Making sense of students’ talk and action for teaching mathematics

1. Making sense of students’ talk and action for teaching mathematics Ruhama Even Weizmann Institute of Science Israel ICME-11,Monterrey, Mexico

2. Difficulties in attending to, and making sense of, students’ talk and action 4 episodes Even, 2005; Even & Wallach, 2004; Even & Tirosh, 2002; Tirosh, Even, & Robinson,1998

3. Episode 1 Having a plan

4. The plan • To motivate learning to simplify algebraic expressions, • by experiencing substitutions in complicated and simple equivalent algebraic expressions. • Example: Substituting a = 1 2 in 4a+3 and in3a+6+5a2

5. Executing the plan The teacher writes on the board: 4a+3 , 3a+6+5a 2 and asks the students to substitute a = 1 2 forgetting to state that the two expressions are equivalent.

6. 4a+3 , 3a+6+5a2 T: Substitute a = 1/2 S1: You get the same result. T: Are the algebraic expressions equivalent? S2: No, because we substituted only one number. S1: Yes. S3: It is impossible to know. We need all the numbers. S4: One example is not enough.

7. The teacher’s conclusion T: We can conclude – it is difficult to substitute numbers in a complicated expression and therefore we should find a simpler equivalent expression.

8. Having a plan • “I prepare my objective and the exercises I want to give the students, and it is very confusing for me when they suddenly ask something not according to my planning.” • Listening forsomething rather than tothe students’ discussion.

9. Episode 2: Lacking knowledge about students’ ways of learning mathematics

10. Simplifying algebraic expressions • 10 + 2b = 12b • 5t + 3t + t + 2 = 11t • 3m + 2 + 2m = 5m + 2 = 7m • 3 + 4x = 7x = 7

11. Research-based explanations • Conventions in natural language. "ab" is read as "a and b" and interpreted as "a+b". • Previous learning from other areas that do not differentiate between conjoining and adding. In chemistry adding oxygen to carbon produces CO2.

12. Research-based explanations • Previous learning in mathematics: The ‘behavior’ of algebraic expressions is expected to be similar to that of arithmetic expressions. Students expect a final, single-termed answer or interpret symbols such as ‘+’ only in terms of actions to be performed.

13. Research-based explanations • The dual nature of mathematical notations: process and object. 5x + 8 stands both for the process ‘add five times x and eight’ and for an object that can be manipulated.

14. Teaching how to simplify algebraic expressions The teacher writes on the board: 3m+2+2m

15. 3m+2+2m = T: What does this equal to? Add the numbers separately and add the letters separately. Let us color the numbers [3m+2+2m]. We get 5m+2.

16. 5m+2 = S1 : And what now? S2 : 7m.

17. 5m+2 = 7m ? T: [Rather surprised] No! 5m+2 does not equal 7m. The rule is - ‘Add the numbers separately and add the letters separately’.

18. 4a+5-2a+7 = T: Here is another example: 4a+5-2a+7. We color the numbers [4a+5-2a+7]. What do we get? 2a+12.

19. The ‘rule’ T: Let us write the rule [dictates]: In an expression in which both numbers and letters appear, we add the numbers separately and add the letters separately. Repeat out loud. S’s: [Repeat the rule out loud.]

20. 6x+2+3x+5 = T: Let’s take another example: 6x+2+3x+5 = We add according to the rule and get 9x+7.

21. 3+2b+7 = Working on the expression: 3+2b+7 … S1:12b. T: No! S2: I got 10+2b. Why isn’t it 12b?

22. Lacking knowledge about students’ ways of learning mathematics • The teacher explained that he sensed there was a problem but he did not understand its sources; he did not understand what his students' difficulties were. • The teacher was not aware of his students’ tendency to conjoin or "finish" algebraic expressions.

23. Episode 3 Not valuing students’ ways of thinking

24. 5th grade quiz 3/5 of a number is 12. Calculate the number. Explain your solution. • The teacher’s expectation:

25. 3/5 of a number is 12. Calculate the number. Explain your solution. Ron’s solution: 12 * 2 = 24 24 : 6 = 4 24 – 4 = 20

26. Teacher’s assessment • “He reached a correct answer but I didn’t understand what he did. It didn’t seem right.” • “Ron is an average-good student, who usually has difficulties with homework.” • Ron’s solution is wrong.

