Developing Number Concepts in K-2 Learners


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Developing Number Concepts in K-2 Learners

  1. 1. Developing and Assessing Number Sense By Michelle Flaming [email_address]
  2. 2. How Does One Build Number Sense <ul><li>There is NO silver bullet. </li></ul><ul><li>It takes time. </li></ul><ul><li>Several components or Building blocks involved. </li></ul>
  3. 3. Design of the class <ul><li> essdack </li></ul><ul><li>Define each building block. </li></ul><ul><li>Discuss Examples. </li></ul><ul><li>View Classroom Vignettes </li></ul><ul><li>Classroom Activities </li></ul><ul><li>Diagnostic Assessment Tool </li></ul>
  4. 4. Rote Counting <ul><li>Knowing how to recite numbers in correct order. It is the simplest of counting concepts to learn. </li></ul><ul><li>Examples: </li></ul><ul><ul><li>1,2,3,4,… </li></ul></ul><ul><ul><li>22, 32, 42, 52, … </li></ul></ul><ul><ul><li>2,4,6,8,… </li></ul></ul>
  5. 5. Rote Counting <ul><li>Tend to memorize through songs, finger plays and rhymes. </li></ul><ul><li>Different groupings are critical. </li></ul><ul><li>Tie in other senses, especially motor. </li></ul>
  6. 6. One-to-One Correspondence <ul><li>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. </li></ul>
  7. 7. One-to-One Correspondence
  8. 8. Subsidizing <ul><li>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. </li></ul>
  9. 9. Subsidizing <ul><li>Example: A student can look at the group of objects and say “four” without actually counting the objects. </li></ul>
  10. 10. Subsidizing <ul><li>U.S. textbooks often do not address this skill. </li></ul><ul><li>Minilessons </li></ul><ul><ul><li>Tens frames </li></ul></ul><ul><ul><li>Arrays </li></ul></ul><ul><ul><li>Dominoes </li></ul></ul><ul><ul><li>Sticky Dots </li></ul></ul>
  11. 11. Tens Frame
  12. 12. 5+ 1 10-4
  13. 14. 10 - 2 5 + 3
  14. 16. 1 2 3 4 5 6 1 2 3
  15. 17. Keeping Track <ul><li>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. </li></ul>
  16. 18. Keeping Track <ul><li>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. </li></ul>
  17. 19. Keeping Track <ul><li>Strategies </li></ul><ul><ul><li>Place in a pile (in groups of five, or ten, etc.) </li></ul></ul><ul><ul><li>Sliding objects to other side of page </li></ul></ul><ul><li>Random order: </li></ul><ul><ul><li>Line, circle, random </li></ul></ul>
  18. 20. Conservation of Number <ul><li>Definition: A “number” means an “amount” and that amount does not change no matter how you arrange the objects. </li></ul>
  19. 21. Conservation of Number <ul><li>The amount is “seven” and doesn’t change. </li></ul>
  20. 22. Conservation of Number <ul><li>NOT dependent upon spatial arrangement. </li></ul><ul><li>Sometimes referred to as “invariance of number”. </li></ul><ul><li>Further enhance this skill - Hidden Numbers </li></ul>
  21. 23. Hierarchal Inclusion <ul><li>Definition: An understanding that 19 is inside of twenty, the numbers are nested inside each other and that the numbers grow one each time. </li></ul><ul><li>Example: 20 is the same as 19+1. If you remove one the number goes back to 19. </li></ul>
  22. 24. Hierarchal Inclusion <ul><li>“1” “2” “3” </li></ul>
  23. 25. Hierarchal Inclusion <ul><li>“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. </li></ul><ul><ul><ul><ul><ul><li>- Labinowics 1980. </li></ul></ul></ul></ul></ul>
  24. 26. Hierarchal Inclusion <ul><li>“Constructing the number as a unit.” </li></ul><ul><li>Child is able to “see” the number as a unit, while at the same time “seeing it made up of it’s parts”. </li></ul><ul><ul><ul><ul><li>Richards, Steffe, and von Glaserfeld </li></ul></ul></ul></ul>
