Tasks, Tools, and Talk:
A Framework for Enacting the CCSS
Mathematical Practices
PEG SMITH
UNIVERSITY OF PITTSBURGH
North Carolina Council of Teachers of Mathematics Leadership Seminar
October 24, 2012
Position
Developing students’ capacity to engage in the
mathematical practices specified in the Common
Core State Standards will ONLY be accomplished
by engaging students in solving challenging
mathematical tasks, providing students with tools
to support their thinking and reasoning, and
orchestrating opportunities for students to talk
about mathematics and make their thinking public.
It is the combination of these three
dimensions of classrooms, working in
unison, that promote understanding.
Standards for Mathematical Practice
1.
2.
3.
4.
5.
6.
7.
8.
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of
others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning
Common Core State Standards for Mathematics, 2010, pp.6-7
Overview
 Discuss the task, tools, and talk framework
 Review and discuss examples of tasks that support
engagement in the mathematical practices
 Analyze and discuss a narrative case with respect
to the task, tools, and talk
 Discuss the potential of the task, tools, and talk
framework for supporting your work with teachers
related to the CCSS.
Tasks, Tools, and Talk
Framework
 the tasks or activities in which students engage should
provide opportunities for them to “figure things out for
themselves” (NCTM, 2009, p.11), and to justify and
communicate the outcome of their investigation;
 tools (i.e., language, materials, and symbols) should be
available to provide external support for learning (Hiebert,
et al, 1997); and
 productive classroom talk should make students’
thinking and reasoning public so that it can be refined
and/or extended (Chapin, O’Conner, & Anderson, 2009).
Comparing Two Versions of a Task
 Compare the two versions of the Adding Odd
Numbers Task and consider how they are the same
and how they are different
 Consider the opportunities each task provides to
engage in the Standards for Mathematical Practice
Comparing Two Versions of a Task
Adding Odds - Version 1
MAKING CONJECTURES Complete the
conjecture based on the pattern you observe
in the specific cases.
29. Conjecture: The sum of any two odd
numbers is ______?
1+1=2
1+3=4
3+5=8
7 + 11 = 18
13 + 19 = 32
201 + 305 = 506
30. Conjecture: The product of any two odd
numbers is ____?
1x1=1
1x3=3
3 x 5 = 15
7 x 11 = 77
13 x 19 = 247
201 x 305 = 61,305
Adding Odds - Version 2
For problems 29 and 30, complete the
conjecture based on the pattern you observe in
the examples. Then explain why the conjecture is
always true or show a case in which it is not true.
MAKING CONJECTURES Complete the
conjecture based on the pattern you observe
in the specific cases.
29. Conjecture: The sum of any two odd
numbers is ______?
1+1=2
1+3=4
3+5=8
7 + 11 = 18
13 + 19 = 32
201 + 305 = 506
30. Conjecture: The product of any two odd
numbers is ____?
1x1=1
7 x 11 = 77
1x3=3
13 x 19 = 247
3 x 5 = 15
201 x 305 = 61,305
Comparing Two Versions of a Task
Same
Different
 Both ask students to
 V2 asks students to
complete a conjecture
about odd numbers based
on a set of finite examples
that are provided
develop an argument
that explains why the
conjecture is always true
(or not)
 V1 can be completed with
limited effort; V2 requires
considerable effort –
students need to figure
out WHY this conjecture
holds up
 The number of ways to
enter and solve the
problem
Comparing Two Versions of a Task
Opportunities to Engage in the Mathematical
Practices
Version 1
Version 2
 MP 7 – look for and
make use of structure
 MP 7 – look for and make
use of structure
 MP1 – Make sense of
problems and persevere in
solving them
 MP3 – Construct viable
arguments and critique
the reasoning of others
 MP5 – Use appropriate
tools strategically
Characteristics of Tasks Aligned with SMP
 High cognitive demand (Stein et. al, 1996; Boaler & Staples,





2008)
Significant content (i.e., they have the potential to leave
behind important residue) (Hiebert et. al, 1997)
Require justification or explanation (Boaler & Staples,
2008)
Make connections between two or more representations
(Lesh, Post & Behr, 1988)
Open-ended (Lotan, 2003; Borasi & Fonzi, 2002)
Multiple ways to enter the task and to show competence
(Lotan, 2003)
Comparing Two Versions of a Task
Compare the two versions of the Tiling a Patio Task
and consider the extent to which each exemplifies
the characteristics of tasks that align with the
Standards for Mathematical Practice.
Tiling a Patio
Alfredo Gomez is designing patios. Each patio has a rectangular
garden area in the center. Alfredo uses black tiles to represent the soil
of the garden. Around each garden, he designs a border of white tiles.
The pictures shown below show the three smallest patios that he can
design with black tiles for the garden and white tiles for the border.
Patio 1
5
10
Patio 2
Patio 3
a. Draw Patio 4 and Patio 5. How many white tiles are in Patio 4? Patio
5?
b. Make some observations about the pat ios that could h elp you
describe larger patios.
c. Describe a m ethod for finding the total number of white tiles needed
for Patio 50 (without constructing it).
d. Write a rule that could be used to determine the number of white tiles
needed for any patio. Explain how your rule relates to t he visual
representation of the patio.
e. Write a different rule that could be used to d etermine the number of
white tiles needed for any patio. Explain how your rule relates to the
visual representation of the patio.
Tiling a Patio: Aligned with SMP?
 High cognitive demand - no specified pathway to follow,




