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 [email protected]
Peg Smith [email protected]
Mary Kay Stein [email protected]
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