Overview of SessionOverview of Session

• Challenge and Goals

• Resources: Tasks and Cases

• Task Analysis

• Research That Informs Our Work

• Mathematical Task Framework (MTF)

Goals of the SessionGoals of the Session

• To understand what characterizes a

cognitively demanding task

• To become familiar with findings from

research on the critical role tasks

play in instruction

Core ChallengeCore Challenge

• Teachers must decide “what aspects of

a task to highlight, how to organize

and orchestrate the work of the

students, what questions to ask to

challenge those with varied levels of

expertise, and how to support students

without taking over the process of

thinking for them and thus eliminating

the challenge.”

• NCTM, 2000, p.19

Goals for Professional Goals for Professional DevelopmentDevelopment

• To provide opportunities for teachers to deepen

the mathematical knowledge used in teaching

mathematics in the middle grades

• To build a collective vision of meaningful

interactions among teachers, students, and

mathematical tasks that positively influence

student learning

• To develop teachers who will have the skills and

knowledge to create more effective mathematics

learning environments for their students

Central Resources Central Resources

• Tasks - complex mathematical tasks

that provoke examination of

underlying mathematical concepts

• Cases - accounts of mathematics

instructional episodes that depict

interactions that occur when a

teacher uses a complex mathematical

task in the classroom

Why Focus on Tasks?Why Focus on Tasks?

• Classroom instruction is generally organized and

orchestrated around mathematical tasks.

• The tasks with which students engage determine

the mathematics they learn.

• Teachers’ facilitation of tasks determine how

students learn the mathematics within the tasks.

• 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.

Tasks & CasesTasks & Cases

• Emerge from the activity of classrooms

• Provide opportunities for teachers to

become involved in critical discussions

of actual teaching situations (Loucks-

Horsley, 1998)

• Promote reexamining our assumptions

about what “understanding mathematics”

really means

(Schifter, Russell, & Bastable, in press)

Mathematical TasksMathematical Tasks

“Not all tasks are created equal,

and different tasks will provoke

different levels and kinds of

student thinking.”•(Stein, Smith, Henningsen, & Silver, 2000)

Comparing TasksComparing Tasks

• Task 1 - Shade 6 small squares in a 4 x 10

grid. Using the grid, explain how to

determine each of the following: a) the

percent area that is shaded, b) the decimal

part that is shaded, and c) the fractional

part that is shaded.

• Task 2 - Convert the fraction 3/8 to a

decimal and percent.

• What similarities and differences do you

notice about the two tasks?

Levels of Cognitive Levels of Cognitive Demand:Demand:Low levelLow level

• Procedures without connections

•Algorithmic and use of procedureis specifically called for

•Have no connection to the concepts or meaning that underlie the procedure being used

• Memorization

•Reproducing previously learned facts

•Committing facts, rules, formulae to memory

Levels of Cognitive Levels of Cognitive Demand: High levelDemand: High level

• ‘Doing’ mathematics

• Requires complex and nonalgorithmic thinking

• Requires students to explore and understand the nature of mathematical concepts, processes, or relationships

• May involve some level of anxiety

• Procedures with connections

• Focuses student’s attention to the use of procedures for the purposes of developing deeper levels of understanding

• Suggest pathways that have close connections to underlying mathematical ideas

• Usually represented in multiple ways (e.g. visual diagrams)

Quantitative Quantitative Understanding:Understanding:

Quantitative Understanding: Amplifying

Student Achievement and Reasoning (QUASAR)

&

Trends in International Mathematics and

Science Study (TIMSS)

TASKS as

presented in

curriculum/ materials

TASKS as

presented in

curriculum/ materials

TASKS as set up by the teachers

TASKS as set up by the teachers

TASKS as

enacted by

students and

teacher in the

classroom

TASKS as

enacted by

students and

teacher in the

classroom

Student

Learning

Student

Learning

(Adapted from Stein, Smith, Henningsen, & Silver, 2000)

The Mathematics Task The Mathematics Task FrameworkFramework

MTF - The Bottom MTF - The Bottom LineLine

• Tasks are important, but teachers also

matter!

• Teacher actions and reactions …

•influence the nature and extent of

student engagement with challenging

tasks,

• and

•affect students’ opportunities to

learn from and through task

engagement.

Challenges in Using Challenges in Using HighHighLevel TasksLevel Tasks

• Resisting the persistent urge to tell and

to direct

• Allowing time for student thinking

• Knowing when/how to ask questions and to

provide information to support rather than

replace student thinking

• Helping students accept the challenge of

solving worthwhile problems and sustaining

their engagement at a high level

(Stein, Smith, Henningsen, & Silver, 2000)

Factors that Undermine Factors that Undermine the the Cognitive Demand of a Cognitive Demand of a TaskTask• Routinizing problematic aspects of the

task

• Shifting the emphasis from meaning,

concepts, or understanding, to the

correctness or the completeness of the

answer

• Not providing enough time

• Not holding students accountable for

high-level products or processes

(Stein, Smith, Henningsen, & Silver,

2000)

Factors that Support Factors that Support Maintaining Maintaining High Cognitive DemandHigh Cognitive Demand

• Building on students’ prior knowledge of

concepts and meanings

• Scaffolding students’ thinking

• Providing appropriate amount of time

• Modeling high-level performance

• Sustaining pressure for explanation and

meaning

• Drawing frequent conceptual conceptions

(Stein, Smith, Henningsen, & Silver,

2000)

ReflectionReflection

• What were the important mathematical

ideas you encountered today?

• Did this experience generate any

insights/connections relate to

teaching? (What about the day

prompted these?)

The MMSTLC ProgramThe MMSTLC Program

The MMSTLC program and these materials

were developed under a Mathematics and

Science Partnership grant awarded by

the Michigan Department of Education.