NCSM April 2013

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  • In her work on strategies for learning the number combinations, Steinberg (1985) states that it appears that the counting-on procedure is not a necessary one for the learning of strategies. She also noted that the use of strategies was accompanied by a decrease in counting.
  • In her work on strategies for learning the number combinations, Steinberg (1985) states that it appears that the counting-on procedure is not a necessary one for the learning of strategies. She also noted that the use of strategies was accompanied by a decrease in counting.
  • In her work on strategies for learning the number combinations, Steinberg (1985) states that it appears that the counting-on procedure is not a necessary one for the learning of strategies. She also noted that the use of strategies was accompanied by a decrease in counting.
  • In her work on strategies for learning the number combinations, Steinberg (1985) states that it appears that the counting-on procedure is not a necessary one for the learning of strategies. She also noted that the use of strategies was accompanied by a decrease in counting.
  • In her work on strategies for learning the number combinations, Steinberg (1985) states that it appears that the counting-on procedure is not a necessary one for the learning of strategies. She also noted that the use of strategies was accompanied by a decrease in counting.
  • NCSM April 2013

    1. 1. © Joan A. Cotter, Ph.D., 2013Teaching Primary Mathematics with MoreUnderstanding and Less CountingNational Council of Supervisors of MathematicsMonday, April 16, 2013Denver, ColoradoJoan A. Cotter, Ph.D.JoanCotter@RightStartMath.comandTracy Mittleider, MESdTracy@RightStartMath.com1
    2. 2. © Joan A. Cotter, Ph.D., 20132ObjectivesI. Review the traditional counting trajectory.
    3. 3. © Joan A. Cotter, Ph.D., 20133ObjectivesI. Review the traditional counting trajectory.II. Experience traditional counting like a child.
    4. 4. © Joan A. Cotter, Ph.D., 20134ObjectivesI. Review the traditional counting trajectory.II. Experience traditional counting like a child.III. Group in 5s and 10s: an alternative tocounting.
    5. 5. © Joan A. Cotter, Ph.D., 20135ObjectivesI. Review the traditional counting trajectory.II. Experience traditional counting like a child.III. Group in 5s and 10s: an alternative tocounting.IV. Meet CCSS without counting.
    6. 6. © Joan A. Cotter, Ph.D., 20136Traditional Counting Model1. Memorizing counting sequence.
    7. 7. © Joan A. Cotter, Ph.D., 20137Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.
    8. 8. © Joan A. Cotter, Ph.D., 20138Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.3. Cardinality principal.
    9. 9. © Joan A. Cotter, Ph.D., 20139Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.3. Cardinality principal.4. Adding by counting all.
    10. 10. © Joan A. Cotter, Ph.D., 201310Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.3. Cardinality principal.4. Adding by counting all.5. Adding by counting on.
    11. 11. © Joan A. Cotter, Ph.D., 201311Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.3. Cardinality principal.4. Adding by counting all.5. Adding by counting on.6. Adding by counting from the larger number.
    12. 12. © Joan A. Cotter, Ph.D., 201312Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.3. Cardinality principal.4. Adding by counting all.5. Adding by counting on.6. Adding by counting from larger number.7. Subtracting by counting backward.
    13. 13. © Joan A. Cotter, Ph.D., 201313Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.3. Cardinality principal.4. Adding by counting all.5. Adding by counting on.6. Adding by counting from larger number.7. Subtracting by counting backward.8. Multiplying by skip counting.
    14. 14. © Joan A. Cotter, Ph.D., 201314Traditional Counting Model1. Memorizing counting sequence.• String level• Unbreakable list• Breakable chain• Numerable chain• Bidirectional chain
    15. 15. © Joan A. Cotter, Ph.D., 2013Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.• Requires stable order for countingwords• Common errors: double counting andmissed count15
    16. 16. © Joan A. Cotter, Ph.D., 201316Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.3. Cardinality principal.• Unlike anything else in child‘sexperience (e.g. in naming family, baby≠ all others).
