c r e s s t / u c l a 1 productive collaborative problem solving: noreen webb and ann mastergeorge...

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


Issues Guiding This Study

The promise of collaborative learning environments—Students can learn by:

Seeking and receiving help from others

Co-constructing new knowledge, skills, understanding

Explaining and justifying their positions

Recognizing and resolving conflicts and disagreements

Questions of this study:

What processes help students make the transition from misconception to correct conception?

What are the group dynamics that facilitate or hinder those processes?


Method

Sample: Four 7th-grade general mathematics classes worked in heterogeneous cooperative groups for four weeks

Preparation for group work: (1) Inclusion activities (learning classmates’ interests, backgrounds); (2) Developing communication skills (norms for group behavior, social skills, helping behavior)

Topic: Operations with decimals

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)

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


Coding of Group Work: Levels of Help Given or Received

Level Description Example

High

8 Fully 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.”


7 Explanation 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.”

6 Explanation 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.”

5 Explanation that includes some relevant numbers, an explicitly stated or implied numerical rule, but no verbal labeling

“Because 30 subtract 1 is 29.”


Low

4 Numerical expression or equation

“13 times 29.”

3 Numbers to write or copy

“Put 13 on top, 29 on the bottom. Then you times it.”

2 Answer to part or all of the problem

“I got $3.77.”

1 Non-content or non-informational response

“Just do it the way she said.”

0 No response

Coding of Group Work: Levels of Help Given or Received


Indications of a Need for Help

Indication of Need for Help M Percent

Number of questions 4.81 53%

General request for help/statements ofconfusion

2.37 26%

Request for specific explanation 1.00 11%

Request for information 1.40 15%

Number of errors 4.25 47%


Frequency of Help Received

Help Received

From AnotherStudent

From theTeacher

Level M % M %

High 1.85 20% .19 38%

Low 7.36 80% .31 62%


Frequency of Help Received from Another Student Over Time

Problem

Level of Help 1 2 3 4 5 6 7 8 9 10 11 12

High 1.21 .18 .29 .29 .14 .05 .00 .00 .00 .08 .00 .00

Low 1.40 .64 1.10 1.29 .73 .67 .67 .17 .50 .42 .33 .20


Frequency of Immediate Responses to Help Received

Response to Help Received

Level M %

High .41 6%

Low 6.80 94%


Significant Partial Correlations Between Behavior Variables and Posttest

PerformanceBehavior Variable partial r

Indication of a need for help

Request for specific explanation .48**

Help received

Highest level of help received .38**

Received high-level help (yes/no) .34*

Further problem-solving activity: Solved problems correctly withoutassistance

Proportion of problems solved .64***

Solved at least one problem without assistance (yes/no) .66***


Posttest Performance at Each Level of Highest Help Received

PosttestPerformance

Highest Level of Help Received Number ofStudents

Correct Incorrect Probability ofCorrect Posttest

High 37 13 24 35%

Low 11 0 11 0%


Relationship Between Unassisted Problem Solving during Group Work and Posttest

Performance

Posttest Performance

Solved Group Work ProblemCorrectly Without Assistance

Number ofStudents

Correct Incorrect Probability ofCorrect Posttest

Yes 21 13 8 62%

No 27 0 27 0%


Posttest Performance by Level of Help Received (High/Low) and Unassisted Problem Solving

Posttest Performance

Number ofStudents

Correct Incorrect Probability ofCorrectPosttest

Received High-Level Help

Unassisted problemsolving

Yes 17 13 4 76%

No 20 0 20 0%

Received Low-Level Help Only

Unassisted problemsolving

Yes 4 0 4 0%

No 7 0 7 0%


Questions Driving Microgenetic Qualitative Analyses

Why were some students able to obtain high-level help while others were not?

Student-level factors:

Persistence in asking questions

Nature of questions asked

Group-level factors:

Willingness to give help

Perception of the task: task completion vs. conceptual understanding

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


Students Who Learned (13 of 48):

Asked specific questions

Received high-level help

Changed their help-seeking strategies

Persisted in asking specific questions until they understood

Attempted subsequent problems on their own


Successful student who asked increasingly focused questions (Student 1)

1 I don’t get this.

2 4 Ok, 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 4 It says 22 minutes per minute.

2 For the first minute, it’s 22 cents.

4 Yeah. And then, it’s 13 cents for each additional minute. So, 13 times, times 29.

5 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…

6 4 Ah. There is the first minute, 22 cents. Now multiply 13 cents times 29. Because 29 minutes are left from the first minute.

