franke std explan_11amtalk
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Unpacking the relationship between classroom practice and student learning in
mathematics: Examining the power of student explanations
Megan Franke, Mathematics Classroom Practice
Study Group UCLA
Session Overview • Understanding the rela.onship between classroom prac.ce and student outcomes
• Prior research on students’ explana.ons and teachers’ support of those explana.ons
• Engaging students in each other’s mathema.cal ideas
• Findings related to student par.cipa.on, teaching and student learning
Results of a large-‐scale intervention study
• Recruited volunteer teachers at 19 schools in low-‐performing, urban school district
• On-‐site professional development focused on algebraic reasoning
• Thinking Mathema-cally: Integra-ng Arithme-c and Algebra in Elementary School
• Equal sign, Rela.onal thinking • Orchestrate conversa.ons
Jacobs, V., Franke, M.., Carpenter, T., Levi, L. & Battey, D. (2007). Exploring the impact of large scale professional development focused on children’s algebraic reasoning. Journal for Research in Mathematics Education 38 (3), pp. 258-288.
The schools and district
Overview of Classrooms: Mean Achievement
01020304050607080
Low (2 classes) Medium (1class)
High (3 classes)
Ach
Collecting data around interactions
Student communication
• Explaining to other students is posi.vely related to achievement outcomes, even when controlling for prior achievement (Brown & Palincsar, 1989; Fuchs, Fuchs, HamleG, Phillips, Karns, & Dutka, 1997; King, 1992; NaNv, 1994; Peterson, Janicki, & Swing, 1981; Saxe, Gearhart, Note, & Paduano, 1993; Slavin, 1987; Webb, 1991; Yackel, Cobb, Wood, Wheatley, & Merkel, 1990).
• When describing their thinking, students must be precise and explicit in their talk, especially providing enough detail and making referents clear so that the teacher and fellow classmates can understand their ideas (Nathan & Knuth, 2003; Sfard & Kieran, 2001).
Potential Benefits of Explaining Your Own Thinking
• Transform what you know into an explana.on that is relevant, coherent, complete, and understandable to others
• Bring concepts/details together in ways that you hadn’t thought of previously
• Recognize misconcep.ons, contradic.ons, incompleteness in your idea
• Develop a sense of yourself as someone who can do mathema.cs and communicate mathema.cally
8
Coding Student Participation • Accuracy of answer given • Correct • Incorrect • No answer
• Nature of explana.on given • Correct and complete • Ambiguous or incomplete • Incorrect
• Further elabora.on aPer teacher’s ques.ons
10
Types of Student Explaining
• Gives correct/complete explana.ons
10 + 10 – 10 = 5 + □
Five? ‘Cause 10 plus 10 equals 20, huh? And then it says minus 10 equals 5 plus blank. So it go\a be 10, so 5 plus 5 equals 10. And that’s how I got it.
• Gives incorrect or incomplete explana.ons
50 + 50 = 50 + □ + 25
[50]. It’s just like 50 plus 50. They are kind of partners because they are the same. 8 + 2 = 7 + 3 (True or False?)
[True] because there’s a 2 and a 3 and a 7 and an 8. They’re like an order.
Table 3. Correlations between Student Participation and Achievement Scores
Highest level of student participation on a problema
Achievement Scoreb
Gives explanation .69*
Correct and complete .73*
Ambiguous, incomplete, or incorrect -.01
Gives no explanation -.69*
aPercent of problems in which a student displayed this behavior. Problems discussed during pairshare and whole-class interaction are included. bPercent of problems correct. Note: Number of students = 35.