    17. 17. © Joan A. Cotter, Ph.D., 201317Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.3. Cardinality principal.• Unlike anything else in child‘sexperience (e.g. in naming family, baby≠ all others).• ―How many‖ not a good test; take n isbetter.
    18. 18. © Joan A. Cotter, Ph.D., 201318Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.3. Cardinality principal.4. Adding by counting all.• Focuses more on counting than adding.
    19. 19. © Joan A. Cotter, Ph.D., 201319Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.3. Cardinality principal.4. Adding by counting all.5. Adding by counting on.• Leads to counting words.
    20. 20. © Joan A. Cotter, Ph.D., 201320Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.3. Cardinality principal.4. Adding by counting all.5. Adding by counting on.• Leads to counting words.• No need to learn strategies.
    21. 21. © Joan A. Cotter, Ph.D., 201321Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.3. Cardinality principal.4. Adding by counting all.5. Adding by counting on.• Leads to counting words.• No need to learn strategies.• Very difficult. (article in Nov. 2011, JRME)
    22. 22. © Joan A. Cotter, Ph.D., 201322Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.3. Cardinality principal.4. Adding by counting all.5. Adding by counting on.6. Adding by counting from larger number.• First need to determine larger number.
    23. 23. © Joan A. Cotter, Ph.D., 201323Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.3. Cardinality principal.4. Adding by counting all.5. Adding by counting on.6. Adding by counting from the larger number.7. Subtracting by counting backward.• Extremely difficult. (Easier to go forward.)
    24. 24. © Joan A. Cotter, Ph.D., 201324Traditional Counting Model1. Memorizing counting sequence.2. One-to-one correspondence.3. Cardinality principal.4. Adding by counting all.5. Adding by counting on.6. Adding by counting from larger number.7. Subtracting by counting backward.8. Multiplying by skip counting.• Tedious for finding multiplication facts.
    25. 25. © Joan A. Cotter, Ph.D., 201325Traditional CountingFrom a childs perspective
    26. 26. © Joan A. Cotter, Ph.D., 201326Traditional CountingFrom a childs perspectiveBecause were so familiar with 1, 2, 3, we‘ll useletters.A = 1B = 2C = 3D = 4E = 5, and so forth
    27. 27. © Joan A. Cotter, Ph.D., 201327Traditional CountingFrom a childs perspectiveF + E =
    28. 28. © Joan A. Cotter, Ph.D., 201328Traditional CountingFrom a childs perspectiveAF + E =
    29. 29. © Joan A. Cotter, Ph.D., 201329Traditional CountingFrom a childs perspectiveA BF + E =
    30. 30. © Joan A. Cotter, Ph.D., 201330Traditional CountingFrom a childs perspectiveA CBF + E =
    31. 31. © Joan A. Cotter, Ph.D., 201331Traditional CountingFrom a childs perspectiveA FC D EBF + E =
    32. 32. © Joan A. Cotter, Ph.D., 201332Traditional CountingFrom a childs perspectiveAA FC D EBF + E =
    33. 33. © Joan A. Cotter, Ph.D., 201333Traditional CountingFrom a childs perspectiveA BA FC D EBF + E =
    34. 34. © Joan A. Cotter, Ph.D., 201334Traditional CountingFrom a childs perspectiveA C D EBA FC D EBF + E =
    35. 35. © Joan A. Cotter, Ph.D., 201335Traditional CountingFrom a childs perspectiveA C D EBA FC D EBWhat is the sum?(It must be a letter.)F + E =
    36. 36. © Joan A. Cotter, Ph.D., 201336Traditional CountingFrom a childs perspectiveG I J KHA FC D EBF + E =K
    37. 37. © Joan A. Cotter, Ph.D., 201337Traditional CountingFrom a childs perspectiveE + D =Find the sum without counters.
    38. 38. © Joan A. Cotter, Ph.D., 201338Traditional CountingFrom a childs perspectiveG + E =Find the sum without fingers.
