numerical fluency developing abstract thinking in … content...addition/subtraction strategies...

Numerical FluencyDeveloping Number Sense in Mathematics

Presented by

Tracey RamirezSenior Program Coordinator

The Charles A. Dana Center

The University of Texas at Austin

Expected Outcomes

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?

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.


Texas Essential Knowledge and SkillsIntroduction 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 numbersin 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.


Numerical Fluency is the ability to

compose and decompose numbers

flexibly, efficiently, and accurately

within the context of meaningful

situations.


A student who is numerically fluent

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.

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.

Composing and Decomposing Numbers

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

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)

A Sample Activity for PK - 1st Grades:Accomplish the following in as many ways as possible.


Show me 8

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.

“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

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

One-to-One Correspondence

Inclusion of Set– Cardinality of Set

– Conservation of Number

Counting On/Counting Back

Subitizing

More Than/Less Than/Equal To

Part/Part/Whole

Unitizing

One-to-One CorrespondenceMatching 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.


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.


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.


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..


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.



SubitizingThe 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.

More Than/Less Than/Equal ToStudents 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.


Part/Part/WholeOne 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.


Part/Part/WholeGiven unifix cubes, what would a 1st grader build if

he/she was asked, “What numbers make up the number 5?”


5 + 0 1 + 4 0 + 53 + 24 + 1 2 + 3

Part/Part/WholeGiven jewels, what would a 2nd grader build if asked,

“What numbers make up the number 4?”


1 + 1 + 1 + 1 2 + 2

2 + 1 + 1

1 + 2 + 1

1 + 1 + 2

1 + 3

3 + 1

0 + 4

4 + 0

UnitizingUnitizing 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.



UnitizingGiven 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 1 x 4 = 4

2 4 8 2 x 4 = 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


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.


Near DoublesOnce 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 + 3 + 1 = 7

3 + 2 = 5

3 + 3 – 1 = 5


Make TenThis 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.


Make Ten

At first, AVOID doubles...

Leads To...

6 + 7 = 8 + 5 = 4 + 9 =

6 + 4 + 3 = 8 + 2 + 3 = 4 + 6 + 3 =

7 + 6 = 7 + 8 =

7 + 3 + 3 = 7 + 4 + 4 =


Make Ten

Let’s Practice...


Make Ten

Let’s assess...

8 + 5

7 + 6

7 + 9

5 + 9

9 + 8

8 + 8


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


SplittingThis 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.


Splitting

28 + 44

20 + 8 + 40 + 4

60 + 12

60 + 10 + 2

70 + 2 = 72

Numerical Fluency

Let’s review what we have learned...

web.doc

Activities that Develop Numerical Fluency

Say It Fast• Single and double Dice

• Double-Six and Double-Nine Dominoes

• 5-frames / 10-frames

• Dot Plates


Frame It

One More/One Less

Counting On/Counting Back


Doubles Snap • Doubles Plus One

• Doubles Minus One

Addition Snap

Subtraction Snap

It’s a Fact Snap

3-Addend Snap


Sum of Ten

Ten Plus / Minus Ten

Nine Plus / Minus Nine


Multiplication Snap

Deluxe Multiplication Snap

It’s a Fact Snap “2”

Multiplication Dice Toss

Multiplication War

Learning the Basic Facts

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 PracticeAdapted 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

FeedbackAdapted from Classroom Instruction That Works

by Marzano, Pickering, and Pollack1. 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.

Tracey RamirezSenior Program Coordinator

The Charles A. Dana Center

The University of Texas at Austin

(512) 471-5055

[email protected]

Questions…

Contact Information

mailto:[email protected]

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