Supporting Rigorous Mathematics
Teaching and Learning
A Performance-Based Assessment:
A Means to High-Level Thinking and Reasoning
Tennessee Department of Education
Elementary School Mathematics
Grade 3
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Forms of Assessment
Assessment as Learning
Assessment of Learning
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Assessment for Learning
2
Session Goals
•  Deepen understanding of the Common Core State
Standards (CCSS) for mathematical content and
mathematical practices.
•  Understand how Performance-Based Assessments
(PBAs) assess the CCSS for both mathematical
content and practices.
•  Understand the ways in which PBAs assess
students’ conceptual understanding.
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Overview of Activities
•  Analyze and discuss the CCSS for mathematical
content and mathematical practices.
•  Analyze PBAs in order to determine the way the
assessments are assessing the CCSSM.
•  Discuss the CCSS related to the tasks and the
implications for instruction and learning.
•  Discuss what it means to develop and assess
conceptual understanding.
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The Common Core State Standards
The standards consist of:
  The CCSS for mathematical content
  The CCSS for mathematical practices
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Analyzing a
Performance-Based Assessment
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2012 – 2013 Tennessee Focus Clusters
Grade 3
•  Represent and solve problems involving
multiplication and division.
•  Understand properties of multiplication and the
relationship between multiplication and division.
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Analyzing Assessment Items
(Private Think Time)
Four assessment items have been provided:
  The Bakery Task
  The Box of Candies Task
  Helping Amber Task
  How Many More? Task
For each assessment item:
•  Solve the assessment item.
•  Make connections between the standard(s) and the
assessment item.
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1. The Bakery Task
§♦
On Monday morning the baker baked 4 full trays of cookies to
sell in his shop.
Each tray had the same number of cookies on it.
Here is what the trays looked like on Monday evening.
a.  How many cookies did the baker sell on Monday? Show how to
use multiplication equations and other operations, if needed, to
show how you solved the problem. Refer to the trays of cookies
in your explanation.
b.  Use words or equations to explain how you know your total is
correct.
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2. The Box of Candies Task
§ ♦
Juan claims that he knows the number of candies in the box
shown below. He wrote the equations 3 x 6 and 6 x 3 and he
claims that he can write either expression. Do you agree or
disagree with Juan, and why?
3x6=6x3
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3. Helping Amber Task
§
Amber says:
I can use multiplication facts that I know to find the product of
problems that I do not know.
­  I know all of the two multiplication facts up to 2 x 10.
­  I know all of the three multiplication facts up to 3 x 10.
­  I know all of the four multiplication facts up to 4 x 10.
Use diagrams, numbers, and words to show Amber how other
multiplication facts that she already knows can be used to find
the answer to 7 x 8. Convince her that the facts that she
knows will give her the product of 7 x 8.
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4. How Many More? Task
§
Sam has 3 boxes of donuts. He has 9 donuts in each box.
Anne has 5 boxes of donuts. She has 9 donuts in each
box.
Anne has more donuts than Sam does.
She claims that she can figure out her amount of donuts by
using what she knows about Sam’s 3 boxes of donuts with 9
donuts in each box.
Draw a picture and write an equation to show how Anne can
use Sam’s situation to find the number of donuts that she has
in 5 boxes.
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Discussing Content Standards
(Small-Group Time)
For each assessment item:
With your small group, discuss the connections
between the content standard(s) and the assessment
item.
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Deepening Understanding of the Content
Standards via the Assessment Items
(Whole Group)
As a result of looking at the assessment items, what
do you better understand about the specifics of the
content standards?
What are you still wondering about?
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The CCSS for Mathematical Content: Grade 3
Operations and Algebraic Thinking
3.OA
Represent and solve problems involving multiplication and division.
3.OA.1
Interpret products of whole numbers, e.g., interpret 5 x 7 as the
total number of objects in 5 groups of 7 objects each. For
example, describe a context in which a total number of objects can
be expressed as 5 x 7.
3.OA.2
Interpret whole-number quotients of whole numbers, e.g., interpret
56 ÷ 8 as the number of objects in each share when 56 objects are
partitioned equally into 8 shares, or as a number of shares when
56 objects are partitioned into equal shares of 8 objects each. For
example, describe a context in which a number of shares or a
number of groups can be expressed as 56 ÷ 8.