    39. 39. © Joan A. Cotter, Ph.D., 201339Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D
    40. 40. © Joan A. Cotter, Ph.D., 201340Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D
    41. 41. © Joan A. Cotter, Ph.D., 201341Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ DD+ C
    42. 42. © Joan A. Cotter, Ph.D., 201342Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ DC+ GD+ C
    43. 43. © Joan A. Cotter, Ph.D., 201343Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ DC+ GD+ C
    44. 44. © Joan A. Cotter, Ph.D., 201344Traditional CountingFrom a childs perspectiveSubtract counting backward by using your fingers.H – C =
    45. 45. © Joan A. Cotter, Ph.D., 201345Traditional CountingFrom a childs perspectiveSubtract by counting backward without fingers.J – F =
    46. 46. © Joan A. Cotter, Ph.D., 201346Traditional CountingFrom a childs perspectiveTry skip counting by Bs to T:B, D, . . . , T.
    47. 47. © Joan A. Cotter, Ph.D., 201347Traditional CountingFrom a childs perspectiveTry skip counting by Bs to T:B, D, . . . , T.What is D x E?
    48. 48. © Joan A. Cotter, Ph.D., 201348Traditional Counting―Special cases‖ of place value (1.NBT.2)Lis a ―bundle‖ of J A‘sand B As.
    49. 49. © Joan A. Cotter, Ph.D., 201349Traditional Counting―Special cases‖ of place value (1.NBT.2)Lis a ―bundle‖ of J A‘sand B As.huh?
    50. 50. © Joan A. Cotter, Ph.D., 201350Traditional Counting―Special cases‖ of place value (1.NBT.2)Lis a ―bundle‖ of J A‘sand B As.(12)
    51. 51. © Joan A. Cotter, Ph.D., 201351Traditional Counting―Special cases‖ of place value (1.NBT.2)Lis a ―bundle‖ of J A‘sand B As.(ten ones)(12)
    52. 52. © Joan A. Cotter, Ph.D., 201352Traditional Counting―Special cases‖ of place value (1.NBT.2)Lis a ―bundle‖ of J A‘sand B As.(ten ones)(two ones)(12)
    53. 53. © Joan A. Cotter, Ph.D., 2013Grouping in Fives
    54. 54. © Joan A. Cotter, Ph.D., 2013Grouping in FivesChinese abacus
    55. 55. © Joan A. Cotter, Ph.D., 2013Grouping in FivesIIIIIIIIIIVVIII123458Early Roman numerals
    56. 56. © Joan A. Cotter, Ph.D., 201356Grouping in FivesMusical staff
    57. 57. © Joan A. Cotter, Ph.D., 2013Clocks and nickelsGrouping in Fives
    58. 58. © Joan A. Cotter, Ph.D., 2013Grouping in FivesClocks and nickels
    59. 59. © Joan A. Cotter, Ph.D., 2013Grouping in FivesTally marks
    60. 60. © Joan A. Cotter, Ph.D., 2013Grouping in FivesSubitizing• Instant recognition of quantity is called subitizing.
    61. 61. © Joan A. Cotter, Ph.D., 2013Grouping in FivesSubitizing• Instant recognition of quantity is called subitizing.• Grouping in fives extends subitizing beyond five.
    62. 62. © Joan A. Cotter, Ph.D., 2013Subitizing• Five-month-old infants can subitize to 1–3.
    63. 63. © Joan A. Cotter, Ph.D., 2013Subitizing• Three-year-olds can subitize to 1–5.• Five-month-old infants can subitize to 1–3.
    64. 64. © Joan A. Cotter, Ph.D., 2013Subitizing• Three-year-olds can subitize to 1–5.• Four-year-olds can subitize 1–10 bygrouping with five.• Five-month-old infants can subitize to 1–3.
    65. 65. © Joan A. Cotter, Ph.D., 2013Subitizing• Three-year-olds can subitize to 1–5.• Four-year-olds can subitize 1–10 bygrouping with five.• Five-month-old infants can subitize to 1–3.• Counting is analogous to sounding out aword; subitizing, recognizing the word.
