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

A fourth-grade class needs five leaves each day to feedits 2 caterpillars. How many leaves would they needeach day for 12 caterpillars?

Use drawings, words, or numbers to show how you gotyour 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.

Orchestrating Productive Mathematical Discussions of

Student Responses:Helping Teachers Move Beyond

“Showing and Telling”

Mary Kay SteinUniversity of Pittsburgh

Northwest Mathematics Conference October 12, 2007

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

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

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

Levels of Cognitive Demand

High LevelDoing MathematicsProcedures with Connections to Concepts,

Meaning and Understanding

Low LevelMemorizationProcedures without Connections to

Concepts, Meaning and Understanding

Procedures without Connection to

Concepts, Meaning, or Understanding

Convert the fraction to a decimal and percent3

8

3.008 .375 = 37.5%2 4

60

.375

564040

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

“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%

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

“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

Other Possible Shading Configurations

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.

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

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

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

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?

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

Leaves and Caterpillar VignetteWhat 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.

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

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.

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

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

1. Anticipatinglikely 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

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

+10

2. Monitoringstudents’ 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)

3. Selectingstudent 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

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

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)

5. Connectingstudent 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

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)

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

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

For more information about the 5 Practices

Randi Engle raengle@berkeley.edu

Peg Smith pegs@pitt.edu

Mary Kay Stein mkstein@pitt.edu

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

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