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WARM UP PROBLEM A 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: Helping Teachers Move Beyond “Showing and Telling” Mary Kay Stein University of Pittsburgh Northwest Mathematics Conference October 12, 2007

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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 appears in curricular materials Task as it is set up in the classroom 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 = 37.5%

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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 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 a)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%. b)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. c)Six shaded squares out of 40 squares is 6/40 which reduces to 3/20. ONE POSSIBLE RESPONSE

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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 appears in curricular materials Task as it is set up in the classroom 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 student’s 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. Crane’s instruction do you see as promising? What aspects of Mr. Crane’s instruction would you want to help him improve?

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Leaves and Caterpillar Vignette What 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 Vignette What Can Be Improved Beyond having students use different strategies, Mr. Crane’s 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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1.Anticipating (e.g., Fernandez & Yoshida, 2004; Schoenfeld, 1998) 2.Monitoring (e.g., Hodge & Cobb, 2003; Nelson, 2001; Shifter, 2001) 3.Selecting (Lampert, 2001; Stigler & Hiebert, 1999) 4.Sequencing (Schoenfeld, 1998) 5.Connecting (e.g., Ball, 2001; Brendehur & Frykholm, 2000) The Five Practices

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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 Kate’s Solution They added 10 caterpillars, and so I added 10 leaves. 2 caterpillars 12 caterpillars 5 leaves 15 leaves +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: 1.Martin – picture (scaling up) 2.Jamal – table (scaling up) 3.Janine -- picture/written explanation (unit rate) 4.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: 1.Martin – picture (scaling up) 2.Jamal – table (scaling up) 3.Janine -- picture/written explanation (unit rate) 4.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 doesn’t 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 Peg Mary Kay Stein

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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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