Numerical Fluency
Developing Number Sense in Mathematics
Presented by
Tracey Ramirez
Senior Program Coordinator
The Charles A. Dana Center
The University of Texas at Austin
Expected Outcomes
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Increased understanding of numerical
fluency.
Increased understanding of the
developmental stages of numerical
fluency.
Increased understanding of strategies
that develop numerical fluency.
How Do You Use Math?
Solve the following problem mentally:
Ms. Hill wants to carpet her rectangular living
room, which measures 14 feet by 11 feet. If the
carpet she wants to purchase costs $1.50 per
square foot, including tax, how much will it cost
to carpet her living room?
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How did you solve this problem?
Turn to someone next to you and share your
problem solving strategies.
What is Mathematics?
Math
Just Patterns
Waiting to be found
From 50 Problem-Solving Lessons, Grades 1-6, by Marilyn Burns.
Copyright © 1996 by Math Solutions Publications
What is Mathematics?
“The ability to think about a number in
MANY different ways is what makes a
person good in math.”
Greg Tang
Mathematics is about patterns and
relationships that exist between numbers.
What is Numerical Fluency?
Texas Essential Knowledge and Skills
Introduction Statement (4)
Grades K-2
Throughout mathematics in K-2, students develop
numerical fluency with conceptual understanding and
computational accuracy. Students in Kindergarten
through grade two use basic number sense to
compose and decompose numbers in order to solve
problems requiring precision, estimation, and
reasonableness. By the end of Grade 2, students
know basic addition and subtraction facts and are
using them to work flexibly, efficiently and
accurately with numbers during addition and
subtraction computation.
What is Numerical Fluency?
Texas Essential Knowledge and Skills
Introduction Statement (4)
Grades 3-5
Throughout mathematics in Grades 3-5, students
develop numerical fluency with conceptual
understanding and computational accuracy. Students
in Grades 3-5 use knowledge of the base-ten place
value system to compose and decompose numbers
in order to solve problems requiring precision,
estimation, and reasonableness. By the end of
Grade 5, students know basic addition,
subtraction, multiplication, and division facts and
are using them to work flexibly, efficiently and
accurately with numbers during addition,
subtraction, multiplication, and division
computation.
What is Numerical Fluency?
Numerical Fluency is the ability to
compose and decompose numbers
flexibly, efficiently, and accurately
within the context of meaningful
situations.
What is Numerical Fluency?
A student who is numerically fluent
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Composes and decomposes numbers in multiple
ways.
Sees patterns in numbers.
Is fluent with the basic facts.
Works quickly and efficiently with numbers in order to
solve problems.
Works flexibly with numbers.
Works flexibly with the place value system.
Composing and Decomposing
Numbers
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Building and taking apart numbers
Looking for patterns and relationships
between numbers
Unitizing numbers
Using numbers as reference points - Using
numbers as reference points is important in being
able to compose and decompose numbers quickly
by creating compatible numbers that are easily
manipulated.
Why is Numerical Fluency
Important?
Why do students need to be
Numerically Fluent?
A Look at Reading Fluency...
Fluency in reading is important because it provides a
bridge between word recognition and
comprehension. Because fluent readers do not
have to concentrate on decoding words, they can
focus their attention on what the text means.
They can make connections among the ideas in the
text and between the text and their background
knowledge. In other words, fluent readers
recognize words and comprehend at the same
time. Less fluent readers, however, must focus
their attention on figuring out the words, leaving
them little attention for understanding the text.
Institute for Literacy. Put Reading First – K-3.
http://www.nifl.gov/partneshipforreading/publications/reading_first1fluency.html
A Look at Numerical Fluency
Fluency in Mathematics is important because it provides a
bridge between number recognition and problem
solving comprehension. Because people who are
numerically fluent do not have to concentrate on
operation facts, they can focus their attention on what
the problem means. They can make connections among
the ideas in the problem and their background
knowledge. In other words, people who are
numerically fluent recognize how to compose and
decompose numbers based on patterns and
comprehend how to use those numerical patterns to
solve problems. People who are less fluent,
however, must focus their attention on the
operations, leaving them little attention for
understanding the problem.
Smith, K. H., and Schielack, J. (2006)
Composing and Decomposing
Numbers
A Sample Activity for PK - 1st Grades:
Accomplish the following in as many ways as possible.
Show me 8
Composing and Decomposing
Numbers
A Sample Activity for 2nd - 6th Grades:
Please answer the following statement by filling as much of the
page a possible in an organized manner.
Show what 24 means to
you.
Composing and Decomposing
Numbers
“When a primary goal is the development of sound
understanding of the number system, students will
spend much of their math time putting together and
pulling apart different numbers as they explore the
relationships among them.”
Beyond Arithmetic
What are some activities that you do in your
classroom to develop this understanding?
How is Numerical Fluency
developed?
