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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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**Northwest Mathematics Conference**

October 12, 2007 Orchestrating Productive Mathematical Discussions of Student Responses: Helping Teachers Move Beyond “Showing and Telling” Mary Kay Stein University of Pittsburgh A key challenge mathematics teachers face in enacting current reforms is to orchestrate discussions that use students’ responses to instructional tasks in ways that advance the mathematical learning of the whole class. In particular, teachers are often faced with a wide array of student responses to complex tasks and must find a way to use them to guide the class towards deeper understandings of significant mathematics. We are proposing a model for effective use of student thinking in whole-class discussions that we think has the potential to make such teaching manageable for more teachers AND HELP TEACHERS ORCHESTRATE DISCUSSIONS THAT MOVE BEYOND SHOWING AND TELLING. Talk is based on a paper that we have written in collaboration with Mary Kay Stein and Elizabeth Hughes.

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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 Here, we draw on my earlier work (Stein, Grover & Henningsen, 1996) to frame the sets of possibilities regarding how and where curriculum might be altered as it makes its way from the pages of a textbook to student learning. This framework depicts the various phases that an instructional task goes through: first as it appears on the pages of a written curriculum, then as the teacher announces or sets up the task inside the classroom, and, finally, as the task is actually enacted in the classroom by students and the teacher. It is this final phase--enactment in the classroom--that is most critical, because this phase represents how students actually engage with the task and hence their opportunities to learn what was intended. Often, the features of an instructional task, especially its cognitive demands, change as the task passes through these phases. A common change is for tasks to begin with a high level of cognitive demand but to then slip--in terms of the levels of cognitive processing that kids actually engage in--during the enactment phase. In order to track changes in cognitive demand, tasks can be classified as shown in the next slide. 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 Tasks can be identified as either high or low-level. There are two kinds of high level tasks: Doing mathematics tasks and Procedure-with-Connections tasks. The differences between these two will be explained shortly. The two kinds of low-level cognitive demand are procedures-withOUT-connections to underlying meaning or concepts and memorization. Memorization tasks are self-explanatory. Procedures-without-connections tasks are activities that ask students to perform a set of routinized procedures without knowing why they are doing them or anything about the underlying meaning associated with the operations that they are performing.

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**Procedures without Connection to Concepts, Meaning, or Understanding**

3 8 Convert the fraction to a decimal and percent .375 8 3.00 .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 Here, we draw on my earlier work (Stein, Grover & Henningsen, 1996) to frame the sets of possibilities regarding how and where curriculum might be altered as it makes its way from the pages of a textbook to student learning. This framework depicts the various phases that an instructional task goes through: first as it appears on the pages of a written curriculum, then as the teacher announces or sets up the task inside the classroom, and, finally, as the task is actually enacted in the classroom by students and the teacher. It is this final phase--enactment in the classroom--that is most critical, because this phase represents how students actually engage with the task and hence their opportunities to learn what was intended. Often, the features of an instructional task, especially its cognitive demands, change as the task passes through these phases. A common change is for tasks to begin with a high level of cognitive demand but to then slip--in terms of the levels of cognitive processing that kids actually engage in--during the enactment phase. In order to track changes in cognitive demand, tasks can be classified as shown in the next slide.

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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? We are going to use a vignette to help ground our discussion of the model we are proposing. We will begin by reading a short vignette and considering these two questions. Read Leaves and Caterpillar Vignette

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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 Many reform-like aspects of the instruction in Mr. Crane’s class. -- go through the bullets One aspect of his instruction that we might want to help him think about is the whole class discussion. The sharing out of solutions has the feel of “show and tell” where there is little filtering on the part of the teacher regarding what was selected for presentation; no assistance is provided with respect to drawing connections among methods or tying them to widely shared disciplinary methods or concepts. Ball has argued that the field needs to take responsibility for helping teachers learn how to continually “size up” whether important mathematical ideas are being developed and when and how to step in and redirect the conversation if needed.

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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. Many aspects of the instruction in Mr. Crane’s class that need work…. One aspect of his instruction that we might want to help him think about is the whole class discussion. The sharing out of solutions has the feel of “show and tell” where there is little filtering on the part of the teacher regarding what was selected for presentation; no assistance is provided with respect to drawing connections among methods or tying them to widely shared disciplinary methods or concepts. Ball has argued that the field needs to take responsibility for helping teachers learn how to continually “size up” whether important mathematical ideas are being developed and when and how to step in and redirect the conversation if needed.

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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. The practices we have identified are meant to make student-centered instruction more manageable by moderating the degree of improvisation required by the teacher during a discussion. Rather than focusing on in-the-moment responses to students contributions, the practices instead emphasize the importance of planning. Through planning, teachers can anticipate likely student contributions, prepare responses they might make to them, and make decisions about how to structure students’ presentations to further their mathematical agenda for the lesson.

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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) The five practices are: Anticipating likely student responses to mathematical tasks While students working on the tasks (in pairs or small groups), Monitoring students’ actual responses to the tasks Selecting particular students to present their mathematical responses during the whole class discussion Purposefully sequencing when these student responses are shared during the discussion Helping the class make mathematical connections between different students’ responses As you can see, each of these has been discussed separately by various authors; our contribution here is to integrate them into a single package.