Common Core State Standards, 2010, p. 23, NGA Center/CCSSO
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The CCSS for Mathematical Content: Grade 3
Operations and Algebraic Thinking
3.OA
Represent and solve problems involving multiplication and division.
3.OA.3
Use multiplication and division within 100 to solve word problems
in situations involving equal groups, arrays, and measurement
quantities, e.g., by using drawings and equations with a symbol for
the unknown number to represent the problem.
3.OA.4
Determine the unknown whole number in a multiplication or
division equation relating three whole numbers. For example,
determine the unknown number that makes the equation true in
each of the equations 8 x ? = 48, 5 = __ ÷ 3, 6 x 6 = ?.
Common Core State Standards, 2010, p. 23, NGA Center/CCSSO
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The CCSS for Mathematical Content: Grade 3
Operations and Algebraic Thinking
3.OA
Understand properties of multiplication and the relationship between
multiplication and division.
3.OA.5
Apply properties of operations as strategies to multiply and
divide. Examples:
  If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known.
(Commutative property of multiplication.)
  3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by
5 x 2 = 10, then 3 x 10 = 30. (Associative property of
multiplication.)
  Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as
8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive
property.)
Common Core State Standards, 2010, p. 23, NGA Center/CCSSO
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The CCSS for Mathematical Content: Grade 3
Operations and Algebraic Thinking
3.OA
Understand properties of multiplication and the relationship between
multiplication and division.
3.OA.6
Understand division as an unknown-factor problem. For example,
find 32 ÷ 8 by finding the number that makes 32 when multiplied
by 8.
Multiply and divide within 100.
3.OA.7
Fluently multiply and divide within 100, using strategies such as
the relationship between multiplication and division (e.g., knowing
that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations.
By the end of Grade 3, know from memory all products of two onedigit numbers.
Common Core State Standards, 2010, p. 23, NGA Center/CCSSO
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Determining the Mathematical Practices
Associated with the Performance-Based
Assessment
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Getting Familiar with the CCSS for
Mathematical Practices
(Private Think Time)
•  Count off by 8. Each person reads one of the CCSS
for mathematical practices.
•  Read your assigned mathematical practice. Be
prepared to share the “gist” of the mathematical
practice.
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The CCSS for Mathematical Practices
1.
Make sense of problems and persevere in solving
them.
2.
Reason abstractly and quantitatively.
3.
Construct viable arguments and critique the reasoning
of others.
4.
Model with mathematics.
5.
Use appropriate tools strategically.
6.
Attend to precision.
7.
Look for and make use of structure.
8.
Look for and express regularity in repeated reasoning.
Common Core State Standards for Mathematics, 2010, NGA Center/CCSSO
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Discussing Practice Standards
(Small-Group Time)
Each person has 2 minutes to share important
information about his/her assigned mathematical
practice.
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Discussing Practice Standards
(Small-Group Time)
For each assessment item:
With your small group, discuss the connections
between the practice standards and the assessment
item.
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Deepening Understanding of the Practice
Standards via the Assessment Items
(Whole Group)
Which mathematical practices do you better
understand?
What are you still wondering about?
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Assessing Conceptual Understanding
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Rationale
We have now examined assessment items and
discussed their connection to the CCSS for
mathematical content and practice. A question that
needs considering, however, is if and how these
assessments will give us a good means of measuring
the conceptual understandings our students have
acquired.
In this activity, you will have an opportunity to
consider what it means to develop conceptual
understanding as described in the CCSS for
Mathematics and what it takes to assess for it.
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Assessing for Conceptual Understanding
The set of PBA items are designed to assess student
understanding of multiplication and division.
Look across the set of related items. What might a
teacher learn about a student’s understanding by
looking at the student’s performance across the set of
items as a whole?
What is the variance from one item to the next?
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Developing and Assessing Understanding
Why is it important, when assessing a student’s
conceptual understanding, to vary items in these
ways?
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Conceptual Understanding
•  What do the authors mean by conceptual
understanding?