    66. 66. © Joan A. Cotter, Ph.D., 201366Research on Subitizing
    67. 67. © Joan A. Cotter, Ph.D., 2013Research on SubitizingKaren Wynns research
    68. 68. © Joan A. Cotter, Ph.D., 2013Research on SubitizingKaren Wynns research
    69. 69. © Joan A. Cotter, Ph.D., 201369Research on SubitizingKaren Wynns research
    70. 70. © Joan A. Cotter, Ph.D., 201370Research on SubitizingKaren Wynns research
    71. 71. © Joan A. Cotter, Ph.D., 201371Research on SubitizingKaren Wynns research
    72. 72. © Joan A. Cotter, Ph.D., 201372Research on SubitizingKaren Wynns research
    73. 73. © Joan A. Cotter, Ph.D., 201373Research on SubitizingKaren Wynns research
    74. 74. © Joan A. Cotter, Ph.D., 201374Research on SubitizingKaren Wynns research
    75. 75. © Joan A. Cotter, Ph.D., 201375Research on SubitizingOther research
    76. 76. © Joan A. Cotter, Ph.D., 201376Research on Subitizing• Subitizing ―allows the child to grasp thewhole and the elements at the same time.‖—BenoitOther research
    77. 77. © Joan A. Cotter, Ph.D., 201377Research on Subitizing• Subitizing ―allows the child to grasp thewhole and the elements at the same time.‖—Benoit• Subitizing seems to be a necessary skill forunderstanding what the counting processmeans. —GlasersfeldOther research
    78. 78. © Joan A. Cotter, Ph.D., 201378Research on Subitizing• Children who can subitize perform better inmathematics long term.—Butterworth• Subitizing ―allows the child to grasp thewhole and the elements at the same time.‖—Benoit• Subitizing seems to be a necessary skill forunderstanding what the counting processmeans. —GlasersfeldOther research
    79. 79. © Joan A. Cotter, Ph.D., 2013Other researchResearch on Subitizing• Australian Aboriginal children from two tribes.Brian Butterworth, University College London,2008.79
    80. 80. © Joan A. Cotter, Ph.D., 2013Other researchResearch on Subitizing• Australian Aboriginal children from two tribes.Brian Butterworth, University College London,2008.• Adult Pirahã from Amazon region.Edward Gibson and Michael Frank, MIT, 2008.80
    81. 81. © Joan A. Cotter, Ph.D., 2013Other researchResearch on Subitizing• Australian Aboriginal children from two tribes.Brian Butterworth, University College London,2008.• Adult Pirahã from Amazon region.Edward Gibson and Michael Frank, MIT, 2008.• Adults, ages 18-50, from Boston.Edward Gibson and Michael Frank, MIT, 2008.81
    82. 82. © Joan A. Cotter, Ph.D., 2013Other researchResearch on Subitizing• Australian Aboriginal children from two tribes.Brian Butterworth, University College London,2008.• Adult Pirahã from Amazon region.Edward Gibson and Michael Frank, MIT, 2008.• Adults, ages 18-50, from Boston.Edward Gibson and Michael Frank, MIT, 2008.• Baby chicks from Italy.Lucia Regolin, University of Padova, 2009.82
    83. 83. © Joan A. Cotter, Ph.D., 2013Research on SubitizingIn Japanese schools• Children are discouraged from usingcounting for adding.83
    84. 84. © Joan A. Cotter, Ph.D., 2013Research on SubitizingIn Japanese schools• Children are discouraged from usingcounting for adding.• They consistently group in 5s.84
    85. 85. © Joan A. Cotter, Ph.D., 2013Research on SubitizingFinger gnosia• Finger gnosia is the ability to know whichfingers can been lightly touched withoutlooking.85
