Understanding classroom practice: Links among student outcomes, mathematical tasks, teacher practice

1
Understanding classroom practice Links among
student outcomes, mathematical tasks, teacher
practice, and student participation
2
Our Goals

• comprehensive approach to studying classrooms
  will help teachers and researchers continue to
  make sense of classroom practices that better
  support student learning
• (1) student achievement
• (2) the mathematical tasks posed by the teacher
• (3) the practices the teacher uses to move the
  class forward
• (4) student talk in response to the teacher and
  with each other
• and the relationships among these components

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Studying Three Classrooms

• Teachers engaged in two years of ongoing PD
  focused on the development of students algebraic
  thinking
• Videotaped two days, 2 cameras, audio 12 students
• Content focused on equality and relational
  thinking
• Assessed students on content
• Interview and paper/pencil
• Created transcripts with all student talk
  embedded

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Algebra as generalized arithmetic and the study
of relations

• Viewing the equal sign as a relation
• 57 36 ? 34
• Using number relations to simplify calculations
• 5 x 499 ?
• Making explicit general relations based on
  fundamental properties of arithmetic
• 768 39 39 ?

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True/False and Open Number Sentences

• Equality 7 7
• Number Facts 5 5 4 6
• Place Value 250 150 ? 100
• Number Sense 45 100 20 ?
• Mathematical Properties 5 6 6 ?
• Multiplication 3 ? 7 7 7 7
• Equivalence ½ ¼ ¼

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Professional Development Principles

• Learning over sustained periods of time in
  professional communities
• Explicitly connected to teachers work with their
  students
• Focus on students mathematical thinking
• Focus on what students can do.
  Counter-storytelling

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Teacher Findings
Generating strategies for 8 4 ? 5

• No differences in teachers perceptions on time
  spent on algebraic thinking tasks in classrooms
• No differences on knowledge of algebra
• Differences in teachers knowledge of student
  thinking- strategies and relational thinking

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Student Findings

• Students in algebraic thinking classrooms scored
  significantly better on the equality written
  assessment.
• Students in 3rd and 5th grades were twice as
  likely to use relational thinking

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Using Student Outcome Data to Describe Student
Understanding to Make Inferences About Classroom
Practice
Marsha IngGraduate School of Education
Information Studies University of California, Los
Angeles
Paper presented at the 2006 Annual Meeting of the
American Educational Research Association in San
Francisco, CA
This research is funded by CAESL and CRESST.
10
Student Outcome Measures

• School level outcomes consistently low for both
  schools
• But, on measures more closely related to
  instruction, differences in performance between
  the three classrooms
• Written Assessment 13 items, administered to all
  students (n55)
• Individual Interview 3 items, administered to
  target students each classroom (n33)

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Results Written Assessment
12
Results Individual Interview
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Consistency of Student Outcomes

• Similar pattern when focusing on four items
  measuring equality
• 3 4 ? 5
• ? 2 6 1
• 3 6 ? 8
• 5 ? 6 2

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Consistency of Student Outcomes

• Similar pattern when comparing performance on
  other general measures of math achievement
• All four district period assessments quarterly
  assessments designed to measure particular
  grade-specific content standards
• California Standards Test end of year assessment
  designed to measure content standards

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Inconsistency of Student Outcomes

• Teacher 2s students perform higher than Teacher
  1s students on 4 items on the written assessment
  (75 47 25 ?, 98 69 2 ?, 203 105
  ?, 200 99 ?) and 1 item on the individual
  interview (98 69 2 ?)
• These items are computationally challenging
  items.
• Note similar pattern for item on written and
  interview.

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Strategies on 4 written equality items
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Relational Thinking Strategy on 3 Interview Items
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Using Planned and Emergent Mathematical Tasks to
Describe Opportunities for Student Learning and
Classroom Practice
Dan Battey College of Education Arizona State
University Angela ChanGraduate School of
Education Information Studies University of
California, Los Angeles
Paper presented at the 2006 Annual Meeting of the
American Educational Research Association in San
Francisco, CA
This research is funded by DiME.
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Teacher 2

• All tasks were planned
• Disconnected mathematics across tasks
• Most tasks focus on computation
• Multi-digit Multiplication
• Long Division
• Place Value
• Subtraction with Regrouping
• Most tasks require a relational understanding of
  the equal sign
• Given these tasks, its not surprising that her
  students do well on items that require heavier
  computation