Developing Numerical Fluency
with Conceptual Understanding
and Computational Accuracy
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First the student MUST build an
understanding of composing and
decomposing number through meaningful
problems.
Then, through much meaningful practice,
children build automaticity, which is the fast,
effortless composing and decomposing of
numbers.
Fosnot, C., & M. Dolk, (2001). Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction.
In order to better understand
how to develop Numerical
Fluency, let’s first look at the
Developmental Foundations of
Numerical Fluency.
Developmental Foundations of
Numerical Fluency
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One-to-One Correspondence
Inclusion of Set
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–
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Cardinality of Set
Conservation of Number
Counting On/Counting Back
Subitizing
More Than/Less Than/Equal To
Part/Part/Whole
Unitizing
Developmental Foundations of
Numerical Fluency
One-to-One Correspondence
Matching two groups so each member of one group is
matched up with an object from the second group
and vice-versa. Children also use one-to-one
correspondence when they count objects so that
each object counted is matched with one number
word. When working on this concept, teachers need
to understand that it is imperative not to always
position the objects in the same arrangements when
students are practicing counting objects.
Developmental Foundations of
Numerical Fluency
Inclusion of Set

Cardinality of Set - The principle of cardinality of set
is the understanding that when a set of objects is
counted, the last number counted is the number of
objects in the set. A student should be able to count
a set of objects and when asked how many are in the
set, the student can say the number of objects
without recounting the objects.
Developmental Foundations of
Numerical Fluency
Inclusion of Set

Conservation of Number - The principle of
conservation of number is the understanding that
changing the position of the objects in a set does not
change the number of objects. A student should be
able to count the objects, tell how many, then after
mixing up the objects, the child can still tell how many
without having to recount the objects.
Developmental Foundations of
Numerical Fluency
Counting On/Counting Back

Counting On - An addition strategy in which a student
starts the counting sequence with one and continues
until the answer is reached. This strategy requires
that the student has a method of keeping track of the
number of counting steps in order to know when to
stop. As the student becomes more proficient with
this strategy, the student recognizes that it is not
necessary to reconstruct the entire counting
sequence and begins “counting on” from either the
first addend or from the largest addend..
Developmental Foundations of
Numerical Fluency
Counting On/Counting Back

Counting Back - A subtraction strategy in which a
student initiates a backward counting sequence
beginning at the largest number in a given equation.
The student can use a backward counting sequence
that contains as many numbers as the given smaller
number OR the student can use a backward counting
sequence until the smaller number is reached.
Developmental Foundations of
Numerical Fluency
Subitizing
The ability to name the number of objects in a set
without counting but rather by identifying the
arrangement of objects. It is a perceptual
understanding of the magnitude rather than counting.
Developmental Foundations of
Numerical Fluency
More Than/Less Than/Equal To
Students can look at a set of objects or are given a
number name and can build a set with either one
more than, one less than, or equal to the original set
or number. The student should also be able to look
at two sets of objects and tell whether the second set
is more than, less than, or equal to the first set.
Developmental Foundations of
Numerical Fluency
Part/Part/Whole
One of the most important concepts in number sense,
this concept allows children to compose and
decompose numbers by looking at the whole and the
parts that make up the whole.
Developmental Foundations of
Numerical Fluency
Part/Part/Whole
Given unifix cubes, what would a 1st grader build if
he/she was asked, “What numbers make up the
number 5?”
5+0
4+1
3+2
2+3
1+4
0+5
Developmental Foundations of
Numerical Fluency
Part/Part/Whole
Given jewels, what would a 2nd grader build if asked,
“What numbers make up the number 4?”
2+1+1
2+2
1+1+1+1
1+2+1
1+1+2
1+3
0+4
3+1
4+0
Developmental Foundations of
Numerical Fluency
Unitizing
Unitizing involves identification of a group or set of
objects as a unit. For example, unitizing is involved
when a student counts by 2’s or counts by 10’s
instead of counting by 1’s. This is a difficult concept
for children to understand because students spend
so much time counting by 1’s. In order to develop the
concept of unitizing, students must now count by sets
or groups. This concept is necessary for
understanding place value, multiplication, and
division.
Developmental Foundations of
Numerical Fluency
Unitizing
Given jewels, what would a 3rd grader build if asked,
“What are the multiples of 4?”
# of
cups
# of jewels
in a cup
Total # of
jewels
Equation
1
4
4
1x4=4
2
4
8
2x4=8
3
4
12
3 x 4 = 12
4
4
16
4 x 4 = 16
5
4
20
5 x 4 = 20
Extending Numerical
Fluency
Once students have the foundation
concepts in place, there are specific
strategies that can help students
develop fluency with basic facts.
Addition/Subtraction Strategies
Counting On/Counting Back*
 Doubles
 Near Doubles (Doubles + 1, Doubles - 1)
 Make Ten
 Related Facts (Fact families)
 Splitting
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Addition/Subtraction Strategies
Doubles
Students need to come to the
understanding that doubles is a way of
unitizing addition. This is an important
prerequisite to understanding repeated
addition, multiplication, and division.