7 Well, it’s 30. 30, 30 minutes. But you are saying, do what?

4 Multiply 29 times 13 cents.

5 29? Why 29? This is 30.

4 Because they already got a minute. That’s the first minute.

5 1 Thank you!


Students Who Didn’t Learn

1. Never received high-level help (11 of 48):

Asked for help before attempting to solve the problems

Asked general questions or generally declared confusion

Were given numbers and calculations to copy

2. Received high-level help but didn’t understand it (24 of 48):

Tended to ask general questions

Did not persist in seeking help

Accepted explanations given without asking further questions

Gave up trying to understand and resorted to copying answers, or never tried to understand


Unsuccessful student who did not receive high-level help (Student 2)

Problem 6:

2 I don’t know where the heck we are.

1 OK, look. Just copy it. Number 2, and number 4.

Problem 10:

2 I got this wrong. Look.

1 Which one do you need? I got 87 cents.

Problem 6, Revisited

2 What is it?

3 First minute.

2 Now what? How many minutes did you say?

3 30 minutes. So, you have to times 13 times 29.

2 Mm-hmm.

3 And whatever it equals, add with 22.

2 OK. I’m done


Unsuccessful student who received high-level help but didn’t understand it (Example 1:

Student 1)Student 1 (2129)

Problem 1:

1 95 cents. It’s, like, 5 minutes. This is the first minute [cost]. So you times, ah, 19?

I got 67.

T How did you get it? Tell them.

2 Alright. 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.

1 Oh, OK, I got it. Wait a minute

Problem 3:

2 How much did you get? Did you get the right answer?

1 I don’t know. I don’t understand it


Problem 7:

2 You got to take one off for the first minute. …That’s what you plus at the bottom, because you have to take it off.

1 What’d you take off?

4 You take one off from the 9, and then times the 9… I mean 8.

2 And then you use the 22, uh, the 19 to plus at the end. That would be 9 minutes.

3 So it’s supposed to be a dollar and 15, right?

4 We got to help her (referring to Student 1).

1 Is that, is that a dollar and 15?

4 Yeah.

1 Alright.

Problem 8:

1 Why’d you put 13?

2 Alright, 13. It’s 13 times 10. You got to take one of the elevens off. Because then you add the additional minute.

4 It’s a dollar 52.


Student 1)


1 I got a dollar 54. Why is it a dollar 52?

4 (calling the teacher) (She)’s not getting it. (She)’s not getting it.

1 I got a dollar 54.

T Well, you multiplied the wrong thing. I don’t know where you even got 12.

1 I got 11 right here. Then I get, that’s what I hear, they were saying to take off one.

T Not away from the money. You have to take away from the time.

1 Then I take away from that, and then…

T OK, now you know what to do. Alright?

Problem 10:

1 I didn’t do it. I don’t understand it.

4 Look 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

1 I don’t understand it.


Student 1)



Student 3)3 How come you got 29?

1 29 for what?

3 For number 6.

1 See, 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.

3 29 times 13?

1 Yeah, ‘cause you already have made another minute right there. That’s a minute right there.

3 Mmm-hmm.

1 That’s how it is.

3 Sure it’s 13 times 29?

1 Yeah.


Problem 1

2 Where’d you get 67 cents from?

3 You 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.

2 And what’d you multiply by? … Is it right like this?

3 Times 12.

2 Oh, so I just time 4 by the additional minutes?

3 Yeah. And then when you get the answer, under it put what the cost for the first minute is, and then add them.

2 Now what do I do? 0.48, then what do I do?

3 Then put 0.19 under it.

2 So it’s 0.67?

3 Yes.


Student 2)


Problem 2

2 Times them all by 4? … You add them all by 4? You multiply by 4

3 No…. 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.

2 Ok.

3 Then add it like this.

Problem 8

2 (Looking at another student’s paper) By 10?

1 Yeah, by 10. Not by 11. By 10, just try it. See what you get.

3 You’re supposed to take one off.

2 Off this?

1 No, off the first minute.

2 Oh.


Student 2)


Attempt to solve a problem to identify areas of confusion

Ask specific questions that focus on areas of confusion

Ask for explanations rather than calculations and answers

Persist in seeking help

Change help-seeking strategies where necessary

Apply help received to try to solve problems without assistance

Responsibilities of the Help-Seeker:


Provide labeled, elaborated explanations

Continue giving help until the help-seeker understands

Monitor the help-seeker’s level of understanding

Provide help-seekers with opportunities to solve problems for themselves

Responsibilities of the Help-Giver:


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

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

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

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

c r e s s t / u c l a 1 productive collaborative problem solving: noreen webb and ann mastergeorge...

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