noreen m. webb and ann m. mastergeorge

C R E S S T / U C L A

UCLA Graduate School of Education & Information StudiesCenter for the Study of Evaluation

National Center for Research on Evaluation, Standards, and Student Testing

Understanding Collaborative Learning Environments:

The Development of Students’ Mathematical Understanding

Noreen M. Webb and Ann M. Mastergeorge

Annual CRESST ConferenceSeptember 15, 2000 UCLA, Los Angeles, CA


Issues Guiding this Study

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

Co-constructing new knowledge, skills, understanding

Explaining and justifying their positions

Seeking and receiving elaborated help from others

Recognizing and resolving conflicts and disagreements


Issues Guiding this Study (con’t)

Receiving help is not always beneficial for Receiving help is not always beneficial for learning—students are more likely to learn learning—students are more likely to learn whenwhen::

Explanations are detailed, elaborated and relevant to a student’s misconception

Students use the help received in a dynamic way to try to solve problems without assistance


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 cooperative groups for four weeks

Preparation for group work

Inclusion activities (learning classmates’ names, interests, backgrounds)

Developing communication skills (norms for group behavior, social skills, helping behavior)


Method (con’t)

TopicTopic

Operations with decimals

Data CollectedData Collected

Audiotapes of all groups for class period on cost of telephone calls

Pretest and posttest on general mathematics skills and cost of telephone calls


Method (con’t)

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)


Coding of Group Work:The Steps in the Problem

Six steps for solving the problem:

1. Identify the first-minute cost ($0.22)

2. Identify the additional -minute cost ($0.13)

3. Determine the number of additional minutes (30 - 1 = 29)

4. Determine the correct cost for the number of additional minutes ($0.13*29=$3.77)

5. Add the cost for the first minute (+$0.22)

6. Determine the final answer ($3.77 + $0.22 = $3.99)


Coding of Group Work: Levels of Help Given or Received

Level Description Example

6 Verbally labeled explanation of how to

solve part or all of the problem

“Multiply 13 cents by 29, because 29

minutes are left after the first minute.”

5 Numerical rule with no verbal labels for

the numbers

“This is 30, so you minus 1.”

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 Student Response to Help Received

Level Description Example

6 Explains problem or reworks problem

and produces correct answer

“One minute costs 22 cents. This is the first minute.

Then there is 29 more minutes. So we have to find outhow much that is. So then you times 29 times 13.

That’s 377.”

5 Applies another student’s numerical

rule and completes the problem

“So, on the first one, I need to minus 1? So, 30 minus 1

is 29.”

4 Finish another student’s calculations Student 1: “Then you times it 29 times 13.”

Student 2: “OK, that’s 370…377.”

3 Copies numbers or writes downnumbers that are dictated

Student 1: “It’s 30 minutes so you times 13 times 29.And then you get 377, right?”

Student 2: “Wait up. 13 times 29? Gives what?”

2 Acknowledges help received “OK, OK, I got it.”

1 Non-content response “May I borrow your eraser?”

0 No response Says and does nothing


A Simple Solution

1 2

6

3

4

5

+ + +

+++


What Actually Happened

1h+62h+6,

5RW+6

2h+3, 3R+45RW+6

4h-2

5RW+6

5RW+4

5RW+4

5RW+4,6R+4

7R+2

?, 0 0, 0 ag+2

0, ag+nv1-0, ??, 0, ?,

ag+2


Results

Predictors of Posttest Performance and Solving Problems

Correctly During Group Work (Correlations)

Predictor Correct on Posttest

(yes/no)

Unassisted Problem

Solving (yes/no)

Help Received

Number of high-level explanations received .21 .10

Received high-level explanation (yes/no) .37* .26

Highest level of help received .33* .34*

Solved problem correctly during group work

without assistance

.78*** --

Pretest score .24 .32*


Posttest Performance at Each Level of Highest Help Received

Posttest Performance

Highest Level of Help Received Incorrect

(n)

Correct

(n)

Probability of

Correct Posttest

Low 20 2 9%

None 4 0 0%

Answer only 2 0 0%

Numbers or calculations 3 0 0%

Numerical equation 11 2 15%

High: Rule or Explanation 15 11 42%


Posttest Performance and Unassisted Problem SolvingDuring Group Work

Posttest

Performance

Solved Group WorkProblem CorrectlyWithout Assistance

Incorrect

(n)

Correct

(n)

