noreen m. webb and ann m. mastergeorge
DESCRIPTION
Understanding Collaborative Learning Environments: The Development of Students’ Mathematical Understanding. Noreen M. Webb and Ann M. Mastergeorge. - PowerPoint PPT PresentationTRANSCRIPT
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
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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
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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
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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?
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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)
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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
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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)
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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)
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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
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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
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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
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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*
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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%
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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%
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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%
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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
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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
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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
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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
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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)
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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.
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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.
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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.
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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
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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.
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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?
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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.
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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.
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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!
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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.
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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.
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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
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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.
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Microgenetic Analysis Examples (con’t)
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
Level of Assistance and Frequency of Response for Problem 2, Component 1.
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Microgenetic Analysis Examples (con’t)
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
Level of Assistance and Frequency of Response for Problem 3, Component 1.
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Microgenetic Analysis Examples (con’t)
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.
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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
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Conclusion (con’t)
The Role of Apprenticeship In Collaborative Environments
Role of Effective Scaffolding
Role of Teacher Feedback
Parameters of Peer Learning
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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
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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
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Website:
WWW.CSE.UCLA.EDUWWW.CSE.UCLA.EDUE-Mail:
Ann M. Mastergeorge:[email protected]
Noreen M. Webb: