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

Focusing on Challenging Focusing on Challenging Mathematical Tasks:Mathematical Tasks:

A Strategy for Improving A Strategy for Improving Teaching and LearningTeaching and Learning

Peg SmithPeg Smith

University of PittsburghUniversity of Pittsburgh

February 15, 2007February 15, 2007

Teachers’ Development Group

Leadership Seminar on Mathematics Professional Development

OverviewOverview

Argue for focusing on mathematical tasksArgue for focusing on mathematical tasks

Discuss the components of the task-based model Discuss the components of the task-based model for Professional Developmentfor Professional Development

Discuss the role of tools in the modelDiscuss the role of tools in the model

Present evidence of teacher learningPresent evidence of teacher learning

Why Focus on Tasks?Why Focus on Tasks?

Classroom instruction is generally organized Classroom instruction is generally organized and orchestrated around mathematical tasksand orchestrated around mathematical tasks

The tasks with which students engage The tasks with which students engage determines what they learn about mathematics determines what they learn about mathematics and how they learn it and how they learn it

The inability to enact The inability to enact challengingchallenging tasks well is tasks well is what distinguished teaching in the U. S. from what distinguished teaching in the U. S. from teaching in other countries that had better teaching in other countries that had better student performance on TIMSS student performance on TIMSS

“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

The Importance of Mathematical The Importance of Mathematical TasksTasks


““Not all tasks are created equal, and Not all tasks are created equal, and different tasks will provoke different different tasks will provoke different levels and kinds of student thinkinglevels and kinds of student thinking.”.”

Stein, Smith, Henningsen, & Silver, 2000Stein, Smith, Henningsen, & Silver, 2000


““The level and kind of thinking in which The level and kind of thinking in which students engage determines what they students engage determines what they will learn.”will learn.”

Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human,Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human,19971997

Task-Focused ActivitiesTask-Focused Activities

DistinguishingDistinguishing between high- and low- between high- and low-level mathematics taskslevel mathematics tasks

SolvingSolving high-level mathematical taskshigh-level mathematical tasks AnalyzingAnalyzing high-level mathematics tasks high-level mathematics tasks

and work produced by students on these and work produced by students on these taskstasks

MaintainingMaintaining the cognitive demands of the cognitive demands of high-level tasks during instructionhigh-level tasks during instruction


DistinguishingDistinguishing between high- and low-level between high- and low-level taskstasks Develop teachers’ capacity to determine the Develop teachers’ capacity to determine the

kind and level of thinking required to solve a kind and level of thinking required to solve a particular mathematics taskparticular mathematics task Comparing pairs of tasksComparing pairs of tasks that focus on the same that focus on the same

mathematics content but different with respect to the mathematics content but different with respect to the thinking demands thinking demands

Analyzing a set of tasks that differ with respect to Analyzing a set of tasks that differ with respect to their cognitive demands and task features (e.g., their cognitive demands and task features (e.g., require an explanation, utilize a diagram, provide require an explanation, utilize a diagram, provide tools such as calculators)tools such as calculators)

DistinguishingDistinguishing

Martha’s Carpeting TaskMartha’s Carpeting Task

The Fencing TaskThe Fencing Task

Martha’s Carpeting TaskMartha’s Carpeting Task

Martha was recarpeting her bedroom Martha was recarpeting her bedroom which was 15 feet long and 10 feet wide. which was 15 feet long and 10 feet wide. How many square feet of carpeting will How many square feet of carpeting will she need to purchase?she need to purchase?

Stein, Smith, Henningsen, & Silver, 2000, p. 1Stein, Smith, Henningsen, & Silver, 2000, p. 1

The Fencing TaskThe Fencing Task

1.1. If Ms. Brown's students want their rabbits to have as much room 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?as possible, how long would each of the sides of the pen be?

2.2. How long would each of the sides of the pen be if they had only How long would each of the sides of the pen be if they had only 16 feet of fencing?16 feet of fencing?

3.3. How would you go about determining the pen with the most How would you go about determining the pen with the most room for any amount of fencing? Organize your work so that room for any amount of fencing? Organize your work so that someone else who reads it will understand it.someone else who reads it will understand it.

Stein, Smith, Henningsen, & Silver, 2000, p. 2Stein, Smith, Henningsen, & Silver, 2000, p. 2

Ms. Brown’s 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.

Comparing Two TasksComparing Two Tasks

Both require prior Both require prior knowledge of areaknowledge of area

Area problemsArea problems

Way in which the area Way in which the area formula is used formula is used

The need to generalizeThe need to generalize The amount of thinking The amount of thinking

and reasoning required and reasoning required The number of ways The number of ways

the problem can be the problem can be solvedsolved

The range of ways to The range of ways to enter the problementer the problem

Importance of DistinguishingImportance of Distinguishing

Low-Level TasksLow-Level Tasks

High-Level TasksHigh-Level Tasks


Low-Level TasksLow-Level Tasks memorizationmemorization procedures without connectionsprocedures without connections

High-Level TasksHigh-Level Tasks procedures with connectionsprocedures with connections doing mathematicsdoing mathematics


Low-Level TasksLow-Level Tasks memorizationmemorization procedures without connections procedures without connections

(e.g., (e.g., Martha’s Carpeting TaskMartha’s Carpeting Task))

High-Level TasksHigh-Level Tasks procedures with connectionsprocedures with connections doing mathematics doing mathematics

(e.g., (e.g., The Fencing TaskThe Fencing Task))


Solving Solving high-level mathematical taskshigh-level mathematical tasks Develop teachers’ understanding of mathematical Develop teachers’ understanding of mathematical

ideas, processes, and tools that support learning ideas, processes, and tools that support learning Solving challenging mathematical tasks that focus on Solving challenging mathematical tasks that focus on

developing understanding of key ideas, that use a range of developing understanding of key ideas, that use a range of tools, that feature different representational forms, and that tools, that feature different representational forms, and that connect procedures with meaningconnect procedures with meaning Multiplying binomials using algebra tilesMultiplying binomials using algebra tiles

Using rectangular grids to make sense of the connections Using rectangular grids to make sense of the connections between fractions, decimals, and percentsbetween fractions, decimals, and percents

Exploring visual patterns and determining the connection Exploring visual patterns and determining the connection between the physical and symbolic representationsbetween the physical and symbolic representations


AnalyzingAnalyzing high-level mathematics tasks and high-level mathematics tasks and work produced by students on these taskswork produced by students on these tasks Develop teachers’ ability to identify the mathematical Develop teachers’ ability to identify the mathematical

potential of a task and to determine what students’ potential of a task and to determine what students’ responses communicate about their current mathematical responses communicate about their current mathematical understandingsunderstandings

Specifying what mathematical ideas could be learned Specifying what mathematical ideas could be learned from engaging with a particular task and what standards from engaging with a particular task and what standards could be addressedcould be addressed

Analyzing students’ written responses and determining Analyzing students’ written responses and determining what students appear to understand about mathematics, what students appear to understand about mathematics, and developing questions to and developing questions to assessassess and and advanceadvance student thinkingstudent thinking


MaintainingMaintaining the cognitive demands of high- the cognitive demands of high-level tasks during instructionlevel tasks during instruction Develop teachers’ awareness of how high-level tasks Develop teachers’ awareness of how high-level tasks

“play out” in the classroom and the factors that support “play out” in the classroom and the factors that support and inhibit students engagement at a high leveland inhibit students engagement at a high level Solving tasks and reflecting on and Solving tasks and reflecting on and discussing how discussing how

the facilitator supported their learning the facilitator supported their learning Analyzing narrative cases and identifying what the Analyzing narrative cases and identifying what the

teacher featured in the case did to support or inhibit teacher featured in the case did to support or inhibit her students’ learning of mathematics her students’ learning of mathematics

Consider and DiscussConsider and Discuss

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


UsingMaintaining

Analyzing

Distinguishing

Solving

TOOLSTOOLS

Practice-based Professional Development Classroom Teaching

UsingUsing

Planning

ClassroomPractice

ReflectingOn Practice

Frameworks and ToolsFrameworks and Tools

provide a provide a focusfocus for professional development; for professional development;

bring bring coherencecoherence to PD across sessions; to PD across sessions;

provide a provide a shared languageshared language for talking about for talking about teaching and learning; and teaching and learning; and

bridgebridge the professional development and K- the professional development and K-12 classroom environments.12 classroom environments.

