Focusing on Challenging Mathematical Tasks: A Strategy for Improving Teaching and Learning

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Focusing on Challenging Mathematical TasksA
Strategy for Improving Teaching and Learning

• Peg Smith
• University of Pittsburgh
• February 15, 2007
• Teachers Development Group
• Leadership Seminar on Mathematics Professional
  Development

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Overview

• Argue for focusing on mathematical tasks
• Discuss the components of the task-based model
  for Professional Development
• Discuss the role of tools in the model
• Present evidence of teacher learning

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Why Focus on Tasks?

• Classroom instruction is generally organized and
  orchestrated around mathematical tasks
• The tasks with which students engage determines
  what they learn about mathematics and how they
  learn it
• The inability to enact challenging tasks well is
  what distinguished teaching in the U. S. from
  teaching in other countries that had better
  student performance on TIMSS

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The Importance of Mathematical Tasks

There is no decision that teachers make that has
a greater impact on students opportunities to
learn, and on their perceptions about what
mathematics is, than the selection or creation of
the tasks with which the teacher engages students
in studying mathematics. Lappan and Briars,
1995
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The Importance of Mathematical Tasks

• Not all tasks are created equal, and different
  tasks will provoke different levels and kinds of
  student thinking.
• Stein, Smith, Henningsen, Silver, 2000

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The Importance of Mathematical Tasks

• The level and kind of thinking in which
  students engage determines what they will
  learn.
• Hiebert, Carpenter, Fennema, Fuson, Wearne,
  Murray, Oliver, Human,1997

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Task-Focused Activities

• Distinguishing between high- and low-level
  mathematics tasks
• Solving high-level mathematical tasks
• Analyzing high-level mathematics tasks and work
  produced by students on these tasks
• Maintaining the cognitive demands of high-level
  tasks during instruction

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Task-Focused Activities

• Distinguishing between high- and low-level tasks
• Develop teachers capacity to determine the kind
  and level of thinking required to solve a
  particular mathematics task
• Comparing pairs of tasks that focus on the same
  mathematics content but different with respect to
  the thinking demands
• Analyzing a set of tasks that differ with respect
  to their cognitive demands and task features
  (e.g., require an explanation, utilize a diagram,
  provide tools such as calculators)

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Distinguishing

• Marthas Carpeting Task
• The Fencing Task

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Marthas Carpeting Task

• Martha was recarpeting her bedroom which was 15
  feet long and 10 feet wide. How many square feet
  of carpeting will she need to purchase?
• Stein, Smith, Henningsen, Silver, 2000, p. 1

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The Fencing Task
Ms. Browns class will raise rabbits for their
spring science fair. They have 24 feet of
fencing with which to build a rectangular rabbit
pen in which to keep the rabbits.

• If Ms. Brown's students want their rabbits to
  have as much room as possible, how long would
  each of the sides of the pen be?
• How long would each of the sides of the pen be if
  they had only 16 feet of fencing?
• How would you go about determining the pen with
  the most room for any amount of fencing?
  Organize your work so that someone else who reads
  it will understand it.
• Stein, Smith, Henningsen, Silver, 2000, p. 2

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Comparing Two Tasks

• Both require prior knowledge of area
• Area problems

• Way in which the area formula is used
• The need to generalize
• The amount of thinking and reasoning required
• The number of ways the problem can be solved
• The range of ways to enter the problem

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Importance of Distinguishing

• Low-Level Tasks
• High-Level Tasks

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Importance of Distinguishing

• Low-Level Tasks
• memorization
• procedures without connections
• High-Level Tasks
• procedures with connections
• doing mathematics

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Importance of Distinguishing

• Low-Level Tasks
• memorization
• procedures without connections (e.g., Marthas
  Carpeting Task)
• High-Level Tasks
• procedures with connections
• doing mathematics (e.g., The Fencing Task)

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Task-Focused Activities

• Solving high-level mathematical tasks
• Develop teachers understanding of mathematical
  ideas, processes, and tools that support learning
  
• Solving challenging mathematical tasks that focus
  on developing understanding of key ideas, that
  use a range of tools, that feature different
  representational forms, and that connect
  procedures with meaning
• Multiplying binomials using algebra tiles
• Using rectangular grids to make sense of the
  connections between fractions, decimals, and
  percents
• Exploring visual patterns and determining the
  connection between the physical and symbolic
  representations

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Task-Focused Activities

• Analyzing high-level mathematics tasks and work
  produced by students on these tasks
• Develop teachers ability to identify the
  mathematical potential of a task and to determine
  what students responses communicate about their
  current mathematical understandings
• Specifying what mathematical ideas could be
  learned from engaging with a particular task and
  what standards could be addressed
• Analyzing students written responses and
  determining what students appear to understand
  about mathematics, and developing questions to
  assess and advance student thinking

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Task-Focused Activities

• Maintaining the cognitive demands of high-level
  tasks during instruction
• Develop teachers awareness of how high-level
  tasks play out in the classroom and the factors
  that support and inhibit students engagement at a
  high level
• Solving tasks and reflecting on and discussing
  how the facilitator supported their learning
• Analyzing narrative cases and identifying what
  the teacher featured in the case did to support
  or inhibit her students learning of mathematics

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Consider and Discuss

• How do you help teachers apply the ideas that
  emerge in professional development sessions in
  their own classrooms?

