C R E S S T / U C L A UCLA Graduate School of Education & Information Studies Center 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 Conference September 15, 2000 UCLA, Los Angeles, CA

C R E S S T / U C L A Issues Guiding this Study The promise of collaborative learning 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

C R E S S T / U C L A Issues Guiding this Study (con’t) Receiving help is not always beneficial for learning—students are more likely to learn when : Receiving help is not always beneficial for learning—students are more likely to learn when :  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

C R E S S T / U C L A 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?

C R E S S T / U C L A 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)

C R E S S T / U C L A Method (con’t) Topic  Operations with decimals Data 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

C R E S S T / U C L A 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)

C R E S S T / U C L A 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 ( = 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 (+$0.22) 6. Determine the final answer ($ $0.22 = $3.99)

C R E S S T / U C L A Coding of Group Work: Levels of Help Given or Received

C R E S S T / U C L A Coding of Group Work: Levels of Student Response to Help Received

C R E S S T / U C L A A Simple Solution

C R E S S T / U C L A What Actually Happened

C R E S S T / U C L A

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

C R E S S T / U C L A 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

C R E S S T / U C L A 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

C R E S S T / U C L A 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

C R E S S T / U C L A Examples of Students Who Received High Level Help and Performed Posttest Correctly: Persistent in Seeking Help 1 ( ) I don’t get this. 2I don’t know how to do it myself. 4Ok, see, like. Ok. See, like, right here it says 771. So you look for minutes, 22 cents per minute. 2Oh, hold on. 1( ) I don’t know. Could you ( ) repeat that again? 4It says 22 minutes per minute. 2For the first minute, it’s 22 cents. 4Yeah. And then, it’s 13 cents for each additional minute. So, 13 times, times 29. 1I don’t know how to do it. (6,20)

C R E S S T / U C L A 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 minutes. But you are saying do what? 4 Multiply 29 times 13 cents. 1 29? Why 29? This is Because they already go a minute. That’s the first minute. 1 Thank you.

C R E S S T / U C L A Examples of Students Who Received High Level Help and Performed Posttest Incorrectly Persistent in Seeking Help 3What did you get? I made my answer the same, I don’t care. 3No, look! You see? Look. She told us the price for the first minutes is 13 cents. 13 times 29 minutes. 4Where did you get 29 from? 3Because it is 30 minutes, and the first minute is 22 cents. 4Uh-huh (yes). 3And then the next minute is 13 cents. 4Uh-huh (yes). 3Then you go 13 times 29, then you get the answer plus 22 cents. 4I got it.

C R E S S T / U C L A 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.

C R E S S T / U C L A Explanation of Problem Solving Process to Other Students 1It’s 79 cents. (25, 15) 4No, it’s 67. 1It’s 12 times 5, which is 60. 4No, that will be 12 times 4. 1( ) 12 cents in 5 minutes, right? Right? 4Yeah, but… 1And then, and then plus 19. 4But it will be 4 times 12, because there’s one minute away. 1(Whispering) 4Understand? 4Look it, look it. Look, if you do… 3I got 79. 1No, it’s 12 times 4. I did it wrong. 3Where’d you get 4 at? 14 minutes and the additional minute is 19 cents. I got minute call, umm… Where’d you 4…? 1Look, 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. 1Because 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. 1Yeah. 167 cents. I get it now

C R E S S T / U C L A 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.

C R E S S T / U C L A Willingness of Students to Give Help TWhat did you get for number 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?

C R E S S T / U C L A Willingness of Students to Give Help 3What did you get? I made my answer the same, I don’t care. 3No, look! You see? Look. She told us the price for the first minutes is 13 cents. 13 times 29 minutes. 4Where did you get 29 from? 3Because it is 30 minutes, and the first minute is 22 cents. 4Uh-huh (yes). 3And then the next minute is 13 cents. 4Uh-huh (yes). 3Then you go 13 times 29, then you get the answer plus 22 cents. 4I got it.

C R E S S T / U C L A Examples of Students Who Did Not Receive High- Level Help and Did Not Perform Posttest Correctly Requested Help and Told to Copy 2Which telephone call cost more? 7-minute call to…Oh man! Oh, Ok, I get it. 1I don’t. 3What do you want? 1I don’t understand it. 3We are supposed to discuss this among ourselves. 3Today is Thursday, ( ) Did you do it? Did you do your class work? Did you do it? 1I didn’t do the page. I can’t understand it. 3You should just copy it.

C R E S S T / U C L A Requested Help and Ignored 1How much you get? What’d you get? 3I don’t know. 1Why are you doing all of it? 3I don’t understand. 1Don’t worry. 3What do you have for 7? (9,13) 1Don’t worry about it. A dollar 15. 3What do you have for 7? A dollar 15? 1Yes!

C R E S S T / U C L A Requested Help and Ignored 3 You go 13 times 11, right? 2No, 10. 3How could it be 10? 1Because you got to take away 1. 2You are supposed to take away 1 and then add it. 1Yeah, you got to multiply it. 2Add 22 then. Look at the problem. 3Oh. Here. So you go 13 times 10? 2Yeah. 1You are doing it right, just that you don’t know how to multiply it right.

C R E S S T / U C L A Confused and Did Not Seek Help TAh-ha. How did you get, wait a minute, how do you get 128? 3I multiplied. TThe first one is 19 cents and then what? 3And that’s ( ) each additional minute was 12 cents. TAnd how many additional minutes are there? 312. TNo. Can’t be…. It costs 12 cents, but how many minutes were there? 3There are 6, I mean 7 minutes. TTotal. 7 minutes total. First was 19, then you have to multiply 12 times what? 37. TYou already took away the first minute. How many do you have left? 312. TNo. Minutes. Minutes. 3Oh. So you have no minutes. T7 minutes, the first minute is how much? 3First minute is 19 cents. TOK, how many minutes do I have left now? 36. TAnd they cost 12 cents each. How much is that? 36 times 12? TMmm (yes). 6 dozen eggs. 3Ah, 6 times… TMultiply it out….Yeah. Then you have to find the difference T72, then you have to add it up with 19. 3So, I add 72 with 19 cents. TThat’s right.

C R E S S T / U C L A Microgenetic Analysis Examples Level of Assistance Received and Frequency of Occurrence

C R E S S T / U C L A Microgenetic Analysis Examples (con’t) Level of Assistance and Frequency of Response for Problem 1, Component 1.

C R E S S T / U C L A Microgenetic Analysis Examples (con’t) Level of Assistance and Frequency of Response for Problem 2, Component 1.

C R E S S T / U C L A Microgenetic Analysis Examples (con’t) Level of Assistance and Frequency of Response for Problem 3, Component 1.

C R E S S T / U C L A Microgenetic Analysis Examples (con’t) Level of Assistance for Problem 1, Students 1, 2 and 3.

C R E S S T / U C L A 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

C R E S S T / U C L A Conclusion (con’t)  The Role of Apprenticeship In Collaborative Environments Role of Effective Scaffolding Role of Teacher Feedback Parameters of Peer Learning

C R E S S T / U C L A 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

C R E S S T / U C L A 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

C R E S S T / U C L A Website: Ann M. Mastergeorge : Noreen M. Webb :