*p <.05
Profiles of Students’ Contributions
0%10%20%30%40%50%60%70%80%90%
100%
LowClasses
MediumClass
HighClasses
CorrectexplanationsIncorrectexplanationCorrect answeronlyIncorrect answeronly
Moving toward understanding the details of practice in relation to student outcomes
• Teachers’ support explaining (Lampert, 2001), Revoicing (Forman et al., 1998; O’Connor & Michaels, 1993, 1996; Strom et al., 2001) Press (Kazemi & S-pek, ) Teachers’ ques.oning (Wood, 1998)
Filtering approach (Sherin, 2002)
• Teachers’ prac.ce supports students’ produc.ve explana.ons (Gillies, 2004; Rosja-‐Drummond and Mercer, 2003)
• And while evidence shows these prac.ces are not likely in many classrooms, they are even less likely in classrooms of low-‐income students of color (Anyon, 1981, Ladson-‐Billings, 1997; Lubienski, 2002; Means & Knapp, 1991).
Teachers’ Supporting of Students to Share their Thinking
• 98% of segments: Teachers asked the target students to explain their thinking
• 91% of segments: Teachers requested an explana.on at the outset of the segment, or aPer an answer was given
• 76% of segments: Teachers asked the student to elaborate further on their explana.on
• Frequent reminders about listening to explana.ons: • “Give [name] a chance [to explain]” • “I like the way [Student] is paying close a\en.on to what [Students] are about to share”
• “Let’s understand [Student’s] thinking.”
Whether teachers elicited student thinking beyond initial explanations and how the engagement ended
0%10%20%30%40%50%60%70%80%90%
100%
Low Medium High
Yes: correctexplanationYes: incorrectexplanationNo: correctexplanationNo: incorrectexplanationNo: correct answer
No: incorrectanswer
Example: General Question
Problem: 375 = __ + (3 x 10) • Student: 345 • Teacher: I’m just a li\le unsure of how you came up with 345. Can you show me what you did?
Example: Speci?ic Question
• Problem: 100 + __ = 100 + 50 Student: The 50 will go right there because it has to be the same number.
Teacher: What has to be the same number?
Probing Sequence
• Used when teacher was unclear about a student’s explana.on
• Used to highlight, clarify or make explicit por.ons of a student’s strategy
• Used when teacher is trying to help a student understand a problem
Engaging with each others’ ideas
• New study: • K-‐5 teachers • Mul.-‐age school • School describes itself as a learning environment that values diversity, encourages crea.vity and innova.on, supports disciplined inquiry, involves families and their communi.es, and makes a commitment to mee.ng the needs of the whole child.
• All teachers par.cipated (12 who taught mathema.cs) • 25-‐35 students per classroom
• Classroom observa.ons • Spent the year in classrooms approximately once a week • Video and audiotaped 2-‐3 days in March, April each class • Collected student work • Researcher designed assessment and standardized test
Collecting observation data • One sta.onary video camera with two flat microphones captured the ongoing flow of the class and the interac.on of up to two groups of students.
• Four Flip video cameras captured the interac.on of the remaining students. • one Flip video camera was sta.onary and the other three were operated by research team members.
• Distributed six digital audio recorders to pick up sound not captured by Flip
• Created a single movie for each classroom observa.on by combining all of the video and sound sources
• Movie analyzed using Studiocode so that we could code the details within the context
The above figure illustrates the way the codes are applied to a video timeline. Instances of codes are represented by rows. In addition, codes have labels as can be seen in the table in the upper right hand corner.
Relationship between Student Participation and Achievement
25
Par%al correla%on with achievement
Provided fully-‐detailed explana.ons of how to solve the problem
.30*
Highest level at which you engaged with other students’ ideas
.44*
Highest level at which other students engaged with your ideas
.41*
• Explain your thinking
• Engage with others’ ideas to a high degree
• Have others engage with your idea to a high degree
26
Teacher Support of Students’ Engagement with Each Other’s Ideas
27
Why the invitation was not enough • Student had no readily available response or a response that provided any detail, and so the teacher had to find ways to work with the student to elaborate and extend their engagement with the other student’s idea
• Student did not discuss the mathema.cal idea in what had been shared or did not address the par.cular mathema.cal idea that the teacher wanted to address
• Students did not know how to take up the teacher’s invita.on
Teacher support for engaging in other’s ideas
• Student did not have much of a detailed response
Jack ate 6 peanut butter sandwiches. He ate 1/6 of a sandwich and decided he didn't want more. How much does Jack have left?
