1. C R E S S T / U C L A 1 Productive Collaborative Problem Solving: Noreen Webb and Ann Mastergeorge Responsibilities of Teachers and Students

2. C R E S S T / U C L A 2 Issues Guiding This Study The promise of collaborative learning environments— Students can learn by: ä Seeking and receiving help from others ä Co-constructing new knowledge, skills, understanding ä Explaining and justifying their positions ä Recognizing and resolving conflicts and disagreements Questions of this study: ä What processes help students make the transition from misconception to correct conception? ä What are the group dynamics that facilitate or hinder those processes?

3. C R E S S T / U C L A 3 Method ä Sample: Four 7 th -grade general mathematics classes worked in heterogeneous cooperative groups for four weeks ä Preparation for group work: (1) Inclusion activities (learning classmates’ interests, backgrounds); (2) Developing communication skills (norms for group behavior, social skills, helping behavior) ä Topic: Operations with decimals ä Sample problem: Find the cost of a 30-minute telephone call to prefix 771 (first minute costs $0.22; each additional minute costs $0.13) ä Data collected: (1) Audiotapes of all groups for class period on cost of telephone calls; (2) Pretest and posttest on general mathematics skills and cost of telephone calls

4. C R E S S T / U C L A 4 Coding of Group Work: Levels of Help Given or Received LevelDescriptionExample High 8Fully elaborated explanation that includes all relevant numbers numbers, explicitly stated numerical rule, and verbal labels for all numbers “29 cents, right? That’s for the first minute. And then each additional minute is 12 cents. And then you can put for 5 minutes, since the first minute is 19 cents, then the next 4 will be 12. I just plus that all together.”

5. C R E S S T / U C L A 5 7Explanation that includes all relevant numbers, explicitly stated numerical rule, and verbal labels for some numbers “And you talk eleven minutes on the phone. See, just take one, one minute away for the first minute, and then go 10 times 13.” 6Explanation that includes some relevant numbers, implied numerical rule, and minimal verbal labeling “It’s for the first minute it’s 22 cents, and then the additional is 13. So we took care of one, that means we need 29 more.” 5Explanation that includes some relevant numbers, an explicitly stated or implied numerical rule, but no verbal labeling “Because 30 subtract 1 is 29.”

6. C R E S S T / U C L A 6 Low 4Numerical expression or equation “13 times 29.” 3Numbers to write or copy “Put 13 on top, 29 on the bottom. Then you times it.” 2Answer to part or all of the problem “I got $3.77.” 1Non-content or non- informational response “Just do it the way she said.” 0No response Coding of Group Work: Levels of Help Given or Received

7. C R E S S T / U C L A 7 Indications of a Need for Help

8. C R E S S T / U C L A 8 Frequency of Help Received

9. C R E S S T / U C L A 9 Frequency of Help Received from Another Student Over Time

10. C R E S S T / U C L A 10 Frequency of Immediate Responses to Help Received

11. C R E S S T / U C L A 11 Significant Partial Correlations Between Behavior Variables and Posttest Performance

12. C R E S S T / U C L A 12 Posttest Performance at Each Level of Highest Help Received

13. C R E S S T / U C L A 13 Relationship Between Unassisted Problem Solving during Group Work and Posttest Performance

14. C R E S S T / U C L A 14 Posttest Performance by Level of Help Received (High/Low) and Unassisted Problem Solving

15. C R E S S T / U C L A 15 Questions Driving Microgenetic Qualitative Analyses Why were some students able to obtain high-level help while others were not? ä Student-level factors: ¥ Persistence in asking questions ¥ Nature of questions asked ä Group-level factors: ¥ Willingness to give help ¥ Perception of the task: task completion vs. conceptual understanding ä Why did some students who received high-level help go on to solve problems without assistance while others did not? ¥ Adequacy, completeness, and relevance of the help received ¥ Comprehension of the help received ¥ Persistence in seeking understandable and relevant help

16. C R E S S T / U C L A 16 Students Who Learned (13 of 48): ä Asked specific questions ä Received high-level help ä Changed their help-seeking strategies ä Persisted in asking specific questions until they understood ä Attempted subsequent problems on their own