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Teacher 1

• Day 1 4 planned tasks / Day 2 10 planned tasks
• Connected mathematics across tasks
• Tasks focus on relational thinking using
  fundamental mathematical properties
• Identity
• Commutativity
• Variables
• Day 1 15 emergent tasks / Day 2 1 emergent task
• All tasks require a relational understanding of
  the equal sign

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Teacher 3

• Day 1 4 planned tasks / Day 2 6 planned tasks
• Connected mathematics across tasks (with
  exceptions)
• Tasks focus on relational thinking using
  fundamental mathematical properties
• Identity
• Commutativity
• Properties of Zero
• Day 1 3 emergent tasks / Day 2 1 emergent task
• All tasks require a relational understanding of
  the equal sign
• All but 3 of the 14 tasks were student generated

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Mathematics Task Summary

• Teachers Varied in Mathematical Focus and
  Consistency Across Tasks
• We might have expected Students in Classrooms 1
  and 3 to perform similarly on test items
• Teacher Generated Tasks for Teachers 1 and 2
  Student Generated for Teacher 3
• Difference in Task Engagement
• e.g. Teacher 1 averaged 230 minutes per task
• How did teachers pose the mathematical tasks?
• What mathematics did teachers and students draw
  out of these tasks?
• What did the mathematical discussion center on?

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Using Teacher Participation Data to Describe
Opportunities for Classroom Practice and Student
Learning
Megan Franke, Julie Kern Schwedtfeger, John
Iwanaga, Deanna FreundGraduate School of
Education Information Studies University of
California, Los Angeles
Paper presented at the 2006 Annual Meeting of the
American Educational Research Association in San
Francisco, CA
This research is funded by DiME
24
Opportunities teachers created for students to
learn mathematics with understanding

• (1) norms each teacher developed
• (2) the ways they posed the mathematical problems
  
• (3) the questions each asked and
• (4) the ways they engaged with students around
  mathematical work

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Developing sociomathematical classroom norms

• Teacher 1 Students talk about their thinking
• Teacher 2 Students provide complete
  explanations and listen to each other
• Teacher 3 Students detail complete
  explanation, understand each others

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Problem Posing

• Teacher 1 poses problems by reading it or saying
  it to them and sometimes having the class read it
  together. She will rephrase the problem in a
  quick sequence for the students. In this quick
  sequence she at times changes the task from
  something more general to a more specific problem
• Teacher 2 begins by asking the students, as a
  class, to read the question out loud. Typically
  it is read once. If the problem is misread in
  some way then she further unpacks the problem.
• Teacher 3 asks students to read the problem to
  her. She calls on individuals and she wants to
  hear all of the different ways you could read the
  problem and she at times follows the different
  versions rendered by the student with questions
  about whether they are getting at the same ideas,
  are these asking the same thing? And at times
  their reading of the problems leads her to ask
  about a particular mathematical idea that a
  reading might raise.

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Questions asked
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Teacher 1 Typical Interaction___ 1 1 200
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Teacher 2 Response to student struggle 50 50
25 ___ 50
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Teacher 2 Typical interaction

• T Ok. Lets come back together. Raise your hand
  if you were able to successfully complete this
  problem. Ok. Alonso and Andre? Do you want to
  come up and tell us how you solved number one?
  Remember that you need to make sure your audience
  is listening and that they can hear you.
• Ss The answer is two.
• T Ok, the missing number is two?
• S Because two ( ). Because seven times two is
  fourteen so if that equals seven this side has to
  equal seven. So three times a number plus one.
  So, three times two equals six plus one equals
  seven. (From notes so if this side equals 7 then
  this side equals 7, 2 goes in the box)
• T Ok. So is there anyone who disagrees with
  Andres and Alonsos explanation on how they
  solved number 1? Wow. Lets give them a silent
  round of applause. (students clap their hands
  silently) Ok. Thank you boys.

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(No Transcript)
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Teacher 3. Typical interaction11 2 10 ___
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Teacher 3 Typical interaction student
incorrect/incomplete response 4 9 5 x 3 -2
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(No Transcript)
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Summary of Teacher Created Opportunities

• All three teachers expect students to solve
  problems in multiple ways and to explain their
  mathematical thinking.
• Teacher 1 asks students to talk to her and to
  each other. Yet, she leads.
• She wants students to talk and share different
  ideas but does not follow through in asking
  students to share different ideas, asking
  specific questions, or completely detailing their
  thinking.
• She listens to what students say and asks genuine
  questions as she often seems to be working to
  figure out the student thinking.
• She doesnt always frame her questions in
  mathematically helpful ways which leads to
  confusion.
• So even though she poses a mathematically sound
  set of problems for her student, many of which
  she creates on the spot, she does not follow
  through with the students around these problems
  in ways that make explicit their mathematical
  thinking or clear up misunderstandings.

