thinking about the ways students encounter mathematics

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

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

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.

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

“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

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

MaintainingMaintaining the cognitive demands of high-level tasks during instruction

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

Distinguishing

Popsicle SticksPopsicle Sticks

Empty Number LineEmpty Number Line

Popsicle SticksExample

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.”

Popsicle SticksExample 

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.

Popsicle SticksExample

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.”

Popsicle SticksSolve 

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

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?

Reflecting…..

How are the two tasks similar?

How are the two tasks different?

Importance of DistinguishingLow Cognitive Demand Tasks

Popsicles Sticks

High Cognitive Demand Tasks Empty Number Line

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.

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.

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.

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.

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

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

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

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

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

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

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