The Most Powerful Tool You’ve Probably Never Heard Of…

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The Most Powerful Tool Youve Probably Never
Heard Of
Christina Tondevold Lynn Rule

• Conference for the Advancement of Mathematics
  Teaching
• Texas-2011

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• A mathematician, like a painter or a poet, is a
  maker of patterns.
• If his patterns are more permanent than theirs,
  it is because they are made with ideas.
• Godfrey Harold Hardy
• A Mathematicians Apology

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Urgency in the teaching of mathematics

• The United States suffers from innumeracy in its
  general population, math avoidance among high
  school students, and 50 failure among college
  calculus students (Reuben Hersh ) Too many
  children choose their college major and their
  career paths based upon how many math courses
  they need to take. (Boaler, 2008)

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Urgency in the classroom

• Teachers need to see themselves as
  mathematicians. If we foster environments in
  which teachers can begin to see mathematics as
  mathematizing-as constructing mathematical
  meaning in their lived world-they will be better
  able to facilitate the journey for the young
  mathematicians with whom they work. (Fosnot)

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Brain research effecting teaching and learning
(Sousa)

• Creating and using conceptual subitizing patterns
  help young students develop the abstract number
  and arithmetic strategies they will need to
  master counting.
• Information is most likely to be stored if it
  makes sense and has meaning

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Brain Research contd

• Too often, mathematics instruction focuses on
  skills, knowledge and performance but spends
  little time on reasoning and deep understanding
• Just as phonemic awareness is a prerequisite to
  learning phonics and becoming a successful
  reader, developing number sense is a prerequisite
  for succeeding in mathematics

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What is Number Sense?

• 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.
  (Howden)

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How We Learn Best

• Memorize this eleven digit number
• 25811141720
• Now look for a connection (relationship) within
  numbers
• 2 5 8 11 14 17 20

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How many dots are there?
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How many dots are there?
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The rekenrek

• is a tool developed at the Freudenthal Institute
  in the Netherlands by Adrian Treffers to support
  the natural mathematical development of children
• in Dutch means calculating frame or arithmetic
  rack.
• looks like a counting frame but is designed to
  move children away from counting each bead.
• looks like an abacus but it is not based on place
  value.

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Features of the rekenrek

• The beads are red and white.
• There are two rows of beads.
• There are five red beads and five white beads on
  the top row, and the same on the bottom.
• There are ten beads total on the top row, and
  ten beads on the bottom row.
• There are ten red beads and ten white beads on
  the rack.
• There are twenty beads altogether.

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• In the United States, the manipulatives most
  commonly used with young children are single
  objects that can be counted-Unifix cubes, bottle
  caps, chips, or buttons, while these
  manipulatives have great benefits in the very
  early stages of counting and modeling problems,
  they do little to support the development of the
  important strategies needed for automaticity.
  (Fosnot)

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MathRack

• The MathRack has a built-in structure that
  encourages children to use their knowledge about
  numbers instead of counting one to one.
• The built-in structure allows children the
  flexibility to develop more advanced strategies
  as well.

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Using the MathRack to build Early Numeracy

• What do you notice? let the children explore the
  tool and learn the built in structure before you
  have them use the tool.
• Builds counting, enumerating, and cardinality
• Show Me on one row show me___. Have them show
  a certain number. Some may count one-by-one to
  show the number but the structure of the tool
  allows for more advanced strategies.
• Builds counting, enumerating, cardinality and
  subitizing
• Flash forward once children become more
  confident with the tool, show the MathRack of a
  certain number (1-10) for a few seconds and have
  them determine which number was flashed. When
  first starting allow enough time that children
  who need to can still count one-by-one, gradually
  shorten the time so that it encourages children
  to see groupings.
• Builds subitizing, but some kids may still be
  working on enumerating and cardinality

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Quick Images

• How many beads are there?
• How do you know?

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How many beads?
Read this side
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How many beads?
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How many beads? How do you know?
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How many beads? How do you know?
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How many beads? How do you know?
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How many beads? How do you know?
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Turn and talk
What are all the possible ways children will
figure out how many?
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Developing the landmark strategies

• Subitizing
• Using the 5-structure
• Using the 10-structure
• Counting on
• Doubles and near-doubles
• Compensation
• Skip counting
• Part/whole

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Contexts for the MathRack

• mathematical meaning in their lived world
• Attendance chart
• Bunk beds
• Double-decker bus
• Bookshelves

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Taking attendance
How many children are here today? How did you
figure it out?
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The Double-Decker Bus
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Games with the MathRack

• How many empty seats on top?
• 3 on top
• 7 on top
• 2 on top
• 8 on top
• 6 on top
• Day 5

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Games with the MathRack

• Passenger Pairs matching game
• Moving from the bus story to a model of the
  context

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Games with the MathRack

• Rack Pairs matching game
• Moving away from the context

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Games with the MathRack

• Bus Stops game
• How many on the bus as it pulls away from the
  bus stop?
• How do you know?