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Summary of Teacher Created Opportunities

• Teacher 2 asks for students to detail their
  thinking.
• When students cannot complete an explanation or
  are incorrect she questions, probes moving them
  toward a particular way of thinking about the
  problem.
• She connects mathematical language to what
  students are doing.
• She works within problems on particular
  mathematical ideas, not across problems.
• She poses challenging mathematical tasks that are
  not necessarily connected and works within those.
  
• She provides the most support when students
  struggle and otherwise works to get students to
  listen to each other. She does more with her
  problems than Teacher 1 but she poses few
  problems that actually help with the equal sign
  and relational thinking.

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Summary of Teacher Created Opportunities

• Teacher 3 engages students in the problem posing,
  asks more questions, asks specific questions of
  students mathematical thinking, and expects
  students to not just listen but to understand
  each other.
• She asks questions to get at fundamental
  mathematical ideas.
• She notices in what students say opportunities
  for following up on the ideas. So it is not just
  the specificity of her question but that she
  focuses on or pulls out the central mathematical
  ideas.
• She takes the problems posed and uses those
  problems while engaging with her students to pull
  out both the students thinking and the central
  mathematical ideas.

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Student Discourse, Teacher Practices, and Student
Learning in Mathematics Classrooms
Noreen WebbGraduate School of Education
Information Studies University of California, Los
Angeles
Paper presented at the 2006 Annual Meeting of the
American Educational Research Association in San
Francisco, CA
This research is funded by CAESL and CRESST.
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Student Discourse, Teacher Practices, and Student
Learning in Mathematics Classrooms

• How did student participation in pairshare and
  whole-class settings differ across classrooms?
• How were differences in student participation
  across classrooms linked to differences in
  teacher practices?
• How does examination of student participation
  help us make sense of classroom practices that
  support student learning?

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Student Participation Variables

• General form of interaction among students in a
  pair
• Accuracy of answer
• Level of elaboration of mathematical talk
• Nature of explanation given
• Discrepancies in suggestions and how they were
  resolved
• Questions
• Monitoring of each others work and understanding

3
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Examples of Student Explanations

• Correct (computational) for 9 4 5 x 3 2
• I told him we took 5 times 3 yeah, 5 times 3 and
  then we found it and it made it 15, and 15 take
  away, 15 take away 2 equals 13.
• Correct (relational) for 20 10 10 ?
• I knew that 10 and 10 are the same, and I knew
  that 20 and 20 have to be there. So its like a
  mirror. 10 and 10 are the same and 20 and 20 are
  the same, so theyre equal.
• Ambiguous for ? 1 1 200
• Say this was this one. And this one you have to,
  like, these two are the same sides. And this one
  is going to have to go like that one.
• Incorrect or faulty for 50 50 50 ? 25
• 50. Its just like 50 plus 50. They are kind
  of partners because they are the same but they
  are (unclear).

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General Form of Interaction Between Students in
Pairshare
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Student Thinking Verbalized During Pairshare
6
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Nature of Explanations Verbalized During Pairshare
7
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Nature of Explanations Verbalized During Pairshare
Complete, correct
Incomplete, incorrect
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Highest Degree of Elaboration During Pairshare
9
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Categories of Student Participation During
Pairshare
10
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Student Performance on Written Assessment for
each Student Participation Category
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Student Performance on Individual Interview for
each Student Participation Category
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Typical Pairshare Explanation in Teacher 1s
Class

• Problem 50 50 25 ? 50
• S1 The second one (unclear). I dont know the
  first one. I dont know.
• S2 I know the answer.
• S1 I know the answer.
• S2 I know the answer is 50its 25.
• S1 Its 25 for real. (unclear) And its not 56,
  And not 56. 56 and not56 and not 56. They
  are playing together. They are playing
  together.

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Typical Pairshare Explanation in Teacher 2s
Class

• Problem 14/2 (3 ?) 1
• S1 Three. This isThis is 3. Its a 3 plus 2
  times 7 equals 14. Its a 3 because two times
  seven equals fourteen. So three times three
  equalsno wait.
• S2 I think its four. It equals 1.
• S1 Its a two. Three times two equals six.
  Plus 1 is 7.
• S2 Plus 1 is 7.
• S1 Look, 2 times 7 equals 14 so we put 3 times 2
  equals 6 plus 1 is 7. So right here the answer
  is 7.
• S2 Lets do number two.