17. C R E S S T / U C L A 17 Successful student who asked increasingly focused questions (Student 1) 1 I don’t get this. 2 4Ok, see, like. Ok. See, like, right here it says 771. So you look for 771. 22 minutes, 22 cents per minute. 3 ( ) I don’t know. Could you ( ) repeat that again? 4 4It says 22 minutes per minute. 2For the first minute, it’s 22 cents. 4 Yeah. And then, it’s 13 cents for each additional minute. So, 13 times, times 29. 1 I don’t know how to do it. (pause) Prefix. So you, look, 771 right here, and there is 30 minutes. So why do you… 2 4Ah. There is the first minute, 22 cents. Now multiply 13 cents times 29. Because 29 minutes are left from the first minute. 3 Well, it’s 30. 30, 30 minutes. But you are saying, do what? 4 Multiply 29 times 13 cents. 1 29? Why 29? This is 30. 4 Because they already got a minute. That’s the first minute. 5 1Thank you!

18. C R E S S T / U C L A 18 Students Who Didn’t Learn ä 1. Never received high-level help (11 of 48): u Asked for help before attempting to solve the problems u Asked general questions or generally declared confusion u Were given numbers and calculations to copy ä 2. Received high-level help but didn’t understand it (24 of 48): u Tended to ask general questions u Did not persist in seeking help u Accepted explanations given without asking further questions u Gave up trying to understand and resorted to copying answers, or never tried to understand

19. C R E S S T / U C L A 19 Unsuccessful student who did not receive high-level help (Student 2) Problem 6: 2I don’t know where the heck we are. 1OK, look. Just copy it. Number 2, and number 4. Problem 10: 2 I got this wrong. Look. 1Which one do you need? I got 87 cents. Problem 6, Revisited 2What is it? 3First minute. 2Now what? How many minutes did you say? 330 minutes. So, you have to times 13 times 29. 2Mm-hmm. 3And whatever it equals, add with 22. 2OK. I’m done

20. C R E S S T / U C L A 20 Unsuccessful student who received high-level help but didn’t understand it (Example 1: Student 1) Student 1 (2129) Problem 1: 195 cents. It’s, like, 5 minutes. This is the first minute [cost]. So you times, ah, 19? I got 67. THow did you get it? Tell them. 2Alright. First, 756 is what you are telephoning. And 19 cents is for the first minute. And then the next 4 minutes are 12 cents. And I just, 1, 9 plus 4 (times) 12. That’s how I got 67 cents. You just, first, one is 19, first minute is 19 cents. And next is 12…so that’s how I got mine, 67 cents. … 19 cents, right? That’s for the first minute. And then each additional minute is 12 cents. And then you can put for 5 minutes, since the first minute is 19 cents. Then the next 4 will be 12. I just plus that all together. 1Oh, OK, I got it. Wait a minute Problem 3: 2How much did you get? Did you get the right answer? 1I don’t know. I don’t understand it

21. C R E S S T / U C L A 21 Problem 7: 2You got to take one off for the first minute. …That’s what you plus at the bottom, because you have to take it off. 1What’d you take off? 4You take one off from the 9, and then times the 9… I mean 8. 2And then you use the 22, uh, the 19 to plus at the end. That would be 9 minutes. 3So it’s supposed to be a dollar and 15, right? 4We got to help her (referring to Student 1). 1Is that, is that a dollar and 15? 4Yeah. 1Alright. Problem 8: 1Why’d you put 13? 2Alright, 13. It’s 13 times 10. You got to take one of the elevens off. Because then you add the additional minute. 4It’s a dollar 52. Unsuccessful student who received high-level help but didn’t understand it (Example 1: Student 1)