requires students to explore relationships Significant
content - equivalence, rate of change
Require justification or explanation - explain in d and e
Make connections between two or more representations connect rule to visual; could also connect with tables and
graphs
Open-ended - different descriptions and rules can be
written and in different forms
Multiple ways to enter the task and to show competence
(Lotan, 2003)-- build patios, draw pictures, make tables,
write equations, draw graphs
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
Mathematical Tasks:
A Critical Starting Point for Instruction
The level and kind of thinking in which
students engage determines what they will
learn.
Hiebert, Carpenter, Fennema, Fuson, Wearne, Murray, Oliver, & Human, 1997
Mathematical Tasks:
A Critical Starting Point for Instruction
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
Mathematical Tasks:
A Critical Starting Point for Instruction
There is no decision that teachers make that
has a greater impact on students’
opportunities to learn and on their
perceptions about what mathematics is than
the selection or creation of the tasks with
which the teacher engages students in
studying mathematics.
Lappan & Briars, 1995
Mathematical Tasks:
A Critical Starting Point for Instruction
If we want students to develop the capacity to
think, reason, and problem solve then we need
start with high-level, cognitively complex
to
tasks.
Stein & Lane, 1996
Tools
 Tools can be thought of as “amplifiers of human
capacities” (Brunner, 1966, p.81).
 “Tools should help students do things more easily or
help students do things they could not do alone”
(Hiebert, et al, 1997, p.53).
Representations as Tools
Pictures
Manipulative
Models
Real-world
Situations
Written
Symbols
Oral
Language
Lesh, Post, and Behr, 1987
Tools
 Adding Odds Task
 Square tiles that can be used to build the rectangular model

Drawing of dots that can be group by two

Use of symbolic notation
2x is even; 2x + 1 odd
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 in which to keep
the rabbits.
1.
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?
1.
How long would each of the sides of the pen be if they had only 16 feet of
fencing?
2.
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.
 Stein, Smith, Henningsen, & Silver, 2009, p. xvii
The Fencing Task
 What tools could you provide that would help
students engage in this task?
 What difference do you think the tools would make?
Fencing Task Approaches
 Build pens with physical materials
 Draw pens on grid paper
 Make a table of the dimensions of possible pens
 Make a graph that shows the relationship between
one linear dimension and the area
 Set up an algebraic equation and solve
Fencing Task Approaches
 Build pens with physical materials (linear and




area pieces)
Draw pens on grid paper (grid paper)
Make a table of the dimensions of possible pens
Make a graph that shows the relationship
between one linear dimension and the area
(graph paper or graphing calculator)
Set up an algebraic equation and solve
Talk
Students must talk, with one another as well as in
response to the teacher. When the teacher talks
most, the flow of ideas and knowledge is primarily
from teacher to student. When students make public
conjectures and reason with others about
mathematics, ideas and knowledge are developed
collaboratively, revealing mathematics as
constructed by human beings within an intellectual
community.
NCTM, 1991, p.34
The Case of Darcy Dunn
 Read the Case of Darcy Dunn
 Consider the way tasks, tools, and talked supported
students engagement in the Standards for
Mathematical Practice.
The Case of Darcy Dunn
 Teacher selected a task that had the potential to engage





students in SMP (e.g., 1, 2, 3, 4, 5, 7)
Teacher provided students with tools they could use to
explored the problem (tiles, grid paper, colored pencils,
calculators)
Teacher provided students with diagrams of the patios that
helped them explain their reasoning to the class
Teacher pressed for explanations and encouraged students to
questions each other
Teacher engaged the class in creating a mathematical model
that was consistent with the verbal description given by a
student and the diagram of the patio
Teacher gave homework that required providing and
justifying a conclusion to the question, “Can they all be right?”
The Case of Darcy Dunn
 Evidence that students were engaged in the
mathematical practices