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Typical Pairshare Explanation in Teacher 3s
Class

• Problem 11 2 10 ?
• S2 Look.
• S1 No, but I11 plus 2 equals 10 plus 3, huh?
  All I did is, um
• S2 Add 11 plus 2.
• S1 I just added 11 plus 2, and then I, and then
  I saw it was 13, huh. So then I added 10 plus
  3, and I saw it was 13 too. So I pull down a
  number and I put 13, huh. And then suddenly I
  look at they just did one up. Goes from 10,
  next is 11 it goes from 2, next is 3.

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Typical Pairshare Explanation in Teacher 3s
Class (continued)

• S2 Yeah. So you know why I put the lines? Its
  cause if this is a higher number, and this is a
  lower number, and the next one is 2 plus 3,
  this is 11. Cause 11, this is higher, this is
  lower, this is higher, this is lower. So 2, if
  if, it doesnt care if its switched. But 3 plus,
  I put this cause this number is lower and this
  number this number is higher and this number is
  lower.
• S2 And then I say that if this is higher, the
  next one has to be lower. And if this is lower
• S1 11 plus 2 equals 10 plus 3.
• S2 And if this is lower, this has to be higher.
• S1 I know you had told me that cause, cause I
  saw that too. Cause 11 is higher than 10,
  verdad, is higher than 10, y this one, this
  ones lower, this ones gotta be higher than
  this one. So this ones lower and this ones
  higher. And this ones lower and this ones
  higher. Get it?

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Frequency of Questioning, Monitoring, and
Disagreements during Pairshare
How did you know it was 3?
You dont understand it?
The missing number is 7. No, its not.
(see next slide)
55
Resolving Disagreement during Pairshare in
Teacher 3s Class

• Problem 9 4 5 x 3 2
• S1 Thats not true because its 15. Thats 13.
  But this one is not. This is 15.
• S2 I think its not false. You know why?
  Cause look. There is no take away sign.
  Imagine the take away sign there. 9 plus
  4 equals 13 and 5 times 3 equals 15. And 15
  take away 2 equals 13. Get it? Look. That
  equals 13.
• S1 I think its false.
• S2 9 plus 4 equals 13.
• S1 I get it. I get it now. And take away 2
  equals 13 So that equals 13. 15 take away 2
  equals 13. So its true.

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Student Participation in Whole Class Teacher 1
(Typical Interaction Around Correct Student
Response)

• Problem ? 1 1 200
• T Ok student, what do you think needs to go
  inside that box and why?
• S I think 200 is supposed to be in that box.
• T You think 200. Why do you think that?
• S I think 200 is supposed to be in the box
  because 200 on this side and 1 next to it equals
  1 plus 200.
• T Ok, so does it matter where I put So you
  said, 200, right?
• S Yeah.

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Student Participation in Whole Class Teacher 1
(Typical Interaction Around Incorrect/Ambiguous
Student Response)

• Problem 50 50 25 ? 50
• T Who would like to tell me what goes inside
  that box and you need to explain it in front of
  the class. Student? What goes inside that
  box?
• S I think 50.
• T You think 50? Go ahead and write 50 and then
  show us how you know that.
• S I know because it has a pattern. So these
  (unclear).
• T Doesnt need to have a pattern? What do you
  mean?
• S Like it doesnt need to have a partner
  (unclear).

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Student Participation in Whole Class Teacher 1
(Typical Interaction Around Incorrect/Ambiguous
Student Response) continued

• T Well, take a look. Karen helped us. 50 plus
  50 is 100. So isnt this side 100? Can I write
  that? Do you agree? Ok, so how can we get
  to the same amount as 100 on that side? How do
  you know that that side is the same as 100? Can
  you show us by adding or by some sort of
  strategy to show us how you got 50?
• S 50 plus 50 is 100.
• T So this 50 plus this 50 you said is 100. What
  about the 25? Are both sides the same?
• Class No.
• T I see this 100 has a partner but what about
  the 25? It doesnt have a partner, does it? So
  can this answer here be 50 in the box?
• Class No.
• T No. Did someone else think differently?