Frameworks and ToolsFrameworks and Tools

Framework Framework

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

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

The Mathematics Task FrameworkThe Mathematics Task Framework

TASKS


TASKS


TASKS


Student Learning



TASKS


TASKS


TASKS


Student Learning



TASKS


TASKS


TASKS


Student Learning



TASKS


TASKS


TASKS


Student Learning



SummarySummaryTask-Based Activities in which Teachers EngageTask-Based Activities in which Teachers Engage

Characterize mathematical tasks based on their Characterize mathematical tasks based on their cognitive demandscognitive demands

Solve, analyze,and discuss cognitively challenging Solve, analyze,and discuss cognitively challenging mathematical tasksmathematical tasks

Analyze narrative cases w/r/t the MTF and identify Analyze narrative cases w/r/t the MTF and identify the factors that appear to support/inhibit students’ the factors that appear to support/inhibit students’ learninglearning

Use narrative cases to generate issues which Use narrative cases to generate issues which teachers can explore in their own practiceteachers can explore in their own practice

Plan, teach, and reflect on lessons based on Plan, teach, and reflect on lessons based on cognitively challenging taskscognitively challenging tasks

SummarySummary

A task-based approach to professional A task-based approach to professional development provides a focus for work with development provides a focus for work with teachers teachers

By influencing the tasks that teachers use during By influencing the tasks that teachers use during instruction instruction and the ways in which they enact themand the ways in which they enact them, , there is an opportunity to impact student learning there is an opportunity to impact student learning

Tools help the ideas that emerge in professional Tools help the ideas that emerge in professional development “travel” to the classroom and backdevelopment “travel” to the classroom and back

Tools help support enactment of tasks in teachers’ Tools help support enactment of tasks in teachers’ own classrooms and foster conservations about own classrooms and foster conservations about teaching between teachersteaching between teachers

Pulling it All TogetherPulling it All Together

Is there any evidence that Is there any evidence that suggests that a task-focused suggests that a task-focused

approach to professional approach to professional development has an impact on development has an impact on teachers’ classroom practices?teachers’ classroom practices?

YES

ESP:ESP:Setting the ContextSetting the Context

Workshop focused on selecting and enacting high Workshop focused on selecting and enacting high level taskslevel tasks

Intended for practicing mathematics teachers 7-12 Intended for practicing mathematics teachers 7-12 with 3 or more years experience with 3 or more years experience

Use practice-based materials with the Use practice-based materials with the PurplePurple book book (Implementing Standards-Based Reform) as the (Implementing Standards-Based Reform) as the centerpiececenterpiece

Assignments link to teachers’ practices in very Assignments link to teachers’ practices in very specific ways and use tools (e.g., TTLP) to specific ways and use tools (e.g., TTLP) to generalize ideas generalize ideas

ESP:ESP:Data CollectedData Collected

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

ESP:ESP:What Teachers LearnedWhat Teachers Learned

Significant increase in teachers’ ability to distinguish between high Significant increase in teachers’ ability to distinguish between high and low level tasks following their participation in the workshopsand low level tasks following their participation in the workshops

Significant increase from fall to spring the number of high-level Significant increase from fall to spring the number of high-level tasks used per teacher over the 5-day data collectiontasks used per teacher over the 5-day data collection

Significant increase in the percent of high-level tasks that were Significant increase in the percent of high-level tasks that were maintained during implementation from fall to springmaintained during implementation from fall to spring

The PD in which teachers engaged appeared to have an The PD in which teachers engaged appeared to have an influence on teachers' practice, particularly with respect to their influence on teachers' practice, particularly with respect to their ability to use and maintain high-level tasks in their own ability to use and maintain high-level tasks in their own classrooms.classrooms.

Teachers who showed the most growth over time were those who Teachers who showed the most growth over time were those who consistently made connections between the PD and their own consistently made connections between the PD and their own classroom practice.classroom practice.

Boston, 2006Boston, 2006

For More InformationFor More Information

[email protected]@pitt.edu

Results from Data AnalysisResults from Data AnalysisThe Task Sort The Task Sort

Pre- and Post-Workshop Task Sort:Pre- and Post-Workshop Task Sort: During the first (October) and last (May) session During the first (October) and last (May) session

in the workshop, teachers were asked to:in the workshop, teachers were asked to: classify a set of tasks as High-Level or Low-Level;classify a set of tasks as High-Level or Low-Level; justify their classification of each task; andjustify their classification of each task; and provide a set of criteria for High-Level and Low-Level provide a set of criteria for High-Level and Low-Level

tasks.tasks.