27. Solution 12 * 2 = 24 24 : 6 = 4 24 – 4 = 20 Explanation If 3/5 is 12 then 24 is 6/5. The value of 1/5 is: 24:6 = 4. The number is: 24 – 4 = 20. Ron’s explanation

28. Not valuing students’ ways of thinking • The teacher did not believe that there was something to understand. • She was not tuned to understand Ron.

29. Episode 4 Having a specific mathematics solution in mind

30. 4th grade problem The following task does not have a solution: Divide 15 players into two teams, so that in one team there are 4 players less than in the other team. Change the number of players, so that there will be a solution.

31. Students’ solution (video-taped) Here are 7 players and here are 3 so 10 players; 7 and 3 makes 10. And 7 minus 3 is 4. So 10 players.

32. Teacher’s interpretation • The solution just came out of the blue. • She just said 10 off the top of her head.

33. The teacher’s own solution • Changing the number of players to 14, using the strategy of removing a minimal number of players to reach an even number of players. The girls’ strategy: • Buildingup two groups that satisfy the requirement.

34. Hearing through… own solution T: I was really surprised that they changed to 10, [that they] removed 5 shirts. Remove one [pause]… I don’t know, it seemed to me that you need to removeone and try.

35. The 4 episodes: Hearing students’ talk and action “through”... • the teacher’s plan for the lesson, • the limited knowledge about the nature and possible sources of students’ tendency to “finish” algebraic expressions, • the teacher’s low expectation of a specific student, • the teacher’s own way of solving the mathematics problem she presented to her students.

36. In general: Hearing students through... • own knowledge of mathematics, • beliefs about mathematics learning and knowing, • understandings of mathematics teaching, • dispositions toward the teacher’s role, • feelings about students, • expectations from students, • the context in which the hearing takes place, • ...

37. Attending to and understanding what students are saying and doing • is problematic

38. Attention to and understanding what students are saying and doing • is not necessarily associated with what teachers do notdo: • do not listen to students, • do not change their plans, • do not know about students’ learning processes, • do not understand the mathematics, • …

39. Attention to and understanding what students are saying and doing • is also associated with, and bounded by, what teachers do: • make lesson plans, • work out the mathematics, • anticipate students’ answers, • assess their students’ learning, • …

40. Attention to and understanding what students are saying and doing • can be improved.

41. Improving attention to, and sense making of, students’ talk and action “Replicating” a research study Even, 1999, 2005

42. Multi-stage activity • academic knowledge • mini-study • “replication” of a research study • writing a reflective report • presentation to peers, and to other mathematics educators

43. Academic knowledge Reading and discussing research on student conceptions, classroom culture, and ways of learning: • Real numbers • Algebra • Analysis • Geometry • Probability and statistics

44. Academic knowledge • Participants were astonished to learn that students "are able to think that way”. • Developing appreciation of the idea that students construct their knowledge in ways that are not necessarily identical to the instruction. • Conceptualizing and making explicit naive and implicit knowledge.

45. Mini-study “Replicating” a study and comparing the findings with the findings of the original study.

46. Mini-study Two kinds of benefit • Academic/theoretical - Better understanding of theoretical issues (e.g., constructivism). • Practical - Better understanding of real students.

47. Academic/theoretical benefit • “When you read a research article, it is one level of depth. When you have to re-do it and implement it again, it is another level. I mean, what I know now about the study, about its hypothesis, its findings, and the theoretical material, I certainly wouldn't have known after reading it once or twice or even if I had summarized it - it is much more. It became mine.”

48. Practical benefit • “In a mini-research, in contrast to an article which is completely theoretical, you have question marks about the findings. Could it be like this? Is it only a coincidence that this happened? Will it happen to students I know? My students? It is very interesting to see what really happens. To duplicate the study and see, to support the original findings or refute them…”

49. Students could do more than expected “Even though I have worked for 30 years as a teacher, I was surprised by some of the things that we found in the group of students we studied. The students reached much higher levels of thinking than what I would have given them credit for. So it was very interesting.”

50. Students could do less than expected “Simply, I was amazed by the results. I said, well, this is a topic [irrational numbers] that we deal with in grade 9. It was several months after we had taught the material. And I said, OK, no problem. Our students, for sure, would know better than those students at the university. And we were shocked that actually with us it was the same as there.”