•  How might analyzing student performance on this
set of assessments help us determine if students
have a deep understanding of Operations and
Algebraic Thinking?
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Developing Conceptual Understanding
Knowledge that has been learned with understanding
provides the basis of generating new knowledge and
for solving new and unfamiliar problems. When
students have acquired conceptual understanding in
an area of mathematics, they see connections among
concepts and procedures and can give arguments to
explain why some facts are consequences of others.
They gain confidence, which then provides a base
from which they can move to another level of
understanding.
Kilpatrick, J., Swafford, J., & Findell, B. (2001). Adding it up: Helping children learn mathematics.
Washington, DC: National Academy Press
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The CCSS on Conceptual Understanding
In this respect, those content standards which set an
expectation of understanding are potential “points of
intersection” between the Standards for Mathematical
Content and the Standards for Mathematical Practice.
These points of intersection are intended to be weighted
toward central and generative concepts in the school
mathematics curriculum that most merit the time,
resources, innovative energies, and focus necessary to
qualitatively improve the curriculum, instruction,
assessment, professional development, and student
achievement in mathematics.
Common Core State Standards for Mathematics, 2010, p. 8, NGA Center/CCSSO
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Assessing Concept Image
Tall (1992) differentiates between the mathematical definition of
a concept and the concept image, which is the entire cognitive
structure that a person has formed related to the concept. This
concept image is made up of pictures, examples and non-examples,
processes, and properties.
A strong concept image is a rich, integrated, mental representation
that allows the student to flexibly move between multiple
formulations and representations of an idea. A student who has
connected mathematical ideas in this way can create and use a
model to analyze a situation, uncover patterns and synthesize them
to form an integrated picture. They can also use symbols
meaningfully to describe generalizations which then provides a base
from which they can move to another level of understanding.
Brown, Seidelmann, & Zimmermann. In the trenches: Three teachers’ perspectives on moving beyond the math wars.
http://mathematicallysane.com/analysis/trenches.asp
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Analyzing a Student’s Performance
Analyze Roberto’s performance on four tasks.
What do you notice? What does Roberto know?
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1. The Bakery Task: Roberto's Work
On Monday morning the baker baked 4 full trays of cookies to sell
in his shop.
Each tray had the same number of cookies on it.
Here is what the trays looked like on Monday evening.
a.  How many cookies did the baker sell on Monday? Show how to use
multiplication equations and other operations, if needed, to show how you
solved the problem. Refer to the trays of cookies in your explanation.
b.  Use words or equations to explain how you know your total is correct.
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2. The Box of Candies Task: Roberto's Work
Juan claims that he knows the number of candies in the box shown below.
He wrote the equations 3 x 6 and 6 x 3 and he claims that he can write
either expression. Do you agree or disagree with Juan, and why?
3x6=6x3
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3. Helping Amber Task: Roberto's Work
I can use multiplication facts that I know to find the product of problems that
I do not know.
­  I know all of the two multiplication facts up to 2 x 10.
­  I know all of the three multiplication facts up to 3 x 10.
­  I know all of the four multiplication facts up to 4 x 10.
Use diagrams, numbers, and words to show Amber how other multiplication
facts that she already knows can be used to find the answer to 7 x 8.
Convince her that the facts that she knows will give her the product of 7 x 8.
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4. How Many More? Task: Roberto's Work
Sam has 3 boxes of donuts. He has 9 donuts in each box.
Anne has 5 boxes of donuts. She has 9 donuts in each box.
Anne has more donuts than Sam does.
She claims that she can figure out her amount of donuts by using what she
knows about Sam’s 3 boxes of donuts with 9 donuts in each box.
Draw a picture and write an equation to show how Anne can use Sam’s
situation to find the number of donuts that she has in 5 boxes.
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Using the Assessment to Think About
Instruction
In order for students to perform well on the PBA, what
are the implications for instruction?
•  What kinds of instructional tasks will need to be
used in the classroom?
•  What will teaching and learning look like and sound
like in the classroom?
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Step Back
•  What have you learned about the CCSS for
mathematical content that surprised you?
•  What is the difference between the CCSS for
mathematical content and the CCSS for
mathematical practices?