Results from Data AnalysisResults from Data AnalysisThe Task SortThe Task Sort

Pre- and Post-Workshop Task Sort:Pre- and Post-Workshop Task Sort: Highly significant increase between the pre- and Highly significant increase between the pre- and

post-workshop task sort scorespost-workshop task sort scores Teachers in the workshop scored significantly Teachers in the workshop scored significantly

higher on the post-workshop task sort than a higher on the post-workshop task sort than a contrast group of secondary mathematics contrast group of secondary mathematics teachers who did not participate in the teachers who did not participate in the workshop workshop

Teachers improved their ability to Teachers improved their ability to distinguish distinguish between high and low level tasksbetween high and low level tasks following following their participation in the workshop.their participation in the workshop.

Results from Data AnalysisResults from Data AnalysisThe Task SortThe Task Sort

Pre- and Post-Workshop Task Sort: Pre- and Post-Workshop Task Sort: Improvements in teachers’ justifications and Improvements in teachers’ justifications and

criteria for high and low level taskscriteria for high and low level tasks No “inconsistent” criteria identified on the postNo “inconsistent” criteria identified on the post

i.e., “i.e., “DifficultDifficult is High-Level” or “ is High-Level” or “Use of a diagramUse of a diagram is is Low-Level”Low-Level”

Criteria and justifications closely connected to Criteria and justifications closely connected to our work in our work in solvingsolving and and distinguishingdistinguishing tasks in tasks in the sessions the sessions

Results from Data AnalysisResults from Data AnalysisTasks Used in Teachers’ ClassroomTasks Used in Teachers’ Classroom

Teachers were asked to submit tasks used over Teachers were asked to submit tasks used over 1-week period in Fall, Winter, and Spring.1-week period in Fall, Winter, and Spring.

In each data collection, 5 main instructional tasks In each data collection, 5 main instructional tasks scored using IQA Academic Rigor in Mathematics scored using IQA Academic Rigor in Mathematics rubric rubric

Boston & Wolfe, 2004; Matsumura et al., 2004Boston & Wolfe, 2004; Matsumura et al., 2004

Score of 1 or 2 = Low-level cognitive demands as Score of 1 or 2 = Low-level cognitive demands as described on Task Analysis Guidedescribed on Task Analysis Guide

Score of 3 or 4 = High-level cognitive demands as Score of 3 or 4 = High-level cognitive demands as described on Task Analysis guidedescribed on Task Analysis guide

Stein, Smith, Silver & Stein, Smith, Silver & Henningsen, 2000 Henningsen, 2000

Results from Data AnalysisResults from Data AnalysisTasks Used in Teachers’ ClassroomTasks Used in Teachers’ Classroom

Comparisons of Tasks Used From Fall to Comparisons of Tasks Used From Fall to Spring:Spring:

Significant increases in Task ScoresSignificant increases in Task Scores Significant increase in overall % of H-L tasks usedSignificant increase in overall % of H-L tasks used Significant increase in the number of high-level Significant increase in the number of high-level

tasks used per teacher over the 5-day data tasks used per teacher over the 5-day data collectioncollection

Results from Data AnalysisResults from Data AnalysisStudent Work Collected from Student Work Collected from

Teachers’ ClassroomTeachers’ Classroom

Collections of Student work:Collections of Student work: Teachers submitted 3 class-sets of student Teachers submitted 3 class-sets of student

work in the Fall, Winter, and Spring.work in the Fall, Winter, and Spring. Student-work was scored using the IQA rubric Student-work was scored using the IQA rubric

for student work Implementation for student work Implementation Scale of 1 to 4Scale of 1 to 4 Score levels based on Task Analysis GuideScore levels based on Task Analysis Guide Low-Level Low-Level << 2 2 High-Level High-Level >> 3 3

Results from Data AnalysisResults from Data Analysis Student Work Collected from Student Work Collected from


Comparisons of Student-Work Comparisons of Student-Work Implementation Scores from Fall to Spring:Implementation Scores from Fall to Spring: Significant increase in mean scores Significant increase in mean scores Significant increase in number of high-level Significant increase in number of high-level

student work implementations student work implementations Significantly less occurrences of decline of high-Significantly less occurrences of decline of high-

level cognitive demandslevel cognitive demands



In all data collections, In all data collections, Implementation scores were lower than task Implementation scores were lower than task

scores. scores. The number of high-level implementations per The number of high-level implementations per

teacher is lower than the number of high-level teacher is lower than the number of high-level tasks used per teacher.tasks used per teacher.

These findings indicate a persistent trend ofThese findings indicate a persistent trend of

decline in the level of cognitive demands. decline in the level of cognitive demands.


