Transcript Orchestrating Productive Mathematical Discussions

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.
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
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
Stein, Grover, & Henningsen, 1996
Task as it
is enacted
in the
classroom
Student
Learning
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

Procedures without Connection to
Concepts, Meaning, or Understanding
3
Convert the fraction to a decimal and percent
8
.375
8 3.00
24
60
56
40
40
.375 = 37.5%
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
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
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
c) Fractional part of the
area that is shaded
which reduces to 3/20.
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 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
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.
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
The Five Practices
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)
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
Leaves and Caterpillar Vignette
Missy and Kate’s Solution
They added 10 caterpillars, and so I added
10 leaves.
+10
2 caterpillars
12 caterpillars
+10
5 leaves
15 leaves
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)
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
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.
2.
3.
4.
Martin – picture (scaling up)
Jamal – table (scaling up)
Janine -- picture/written explanation (unit rate)
Jason -- written explanation (scale factor)
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
Leaves and Caterpillar Vignette
Possible Connections:
1.
2.
3.
4.
Martin – picture (scaling up)
Jamal – table (scaling up)
Janine -- picture/written explanation (unit rate)
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