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Task-Focused Activities
Distinguishing
Solving
Using
Maintaining
TOOLS
Analyzing
Practice-based Professional Development
Classroom Teaching
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Using
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Frameworks and Tools

• provide a focus for professional development
• bring coherence to PD across sessions
• provide a shared language for talking about
  teaching and learning and
• bridge the professional development and K-12
  classroom environments.

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Frameworks and Tools

• Framework
• Tools for
• Analyzing Cognitive Demands (purple)
• Identifying Classroom Influences (gold)
• Planning Lessons (salmon)
• Conferencing after a Lesson
• Talking about and Sharing Teaching Experiences

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The Mathematics Task Framework
TASKS as they appear in curricular/
instructional materials
TASKS as set up by the teachers
TASKS as implemented by students
Student Learning
Stein, Smith, Henningsen, Silver, 2000, p. 4
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The Mathematics Task Framework
TASKS as they appear in curricular/
instructional materials
TASKS as set up by the teachers
TASKS as implemented by students
Student Learning
Stein, Smith, Henningsen, Silver, 2000, p. 4
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The Mathematics Task Framework
TASKS as they appear in curricular/
instructional materials
TASKS as set up by the teachers
TASKS as implemented by students
Student Learning
Stein, Smith, Henningsen, Silver, 2000, p. 4
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The Mathematics Task Framework
TASKS as they appear in curricular/
instructional materials
TASKS as set up by the teachers
TASKS as implemented by students
Student Learning
Stein, Smith, Henningsen, Silver, 2000, p. 4
28
The Mathematics Task Framework
TASKS as they appear in curricular/
instructional materials
TASKS as set up by the teachers
TASKS as implemented by students
Student Learning
Stein, Smith, Henningsen, Silver, 2000, p. 4
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SummaryTask-Based Activities in which Teachers
Engage

• Characterize mathematical tasks based on their
  cognitive demands
• Solve, analyze,and discuss cognitively
  challenging mathematical tasks
• Analyze narrative cases w/r/t the MTF and
  identify the factors that appear to
  support/inhibit students learning
• Use narrative cases to generate issues which
  teachers can explore in their own practice
• Plan, teach, and reflect on lessons based on
  cognitively challenging tasks

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Summary

• A task-based approach to professional development
  provides a focus for work with teachers
• By influencing the tasks that teachers use during
  instruction and the ways in which they enact
  them, there is an opportunity to impact student
  learning
• Tools help the ideas that emerge in professional
  development travel to the classroom and back
• Tools help support enactment of tasks in
  teachers own classrooms and foster conservations
  about teaching between teachers

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Pulling it All Together
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Is there any evidence that suggests that a
task-focused approach to professional development
has an impact on teachers classroom practices?

YES
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ESPSetting the Context

• Workshop focused on selecting and enacting high
  level tasks
• Intended for practicing mathematics teachers 7-12
  with 3 or more years experience
• Use practice-based materials with the Purple book
  (Implementing Standards-Based Reform) as the
  centerpiece
• Assignments link to teachers practices in very
  specific ways and use tools (e.g., TTLP) to
  generalize ideas

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ESPData Collected

• Pre- and post task sorts
• Pre- and post interviews
• Videotapes of all workshop sessions
• Teachers notebooks and all assignments
• All artifacts generated during the course
• Task packets
• Student work packets
• Classroom observations

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ESPWhat Teachers Learned

• Significant increase in teachers ability to
  distinguish between high and low level tasks
  following their participation in the workshops
• Significant increase from fall to spring the
  number of high-level tasks used per teacher over
  the 5-day data collection
• Significant increase in the percent of high-level
  tasks that were maintained during implementation
  from fall to spring
• The PD in which teachers engaged appeared to have
  an influence on teachers' practice, particularly
  with respect to their ability to use and maintain
  high-level tasks in their own classrooms.
• Teachers who showed the most growth over time
  were those who consistently made connections
  between the PD and their own classroom practice.
• Boston, 2006

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For More Information

• pegs_at_pitt.edu

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Results from Data AnalysisThe Task Sort

• Pre- and Post-Workshop Task Sort
• During the first (October) and last (May) session
  in the workshop, teachers were asked to
• classify a set of tasks as High-Level or
  Low-Level
• justify their classification of each task and
• provide a set of criteria for High-Level and
  Low-Level tasks.

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Results from Data AnalysisThe Task Sort

• Pre- and Post-Workshop Task Sort
• Highly significant increase between the pre- and
  post-workshop task sort scores
• Teachers in the workshop scored significantly
  higher on the post-workshop task sort than a
  contrast group of secondary mathematics teachers
  who did not participate in the workshop
• Teachers improved their ability to distinguish
  between high and low level tasks following their
  participation in the workshop.