5
8
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Games using the MathRack

• Bus Stops game
• How many on the bus as it pulls away from the
  bus stop?
• How do you know?

- 4
11
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Childrens progression to make sense of the
formal symbolism we use in mathematics. (Bruner)

• Enactive-using tangible items to model the
  problem the MathRack, cubes, acting it out, etc
• Iconic-representing what they did in the enactive
  phase with an icon (tally marks, circles, etc. on
  paper
• Symbolic-writing the formal signs and symbols

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Mini-lessons with the MathRack

• Last day of unit
• 5 5
• 5 6
• 7 3
• 7 8
• 8 5
• 8 6
• 9 7

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Modeling Story Problems

• There are 7 people on the double-decker bus. At
  the next stop, 8 people got on the bus. How many
  people are on the bus now?
• At Kylies slumber party, some of the girls were
  playing on her bunk bed. There were 6 girls
  sitting on the top bunk and 8 girls on the bottom
  bunk. How many girls were on the bunk bed?
• There are fifteen people on the double-decker
  bus. At the first stop, 7 people got off the
  bus. How many people are still on the bus?
• Kylie was having a slumber party. There were 13
  girls total at her party and all of them are
  piled on her bunk bed. Eight girls are on the
  top bunk, how many are on the bottom bunk?

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Fluency and Flexibility

• Fluency- efficient and correct
• Flexibility- multiple solution strategies
  determined by the problem
• Fluency is the by-product of flexibility.
  Assessing fluency by occasionally using timed
  tests is acceptable. Using timed tests as an
  instructional tool to build fluency is
  ineffective, inefficient, and damaging to student
  learning.

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Focus on Relationships

• When we focus on relationships, it helps give
  children flexibility when dealing with their
  basic facts and extending their knowledge to new
  task. When we build a childs number sense it
  promotes thinking instead of just computing.

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Cognitively Guided Instruction-characteristics of
each stage

• Direct Modeler
• Follow time sequence, one to one correspondence,
  no quotity
• Counter
• Independent of time sequence, simultaneous
  counting, quotity
• Derived Facts
• Using what you know to solve what you dont know
• Facts
• Known Fact

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Cognitively Guided Instruction Progression

• 7 8
• Direct Modeler counts out 7 things, counts out
  8, pushes them all together and counts the total.
• Counter holds 7 in their head and counts on 8
  more.
• Derived Fact uses a fact they know to help them

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Derived Facts

• 6 7
• What are some derived facts kids might use to
  solve this problem?

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Using Number Sense to Help

• 6 7
• What are some derived facts kids might use to
  solve this problem?
• 661 643 77-1 337
• 1552
• What relationships (spatial, one/two more and
  less, benchmarks of 5 10, and part-whole) would
  students need to have before they can use the
  derived facts for this problem?

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Sample of the Connection to Common Core Standards

• Kindergarten
• Decompose numbers less than or equal to 10 into
  pairs in more than one way, e.g., by using
  objects or drawings, and record each
  decomposition by a drawing or equation (e.g., 5
  2 3 and 5 4 1).
• For any number from 1 to 9, find the number that
  makes 10 when added to the given number, e.g., by
  using objects or drawings, and record the answer
  with a drawing or equation.
• Fluently add and subtract within 5
• 1st Grade
• Add and subtract within 20
• Understand that the two digits of a two-digit
  number represent amounts of tens and ones.
• Add within 100, including adding a two-digit
  number and a one-digit number, and adding a
  two-digit number and a multiple of 10

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Core Standards contd

• 2nd Grade
• Fluently add and subtract within 20 using mental
  strategies
• Fluently add and subtract within 100 using
  strategies based on place value, properties of
  operations, and/or the relationship between
  addition and subtraction
• Explain why addition and subtraction strategies
  work, using place value and the properties of
  operations.

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Like mathematicians, we want kids to

• Look to the numbers
• Make numbers friendly
• Use landmark numbers
• Play with relationships

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• Mathematics can be defined simply as the science
  of patterns

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• For students to become mathematicians they need
  to organize and interpret their world through a
  mathematical lens. (Fosnot) It is the teachers
  job to keep the lens in focusthe actions of
  learning and teaching are inseparable.

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Resources

• Contexts for Learning Mathematics, by Catherine
  Twomey Fosnot and colleagues, is a series of
  two-week units published by Heinemann.
• www.contextsforlearning.com http//books.heinem
  ann.com
• The Double-Decker Bus Early Addition and
  Subtraction
• Bunk Beds and Apple Boxes Early Number Sense
• How the Brain Learns Mathematics, David Sousa
• MathRack ( www.mathrack.com ) makes arithmetic
  rack products, many of which are magnetic.
• Mastering the MathRack-to Build Mathematical
  Minds, Christina Tondevold, www.mathematicallymind
  ed.com
• Teaching Student Centered Mathematics K-3, John
  a. Van De Walle
• Young Mathematicians at Work series by Catherine
  Twomey Fosnot and Maarten Dolk (Heinemann)

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Contact Information

• To download this presentation visit
• www.mathematicallyminded.com