    86. 86. © Joan A. Cotter, Ph.D., 2013Research on SubitizingFinger gnosia• Finger gnosia is the ability to know whichfingers can been lightly touched withoutlooking.• Part of the brain controlling fingers isadjacent to math part of the brain.86
    87. 87. © Joan A. Cotter, Ph.D., 2013Research on SubitizingFinger gnosia• Finger gnosia is the ability to know whichfingers can been lightly touched withoutlooking.• Part of the brain controlling fingers isadjacent to math part of the brain.• Children who use their fingers asrepresentational tools perform better inmathematics.—Butterworth87
    88. 88. © Joan A. Cotter, Ph.D., 2013Research on SubitizingFinger gnosia• Finger gnosia is the ability to know whichfingers can been lightly touched withoutlooking.• Part of the brain controlling fingers isadjacent to math part of the brain.• Children who use their fingers asrepresentational tools perform better inmathematics.—Butterworth88• Children learn subitizing up to 5 beforecounting.—Starkey & Cooper
    89. 89. © Joan A. Cotter, Ph.D., 2013Learning 1–10Using fingers
    90. 90. © Joan A. Cotter, Ph.D., 2013Learning 1–10Using fingers
    91. 91. © Joan A. Cotter, Ph.D., 201391Learning 1–10Using fingers
    92. 92. © Joan A. Cotter, Ph.D., 201392Learning 1–10Using fingers
    93. 93. © Joan A. Cotter, Ph.D., 201393Learning 1–10Using fingers
    94. 94. © Joan A. Cotter, Ph.D., 201394Learning 1–10Using fingers
    95. 95. © Joan A. Cotter, Ph.D., 2013Learning 1–10Subitizing 5
    96. 96. © Joan A. Cotter, Ph.D., 2013Learning 1–10Subitizing 5
    97. 97. © Joan A. Cotter, Ph.D., 2013Learning 1–105 has a middle; 4 does not.Subitizing 5
    98. 98. © Joan A. Cotter, Ph.D., 201398Learning 1–10Tally sticks
    99. 99. © Joan A. Cotter, Ph.D., 201399Learning 1–10Tally sticks
    100. 100. © Joan A. Cotter, Ph.D., 2013100Learning 1–10Tally sticks
    101. 101. © Joan A. Cotter, Ph.D., 2013101Learning 1–10Tally sticksFive as a group.
    102. 102. © Joan A. Cotter, Ph.D., 2013102Learning 1–10Tally sticks
    103. 103. © Joan A. Cotter, Ph.D., 2013103Learning 1–10Tally sticks
    104. 104. © Joan A. Cotter, Ph.D., 2013Learning 1–10Entering quantities
    105. 105. © Joan A. Cotter, Ph.D., 20133Learning 1–10Entering quantities
    106. 106. © Joan A. Cotter, Ph.D., 20131065Learning 1–10Entering quantities
    107. 107. © Joan A. Cotter, Ph.D., 20131077Learning 1–10Entering quantities
    108. 108. © Joan A. Cotter, Ph.D., 2013108Learning 1–1010Entering quantities
    109. 109. © Joan A. Cotter, Ph.D., 2013109Learning 1–10The stairs
    110. 110. © Joan A. Cotter, Ph.D., 2013Learning 1–10Adding
    111. 111. © Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 =Adding
    112. 112. © Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 =Adding
    113. 113. © Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 =Adding
    114. 114. © Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 =Adding
    115. 115. © Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 = 7Adding
    116. 116. © Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 = 7VisualizingJapanese children learn to do this mentally.
    117. 117. © Joan A. Cotter, Ph.D., 2013117Visualizing• Visual is related to seeing.• Visualize is to form a mental image.