•  Why do we say that students must work on both
mathematical content and the mathematical
practices?
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Common Multiplication and Division
Situations
♦ *
Common Core State Standards for Mathematics, 2010, Table 2, p. 89
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Properties of Operations
♦
Common Core State Standards for Mathematics, 2010, Table 3, p. 90
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Representations: Multiplication Situations
Van de Walle, Karp, & Bay-Williams, 2010, p. 154
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Representations: Multiplication Situations *
Van de Walle, Karp, & Bay-Williams, 2010, p. 161
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1. The Bakery Task
§
Operations and Algebraic Thinking
3.OA
Represent and solve problems involving multiplication and division.
3.OA.1
Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of
objects in 5 groups of 7 objects each. For example, describe a context in which a
total number of objects can be expressed as 5 x 7.
Use multiplication and division within 100 to solve word problems in situations
involving equal groups, arrays, and measurement quantities, e.g., by using
drawings and equations with a symbol for the unknown number to represent the
problem.
Multiply and divide within 100.
3.OA.3
3.OA.7
Fluently multiply and divide within 100, using strategies such as the relationship
between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷
5 = 8) or properties of operations. By the end of Grade 3, know from memory all
products of two one-digit numbers.
Solve problems involving the four operations, and identify and explain patterns in
arithmetic.
3.OA.8
Solve two-step word problems using the four operations. Represent these
problems using equations with a letter standing for the unknown quantity. Assess
the reasonableness of answers using mental computation and estimation
strategies including rounding.
Common Core State Standards, 2010, NGA Center/CCSSO
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2. The Box of Candies Task
Operations and Algebraic Thinking
§
3.OA
Represent and solve problems involving multiplication and division.
Interpret products of whole numbers, e.g., interpret 5 x 7 as the
total number of objects in 5 groups of 7 objects each. For
example, describe a context in which a total number of objects
can be expressed as 5 x 7.
Understand properties of multiplication and the relationship between
multiplication and division.
3.OA.1
3.OA.5
Apply properties of operations as strategies to multiply and divide.
Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known.
(Commutative property of multiplication.) 3 x 5 x 2 can be found
by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30.
(Associative property of multiplication.) Knowing that 8 x 5 = 40
and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2)
= 40 + 16 = 56. (Distributive property.)
Common Core State Standards, 2010, NGA Center/CCSSO
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3. Helping Amber Task
Operations and Algebraic Thinking
§
3.OA
Represent and solve problems involving multiplication and division.
3.OA.1
Interpret products of whole numbers, e.g., interpret 5 x 7 as the total
number of objects in 5 groups of 7 objects each. For example, describe a
context in which a total number of objects can be expressed as 5 x 7.
3.OA.3
Use multiplication and division within 100 to solve word problems in
situations involving equal groups, arrays, and measurement quantities,
e.g., by using drawings and equations with a symbol for the unknown
number to represent the problem.
Understand properties of multiplication and the relationship between
multiplication and division.
3.OA.5
Apply properties of operations as strategies to multiply and divide.
Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known.
(Commutative property of multiplication.) 3 x 5 x 2 can be found by 3 x 5 =
15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30. (Associative
property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can
find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive
property.)
Common Core State Standards, 2010, NGA Center/CCSSO
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4. How Many More? Task
Operations and Algebraic Thinking
§
3.OA
Represent and solve problems involving multiplication and division.
3.OA.1
Interpret products of whole numbers, e.g., interpret 5 x 7 as the total
number of objects in 5 groups of 7 objects each. For example, describe a
context in which a total number of objects can be expressed as 5 x 7.
3.OA.3
Use multiplication and division within 100 to solve word problems in
situations involving equal groups, arrays, and measurement quantities, e.g.,
by using drawings and equations with a symbol for the unknown number to
represent the problem.
Understand properties of multiplication and the relationship between
multiplication and division.
3.OA.5
Apply properties of operations as strategies to multiply and divide.
Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known.
(Commutative property of multiplication.) 3 x 5 x 2 can be found by 3 x 5 =
15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30. (Associative
property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can
find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive
property.)
Common Core State Standards, 2010, NGA Center/CCSSO
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