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Developing Number Concepts in K-2 Learners
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Developing Number Concepts in K-2 Learners

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Developing Number Concepts in K-2 Learners Developing Number Concepts in K-2 Learners Presentation Transcript

  • Developing and Assessing Number Sense By Michelle Flaming [email_address]
  • How Does One Build Number Sense
    • There is NO silver bullet.
    • It takes time.
    • Several components or Building blocks involved.
  • Design of the class
    • http://8.6.89.92/classroom/portal/ essdack
    • Define each building block.
    • Discuss Examples.
    • View Classroom Vignettes
    • Classroom Activities
    • Diagnostic Assessment Tool
  • Rote Counting
    • Knowing how to recite numbers in correct order. It is the simplest of counting concepts to learn.
    • Examples:
      • 1,2,3,4,…
      • 22, 32, 42, 52, …
      • 2,4,6,8,…
  • Rote Counting
    • Tend to memorize through songs, finger plays and rhymes.
    • Different groupings are critical.
    • Tie in other senses, especially motor.
  • One-to-One Correspondence
    • Definition: When a student says or thinks one number word for each object. One-to-One correspondence is matching one word with one object. Children who are insecure with this concept will say the number words faster or slower than they point to an object.
  • One-to-One Correspondence
  • Subsidizing
    • Definition: Often referred to as “magnitude of a group”. It is one’s ability to look at a group of objects (usually 2-5 objects) and know how many is in the group, and which group has more without even counting. It is the visual recognition of number size.
  • Subsidizing
    • Example: A student can look at the group of objects and say “four” without actually counting the objects.
  • Subsidizing
    • U.S. textbooks often do not address this skill.
    • Minilessons
      • Tens frames
      • Arrays
      • Dominoes
      • Sticky Dots
  • Tens Frame
  • 5+ 1 10-4
  •  
  • 10 - 2 5 + 3
  •  
  • 1 2 3 4 5 6 1 2 3
  • Keeping Track
    • Definition: Keeping track of which numbers or objects they have already counted. It requires another level of sophistication in children’s conceptual understanding of counting.
  • Keeping Track
    • Example: A student counts the following sets of objects “one”, “two”, “three”, “four”, and “five” and recognized when all objects have been counted, only once, and does not duplicate a count.
  • Keeping Track
    • Strategies
      • Place in a pile (in groups of five, or ten, etc.)
      • Sliding objects to other side of page
    • Random order:
      • Line, circle, random
  • Conservation of Number
    • Definition: A “number” means an “amount” and that amount does not change no matter how you arrange the objects.
  • Conservation of Number
    • The amount is “seven” and doesn’t change.
  • Conservation of Number
    • NOT dependent upon spatial arrangement.
    • Sometimes referred to as “invariance of number”.
    • Further enhance this skill - Hidden Numbers
  • Hierarchal Inclusion
    • Definition: An understanding that 19 is inside of twenty, the numbers are nested inside each other and that the numbers grow one each time.
    • Example: 20 is the same as 19+1. If you remove one the number goes back to 19.
  • Hierarchal Inclusion
    • “1” “2” “3”
  • Hierarchal Inclusion
    • “Beyond labeling individual objects in a collection with a name, counting eventually involves a further mental act of relating the individual objects into wholes of increasing size.
            • - Labinowics 1980.
  • Hierarchal Inclusion
    • “Constructing the number as a unit.”
    • Child is able to “see” the number as a unit, while at the same time “seeing it made up of it’s parts”.
          • Richards, Steffe, and von Glaserfeld
  • Compensation
    • Directly linked to Hierarchal Inclusion.
    • Definition: When working with numbers you can take an amount from one set and add it to another set, the total amount does not change.
  • Compensation
    • Example: 6+1 = 7. I can take one away from 6 and make it 5, as long as I add the 1 back with the other 1 and make it 2, 5 + 2 = 7. The total amount does not change.
  • Compensation
    • IMPORTANT conceptual skill.
    • Referred to as “compose and decompose” numbers.
    • Flexibility with numbers
  • Compensation
    • Suppose the problem is 44 - 28. Many problems with give us the same answer.
      • 43 - 27; 36 - 20
      • 42 - 26;
      • 41 - 25;
      • 40 - 24;
      • 39 - 23;
      • 38 - 22;
      • 37 - 21;
  • Compensation Strategy
    • Shift both numbers to amounts that don’t require regrouping.
    • Students MUST understand a strategy to be competent with it.
  • Part/Whole Relationships
    • Definition: The ability to reason with numbers and to work with numbers flexibly, to chose the most appropriate representation of a number for a given circumstance.
  • Part/Whole Relationships
    • Example:
      • The number “seven” can be represented as: 5 + 2
        • 3 + 4
        • 7 + 0
        • 9 - 2
        • 1 + 6
        • Etc….
  • Unitizing/Place Value
    • Definition: Unitizing is the place value understanding that ten can be represented and thought of as one group of ten or ten individual units.
    • HUGE shift in thinking for children.
    • 47 4 tens and 7 ones;
        • 3 tens and 17 ones;
        • 2 tens and 27 ones;
        • 1 ten and 37 ones;
        • 47 ones.
  • Unitizing/Place Value
    • The number 34 can be represented as:
      • 3 tens, 4 ones
      • 2 tens, 14 ones
      • 1 ten, 24 ones
      • 0 tens, 34 ones
      • Etc…
  • Unitizing/Place Value
    • “Big Idea” in mathematics.
    • Shift in reasoning, perspective, logic, and in mathematical relationships.
    • Connected to part/whole relationships.
    • Important skill for all operations.
  • Relationships
    • Definition: Repeated subtraction is the equivalent to division and repeated addition is equivalent to multiplication. The relationship between the operations is necessary before facts can be automatic.
  • Relationships
    • Research-based Strategy:
      • Cognitively Guided Instruction (Thomas Carpenter)
  • A Numerically Powerful Child:
    • Decompose of break apart numbers in different ways.
    • Knows how numbers are related to other numbers.
    • Understands how the operations are connected to each other.
    • Connects numerals with situations from life experiences.
    • Creates appropriate representation for numbers/operations.
  • What mathematical concept does this child have, what concepts are lacking?
    • Diagnostic Tool - Spreadsheet
    • Contact Information
      • Michelle Flaming - michellef@essdack.org