Probability of

Correct Posttest

No 27 0 0%

Yes 5 12 71%

Can’t tell 3 1 25%


Posttest Performance Predicted by Level of Help Received and

Unassisted Problem Solving

Incorrect

on Posttest

Correct on

Posttest

Total n Probability of

Correct Posttest

Received only low-level help 20 2 22 9%

Unassisted problem solving: No 15 0 15 0%

Unassisted problem solving: Yes 4 1 5 20%

Unassisted problem solving: Can’t tell 1 1 2 50%

Received high-level help 15 11 26 42%

Unassisted problem solving: No 12 0 12 0%

Unassisted problem solving: Yes 1 11 12 92%

Unassisted problem solving: Can’t tell 2 0 2 0%


Questions Driving Microgenetic and 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

Nature of errors made


Questions Driving Microgenetic and Qualitative Analyses (con’t)

Group-level factors:

Willingness to give help

Sufficient competence to help

Perception of the task (task completion vs. conceptual understanding)

Attitudes toward students needing help


Questions Driving Microgenetic and Qualitative Analyses (con’t)

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

Response of group to repeated help-seeking


Three Categories of Students Receiving High Level of Help

(Correct Post Test)

(1.) Persistent In Understanding Reasons For The Numerical Procedures

(2.) Persistent And When Did Not Understand Turned To Teacher For Assistance

(3.) Received Explanations They Understood And Then Gave Explanations Of The Process To

Other Students


Examples of Students Who Received High Level Help and Performed Posttest Correctly:

Persistent in Seeking Help

1 ( ) I don’t get this.

2 I don’t know how to do it myself.

4 Ok, see, like. Ok. See, like, right here it says 771. So you look for 771. 22 minutes, 22 cents per minute.

2 Oh, hold on.

1 ( ) I don’t know. Could you ( ) repeat that again?

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.

1 I don’t know how to do it. (6,20)


Persistent in Seeking Help (con’t)

4 You didn’t understand?

1 None! Ok. Let me get this straight. Okay, it’s a 30 minute call to 771 prefix so look--771 right here and this is 30 minutes. So why do you….

4 There is the first minute 22 cents, now multiply 13 cents times 29 because 29 minutes are left from the first minute.

1 Well, it’s 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 go a minute. That’s the first minute.

1 Thank you.


Examples of Students Who Received High Level Help and Performed Posttest IncorrectlyPersistent in Seeking Help

3 What did you get?

4 6.60. I made my answer the same, I don’t care.

3 No, look! You see? Look. She told us the price for the first minutes

is 13 cents. 13 times 29 minutes.

4 Where did you get 29 from?

3 Because it is 30 minutes, and the first minute is 22 cents.

4 Uh-huh (yes).

3 And then the next minute is 13 cents.

4 Uh-huh (yes).

3 Then you go 13 times 29, then you get the answer plus 22 cents.

4 I got it.


Two Categories of Students Who Received High Level of Help

(Incorrect Post test)

(1) Students asked questions that suggested that they were

trying to solve the problem, and when did not understand resorted to copying.

(2) Students did not respond or responded minimally to the

explanation they received.


Explanation of Problem Solving Process to Other Students

1 It’s 79 cents. (25, 15)

4 No, it’s 67.

1 It’s 12 times 5, which is 60.

4 No, that will be 12 times 4.

1 ( ) 12 cents in 5 minutes, right? Right?

4 Yeah, but…

1 And then, and then plus 19.

4 But it will be 4 times 12, because there’s one minute away.

1 (Whispering)

4 Understand?

4 Look it, look it. Look, if you do…3 I got 79.1 No, it’s 12 times 4. I did it wrong.3 Where’d you get 4 at?1 4 minutes and the additional minute is 19 cents. I got 0.67.3 5 minute call, umm… Where’d you 4…?1 Look, look. Ok, 12 times 4, right? And then the

minute, that’s the 19 cents is the extra minute, which makes

it 5.? Uh-huh.1 Because it says after each additional minute, so that means it will be 12 times 4.

And then you add 19 cents, which is the additional time.

1 Yeah.1 67 cents. I get it now


Four Categories of Students Who Did Not Receive High-Level Help

(Incorrect Post Test)

(1) Students asked for help were only given the calculation without explanation of where the numbers came from.

(2) Some students asked for help and were told to copy others work.

(3) Some students asked for help were were ignored.

(4) Some students were confused but did not seek help.


Willingness of Students to Give Help

T What did you get for number 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?


Willingness of Students

to Give Help3 What did you get?

4 6.60. I made my answer the same, I don’t care.

