Number Sense and Numerical Fluency

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Number Sense and Numerical Fluency

• Lise Brown
• Caldwell Elementary

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Number Sense and Numerical Fluency

• What exactly is number sense and how does it
  impact numerical fluency?
• What is the developmental sequence of number
  ideas?
• How do you help children develop mental tools
  in order to master the basic facts if youre a
  primary grade teacher (what is my role and what
  do I do)?
• How do you do this with everything else youre
  supposed to teach?
• What if youre an upper grade teacher and you
  have students who are lacking number sense and/or
  numerical fluency?

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Texas Essential Knowledge and Skills

• Throughout mathematics in Kindergarten-Grade 2,
  students develop numerical fluency with
  conceptual understanding and computational
  accuracy. Students in Kindergarten-Grade 2 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.

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Texas Essential Knowledge and Skills

• 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.

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What does it mean to have number sense?

• Come up with a definition as a group, write it on
  a piece of butcher paper and post.
• 1 person from your group should be ready to share
  your definition.

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Number Sense

• Marilyn Burns in her book Developing Number
  Sense, Grades 3-6 describes it as
• understanding the relationships between and
  among numbers,
• having the ability to think flexibly about
  numbers and to break numbers apart and put them
  back together,
• being familiar with the properties of single
  digit numbers and using this information to
  calculate efficiently using larger numbers,
• having the ability to manipulate numbers in
  their head, and
• having effective ways to estimate.

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Number Sense continued

• Howden (1989) described it as good intuition
  about numbers and their relationships. It
  develops gradually as a result of exploring
  numbers, visualizing them in a variety of
  contexts, and relating them in ways that are not
  limited by traditional algorithms.
• According to John Van de Walle this may be the
  best definition.

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According to Van de Walle

• Number relationships provide the foundation for
  strategies that help students remember basic
  facts.
• For example, knowing how numbers are related to
  5 and 10 helps students master facts such as 3
  5 (think a ten frame) and 8 6 (since 8 is 2
  away from 10, take 2 from 6 to make 10 4 14).

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From Young Mathematicians At Work Constructing
Number Sense, Addition and Subtraction

• Catherine Twomey Fosnot and Maarten Dolk argue
  that children who struggle to commit basic facts
  to memory believe there are hundreds to be
  memorized because they have little or no
  understanding of the relationships among them.
• Children who commit the facts to memory easily
  are able to do so because they have constructed
  relationships among them and between addition and
  subtraction in general, and they use these
  relationships as shortcuts.

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Mastery of basic facts means that a child can
give a quick response (in about 3 seconds)
without resorting to nonefficient means, such as
counting. Van de Walle states, All children
are able to master the basic facts- including
children with learning disabilities. Children
simply need to construct efficient mental tools
that will help them. Our job is to help
children develop those tools.
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BREAK
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What is my role in concept/strategy development
of the mental tools?

• Relationships of More, Less and Same
• These concepts are basic relationships
  contributing to the overall concept of number.
• Activities see handout

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• Early Counting
• Generally, children by midyear kindergarten
  should understand counting. Children must
  construct this idea, it cannot be forced. Only
  the counting sequence is a rote procedure. The
  meaning attached to counting is the key
  conceptual idea on which all other number
  concepts are developed (Van de Walle, 2006).
• Fosnot and Dolk (2001) make it clear that an
  understanding of cardinality and the connection
  to counting is not simple for 4-year olds. They
  will learn how to count (matching counting words
  with objects) before they understand that the
  last word of the count indicates the amount of
  the set.
• Activities see handout

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• Counting on and Counting Back
• Forward sequence of numbers is relatively
  familiar to most young children.
• Counting on and counting back are difficult
  skills for many.
• Frequent short practice drills are recommended.
• Activities see handout

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Early Number Sense and Relationships Among
Numbers 1-10 (and 10-20)

• Very important that we look at the kinds of
  relationships and connections children should be
  making about smaller numbers up to 20. But, good
  intuition about numbers shouldnt end with these
  smaller whole numbers. Children continue to
  develop number sense as they begin to use numbers
  in operations, build an understanding of place
  value, and devise flexible methods of computing
  and making estimates involving very large
  numbers. Flexible, intuitive thinking with
  numbers--number sense--should continue to be
  developed throughout the school years. (Van de
  Walle, 2006)
• Van de Walle states that early number sense
  development should demand significantly more
  attention than it is given in most K-2 programs.

