WARM UP PROBLEM A copy of the problem appears on the blue handout.

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WARM UP PROBLEMA copy of the problem appears on
the blue handout.

• A fourth-grade class needs five leaves each day
  to feed
• its 2 caterpillars. How many leaves would they
  need
• each day for 12 caterpillars?
• Use drawings, words, or numbers to show how you
  got
• your answer.
• Please try to do this problem in as many ways as
  you can, both correct and incorrect. What might
  a 4th grader do?
• If done, share your work with a neighbor or look
  at the student work in your handout.

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Orchestrating Productive Mathematical Discussions
of Student Responses
Northwest Mathematics Conference October 12,
2007

• Helping Teachers Move Beyond Showing and
  Telling
• Mary Kay Stein
• University of Pittsburgh

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Overview

• The challenge of cognitively demanding tasks
• The importance and challenge of facilitating a
  discussion
• A description of 5 practices that teachers can
  learn in order to facilitate discussions more
  effectively

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Overview

• The challenge of cognitively demanding tasks
• The importance and challenge of facilitating a
  discussion
• A description of 5 practices that teachers can
  learn in order to facilitate discussions more
  effectively

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Mathematical Tasks Framework
Task as it is set up in the classroom
Task as it appears in curricular materials
Task as it is enacted in the classroom
Student Learning
Stein, Grover, Henningsen, 1996
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Levels of Cognitive Demand

• High Level
• Doing Mathematics
• Procedures with Connections to Concepts, Meaning
  and Understanding
• Low Level
• Memorization
• Procedures without Connections to Concepts,
  Meaning and Understanding

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Procedures without Connection to Concepts,
Meaning, or Understanding

• Convert the fraction to a decimal and percent

.375
3.00
8
.375 37.5
2 4
60
56
40
40
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Hallmarks of Procedures Without Connections
Tasks

• Are algorithmic
• Require limited cognitive effort for completion
• Have no connection to the concepts or meaning
  that underlie the procedure being used
• Are focused on producing correct answers rather
  than developing mathematical understanding
• Require no explanations or explanations that
  focus solely on describing the procedure that was
  used

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Procedures with Connections Tasks
Using a 10 x 10 grid, identify the decimal and
percent equivalent of 3/5. EXPECTED
RESPONSE Fraction 3/5 Decimal 60/100
.60 Percent 60/100 60
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Hallmarks of PwithC Tasks

• Suggested pathways have close connections to
  underlying concepts (vs. algorithms that are
  opaque with respect to underlying concepts)
• Tasks often involve making connections among
  multiple representations as a way to develop
  meaning
• Tasks require some degree of cognitive effort
  (cannot follow procedures mindlessly)
• Students must engage with the concepts that
  underlie the procedures in order to successfully
  complete the task

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Doing Mathematics Tasks
ONE POSSIBLE RESPONSE
Shade 6 squares in a 4 x 10 rectangle. Using the
rectangle, explain how to determine each of the
following a) Percent of area that is shaded b)
Decimal part of area that is shaded c) Fractional
part of the area that is shaded

• Since there are 10 columns, each column is 10 .
  So 4 squares 10. Two squares would be 5. So
  the 6 shaded squares equal 10 plus 5 15.
• One column would be .10 since there are 10
  columns. The second column has only 2 squares
  shaded so that would be one half of .10 which is
  .05. So the 6 shaded blocks equal .1 plus .05
  which equals .15.
• Six shaded squares out of 40 squares is 6/40
  which reduces to 3/20.

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Other Possible Shading Configurations
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Hallmarks of DM Tasks

• There is not a predictable, well-rehearsed
  pathway explicitly suggested
• Requires students to explore, conjecture, and
  test
• Demands that students self monitor and regulated
  their cognitive processes
• Requires that students access relevant knowledge
  and make appropriate use of them
• Requires considerable cognitive effort and may
  invoke anxiety on the part of students

Requires considerable skill on the part of the
teacher to manage well.
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High Level Tasks often Decline from Set Up to
Enactment Phase
Task as it is set up in the classroom
Task as it appears in curricular materials
Task as it is enacted in the classroom
Student Learning
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Overview

• The challenge of cognitively demanding tasks
• The importance and challenge of facilitating a
  discussion
• A description of 5 practices that teachers can
  learn in order to facilitate discussions more
  effectively

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The Importance of Discussion

• Mathematical discussions are a key part of
  keeping doing mathematics tasks at a high level
  
• Goals of mathematics discussions
• To encourage student construction of mathematical
  ideas
• To make students thinking public so it can be
  guided in mathematically sound directions
• To learn mathematical discourse practices

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Leaves and Caterpillar Vignette

• What aspects of Mr. Cranes instruction do you
  see as promising?
• What aspects of Mr. Cranes instruction would you
  want to help him improve?

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Leaves and Caterpillar VignetteWhat is Promising

• Students are working on a mathematical task that
  appears to be both appropriate and worthwhile
• Students are encouraged to provide explanations
  and use strategies that make sense to them
• Students are working with partners and publicly
  sharing their solutions and strategies with peers
• Students ideas appear to be respected

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Leaves and Caterpillar VignetteWhat Can Be
Improved

• Beyond having students use different strategies,
  Mr. Cranes goal for the lesson is not clear
• Mr. Crane observes students as they work, but
  does not use this time to assess what students
  seem to understand or identify which aspects of
  students work to feature in the discussion in
  order to make a mathematical point
• There is a show and tell feel to the
  presentations
• not clear what each strategy adds to the
  discussion
• different strategies are not related
• key mathematical ideas are not discussed
• no evaluation of strategies for accuracy,
  efficiency, etc.