MP1 – Students were able to connect verbal descriptions with the
diagram and with the equation.
MP2 – Beth, Faith, and Devon were able to make sense of quantities
and their relationships in problem situations (Tamika’s table that
didn’t get shared yet is another example)
MP3 – Beth, Faith, and Devon justified their conclusions and
communicated them to others; all students were asked to consider
the equivalence of the 3 equations for homework and justify their
conclusions)
MP4- The class was able to write algebraic equations for the
situations described by Beth, Faith, and Devon.
MP7 – Beth, Faith, Devon, and others identified the underlying
structure of the pattern that they used to generalize
Reflect
In what ways might the Task, Tools, and Talk
framework help you in your work with teachers?
THANK YOU!
Draw a Picture
Every odd number (like 11 and 13) has one loner number.
Add the two loner numbers and you will get an even
number (24). Now add all together the loner numbers and
the other two (now even) numbers.
Build a Model
If I take the numbers 5 and 11 and organize the counters as
shown, you can see the pattern.
You can see that when you put the sets together (add the
numbers), the two extra blocks will form a pair and the
answer is always even. This is because any odd number will
have an extra block and the two extra blocks for any set of
two odd numbers will always form a pair.
Use Algebra
If a and b are odd integers, then a and b can be written
a = 2m + 1 and b = 2n + 1, where m and n are other
integers.
If a = 2m + 1 and b = 2n + 1, then a + b = 2m + 2n + 2.
If a + b = 2m + 2n + 2, then a + b = 2(m + n + 1).
If a + b = 2(m + n + 1), then a + b is an even integer.
Logical Argument
An odd number = [an] even number + 1. e.g. 9 = 8 + 1
So when you add two odd numbers you are adding an even
no. + an even no. + 1 + 1. So you get an even number. This
is because it has already been proved that an even number
+ an even number = an even number.
Therefore as an odd number = an even number + 1, if you
add two of them together, you get an even number + 2,
which is still an even number.
Tiling a Patio
Using a Visual Model to Find a Pattern of Growth
T = 2p + 6
T = 2(p + 2) + 2
T = 3(p + 2) - p
Tiling a Patio
Making a Table to Find the Pattern of Growth
Patio Number
1
2
3
4
5
6
7
8
9
10
Number of White Tiles
8
10
12
14
16
18
20
22
24
26
Tiling a Patio
Using a Graph to Determine the Pattern of Growth
30
Number of Tiles
25
20
15
10
5
0
0
1
2
3
4
5
6
Patio Number
7
8
9
10
11
The Fencing Task
Building Pens
The Fencing Task
Building Pens
The Fencing Task
Diagrams on Grid Paper
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
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
Length
8
9
10
11
12
13
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
Length
8
9
10
11
12
13
The Fencing Task
Equation and Graph
P = 2l + 2w
24 = 2l + 2w
12 = l + w
l = 12 - w
A=lxw
A = l(12 – l)
A = 12l – l2
The Fencing Task
Equation and Calculus
A = 12l – l2.
This is a quadratic equation of a parabola that has a maximum.
Finding the derivative of the equation, then setting that derivative
equal to zero, will give us the l value for the maximum.
A(l) = 12l – l2
A’(l) = 12 – 2l
12 – 2l = 0
l=6
If l is 6, then the width is 12 – 6 or 6. Thus, the
configuration with the maximum area is 6 x 6,
Task Analysis Guide
Lower-Level Demands
Memorization
• involve either reproducing previously learned facts, rules, formulae
or definitions OR committing facts, rules, formulae or definitions to
memory.
• cannot be solved using procedures because a procedure does not
exist or because the time frame in which the task is being completed
is too short to use a procedure.
• are not ambiguous. Such tasks involve exact reproduction of
previously-seen material and what is to be reproduced is clearly and
directly stated.
• have no connection to the concepts or meaning that underlie the
facts, rules, formulae or definitions being learned or reproduced.
Procedures Without Connections
• are algorithmic. Use of the procedure is either specifically called
for or its use is evident based on prior instruction, experience, or
placement of the task.
• require limited cognitive demand for successful completion. There
is little ambiguity about what needs to be done and how to do it.
• have no connection to the concepts or meaning that underlie the
procedure being used.
• are focused on producing correct answers rather than developing
mathematical understanding.
• require no explanations or explanations that focuses solely on
describing the procedure that was used.
Higher-Level Demands
Procedures With Connections
• focus students' attention on the use of procedures for the purpose of
developing deeper levels of understanding of mathematical concepts
and ideas.
• suggest pathways to follow (explicitly or implicitly) that are broad
general procedures that have close connections to underlying
conceptual ideas as opposed to narrow algorithms that are opaque
with respect to underlying concepts.
• usually are represented in multiple ways (e.g., visual diagrams,
manipulatives, symbols, problem situations). Making connections
among multiple representations helps to develop meaning.
• require some degree of cognitive effort. Although general
procedures may be followed, they cannot be followed mindlessly.
Students need to engage with the conceptual ideas that underlie the
procedures in order to successfully complete the task and develop
understanding.
Doing Mathematics
• require complex and non-algorithmic thinking (i.e., there is not a
predictable, well-rehearsed approach or pathway explicitly
suggested by the task, task instructions, or a worked-out example).
• require students to explore and understand the nature of
mathematical concepts, processes, or relationships.
• demand self-monitoring or self-regulation of one's own cognitive
processes.
• require students to access relevant knowledge and experiences and
make appropriate use of them in working through the task.
• require students to analyze the task and actively examine task
constraints that may limit possible solution strategies and solutions.
• require considerable cognitive effort and may involve some level
of anxiety for the student due to the unpredictable nature of the
solution process required.
Figure 2. 3 Characteristes of mathematical instructional tasks*.
*These characteristics are derived from the work of Doyle on academic tasks (1988), Resnick on high-level thinking skills (1987), and from the examination and categorization of
hundreds of tasks used in QUASAR classrooms (Stein, Grover, & Henningsen, 1996; Stein, Lane, and Silver, 1996).