Teachers’ ClassroomTeachers’ ClassroomWas the decline in the level of cognitive Was the decline in the level of cognitive

demands significant?demands significant? Fall and Winter: Implementation scores were Fall and Winter: Implementation scores were

highly significantly lower than task scores highly significantly lower than task scores Spring: Implementation scores were not Spring: Implementation scores were not

significantly lower than Task scores significantly lower than Task scores Increase in # of high-level implementations per Increase in # of high-level implementations per

teacherteacher Increase in % of high-level tasks that were Increase in % of high-level tasks that were

maintained during implementationmaintained during implementation

Results from Data AnalysisResults from Data Analysis Observation of Teachers’ ClassroomObservation of Teachers’ Classroom

Eleven teachers participated in 1 classroom Eleven teachers participated in 1 classroom observation per data collectionobservation per data collection Marginally significant increase in lesson implementation Marginally significant increase in lesson implementation

from Fall to Spring from Fall to Spring Teachers significantly more likely to maintain high-level Teachers significantly more likely to maintain high-level

cognitive demands during implementation in Spring than cognitive demands during implementation in Spring than in Fall or Winter.in Fall or Winter.

In Spring, teachers implemented tasks at a significantly In Spring, teachers implemented tasks at a significantly higher level than teachers in a contrast group who did higher level than teachers in a contrast group who did not participate in the workshopnot participate in the workshop

Connecting Professional Connecting Professional Development to Teacher’s PracticeDevelopment to Teacher’s Practice

Focusing on a particular factor (light gold sheet) Focusing on a particular factor (light gold sheet) they wanted to work on and use the factor as a they wanted to work on and use the factor as a lens for reflecting on classroom instructionlens for reflecting on classroom instruction

Expectations Expectations Teachers would teach a lesson based on a high-level Teachers would teach a lesson based on a high-level

mathematical task of their choicemathematical task of their choice Teachers would select several pieces of work produced Teachers would select several pieces of work produced

by students during the lesson that they felt accurately by students during the lesson that they felt accurately reflect the lessonreflect the lesson

Teachers would bring blinded copies of the student work Teachers would bring blinded copies of the student work to shareto share

Rick Carson’sRick Carson’sCase StoryCase Story

Teacher:Teacher:• 9 years of experience 9 years of experience • First time student work has been shared First time student work has been shared

in this format in this format

School Setting:School Setting:• 10th grade 10th grade • Integrated curriculum in a probability unitIntegrated curriculum in a probability unit

Case Stories:Case Stories:Storytelling through Student WorkStorytelling through Student Work

Storyteller will distribute a complete set of Storyteller will distribute a complete set of student work to each team member student work to each team member withoutwithout comment.comment.

The team members will individually review The team members will individually review the work in silence.the work in silence.

• 5 minutes allowed5 minutes allowed


The team should share what they saw in their The team should share what they saw in their review of the student work.review of the student work.

Only factual statements can be made. Do not share your Only factual statements can be made. Do not share your evaluation of the work, or statements of personal preference.evaluation of the work, or statements of personal preference.

Start comments with, Start comments with, “I noticed that…”“I noticed that…”

The storyteller is quietly listening and making note of The storyteller is quietly listening and making note of statements. statements.

5 minutes is allowed5 minutes is allowed


The team should share questions they have about The team should share questions they have about the student work.the student work.

Responses should be in the form, Responses should be in the form, “I’m wondering…”“I’m wondering…”

The storyteller should make note of the wonderings, and should The storyteller should make note of the wonderings, and should continue to remain quiet.continue to remain quiet.

5 minutes is allowed.5 minutes is allowed.

Rick Carson Shares His Rick Carson Shares His PerspectivePerspective

Case Stories - What EmergedCase Stories - What Emerged

The Facilitator started:The Facilitator started:

……I’d like to talk about the extent to which the I’d like to talk about the extent to which the student work facilitated your discussions about student work facilitated your discussions about teaching. If you recall, last month when we did teaching. If you recall, last month when we did this, we told stories, but we didn’t have any this, we told stories, but we didn’t have any artifacts, we didn’t have any student work or artifacts, we didn’t have any student work or anything else that came directly from the lesson. anything else that came directly from the lesson. So I’m wondering to what extent, and in what So I’m wondering to what extent, and in what ways, the student work actually might have ways, the student work actually might have facilitated the discussion?facilitated the discussion?

Rick Carson’s ResponseRick Carson’s Response

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

Documents

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