    118. 118. © Joan A. Cotter, Ph.D., 2013118Visualizing―Think in pictures, because thebrain remembers images betterthan it does anything else.‖—Ben Pridmore, World Memory Champion,2009
    119. 119. © Joan A. Cotter, Ph.D., 2013119Visualizing―The role of physical manipulativeswas to help the child form those visualimages and thus to eliminate the needfor the physical manipulatives.‖—Ginsberg and others
    120. 120. © Joan A. Cotter, Ph.D., 2013VisualizingJapanese criteria for manipulatives
    121. 121. © Joan A. Cotter, Ph.D., 2013• Representative of structure ofnumbers.VisualizingJapanese criteria for manipulatives
    122. 122. © Joan A. Cotter, Ph.D., 2013• Representative of structure ofnumbers.• Easily manipulated by children.VisualizingJapanese criteria for manipulatives
    123. 123. © Joan A. Cotter, Ph.D., 2013• Representative of structure ofnumbers.• Easily manipulated by children.• Imaginable mentally.VisualizingJapanese criteria for manipulatives—Japanese Council ofMathematics Education
    124. 124. © Joan A. Cotter, Ph.D., 2013Visualizing• Reading• Sports• Creativity• Geography• Engineering• ConstructionNecessary in:
    125. 125. © Joan A. Cotter, Ph.D., 2013Visualizing• Reading• Sports• Creativity• Geography• Engineering• Construction• Architecture• Astronomy• Archeology• Chemistry• Physics• SurgeryNecessary in:
    126. 126. © Joan A. Cotter, Ph.D., 2013VisualizingTry to visualize 8 identical apples withoutgrouping.
    127. 127. © Joan A. Cotter, Ph.D., 2013VisualizingTry to visualize 8 identical apples withoutgrouping.
    128. 128. © Joan A. Cotter, Ph.D., 2013VisualizingNow try to visualize 8 apples: 5 red and 3 green.
    129. 129. © Joan A. Cotter, Ph.D., 2013VisualizingNow try to visualize 8 apples: 5 red and 3 green.
    130. 130. © Joan A. Cotter, Ph.D., 2013Learning 1–10Partitioning
    131. 131. © Joan A. Cotter, Ph.D., 2013Learning 1–105 = +Partitioning
    132. 132. © Joan A. Cotter, Ph.D., 2013Learning 1–105 = 4 + 1Partitioning
    133. 133. © Joan A. Cotter, Ph.D., 2013Learning 1–105 = 3 + 2Partitioning
    134. 134. © Joan A. Cotter, Ph.D., 2013Learning 1–105 = 2 + 3Partitioning
    135. 135. © Joan A. Cotter, Ph.D., 2013Learning 1–105 = 1 + 4Partitioning
    136. 136. © Joan A. Cotter, Ph.D., 2013Learning 1–105 = 5 + 0Partitioning
    137. 137. © Joan A. Cotter, Ph.D., 2013Learning 1–105 = 0 + 5Partitioning
    138. 138. © Joan A. Cotter, Ph.D., 2013Learning 1–10Place value• Place value is the foundation of modernarithmetic.
    139. 139. © Joan A. Cotter, Ph.D., 2013Learning 1–10Place value• Place value is the foundation of modernarithmetic.• Critical for understanding algorithms.
    140. 140. © Joan A. Cotter, Ph.D., 2013Learning 1–10Place value• Place value is the foundation of modernarithmetic.• Critical for understanding algorithms.• Must be taught, not left for discovery.
    141. 141. © Joan A. Cotter, Ph.D., 2013Learning 1–10Place value• Place value is the foundation of modernarithmetic.• Critical for understanding algorithms.• Children need the big picture, not tinysnapshots.• Must be taught, not left for discovery.
    142. 142. © Joan A. Cotter, Ph.D., 2013Place ValueCCSS (K.NBT.1, 1.NBT.2)Does it make sense that students should:• ―Work with numbers 11–19 to gainfoundations for place value.‖ (They arethe most difficult numbers we have inEnglish.)
