High Cognitive

vs. Low Cognitive

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An effective mathematical task is needed to

challenge and engage students intellectually.

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Read an

excerpt from

the article:

Comparing Two Mathematical Tasks

Solve Two Tasks:

•Martha’s Carpeting Task

• The Fencing Task

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How are Martha’s Carpeting Task and the Fencing Task the same and how are they different?

Comparing Two Mathematical Tasks

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Similarities and Differences

Similarities

• Both are “area” problems

• Both require prior knowledge of area

Differences

• The amount of thinking and reasoning required

• The number of ways the problem can be solved

• Way in which the area formula is used

• The need to generalize

• The range of ways to enter the problem

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Do the differences between the Fencing Task and Martha’s Carpeting Task matter?

Why or Why not?

Comparing Two Mathematical Tasks

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Does Maintaining Cognitive Demand Matter?

YES

Criteria for low cognitive demand tasks

•Recall

•Memorization

•Low on Bloom’s Taxonomy

Criteria for high cognitive demand tasks

•Requires generalizations

•Requires creativity

•Requires multiple representations

•Requires explanations (must be “worth explaining”)

What causes high-level cognitive demand tasks to decline?

Stein & Lane, 2012

A.

B.

C.

High High

Low Low

High LowModerate

High

Low

Task Set Up Task Implementation

Student Learning

Patterns of Set up, Implementation, and Student Learning

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Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands

• Routinizing problematic aspects of the task

• Shifting the emphasis from meaning, concepts, or understanding to the correctness or completeness of the answer

• Providing insufficient time to wrestle with the demanding aspects of the task or so much time that students drift into off-task behavior

• Engaging in high-level cognitive activities is prevented due to classroom management problems

• Selecting a task that is inappropriate for a given group of students

• Failing to hold students accountable for high-level products or processes

(Stein, Grover & Henningsen, 2012)

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• Scaffolding of student thinking and reasoning

• Providing a means by which students can monitor their own progress

• Modeling of high-level performance by teacher or capable students

• Pressing for justifications, explanations, and/or meaning through questioning, comments, and/or feedback

• Selecting tasks that build on students’ prior knowledge

• Drawing frequent conceptual connections

• Providing sufficient time to explore

(Stein, Grover & Henningsen, 2012)

Factors Associated with the Maintenance and Decline of High-Level Cognitive Demands

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

(Stein, Smith, Henningsen, & Silver, 2011)

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

(Hiebert et al., 2011)

Characteristics of good questionsRequire more than remembering a fact or reproducing a skill

Students can learn from answering the questions; teachers can learn about the students

May have several acceptable answers

Sullivan and Lilburn 2004

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Making questions open endedMethod 1: Working backwards

Identify a mathematical topic or concept.

Think of a closed question and write down the answer.

Make up a new question that includes (or addresses) the answer.

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Method 1: Working backwards

What is the product of 90 and 4?

can become …. Two numbers multiply to make 360. One of them has a zero on the end. What might the two numbers be?

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Making questions open endedMethod 2: Adapting a standard question

Identify a mathematical topic or concept.

Think of a standard questionAdapt it to make an open ended question.

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Method 2: Adapting a standard question

I bought a notebook that cost $1.65. How much change did I receive from $2.00?

Can become…I bought an item at a shop and got 35 cents change. What did I buy and how much did it cost?

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