22. C R E S S T / U C L A 22 1I got a dollar 54. Why is it a dollar 52? 4(calling the teacher) (She)’s not getting it. (She)’s not getting it. 1I got a dollar 54. TWell, you multiplied the wrong thing. I don’t know where you even got 12. 1I got 11 right here. Then I get, that’s what I hear, they were saying to take off one. TNot away from the money. You have to take away from the time. 1Then I take away from that, and then… TOK, now you know what to do. Alright? Problem 10: 1I didn’t do it. I don’t understand it. 4Look it. Let me show you. …Ok, look it, here’s what you can do. Look, right here. Look. Ok, 756. Is that 756? Yeah, Ok, then right. So you put 12 times 5. Yeah, put 12 times 5 up there. 12 times 5…. And then look it. Time it together, and then get the answer, Ok? Problem 1, Page 2 1I don’t understand it. Unsuccessful student who received high-level help but didn’t understand it (Example 1: Student 1)

23. C R E S S T / U C L A 23 Unsuccessful student who received high-level help but didn’t understand it (Example 2: Student 3) 3How come you got 29? 129 for what? 3For number 6. 1See, look, look. See what happens is number 6. Watch. It says, it says, a 30 minute call to 771 prefix, right? So first, so it’s 30 minutes, right? First, it’s all. You have to say 29 times 13. And then, times, plus 22. 329 times 13? 1Yeah, ‘cause you already have made another minute right there. That’s a minute right there. 3Mmm-hmm. 1That’s how it is. 3Sure it’s 13 times 29? 1Yeah.

24. C R E S S T / U C L A 24 Problem 1 2Where’d you get 67 cents from? 3You go 4 times 12. … You do the additional minutes. When you get your answer, you put it under the rate for the first minute. Additional minutes. When you get your answer, you put under it the rate for the first minute. It’s 4, isn’t it? Yeah, because that’s the additional minutes. 2And what’d you multiply by? … Is it right like this? 3Times 12. 2Oh, so I just time 4 by the additional minutes? 3Yeah. And then when you get the answer, under it put what the cost for the first minute is, and then add them. 2Now what do I do? 0.48, then what do I do? 3Then put 0.19 under it. 2So it’s 0.67? 3Yes. Unsuccessful student who received high-level help but didn’t understand it (Example 3: Student 2)

25. C R E S S T / U C L A 25 Problem 2 2Times them all by 4? … You add them all by 4? You multiply by 4 3No…. How many minutes they call, they talked on the phone for, take one minute off it, and times it by the [cost of the] additional [minutes]. Then add it by this. 2Ok. 3Then add it like this. Problem 8 2(Looking at another student’s paper) By 10? 1Yeah, by 10. Not by 11. By 10, just try it. See what you get. 3You’re supposed to take one off. 2Off this? 1No, off the first minute. 2Oh. Unsuccessful student who received high-level help but didn’t understand it (Example 3: Student 2)

26. C R E S S T / U C L A 26 ä Attempt to solve a problem to identify areas of confusion ä Ask specific questions that focus on areas of confusion ä Ask for explanations rather than calculations and answers ä Persist in seeking help ä Change help-seeking strategies where necessary ä Apply help received to try to solve problems without assistance Responsibilities of the Help-Seeker:

27. C R E S S T / U C L A 27 ä Provide labeled, elaborated explanations ä Continue giving help until the help-seeker understands ä Monitor the help-seeker’s level of understanding ä Provide help-seekers with opportunities to solve problems for themselves Responsibilities of the Help-Giver:

28. C R E S S T / U C L A 28 ä Establish positive classroom norms for group work: ¥ Encourage understanding rather than only obtaining the correct answer ¥ Support providing elaborated help rather than answers and calculations ä Structure the task to support understanding: ¥ Reduce time pressure to complete numerous problems ¥ Eliminate grades based on accuracy or amount of work completed ¥ Reward effort and understanding rather than speed and accuracy ä Model effective helping behavior: ¥ Invite students’ questions ¥ Respond to questions in terms of concepts instead of calculations ¥ Encourage students to share problem-solving strategies ¥ Encourage discussion of discrepant solutions to problems ä Monitor group interaction ¥ Determine whether groups are giving each other calculations and answers ¥ Help redirect groups to a focus on understanding and providing explanations Responsibility of the Teacher