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**likely student responses to mathematical problems**

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 The first practice is for teachers to make an effort to actively envision how students might mathematically approach the instructional task (s) that they will be asked to work on. This involves much more than simply evaluating whether a task will be at the right level of difficulty or of sufficient interest to students, and it goes beyond considering whether or not they are getting the ‘right answer.’ [Click to INVOLVES] Anticipating students’ responses involves developing considered expectations about how students might mathematically interpret a problem, the array of strategies—both correct and incorrect—they might use to tackle it, and how those strategies and interpretations might relate to the mathematical concepts, representations, procedures, and practices that the teacher would like his or her students to learn. [Click to SUPPORTED and read/explain, making reference to their experience solving the caterpillar problem as helping them make sense of the vignette and the student work] Returning to the Leaves and Caterpillars vignette, [NEXT SLIDE]

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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 caterpillars 5 leaves leaves +10 For example we would want a teacher like Mr. Crane to recognize that the response given by Missy and Kate as a common misconception students have – identifying the relationship between quantities here as additive not multiplicative. Anticipating this in advance would have made it possible for Mr. Crane to have a question ready to ask that might have helped these and other students recognize why this approach, though tempting, doesn’t work. +10

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**students’ actual responses during independent work**

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) [Click INVOLVES] Monitoring student responses involves paying close attention to the mathematical thinking that students actually use as they work on the problem. Commonly, this is done by circulating around the classroom while students work . Lampert summarizes it – “If I watch and listen during small-group independent work, I am able to use my observations to decide what and who to make focal” during the discussion that follows [Click SUPPORTED and read quickly, referring to handout ***REFER TO BY COLOR ONCE I KNOW IT***] Returning to Leaves and Caterpillar Vignette, While the teacher understood who got correct answers and who did not, and that a range of strategies had been used, his sharing at the end of the class suggests he had not particularly monitored the specific mathematical learning potential available in any of the responses. [IF TIME] Targeting responses in advance just makes it easier to hone in on the math during the discussion since it is not totally improvisational.

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**student responses to feature during discussion**

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 [Click INVOLVES] Having monitored the available student strategies in the class, the teacher can then select particular students to share their work with the rest of the class in order to get “particular piece[s] of mathematics on the table,” thus giving the teacher more control over the discussion as well as more time to plan A typical way to do this is to call on specific students (or groups of students) to present their work as the discussion proceeds. Alternatively, the teacher may let students know in advance of the discussion that they will be presenting their work. [If time] In a hybrid variety, a teacher might ask for volunteers but then select a particular student that he or she knows is one of several who has a particularly useful idea to share with the class. This is one way of balancing the tension between “keeping the discussion on track and allowing students to make spontaneous contributions that they consider…to be relevant.” [Click SUPPORTED and quickly say what’s there] Returning to Leaves and Caterpillar Vignette IF we look at the strategies that were shared we note that Kyra and Janine had similar strategies that used the idea of unit rate – finding out the # of leaves needed for one caterpillar. Given that, there may not have been any added mathematical value to sharing both. In fact, if Mr. Crane wanted to students to see the multiplicative nature of the relationship, he might have selected Janine as her approach clearly involved multiplication. Also, there may have been some payoff in sharing the solution produced by Missy and Kate and contrasting it with the solution produced by Melissa which also used addition.

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**student responses during the discussion**

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 [Click INVOLVES & SUPPORTED BY] Having selected particular students to present, the teacher can then make decisions about how to sequence the students’ presentations. By making purposeful choices about the order in which students’ work is shared, teachers can maximize the chances that their mathematical goals for the discussion will be achieved. - For example, the teacher might want to have the strategy used by the majority of students presented before those that only a few students used in order to help validate the work that students did and make the beginning of the discussion accessible to as many students as possible. - Or, if there is a common misconception that underlies a strategy that several students used, the teacher might want to have it addressed first so the class can clear up that misunderstanding in order to be able to work on developing more successful ways of tackling the problem. - In addition, the teacher might want to have related or contrasting strategies be presented right after one another in order to make it easier for the class to mathematically compare them. More research needs to be done to compare the affordances of different sequencing methods, but we want to emphasize here that particular sequences can be used to advance particular goals for a lesson. Returning to the Leaves and Caterpillar Vignette, for example, here’s one possible sequence that could have been used: [NEXT SLIDE]

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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) This ordering begins with the least sophisticated (scaling up) strategy and ends with the most sophisticated (scale factor) strategy, which would help with the goal of accessibility. In addition, by having the same relatively accessible strategy--scaling up--be embodied in two different representations could help with the goal of having a class better understand how the same strategy can be embodied in different representations.

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**student responses during the discussion**

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 [Click INVOLVES] Finally, teachers can help students draw connections between their mathematical ideas as represented in the strategies and representations that they use . They can help students to make judgments about the consequences of different approaches for: the range of problems that can be solved, one’s likely accuracy and efficiency in solving them, and the kinds of mathematical patterns that can be most easily discerned. So, rather than having mathematical discussions consist of separate presentations of different ways to solve a particular problem, the goal is to have student presentations build on each other to develop powerful mathematical ideas. [Click SUPPORTED and read them out] For example in the Leaves and Caterpillar Vignette, NEXT SLIDE]

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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) So if we had this ordering we might also want students to compare Jamal and Janine’s Reponses and see where Janine’s unit rate is in Jamal’s table. Or they could compare Jason’s work with Jamal and Martin’s to see if the scale factor of 6 can be seen in each of their tabular and pictorial representations. So if a teacher’s goal for this lesson was understanding different approaches – scale factor, scaling up, and unit rate-- you could identify each of these ideas in each representation.

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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 [Relate to Mr. Crane ] “teachers like Mr. Crane”

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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 Engle Peg Smith 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 Briefly talk about the findings but don’t get too detailed. ***WE NEED TO SOMEHOW ADDRESS THE ISSUE OF WHETHER OR NOT WE KNOW THAT TEACHERS’ CLASSROOM PRACTICES CHANGED

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