Mathematics Tasks for Cognitive
Instruction
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Based on research from the Quasar Project
found in Implementing Standards-Based
Mathematics Instruction: A Casebook for
Professional Development(Stein, Smith,
Henningsen, & Silver, 2000).
1
NCTM Standards Compared to
Connecticut Scope and Sequence
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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
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NCTM Content Standards
Numbers and Operations
Algebra
Data Analysis and Probability
Geometry
Measurement
NCTM Process Standards
Problem Solving
Reasoning and Proof
Connections
Communication
Representation
2
NCTM and CT Scope and
Sequence
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http://www.nctm.org
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http://www.sde.ct.gov/
sde/cwp/view.asp?a=2
618&q=320872
3
Common Core State
Standards
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http://www.corestandards.org/
4
The Mathematical Tasks
Framework
TASKS
as they
appear in
curricular/
instructional
materials
TASKS
as set up by
teacher
TASKS
as implemented
by students
Student
Learning
A representation of how mathematical tasks unfold in the classroom
during classroom instruction (Stein & Smith, 1998)
5
Levels of Cognitive Demand as
Compared to Bloom’s Taxonomy
Highest Levels
Doing Math
Procedures with Connections
Procedures without Connections
Memorization
Lowest Levels
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Defining Levels of Cognitive
Demand of Mathematical Tasks
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Lower Level Demands
– Memorization
– Procedures without connections
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Higher Level Demands
– Procedures with Connections
– Doing Mathematics
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Verb Examples Associated with Each
Activity
Lower Level of Cognitive Demands
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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
Demands of Mathematical Tasks
Lower Level Demands
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Memorization:
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What are the decimal and percent
equivalents for the fractions ½ and ¼ ?
9
Defining Levels of Cognitive
Demands of Mathematical Tasks
Lower Level Demands
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Memorization:
 What are the decimal and percent
equivalents for the fractions ½ and ¼ ?
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Expected Student Response:
 ½=.5=50%
 ¼=.25=25%
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Defining Levels of Cognitive
Demands of Mathematical Tasks
Lower Level Demands
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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 %
11
Verb Examples Associated with Each
Activity
Higher levels of cognitive demand
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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
Demands of Mathematical Tasks
Higher Level Demands
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Procedure 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
Activity
Highest levels of cognitive demands
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Synthesis: 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
Demands of Mathematical Tasks
Higher Level Demands
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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
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Comparing Two
Mathematical Tasks
Martha’s Carpeting Task
The Fencing Task
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Martha’s Carpeting Task
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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 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
Mathematical Tasks
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Think privately about how you would go
about solving each task
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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 Task
Using the Area Formula
A=lxw
A = 15 x 10
A = 150 square feet
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Martha’s Carpeting Task
Drawing a Picture
10
15
21
The Fencing Task
Diagrams on Grid Paper
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The Fencing Task
Using 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 Task
Graph of Length and Area
40
35
30
Area
25
20
15
10
5
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Length
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Comparing Tasks
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How are Martha’s Carpeting Task and the
Fencing Task the same and how are they
different?
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Similarities and Differences
Similarities
Differences
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Similarities and Differences
Similarities
 Both are “area” problems
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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:
A Critical Starting Point for
Instruction
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 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?
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In what ways will you use your knowledge
and understanding of cognitive demands in
your role as teacher leader?
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