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Student Participation in Whole Class Teacher 2
(Typical Interaction Around Correct Student
Response)

• Problem 14/2 (3 x ?) 1
• T Ok pair of students, do you want to come up
  and tell us how you solved number one? Remember
  that you need to make sure your audience is
  listening and that they can hear you.
• S The answer is 2.
• T Ok, the missing number is 2?
• S Because 2 (unclear). Because 7 times 2 is 14
  so if that equals 7 this side has to equal 7.
• T Ok. So is there anyone who disagrees with
  pair of students explanation on how the
  solved number one? Wow. Lets give them a
  silent round of applause. Ok. Thank you boys.
  Well said. Is there anyone who worked this
  problem out different?

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Student Participation in Whole Class Teacher 2
(Typical Interaction Around Incorrect/Ambiguous
Student Response)

• Problem 14/2 (3 x ?) 1
• T OK, tell us what you are doing right now.
• S1 We are using the tally strategy.
• T OK, you are using the tally strategy. Why do
  you have fourteen tallies?
• S1 14 divided by 2 is 7.
• T OK, so your quotient is 7.
• S2 And theres seven groups

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Student Participation in Whole Class Teacher 2
(Typical Interaction Around Incorrect/Ambiguous
Student Response)

• T OK, so how does that help you? How does
  knowing that 14 divided by 2 help you? Im
  sorry. How does it help you to know that 14
  divided by 2 is 7? How does that help you solve
  number one?
• S2 Because we counted by twos.
• T OK.
• S1 Its divided by two.
• T But how does it help you knowing that 14
  divided by 2 is 7? Look at number one. Now you
  solved the left hand side of that problem. Now
  you know that the left side is seven. The answer
  to, the quotient, on the left hand side is seven
  and how does that help you with the right hand
  side, with that missing number that you are
  looking for?
• S2 The left side
• S1 Seven.
• T Is there anyone out in the audience that can
  help out?

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Student Participation in Whole Class Teacher 3
(Typical Interaction Around Correct Student
Response)

• Problem 11 2 10 ?
• T Okay, lets lift up your papers. One, two,
  three. And I see a lot of people put the number
  3 in here. Okay. Okay, student, would you
  like to come up?
• S I put 11, then I put 2. And I added the 13,
  and I put zero here. I take away (unclear).
• T You want to do it again? Write a little bit
  bigger, please.
• S I put 11 here, put a 2 right here, then I
  plussed it, and it was 13. I put take away
  (unclear) take away holds fingers up. I write
  13 right here. I put right here a caret and put
  13. I put here a 13. Three, 13 take away, take
  away (unclear) and then I minus. I minus 10.
  And then (unclear).
• T Wow. Okay. This is really interesting.
  Okay, lets look at this. Does everybody
  understand how you got 11 plus 2 equals 13?
• Class Yes.

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Student Participation in Whole Class Teacher 3
(Typical Interaction Around Correct Student
Response)

• T Okay. Why did you minus 10? And where did
  you get that 10 from?
• S Cause on the 3 I added firstI went 3, 4, 5,
  6, 7, 8, 9, 10, 11, 12, 13 (unclear).
• T There. Okay, does anyone see that connection?
• Class Yes.
• T So he adds 11 plus 2. He adds this side
  together to get 13. And the way for him to find
  this unknown is he takes away?
• Class Ten.
• T He takes 10 away. So 13 Teacher 3us 10 equals
  3, and he counted up. Okay, is there another way
  that someone else found that is different from
  his answer?

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Student Participation in Whole Class Teacher 3
(Typical Interaction Around Incorrect/Ambiguous
Student Response)

• Problem 4 9 5 x 3 2 (true or false)
• T Okay, did someone find a different way to do
  this? Okay, Student 1 you thought it was
  false. Can you explain what you thought? Do you
  still think it's false now that she did it?
• S1 Yeah.
• T You still think it's false?
• S1 I thought it was false because 4 plus 9 is
  13, and 5 times 3 is 15.
• T Okay.
• S1 Those two do not match.
• T Okay, say it one more time. 4 plus 9 is 13.
• S1 And 5 times 3 is 15.
• T Is 15? You think it makes 15? You wrote 5.
  Write 15 on the bottom. Okay. So do you guys
  understand why Student 1 thinks it's false?
• Class Yes.

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65
Student Participation in Whole Class Teacher 3
(Typical Interaction Around Incorrect/Ambiguous
Student Response)

• T Because he's looking at it like this. He says
  4 plus 9 -- do you agree with this, Student 2?
  Do you agree with Student 1?
• S2 No.
• T You changed your mind?
• S2 Yeah.
• T Okay, so Student 1 said 4 plus 9 is 13. Five
  times 3 is 15. That's not the same number, so
  it's false. Okay, I can agree with that. Student
  1, what about the minus two? What did you do
  with that?
• S1 Oh!
• T Did you see the minus two?
• S1 I thought because 5 times 3 thatll make it 15
  because 5 times 3 take away 2.
• T So it doesn't make sense? Okay, so what could
  Student 1 do? What could be the next step for
  Student 1? Student 3, what can be the next
  step?
• S3 15 take away 2 equals 13.
• S1 Because Ive never done three 5 times 3 take
  away 2.