Addition/Subtraction Strategies
Near Doubles
Once students have the understanding of doubles,
teachers should work on the concepts of
doubles
+ 1 and doubles - 1.
3+3=6
3+4=7
3+2=5
3+3+1=7
3+3–1=5
Addition/Subtraction Strategies
Make Ten
This is probably the MOST IMPORTANT strategy that
can be taught to students, because this strategy will
begin to take students from the strategies of
Counting On and Counting Back to a higher level of
numerical fluency. Students will begin to use their
understanding of Part/Part/Whole, while making
connections to the base ten system.
Addition/Subtraction Strategies
Make Ten
6+7=
8+5=
4+9=
6+4+3=
8+2+3=
4+6+3=
At first, AVOID doubles...
7+6=
7+3+3=
Leads To...
7+8=
7+4+4=
Addition/Subtraction Strategies
Make Ten
Let’s Practice...
Addition/Subtraction Strategies
Make Ten
Let’s assess...
5+9
7+6
8+8
9+8
8+5
7+9
Addition/Subtraction Strategies
Related Facts
Familiarity with specific facts can help
students solve unknown facts.
If a student knows: 8 + 2
Then he/she can solve: 8 + 3
If a student knows: 6 + 5 = 11
Then he/she can solve: 11 - 6 = 5
Addition/Subtraction Strategies
Splitting
This strategy is one that students develop almost on
their own, as soon as they begin to understand place
value. This strategy involves splitting numbers into
friendly pieces, usually into hundreds, tens, and
ones.
When a student uses this strategy, he/she
demonstrates numerical fluency and a comfort in
decomposing and composing numbers.
Addition/Subtraction Strategies
Splitting
28
+
44
20 + 8
+
40 + 4
60
+
12
60
+
10 + 2
70 + 2 = 72
Numerical Fluency
Let’s review what we have learned...
Activities that Develop
Numerical Fluency
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Say It Fast
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•
Single and double Dice
Double-Six and Double-Nine Dominoes
5-frames / 10-frames
Dot Plates
Activities that Develop
Numerical Fluency
Frame It
 One More/One Less
 Counting On/Counting Back
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Activities that Develop
Numerical Fluency
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Doubles Snap
•
•
Doubles Plus One
Doubles Minus One
Addition Snap
 Subtraction Snap
 It’s a Fact Snap
 3-Addend Snap
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Activities that Develop
Numerical Fluency
Sum of Ten
 Ten Plus / Minus Ten
 Nine Plus / Minus Nine
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Activities that Develop
Numerical Fluency
Multiplication Snap
 Deluxe Multiplication Snap
 It’s a Fact Snap “2”
 Multiplication Dice Toss
 Multiplication War
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Learning the Basic Facts
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Assess student’s fluency with basic facts.
Identify which facts are known and unknown.
Provide intervention and acceleration that
includes strategies for mastering facts.
Provide multiple opportunities to practice.
These opportunities should include the use
of technology, games, relational flashcards
and drill.
Research on Practice
Adapted from Classroom Instruction That Works
by Marzano, Pickering, and Pollack
1. Mastering a skill requires a fair amount of practice.
Learning new content does not happen quickly. It requires practice
spread out over time. It is only after a great deal of practice that
students can perform a skill with speed and accuracy. It is not until
students have practiced upwards of 24 times that they reach 80%
competency.
2. While practicing, students should adapt and shape what
they have learned. Students must shape skills as they are
learning them. It is during this time that students attend to their
conceptual understanding of a skill. When students lack conceptual
understanding of skills, they are liable to use procedures in shallow
and ineffective ways. During this phase, students should NOT be
pressed to perform a skill with significant speed. Students FIRST
need to understand how a skill or process works before practicing to
increase speed.
Research on Providing
Feedback
Adapted from Classroom Instruction That Works
by Marzano, Pickering, and Pollack
1. Feedback should be corrective in nature. The best feedback
appears to involve an explanation as to what is accurate and what is
inaccurate in terms of student responses. In addition, asking students
to keep working on a task until they succeed appears to enhance
achievement.
2. Feedback should be timely. In general, the more delay that
occurs in giving feedback, the less improvement there is in
achievement.
3. Feedback should be specific to criterion. Criterion-referenced
feedback tells students where they stand relative to a specific target of
knowledge or skill. Giving students norm-referenced feedback tells
students nothing about their learning. This only tells students where
they stand in relation to other students.
4. Students can effectively provide some of their own
feedback. Students can effectively monitor their own progress by
simply keeping track of their performance as learning occurs. For
example, students might keep a chart of their accuracy, their speed, or
both while learning a new skill.
Questions…
Contact Information
Tracey Ramirez
Senior Program Coordinator
The Charles A. Dana Center
The University of Texas at Austin
(512) 471-5055
[email protected]