Transcript Presentation

C RE SS T/U C LA
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Productive Collaborative Problem Solving:
Responsibilities of Teachers and Students
Noreen Webb and Ann Mastergeorge
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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?
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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
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Coding of Group Work: Levels of Help
Given or Received
Level
Description
Example
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.”
High
8
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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.”
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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.”
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Coding of Group Work: Levels of Help
Given or Received
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 noninformational
response
“Just do it the way she said.”
0
No response
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Indications of a Need for Help
Indication o f Need for Help
M
Percent
4.81
53%
Gene ral reques t for help/statements of
confusion
2.37
26%
Reques t for specifi c expl anation
1.00
11%
Reques t for information
1.40
15%
4.25
47%
Number of questions
Number of errors
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Frequency of Help Received
Help Received
From Ano ther
Studen t
From the
Teache r
Leve l
M
%
M
%
High
1.85
20%
.19
38%
Low
7.36
80%
.31
62%
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Frequency of Help Received from Another
Student Over Time
Proble m
Leve l 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
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Frequency of Immediate Responses to Help
Received
Respons e to Help Received
Leve l
M
%
High
.41
6%
Low
6.80
94%
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Significant Partial Correlations Between
Behavior Variables and Posttest
Performance
Behav ior Variable
partial r
Indication o f a need for help
Reques t for specifi c expl anation
.48**
Help received
Highe st leve l of help received
.38**
Received high -leve l help (yes/no )
.34*
Further prob lem- solving activit y: Solved p roblems correctly w it hout
assistance
Proportion o f problems solv ed
.64***
Solved at least one proble m wit hout assist ance (ye s/no)
.66***
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Posttest Performance at Each Level of
Highest Help Received
Posttest
Performance
Highest Level of Help Received
Number of
Students
Correct
Incorrect
Probability of
Correct Posttest
High
37
13
24
35%
Low
11
0
11
0%
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Relationship Between Unassisted Problem
Solving during Group Work and Posttest
Performance
Posttest Performanc e
Solved Group Work Proble m
Correctly Without Assistanc e
Number of
Studen ts
Correct
Incor rect
Probabilit y of
Correct Posttest
Yes
21
13
8
62%
No
27
0
27
0%
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Posttest Performance by Level of Help Received
(High/Low) and Unassisted Problem Solving
Posttest Performanc e
Number of
Studen ts
Correct
Incor rect
Probabilit y of
Correct
Posttest
Yes
17
13
4
76%
No
20
0
20
0%
Yes
4
0
4
0%
No
7
0
7
0%
Received High- Leve l Help
Una ssis ted problem
solving
Received Low -Level Help Only
Una ssis ted problem
solving
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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
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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
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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
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. (pause) Prefix. So you, look, 771 right here, and there is 30
minutes. So why do you…
2
4
Ah. 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
1
It says 22 minutes per minute.
Thank you!
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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
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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
1
I got this wrong. Look.
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
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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
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Unsuccessful student who received high-level
help but didn’t understand it (Example 1:
Student 1)
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.
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Unsuccessful student who received high-level
help but didn’t understand it (Example 1:
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.
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Unsuccessful student who received high-level
help but didn’t understand it (Example 2:
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.
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Unsuccessful student who received high-level
help but didn’t understand it (Example 3:
Student 2)
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.
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Unsuccessful student who received high-level
help but didn’t understand it (Example 3:
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.
C RE SS T/U C LA
Responsibilities of the Help-Seeker:
 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
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Responsibilities of the Help-Giver:
 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
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Responsibility of the Teacher




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