  25. 27. Compensation <ul><li>Directly linked to Hierarchal Inclusion. </li></ul><ul><li>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. </li></ul>
  26. 28. Compensation <ul><li>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. </li></ul>
  27. 29. Compensation <ul><li>IMPORTANT conceptual skill. </li></ul><ul><li>Referred to as “compose and decompose” numbers. </li></ul><ul><li>Flexibility with numbers </li></ul>
  28. 30. Compensation <ul><li>Suppose the problem is 44 - 28. Many problems with give us the same answer. </li></ul><ul><ul><li>43 - 27; 36 - 20 </li></ul></ul><ul><ul><li>42 - 26; </li></ul></ul><ul><ul><li>41 - 25; </li></ul></ul><ul><ul><li>40 - 24; </li></ul></ul><ul><ul><li>39 - 23; </li></ul></ul><ul><ul><li>38 - 22; </li></ul></ul><ul><ul><li>37 - 21; </li></ul></ul>
  29. 31. Compensation Strategy <ul><li>Shift both numbers to amounts that don’t require regrouping. </li></ul><ul><li>Students MUST understand a strategy to be competent with it. </li></ul>
  30. 32. Part/Whole Relationships <ul><li>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. </li></ul>
  31. 33. Part/Whole Relationships <ul><li>Example: </li></ul><ul><ul><li>The number “seven” can be represented as: 5 + 2 </li></ul></ul><ul><ul><ul><li>3 + 4 </li></ul></ul></ul><ul><ul><ul><li>7 + 0 </li></ul></ul></ul><ul><ul><ul><li>9 - 2 </li></ul></ul></ul><ul><ul><ul><li>1 + 6 </li></ul></ul></ul><ul><ul><ul><li>Etc…. </li></ul></ul></ul>
  32. 34. Unitizing/Place Value <ul><li>Definition: Unitizing is the place value understanding that ten can be represented and thought of as one group of ten or ten individual units. </li></ul><ul><li>HUGE shift in thinking for children. </li></ul><ul><li>47 4 tens and 7 ones; </li></ul><ul><ul><ul><li>3 tens and 17 ones; </li></ul></ul></ul><ul><ul><ul><li>2 tens and 27 ones; </li></ul></ul></ul><ul><ul><ul><li>1 ten and 37 ones; </li></ul></ul></ul><ul><ul><ul><li>47 ones. </li></ul></ul></ul>
  33. 35. Unitizing/Place Value <ul><li>The number 34 can be represented as: </li></ul><ul><ul><li>3 tens, 4 ones </li></ul></ul><ul><ul><li>2 tens, 14 ones </li></ul></ul><ul><ul><li>1 ten, 24 ones </li></ul></ul><ul><ul><li>0 tens, 34 ones </li></ul></ul><ul><ul><li>Etc… </li></ul></ul>
  34. 36. Unitizing/Place Value <ul><li>“Big Idea” in mathematics. </li></ul><ul><li>Shift in reasoning, perspective, logic, and in mathematical relationships. </li></ul><ul><li>Connected to part/whole relationships. </li></ul><ul><li>Important skill for all operations. </li></ul>
  35. 37. Relationships <ul><li>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. </li></ul>
  36. 38. Relationships <ul><li>Research-based Strategy: </li></ul><ul><ul><li>Cognitively Guided Instruction (Thomas Carpenter) </li></ul></ul>
  37. 39. A Numerically Powerful Child: <ul><li>Decompose of break apart numbers in different ways. </li></ul><ul><li>Knows how numbers are related to other numbers. </li></ul><ul><li>Understands how the operations are connected to each other. </li></ul><ul><li>Connects numerals with situations from life experiences. </li></ul><ul><li>Creates appropriate representation for numbers/operations. </li></ul>
  38. 40. What mathematical concept does this child have, what concepts are lacking?
  39. 41. <ul><li>Diagnostic Tool - Spreadsheet </li></ul><ul><li>Contact Information </li></ul><ul><ul><li>Michelle Flaming - </li></ul></ul>