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How Expert Discussion Facilitation is
Characterized

• Skillful improvisation
• Diagnose students thinking on the fly
• Fashion responses that guide students to evaluate
  each others thinking, and promote building of
  mathematical content over time
• Requires deep knowledge of
• Relevant mathematical content
• Student thinking about it and how to diagnose it
• Subtle pedagogical moves
• How to rapidly apply all of this in specific
  circumstances

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Purpose of the Five Practices

• To make student-centered instruction more
  manageable by moderating the degree of
  improvisation required by the teachers and during
  a discussion.

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Overview

• The challenge of cognitively demanding tasks
• The importance and challenge of facilitating a
  discussion
• A description of 5 practices that teachers can
  learn in order to facilitate discussions more
  effectively

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The Five Practices

• Anticipating (e.g., Fernandez Yoshida, 2004
  Schoenfeld, 1998)
• Monitoring (e.g., Hodge Cobb, 2003 Nelson,
  2001 Shifter, 2001)
• Selecting (Lampert, 2001 Stigler Hiebert,
  1999)
• Sequencing (Schoenfeld, 1998)
• Connecting (e.g., Ball, 2001 Brendehur
  Frykholm, 2000)

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1. Anticipating
likely student responses to mathematical problems

• It involves developing considered expectations
  about
• How students might interpret a problem
• The array of strategies they might use
• How those approaches relate to the math they are
  to learn
• It is supported by
• Doing the problem in as many ways as possible
• Doing so with other teachers
• Drawing on relevant research
• Documenting student responses year to year

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Leaves and Caterpillar Vignette

• Missy and Kates Solution
• They added 10 caterpillars, and so I added 10
  leaves.
• 2 caterpillars 12 caterpillars
• 5 leaves 15 leaves

10
10
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2. Monitoring
students actual responses during independent work

• It involves
• Circulating while students work on the problem
• Recording interpretations, strategies, other
  ideas
• It is supported by
• Anticipating student responses beforehand
• Carefully listening and asking probing questions
• Using recording tools (see handout)

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3. Selecting
student responses to feature during discussion

• It involves
• Choosing particular students to present because
  of the mathematics available in their responses
• Gaining some control over the content of the
  discussion
• Giving teacher some time to plan how to use
  responses
• It is supported by
• Anticipating and monitoring
• Planning in advance which types of responses to
  select

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4. Sequencing
student responses during the discussion

• It involves
• Purposefully ordering presentations to facilitate
  the building of mathematical content during the
  discussion
• Need empirical work comparing sequencing methods
• It is supported by
• Anticipating, monitoring, and selecting
• During anticipation work, considering how
  possible student responses are mathematically
  related

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Leaves and Caterpillar Vignette

• Possible Sequencing
• Martin picture (scaling up)
• Jamal table (scaling up)
• Janine -- picture/written explanation (unit rate)
• Jason -- written explanation (scale factor)

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5. Connecting
student responses during the discussion

• It involves
• Encouraging students to make mathematical
  connections between different student responses
• Making the key mathematical ideas that are the
  focus of the lesson salient
• It is supported by
• Anticipating, monitoring, selecting, and
  sequencing
• During planning, considering how students might
  be prompted to recognize mathematical
  relationships between responses

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Leaves and Caterpillar Vignette

• Possible Connections
• Martin picture (scaling up)
• Jamal table (scaling up)
• Janine -- picture/written explanation (unit rate)
• Jason -- written explanation (scale factor)

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Why These Five Practices Likely to Help

• Provides teachers with more control
• Over the content that is discussed
• Over teaching moves not everything improvisation
• Provides teachers with more time
• To diagnose students thinking
• To plan questions and other instructional moves
• Provides a reliable process for teachers to
  gradually improve their lessons over time

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Why These Five Practices Likely to Help

• Honors students thinking while guiding it in
  productive, disciplinary directions (Engle
  Conant, 2002)
• Key is to support students disciplinary
  authority while simultaneously holding them
  accountable to discipline
• Guidance done mostly under the radar so doesnt
  impinge on students growing mathematical
  authority
• At same time, students led to identify problems
  with their approaches, better understand
  sophisticated ones, and make mathematical
  generalizations
• This fosters students accountability to the
  discipline

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For more information about the 5 Practices Randi
Engle raengle_at_berkeley.edu Peg Smith pegs_at_pitt.edu
Mary Kay Stein mkstein_at_pitt.edu
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A Course In Which Teachers Could Learn About the
Five Practices

• Math education course about proportionality
• For 17 secondary and elementary teachers
• Preservice and early inservice
• Learned about content and pedagogy in tandem
• Practice-based materials tasks, student work,
  cases
• Opportunities to learn about the five practices
• Discussion of detailed case illustrating them
• Modeling of practices by instructor
• Lesson planning assignment

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Evidence Teachers May Have Learned About the Five
Practices

• Changes in response to pre/post pedagogical
  scenarios
• References to them in relevant case analysis
  papers
• Salient enough to mention in exit interviews

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