Ms. A: Okay. Who can explain what Yadira did? Who can explain, Cole, who can explain what Yadira did here? Cole, can you come and explain? invita.on
Cole: She took these things (poin.ng to 6 rectangles) and then she did this (mo.oning over the lines dividing one of the rectangle into sixths) so that she can throw away this (poin.ng to the shaded part).
Ms. A: But what did she… what are these? (poin.ng to the 5 wholes in Yadira’s picture) probe
Cole: Wholes. Sandwiches. Ms. A: Those are the sandwiches. Ok. Those are the whole
sandwiches. Yes. And? probe Cole: And then she did that (mo.oning to the lines dividing one
whole into sixths again) so you can see that she colored in one, and that's the one he ate.
Ms. A: So how many does he have leP? Cole: Umm. 5 wholes and 5 sixths sandwiches. Ms. A: Do you agree with Yadira and Cole? (to class)
Teacher support for engaging in other’s ideas
• Student needed support to get to the mathema.cal ideas
The students were in the middle of a conversation about 0/3. The question arose as the students were counting backwards by 1/3 from 4. Ben stated that he thought 0/3 was a whole and should follow 1/3 in counting backwards.
Ms. J: Sara has her hand raised. Sara do you want to add something? Invitation
Sara: I don’t agree with, I wanted to actually kind of come up and, Ben said that 0/3 is a whole. Ben it is not. It is not a whole.
Ms. J: Can you give him any evidence of that? Probe
Sara: It is not a whole because none of the, like like, for 3/3 [she is drawing a picture and all of it is shaded in and walks over to Ben’s picture and shows that none of it is shaded in] None of it is shaded in, in this one.
Ms. J: Well Sara what is this, 3 parts of what [pointing to her picture] Lets make it a real thing, it is easier to talk about scaffold
Sara: 3 part of, pieces of chocolate, a brownie [students in the class are also calling out different things it could be]
Ms. J: So this is chocolate, a, brownie, a tray of brownies, alright brownies. So you are saying that these 3 pieces, this is a whole, a whole brownie that has been what probe
Sara: cut Ms. J: into? Sara: 3 Pieces Ms. J: Ben do you have something to say? invitation Ben: yeah Ms. J: come on up.
Teacher support for engaging in other’s ideas
• Student did not know how to take up the teacher’s invita.on
If Seily has five-thirds liters of soda, what would that look like? Draw and label all parts. Two students have written their solutions on the board
Ms J: Carlos. come on up and explain Daniel’s because you said that yours was more like Daniel’s (she had asked earlier for students to point to the strategy on the board that was like theirs). You said yours was a little more like Daniels. (As Carlos is walking to the board with his paper) Can you explain that one? invitation
Carlos: (walking slowly and pauses) No.
Ms J: You can’t explain this picture (pointing to Daniel’s picture)? Yours is very much like it. positioning
Carlos: (looks at the drawing for about 6 sec) I understand that (points to one part of Daniel’s picture) but not the lines.
Ms J: Oh, Can you ignore the lines and explain the picture? scaffolding
Carlos: Yes
Ms J: Okay
Carlos: What Daniel did, right here (points to his picture) is 5/3 which is one liter (pointing to Daniel’s picture) and 2/3 of a liter (pointing to the picture).
Supporting teachers to engage students in mathematics
• A number of researchers have not only engaged in significant research in this area but they have also begun to support teachers • Mercer and colleagues: rules for par.cipa.on • Gillies and colleagues: communica.on skills • O’Connor and Michaels: Talk moves
• Consistent with • Seymour & Lehrer
• Hueris.c moves vs. specialized version of the move • Kazemi & S.pek
• Norms for press
• Cengiz, Kline & Grant (2011) • Combina.on of instruc.onal ac.ons
• Lampert
Teacher Practice
Student Achievement
0.198 (0.082)
Teacher Practice
Student Achievement
Student Participation 0.323
(0.104) 0.297
(0.092)