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Results from Data AnalysisThe Task Sort

• Pre- and Post-Workshop Task Sort
• Improvements in teachers justifications and
  criteria for high and low level tasks
• No inconsistent criteria identified on the post
• i.e., Difficult is High-Level or Use of a
  diagram is Low-Level
• Criteria and justifications closely connected to
  our work in solving and distinguishing tasks in
  the sessions

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Results from Data AnalysisTasks Used in
Teachers Classroom

• Teachers were asked to submit tasks used over
  1-week period in Fall, Winter, and Spring.
• In each data collection, 5 main instructional
  tasks scored using IQA Academic Rigor in
  Mathematics rubric
• Boston Wolfe, 2004 Matsumura et al., 2004
• Score of 1 or 2 Low-level cognitive demands as
  described on Task Analysis Guide
• Score of 3 or 4 High-level cognitive demands as
  described on Task Analysis guide
• Stein, Smith, Silver Henningsen, 2000

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Results from Data AnalysisTasks Used in
Teachers Classroom

• Comparisons of Tasks Used From Fall to Spring
• Significant increases in Task Scores
• Significant increase in overall of H-L tasks
  used
• Significant increase in the number of high-level
  tasks used per teacher over the 5-day data
  collection

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Results from Data AnalysisStudent Work Collected
from Teachers Classroom

• Collections of Student work
• Teachers submitted 3 class-sets of student work
  in the Fall, Winter, and Spring.
• Student-work was scored using the IQA rubric for
  student work Implementation
• Scale of 1 to 4
• Score levels based on Task Analysis Guide
• Low-Level lt 2
• High-Level gt 3

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Results from Data Analysis Student Work
Collected from Teachers Classroom

• Comparisons of Student-Work Implementation Scores
  from Fall to Spring
• Significant increase in mean scores
• Significant increase in number of high-level
  student work implementations
• Significantly less occurrences of decline of
  high-level cognitive demands

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Results from Data Analysis Student Work
Collected from Teachers Classroom

• In all data collections,
• Implementation scores were lower than task
  scores.
• The number of high-level implementations per
  teacher is lower than the number of high-level
  tasks used per teacher.
• These findings indicate a persistent trend of
• decline in the level of cognitive demands.

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Results from Data Analysis Student Work
Collected from Teachers Classroom

• Was the decline in the level of cognitive demands
  significant?
• Fall and Winter Implementation scores were
  highly significantly lower than task scores
• Spring Implementation scores were not
  significantly lower than Task scores
• Increase in of high-level implementations per
  teacher
• Increase in of high-level tasks that were
  maintained during implementation

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Results from Data Analysis Observation of
Teachers Classroom

• Eleven teachers participated in 1 classroom
  observation per data collection
• Marginally significant increase in lesson
  implementation from Fall to Spring
• Teachers significantly more likely to maintain
  high-level cognitive demands during
  implementation in Spring than in Fall or Winter.
• In Spring, teachers implemented tasks at a
  significantly higher level than teachers in a
  contrast group who did not participate in the
  workshop

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Connecting Professional Development to Teachers
Practice

• Focusing on a particular factor (light gold
  sheet) they wanted to work on and use the factor
  as a lens for reflecting on classroom instruction
• Expectations
• Teachers would teach a lesson based on a
  high-level mathematical task of their choice
• Teachers would select several pieces of work
  produced by students during the lesson that they
  felt accurately reflect the lesson
• Teachers would bring blinded copies of the
  student work to share

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Rick CarsonsCase Story

• Teacher
• 9 years of experience
• First time student work has been shared in this
  format
• School Setting
• 10th grade
• Integrated curriculum in a probability unit

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Case StoriesStorytelling through Student Work

• Storyteller will distribute a complete set of
  student work to each team member without comment.
• The team members will individually review the
  work in silence.
• 5 minutes allowed

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Case StoriesStorytelling through Student Work

• The team should share what they saw in their
  review of the student work.
• Only factual statements can be made. Do not
  share your evaluation of the work, or statements
  of personal preference.
• Start comments with, I noticed that
• The storyteller is quietly listening and making
  note of statements.
• 5 minutes is allowed

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Case StoriesStorytelling through Student Work

• The team should share questions they have about
  the student work.
• Responses should be in the form, Im wondering
• The storyteller should make note of the
  wonderings, and should continue to remain quiet.
• 5 minutes is allowed.

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Rick Carson Shares His Perspective
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Case Stories - What Emerged

• The Facilitator started
• Id like to talk about the extent to which the
  student work facilitated your discussions about
  teaching. If you recall, last month when we did
  this, we told stories, but we didnt have any
  artifacts, we didnt have any student work or
  anything else that came directly from the lesson.
  So Im wondering to what extent, and in what
  ways, the student work actually might have
  facilitated the discussion?

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Rick Carsons Response

• When we saw so many papers, you could see a lot
  of uniformity between what the kids were doing
  and that led us into a discussion about how much
  did the teacher lead the discussion, how many
  hints did you give them, and whether thats a
  good thing or a bad thing.