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Student Participation in Whole Class Teacher 3
(Typical Interaction Around Incorrect/Ambiguous
Student Response)

• T This is the first time you've done this. Okay,
  so while we continue on to operate the different
  symbols, we can continue on with the math. So
  this is our first time doing it, right? So now
  let's take another look at it. What could you do
  now, now that you know that you can continue?
• S1 Its just like this 3 take away 2.
• T Three take away 2? Okay, but then you added
  this one first. You did this one first. Five
  times 3 is 15. So heres your new sum 15 minus
  what?
• S1 Fifteen minus 2.
• T So what's 15 minus 2?
• S1 Ten.
• T Fifteen minus 2 is 10?
• S1 Seventeen.
• T So now you're adding 15 plus 2. But this one
  says 15
• Class Minus 2.
• S4 So that means take away.
• S1 Oh, 15 minus 2.

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Student Participation in Whole Class Teacher 3
(Typical Interaction Around Incorrect/Ambiguous
Student Response)

• T So put 10 fingers up. So, if I have 15 right
  here and you have to take out two. So now how
  many do you have?
• S1 Thirteen.
• T Thirteen. So now does that equation balance?
• S1 Yeah.
• T Yeah? Yes? Okay, we're going to revisit this
  one, Student 1, because this one was tough
  one. Okay, did anyone find another way to do
  this? Ok, Student 5 last one.
• S5 Its because 9 plus 4 equals 13, and 5 times
  3 equals 15, and you take away 2 from the 15 and
  it equals 13.
• T Okay, does everyone understand that?
• Class Yes.

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68
Student Participation in Whole Class Teacher 3
(Typical Interaction Around Incorrect/Ambiguous
Student Response)

• T Okay. Thank you. We're going to talk about
  this. We're going to talk about when we see
  numbers that have two different signs, which ones
  we do first. Because Student 1 said, well, we
  could do this one first -- 3 minus 2 -- but that
  would change the problem. If I did it 3 minus 2
  first 5 times 3 minus 2. Student 1 said we
  could do 3 minus 2 first. If we already did that
  first, what would this be? What's 3 minus 2?
• Class One.
• T One. And 5 times 1 would be what?
• S6 Five times one equals five.
• S7 Would be 5.
• T Would be 5. And this would be 13, so that
  would make it false. But in math, there are some
  rules that we stick to called order of
  operations, which one we do first. If you see a
  multiplication sign or a subtraction sign, there
  are rules in math that they say you have to do
  the multiplication sign first. Okay, I know we
  spent a lot of time on this problem.

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Summary of Student Participation
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Linking mathematical tasks, teacher practice, and
student participation, and student outcomes

• Teacher 1
• Generated connected mathematical tasks focused on
  relational thinking using fundamental
  mathematical properties and required a
  relational understanding of the equal sign
• Provided little time for students to converse
  about mathematics
• Listened to students and asked genuine questions,
  but did not frame her questions in mathematically
  helpful ways
• Did not ask students to share different ideas or
  detail their thinking
• Students tended not to engage with each other
  around mathematics, tended to give ambiguous or
  faulty explanations
• Students showed low performance on written
  assessment and individual interview

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• Teacher 2
• Assigned tasks focused exclusively on computation
  (not connected mathematically)
• Asked students to detail their thinking
• Provided extensive support when students
  struggled prompting them to think in particular
  mathematical ways
• Students tended not to engage with each other
  around mathematics, and relatively few gave
  explanations
• Explanations, when offered, were correct and
  complete
• Students showed higher performance than students
  in Teacher 1s class, especially on items
  requiring heavier computation

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• Teacher 3
• Generated connected mathematical tasks focused on
  relational thinking using fundamental
  mathematical properties
• Tasks required a relational understanding of the
  equal sign
• Tasks were mostly student generated
• Asked specific questions of students
  mathematical thinking expected students to
  listen and understand each other
• Followed up on student thinking with central
  mathematical ideas
• Students engaged with each other around
  mathematics and gave mostly correct and complete
  relational and computational explanations
• Students in this class scored the highest on the
  assessments