinking • Supports situation-specific imagery that facilitates problem solving • Brings significant mathematical ideas to the fore as a focus of discussion

31. Reflective Shifts in Discourse • Reflection viewed as a critical aspect of mathematical learning • Learning as problem solving-- students reflect when they experience perturbations • Teacher’s role limited to posing tasks that are genuinely problematic for students and that give rise to perturbations

32. Reflective Shifts in Discourse • What is said and done in action subsequently becomes an explicit focus of discussion and analysis

33. Reflective Shifts in Discourse • First-grade classroom • Instructional intent: Flexible partitioning of small quantities (e.g., five as four and one, three and two, etc.) • Instructional activity: Five monkeys in two trees

34. Reflective Shifts in Discourse

35. Reflective Shifts in Discourse • Anna: I think that three could be in the little tree and two could be in the big tree. • Teacher: OK, three could be in the little tree, two could be in the big tree [writes 3|2 between the trees]. So, still 3 and 2 but they are in different trees this time; three in the little one and two in the big one. Linda, you have another way? • Linda: Five could be in the big one.

36. Reflective Shifts in Discourse • Teacher: OK, five could be in the big one [writes 5] and then how many would be in the little one? • Linda: Zero. • Teacher: [Writes 0]. Another way? Another way Jan? • Jan: Four could be in the little tree, one in the big tree.

37. Reflective Shifts in Discourse

38. Reflective Shifts in Discourse • Teacher: Are there more ways? Elizabeth. • Elizabeth: I don’t think there are more ways. • Teacher: You don’t think so? Why not? • Elizabeth: Because all the ways that they can be.

39. Reflective Shifts in Discourse • Initial focus of classroom discourse: Generating possible ways the monkeys could be in the two trees • First reflective shift: Determining whether there are more possibilities by checking the table empirically

40. Reflective Shifts in Discourse • Second (teacher initiated) shift: • Teacher: Is there a way that we could be sure and know that we’ve gotten all the ways? • Demonstrating that there were no further possibilities by identifying patterns in the table

41. Reflective Shifts in Discourse • Jordan: See, if you had four in this [big] tree and one in this [small] tree in here, and one in this [big] tree and four in this [small] tree, couldn’t be that no more. If you had five in this [big] tree and none in this[small] tree you could do one more. But you’ve already got it right here [points to 5|0]. And if you get two And if you get two in this [small] tree and three in that [big] tree, but you can’t do that because three in this [small] one and two in that [big] one—there is no more ways, I guess.

42. Reflective Shifts in Discourse • Teacher: What Jordan said is that you can look at the numbers and there are only a certain…there are only certain ways you can make five. • Mark: I know if you already had two up there and then both ways, you cannot do it no more.

43. Reflective Shifts in Discourse • Initially, the teacher and children generated possible ways in which the monkeys could be in two trees • Subsequently, the teacher and children collectively constituted the (symbolically recorded) results of prior activity as an explicit focus of discussion and discerned structural patterns

44. Reflective Shifts in Discourse • The identification of patterns is central to mathematical learning • “How can we know for sure” questions are natural precursors to mathematical proof • Students learn to ask “mathematical” questions

45. Reflective Shifts in Discourse • S: Now we know the terms like mean, median, range, and when you would want to use those terms. • I: You said you know what average is. How is it different from before? Average is something you talked about before the class in other classes? • S: I learned when to use it to describe the data.

46. Reflective Shifts in Discourse • I: When should you use it? • S:Most of the time I don't use the average. I like using the range. I use the range when the points are spread out. If the points are around in a really small area you probably want to use the median since that would be a better way to let someone know about the points.

47. Reflective Shifts in Discourse • Mathematical ideas as tools • Discussing particular mathematical ideas while explaining specific solutions • Discussing when particular mathematical ideas prove to be useful • Identify patterns in (collective) activity of using ideas

48. Reflective Shifts in Discourse • Students do not just happen to spontaneous reflect at the same time • Opportunities for the students to reflect on and identify patterns activity arise as they participate in and contribute to the reflective shifts in discourse • Central role for the teacher in supporting reflection on collective as well as individual activity • Wisdom and judgment

49. Agency Revisited • Disciplinary agency:Involves applying established mathematical methods • Conceptual agency:Involves choosing mathematical methods and developing meanings and relations between concepts and principles

50. Equity in Students’ Access to Significant Mathematical Ideas • All students are able to participate substantially in classroom activities • All students see reason and purpose to engage in classroom activities • Students view classroom activities as worthy of their engagement