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Mathematics Tasks for Cognitive Mathematics Tasks for Cognitive InstructionInstruction

Based on research from the Quasar Project found in Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development(Stein, Smith, Henningsen, & Silver, 2000).

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NCTM Standards Compared to NCTM Standards Compared to Connecticut Scope and SequenceConnecticut Scope and Sequence

Connecticut Scope and Sequence

Number Sense Operations Estimation Ratio, Proportion and Percent Measurement Spatial Relations and Geometry Probability and Statistics Patterns Algebra and Functions Discrete Mathematics

NCTM Content Standards Numbers and Operations Algebra Data Analysis and Probability Geometry Measurement

NCTM Process Standards Problem Solving Reasoning and Proof Connections Communication Representation

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NCTM and CT Scope and NCTM and CT Scope and SequenceSequence

http://www.nctm.org http://www.sde.ct.gov/sde/cwp/view.asp?a=2618&q=320872

Common Core State Common Core State StandardsStandards

http://www.corestandards.org/

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The Mathematical Tasks The Mathematical Tasks FrameworkFramework

TASKSas they

appear incurricular/

instructionalmaterials

TASKS as implemented

by studentsStudent

Learning

TASKSas set up by teacher

A representation of how mathematical tasks unfold in the classroom during classroom instruction (Stein & Smith, 1998)

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Levels of Cognitive Demand as Levels of Cognitive Demand as Compared to Bloom’s TaxonomyCompared to Bloom’s Taxonomy

Doing Math

Procedures with Connections

Procedures without Connections

Memorization

Highest Levels

Lowest Levels

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Defining Levels of Cognitive Defining Levels of Cognitive Demand of Mathematical TasksDemand of Mathematical Tasks

Lower Level Demands– Memorization– Procedures without connections

Higher Level Demands– Procedures with Connections– Doing Mathematics

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Verb Examples Associated with Each Verb Examples Associated with Each Activity Activity

Lower Level of Cognitive DemandsLower Level of Cognitive Demands

Knowledge: arrange, define, duplicate, label, list, memorize, name, order, recognize, relate, recall, repeat, reproduce state.

Comprehension: classify, describe, discuss, explain, express, identify, indicate, locate, recognize, report, restate, review, select, translate.

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Defining Levels of Cognitive Defining Levels of Cognitive Demands of Mathematical TasksDemands of Mathematical Tasks

Lower Level DemandsLower Level DemandsMemorization:

What are the decimal and percent equivalents for the fractions ½ and ¼ ?

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Defining Levels of Cognitive Defining Levels of Cognitive Demands of Mathematical TasksDemands of Mathematical Tasks

Lower Level DemandsLower Level DemandsMemorization:What are the decimal and percent

equivalents for the fractions ½ and ¼ ?

Expected Student Response:½=.5=50%¼=.25=25%

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Defining Levels of Cognitive Defining Levels of Cognitive Demands of Mathematical TasksDemands of Mathematical Tasks

Lower Level DemandsLower Level Demands Procedures without connections: Convert the fraction 3/8 to a decimal and a

percent. Expected Student Response: Fraction 3/8 Divide 3 by 8 and get a decimal equivalent of .375 Move the decimal point two places to the right and

get 37.5 %

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Verb Examples Associated with Each Verb Examples Associated with Each Activity Activity

Higher levels of cognitive demandHigher levels of cognitive demand

Application: apply, choose, demonstrate, dramatize, employ, illustrate, interpret, operate, practice, schedule, sketch, solve, use, write.

Analysis: analyze, appraise, calculate, categorize, compare, contrast, criticize, differentiate, discriminate, distinguish, examine, experiment, question, test.

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Defining Levels of Cognitive Defining Levels of Cognitive Demands of Mathematical TasksDemands of Mathematical Tasks

Higher Level DemandsHigher Level DemandsProcedure with connections:Using a 10 by 10 grid, illustrate the decimal

and percent equivalents of 3/5.

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Verb Examples Associated with Each Verb Examples Associated with Each ActivityActivity

Highest levels of cognitive demandsHighest levels of cognitive demandsSynthesis: arrange, assemble, collect,

compose, construct, create, design, develop, formulate, manage, organize, plan, prepare, propose, set up, write.

Evaluation: appraise, argue, assess, attach, choose, compare, defend estimate, judge, predict, rate, core, select, support, value, evaluate

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Defining Levels of Cognitive Defining Levels of Cognitive Demands of Mathematical TasksDemands of Mathematical Tasks

Higher Level DemandsHigher Level Demands Doing Mathematics: Shade 6 small squares in a 4 X 10 rectangle. Using

the rectangle, explain how to determine each of the following:

A) the percent of area that is shaded B) the decimal part of the area that is shaded C) the fractional part of the area that is shaded

Comparing Two Comparing Two Mathematical TasksMathematical Tasks

Martha’s Carpeting Task

The Fencing Task

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Martha’s Carpeting TaskMartha’s Carpeting Task

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

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The Fencing TaskThe Fencing Task

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 to keep the rabbits.

a) 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?

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

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

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

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Comparing Two Comparing Two Mathematical TasksMathematical Tasks

Think privately about how you would go about solving each task

Talk with your neighbor about how you could solve each of the tasks

–The Fencing Task–Martha’s Carpeting Task

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Martha’s Carpeting TaskMartha’s Carpeting TaskUsing the Area FormulaUsing the Area Formula

A = l x w

A = 15 x 10

A = 150 square feet

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Martha’s Carpeting TaskMartha’s Carpeting TaskDrawing a PictureDrawing a Picture

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10

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The Fencing TaskThe Fencing TaskDiagrams on Grid PaperDiagrams on Grid Paper

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The Fencing TaskThe Fencing TaskUsing a TableUsing a Table

Length Width Perimeter Area

1 11 24 11

2 10 24 20

3 9 24 27

4 8 24 32

5 7 24 35

6 6 24 36

7 5 24 35

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The Fencing TaskThe Fencing TaskGraph of Length and AreaGraph of Length and Area

24

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7 8 9 10 11 12 13

Length

Area

Comparing TasksComparing Tasks

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

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Similarities and DifferencesSimilarities and Differences

Similarities Differences

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Similarities and DifferencesSimilarities 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 Many ways to enter the

problem

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Mathematical Tasks:Mathematical Tasks:A Critical Starting Point for A Critical Starting Point for

InstructionInstruction

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

Stein, Smith, Henningsen, & Silver, 2000

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Mathematical TasksMathematical Tasks

If we want students to develop the capacity to think, reason, and problem solve then we need to start with high-level, cognitively complex tasks.

Stein & Lane, 1996

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What do you think?What do you think?

In what ways will you use your knowledge and understanding of cognitive demands in your role as teacher leader?

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