3 No, look! You see? Look. She told us the price for the first minutes is 13 cents. 13 times 29 minutes.

4 Where did you get 29 from?

3 Because it is 30 minutes, and the first minute is 22 cents.

4 Uh-huh (yes).

3 And then the next minute is 13 cents.

4 Uh-huh (yes).

3 Then you go 13 times 29, then you get the answer plus 22 cents.

4 I got it.


Examples of Students Who Did Not Receive High-Level Help and Did Not Perform Posttest Correctly

Requested Help and Told to Copy

2 Which telephone call cost more? 7-minute call to…Oh man! Oh, Ok, I get it.

1 I don’t.

3 What do you want?

1 I don’t understand it.

3 We are supposed to discuss this among ourselves.

3 Today is Thursday, ( ) Did you do it? Did you do your class work? Did you do it?

1 I didn’t do the page. I can’t understand it.

3 You should just copy it.


Requested Help and Ignored

1 How much you get? What’d you get?

3 I don’t know.

1 Why are you doing all of it?

3 I don’t understand.

1 Don’t worry.

3 What do you have for 7? (9,13)

1 Don’t worry about it. A dollar 15.

3 What do you have for 7? A dollar 15?

1 Yes!


Requested Help and Ignored

3 You go 13 times 11, right?

2 No, 10.

3 How could it be 10?

1 Because you got to take away 1.

2 You are supposed to take away 1 and then add it.

1 Yeah, you got to multiply it.

2 Add 22 then. Look at the problem.

3 Oh. Here. So you go 13 times 10?

2 Yeah.

1 You are doing it right, just that you don’t know how to multiply it right.


Confused and Did Not Seek HelpT Ah-ha. How did you get, wait a

minute, how do you get 128?

3 I multiplied.

T The first one is 19 cents and then what?

3 And that’s ( ) each additional minute was 12 cents.

T And how many additional minutes are there?

3 12.

T No. Can’t be…. It costs 12 cents, but how many minutes were there?

3 There are 6, I mean 7 minutes.

T Total. 7 minutes total. First was 19, then you have to multiply 12 times what?

3 7.

T You already took away the first minute. How many do you have left?

3 12.

T No. Minutes. Minutes.3 Oh. So you have no minutes.T 7 minutes, the first minute is how much?3 First minute is 19 cents.T OK, how many minutes do I have left now?3 6.T And they cost 12 cents each. How much is that?3 6 times 12?T Mmm (yes). 6 dozen eggs.3 Ah, 6 times…T Multiply it out….Yeah. Then you have to find the difference.3 72.T 72, then you have to add it up with 19.3 So, I add 72 with 19 cents.T That’s right.


Microgenetic Analysis Examples

43

0

4 5

13

6

22

64

02

8

14

13

81

0 13

6

1 10

10

20

30

40

50

60

70

80

90

0 1 2 3 4 5 6

Level of Assistance

Problem 1

Problem 2

Problem 3

Level of Assistance Received and Frequency of Occurrence


Microgenetic Analysis Examples (con’t)

10

5

1

22

11

3

4

2

0

0

5

10

15

20

25

0 1 2 3 4 5 6

Level of Assistance

Failed

Passed

Level of Assistance and Frequency of Response for Problem 1, Component 1.



15

0 0

1

5

0 0

6

0 0

3

0 0 00

5

10

15

20

25

0 1 2 3 4 5 6

Level of Assistance

Failed

Passed




21

0 0

1

5

0

11

2

0

5

10

15

20

25

0 1 2 3 4 5 6

Level of Assistance

Failed

Passed




0

1

2

3

4

5

6

Student 1 Student 2 Student 3

Component 1

Component 2

Component 3

Level of Assistance for Problem 1, Students 1, 2 and 3.


Conclusion Quantitative Results and Predictors of

Performance

Level of Assistance

Implications of Qualitative Analyses In Cooperative Group Learning: The Macro and Micro Genesis of Problem-Solving

Mediated group dynamics

Mediated group discourse


Conclusion (con’t)

The Role of Apprenticeship In Collaborative Environments

Role of Effective Scaffolding

Role of Teacher Feedback

Parameters of Peer Learning


Conclusion (con’t) Paths of Variability: Understanding

Change (or No Change) in Context

“Seeing” students’ thinking

Process of variability in understanding problem-solving contexts

The conceptual-procedural divide

The pedagogical divide


Next Steps

Systematic Quantitative Analysis of Concept Maps

Qualitative Analyses of Amount and Type of Difficulty on Problem Components

In-Depth Qualitative Analyses of Student Questions

In-Depth Microgenetic Analysis of Student Performance

Strategies Across Problems and Components

Student Reflections on Group Experiences

Hypothetical and Ideal Explanations From Student and Teacher Perspective


Website:

WWW.CSE.UCLA.EDUWWW.CSE.UCLA.EDUE-Mail:

Ann M. Mastergeorge:[email protected]

Noreen M. Webb:

[email protected]

noreen m. webb and ann m. mastergeorge

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