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Relationships Among Numbers 1-10

• Spatial Relationships
• Children learn to recognize sets of objects in
  patterned arrangements and tell how many without
  counting. For most numbers there are several
  common patterns.
• Activities see handout

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• One and Two More, One and Two Less
• The two-more-than and two-less-than relationships
  involve more than just the ability to count on
  two or count back two. Children should know that
  7, for example, is 1 more than 6 and also 2 less
  than 9.
• Activities see handout

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• Anchors or benchmarks of 5 and 10
• Since 10 plays such a huge role in our number
  system and because two fives make up 10, its
  very useful to develop relationships for the
  numbers 1 to 10 to the important anchors of 5 and
  10.
• Other activities see handout

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• Part-Part-Whole Relationships
• To conceptualize a number as being made up of
  two or more parts is the most important
  relationship that can be developed about numbers.
  For example, 7 can be thought of as a set of 3
  and a set of 4 or a set of 2 and a set of 5.
• Van de Walle says of the four relationships
  discussed, part-whole ideas are EASILY the most
  important!
• (Composing and decomposing numbers EVERY day is
  essential in the primary grades!)
• Activities see handout

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Relationships for Numbers 10 to 20

• Pre-Place Value Relationship with 10
• A set of ten should play a major role in
  childrens initial understanding of numbers
  between 10 and 20. When children see a set of six
  with a set of ten, they should know without
  counting the total is 16.
• Activities see handout

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• Extending More and Less Relationships
• The relationships of one more than, two more
  than, one less than, and two less than are
  important for all numbers. However, these ideas
  are built on or connected to the same concepts
  for all numbers less than 10. The fact that 17 is
  one less than 18 is connected to the idea that 7
  is one less than 8.
• Children may need help in making this connection.
• Activities see handout

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• Double and Near-Double Relationships
• The use of doubles (double 6 is 12) and
  near-doubles (13 is double 6 and 1 more) is
  considered a strategy for memorizing basic
  addition facts. There is no reason why children
  should not begin to develop these relationships
  long before they are concerned with memorizing
  basic facts.
• Relate the doubles to special images (see
  handout).

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• Double 3 is the bug double 3 legs on each side
• Double 4 is the spider double 4 legs on each
  side
• Double 5 is the hand double two hands
• Double 6 is the egg carton double two rows of 6
  eggs
• Double 7 is the two-week double two weeks on the
  calendar
• Double 8 is the crayon double two rows of 8
  crayons in a box
• Double 9 is the 18-wheeler double two sides, 9
  wheels on each side
• Suggestion from Van de Walle
• Have children draw pictures or make posters
  illustrating
• the doubles for each number.
• There is no reason to restrict the images to
  those listed here. Any images that are
  strong ideas for your kids will be good for them.

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• Numbers to 100 Early Introductions
• An exposure to numbers to 100 is important even
  at the K-1 level. It is unlikely they will have
  an understanding of tens and ones or place value,
  but they can learn much about the sequence of
  numbers to 100 if not beyond. Most important at
  this early level is for students to become
  familiar with the counting patterns to 100.
• Activities see handout

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BREAK
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From Number Sense to Numerical Fluency
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Reading Fluency

• Fluency is important because it provides a bridge
  between word recognition and comprehension.
  Because fluent readers do not have to concentrate
  on decoding the 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. (2006, March). Put
reading first - k-3 (fluency) online at
http//www.nifl.gov/partnershipforreading/publicat
ions/reading_first1fluency.html
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Numerical Fluency

• Fluency 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)
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3 Steps to Fact Mastery

• Help children develop a strong understanding of
  number relationships and of the operations.
• Develop efficient strategies for fact retrieval
  through practice.
• Provide drill in the use of and selection of
  those strategies once they have been developed.

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Addition/Subtraction/Multiplication/Division
Strategies

• See handout of list of strategies and order to
  teach them from Teaching Student-Centered
  Mathematics as well as letter our 3rd grade team
  sent home to parents last year.

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How do I do this with everything else?

• Board games
• Dice/Domino games
• Card games
• Warm-up activities
• Mini-lessons to teach strategies
• Routines (attendance, lunch count, etc.)
  compose/decompose s, add strings
• Small, flexible groups according to need (do I
  know who knows facts and strategies and who
  doesnt?)
• Integrate number sense into other math concepts.

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What If Im An Upper Grade Teacher and I Have
Kiddos Who Are Lacking Number Sense/Numerical
Fluency???

• You HAVE to go back and begin building from the
  beginning.
• Use flexible groups in math and have a station
  for number sense and numerical fluency. Make
  yourself a station for those kiddos who need
  direct teach of different strategies.
• Have them play games, games, games that help
  build number sense and fluency!

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Time to PLAY!!!

• Around the room are some games from Nimble With
  Numbers that will support your students in
  developing number sense. Feel free to take copies
  with you!
• Contact me at
• lise.brown_at_pflugervilleisd.net