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Thinking about the ways Thinking about the ways students encounter students encounter mathematics mathematics

Mathematics task framework

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Page 1: Mathematics task framework

Thinking about the ways Thinking about the ways students encounter mathematicsstudents encounter mathematics

Page 2: Mathematics task framework

Why Focus on Tasks?Classroom instruction is generally

organized and orchestrated around mathematical tasks

The tasks with which students engage determine 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

Page 3: Mathematics task framework

The Importance of Mathematical Tasks“…a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity.

Page 4: Mathematics task framework

But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a task for, and some means of, independent thinking.”

Polya, 1973/1945

The Importance of Mathematical Tasks

Page 5: Mathematics task framework

“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, 2000

The Importance of Mathematical Tasks

Page 6: Mathematics task framework

Task-Focused ActivitiesDistinguishingDistinguishing between high and low cognitive demand mathematics tasks

MaintainingMaintaining the cognitive demands of high-level tasks during instruction

Page 7: Mathematics task framework

Task-Focused ActivitiesDistinguishingDistinguishing between high and low cognitive demand mathematics tasks

Page 8: Mathematics task framework

Distinguishing

Popsicle SticksPopsicle Sticks

Empty Number LineEmpty Number Line

Page 9: Mathematics task framework

Popsicle Sticks

Example

Say, “We have nine ones. Let’s put the nine ones using the popsicle sticks in the ones place (cup). The problem is 9 + 3. Let’s count three more popsicle sticks and add them to the nine ones.”

Page 10: Mathematics task framework

Popsicle Sticks

Example 

Say, “Oh no! What’s wrong with the ones place?”Students respond, “There can only be nine ones in the ones place. We have to regroup by making a group of ten.” Count the ten popsicle sticks with the class and make a group of ten.

Page 11: Mathematics task framework

Popsicle Sticks

Example

Say, “How many groups of ten do we have in the tens place? How many ones do we have in the ones place? We have 9 + 3 which equals one group of ten and two ones.”

Page 12: Mathematics task framework

Popsicle Sticks

Solve 

Summarize: “We learned addition with regrouping one digit, how we can do regrouping with popsicle sticks, and how we use regrouping every day.”

Page 13: Mathematics task framework

Jerry made a drawing using an empty number line to solve this problem

Zena also used an empty number line but her solution does not look like Jerry’s solution.

Explain how Jerry solved the problem.Explain how Zena solved the problem.Are both solutions correct? How do you know?

Page 14: Mathematics task framework

Reflecting…..

How are the two tasks similar?

How are the two tasks different?

Page 15: Mathematics task framework

Importance of DistinguishingLow Cognitive Demand Tasks

Popsicles Sticks

High Cognitive Demand Tasks Empty Number Line

Page 16: Mathematics task framework

Sorting TasksYou will need the complete set of the Task Sort Card set (File name: Task Cards (PV)082909).

Before you continue, make sure you have each task as a separate sheet so you can move them individually as you sort them.

Also make sure you have the Task Sort Chart to record your decisions.

Page 17: Mathematics task framework

Sorting TasksMake two piles: those tasks that your group considers to be high cognitive demand tasks and those tasks that your group considers to be low cognitive demand tasks.

As you sort, take notes on your thoughts about what makes a task high or low in terms of cognitive demand.

Page 18: Mathematics task framework
Page 19: Mathematics task framework

Some Features of High Cognitive Demand Tasks A high cognitive demand task begins where the

students are (zone of proximal development; scaffolding)

The problematic or engaging aspect of a high cognitive demand task is due to mathematics that students are to learn.

A high cognitive demand task requires justifications and explanations for answers and methods.

Page 20: Mathematics task framework

Task-Focused ActivitiesHigh cognitive demand tasks or activities are the vehicle through which the curriculum can be developed. MaintainingMaintaining the cognitive demands of high-level tasks during instruction affects the learning that occurs.

Page 21: Mathematics task framework

MaintainingThe Mathematical Tasks Framework

Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 4

TASKS as they appear in curricular/ instructional materials or are designed by teachers

Student LearningTASKS as they are set up by the teachers

TASKS as they are implemented by students

TASKS as they are summarized by teacher and students

Page 22: Mathematics task framework

MaintainingThe Mathematical Tasks Framework

Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 4

TASKS as they appear in curricular/ instructional materials or are designed by teachers

Student LearningTASKS as they are set up by the teachers

TASKS as they are implemented by students

TASKS as they are summarized by teacher and students

Page 23: Mathematics task framework

MaintainingThe Mathematical Tasks Framework

Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 4

TASKS as they appear in curricular/ instructional materials or are designed by teachers

Student LearningTASKS as they are set up by the teachers

TASKS as they are implemented by students

TASKS as they are summarized by teacher and students

Page 24: Mathematics task framework

MaintainingThe Mathematical Tasks Framework

Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 4

TASKS as they appear in curricular/ instructional materials or are designed by teachers

Student LearningTASKS as they are set up by the teachers

TASKS as they are implemented by students

TASKS as they are summarized by teacher and students

Page 25: Mathematics task framework

MaintainingThe Mathematical Tasks Framework

Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 4

TASKS as they appear in curricular/ instructional materials or are designed by teachers

Student LearningTASKS as they are set up by the teachers

TASKS as they are implemented by students

TASKS as they are summarized by teacher and students

Page 26: Mathematics task framework

MaintainingThe Mathematical Tasks Framework

Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 4

TASKS as they appear in curricular/ instructional materials or are designed by teachers

Student LearningTASKS as they are set up by the teachers

TASKS as they are implemented by students

TASKS as they are summarized by teacher and students

Page 27: Mathematics task framework

MaintainingThe Mathematical Tasks Framework

Adapted from Stein, Smith, Henningsen, & Silver, 2000, p. 4

TASKS as they appear in curricular/ instructional materials or are designed by teachers

Student LearningTASKS as they are set up by the teachers

TASKS as they are implemented by students

TASKS as they are summarized by teacher and students