    143. 143. © Joan A. Cotter, Ph.D., 2013Place ValueCCSS (K.NBT.1, 1.NBT.2)Does it make sense that students should:• ―Work with numbers 11–19 to gainfoundations for place value.‖ (They arethe most difficult numbers we have inEnglish.)Are these really ―special cases‖?• ―10 can be thought of as a bundle often ones — called a ‗ten.‘‖• ―100 can be thought of as a bundle often tens — called a ‗hundred.‘‖
    144. 144. © Joan A. Cotter, Ph.D., 2013Place ValueTwo aspectsStatic• Value of a digit is determined by position.• No position may have more than nine.• As you progress to the left, value at eachposition is ten times greater than previousposition.• (Shown by the place-value cards.)Dynamic (Trading)• 10 ones = 1 ten; 10 tens = 1 hundred;• 10 hundreds = 1 thousand, ….• (Represented on the abacus and other
    145. 145. © Joan A. Cotter, Ph.D., 2013Place ValueAsian number-naming(Math way of numbernaming)• Asian children do not struggle with theteens.
    146. 146. © Joan A. Cotter, Ph.D., 2013Place ValueAsian number-naming(Math way of numbernaming)• Their languages are completely ―ten-based.‖• Asian children do not struggle with theteens.
    147. 147. © Joan A. Cotter, Ph.D., 2013Place ValueAsian number-naming(Math way of numbernaming)• Their languages are completely ―ten-based.‖• Asian children do not struggle with theteens.• Asian countries use the ten-based metricsystem.
    148. 148. © Joan A. Cotter, Ph.D., 2013148―Math‖ Way of Number Naming
    149. 149. © Joan A. Cotter, Ph.D., 2013149―Math‖ Way of Number Naming11 = ten 1
    150. 150. © Joan A. Cotter, Ph.D., 2013150―Math‖ Way of Number Naming11 = ten 112 = ten 2
    151. 151. © Joan A. Cotter, Ph.D., 2013151―Math‖ Way of Number Naming11 = ten 112 = ten 213 = ten 3
    152. 152. © Joan A. Cotter, Ph.D., 2013152―Math‖ Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4
    153. 153. © Joan A. Cotter, Ph.D., 2013153―Math‖ Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 9
    154. 154. © Joan A. Cotter, Ph.D., 2013154―Math‖ Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten
    155. 155. © Joan A. Cotter, Ph.D., 2013155―Math‖ Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten21 = 2-ten 1
    156. 156. © Joan A. Cotter, Ph.D., 2013156―Math‖ Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten21 = 2-ten 122 = 2-ten 2
    157. 157. © Joan A. Cotter, Ph.D., 2013157―Math‖ Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten21 = 2-ten 122 = 2-ten 223 = 2-ten 3
    158. 158. © Joan A. Cotter, Ph.D., 2013158―Math‖ Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten21 = 2-ten 122 = 2-ten 223 = 2-ten 3. . . .. . . .99 = 9-ten 9
    159. 159. © Joan A. Cotter, Ph.D., 2013159―Math‖ Way of Number Naming137 = 1 hundred 3-ten 7
    160. 160. © Joan A. Cotter, Ph.D., 2013160―Math‖ Way of Number Naming137 = 1 hundred 3-ten 7or137 = 1 hundred and 3-ten 7
    161. 161. © Joan A. Cotter, Ph.D., 2013161―Math‖ Way of Number Naming01020304050607080901004 5 6Age (yrs.)Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on youngchildrens counting: A natural experiment in numerical bilingualism. International Journalof Psychology, 23, 319-332.Korean formal [math way]Korean informal [not explicit]ChineseU.S.AverageHighestNumberCounted
    162. 162. © Joan A. Cotter, Ph.D., 2013162―Math‖ Way of Number Naming01020304050607080901004 5 6Age (yrs.)Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on youngchildrens counting: A natural experiment in numerical bilingualism. International Journalof Psychology, 23, 319-332.Korean formal [math way]Korean informal [not explicit]ChineseU.S.AverageHighestNumberCounted
    163. 163. © Joan A. Cotter, Ph.D., 2013163―Math‖ Way of Number Naming01020304050607080901004 5 6Age (yrs.)Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on youngchildrens counting: A natural experiment in numerical bilingualism. International Journalof Psychology, 23, 319-332.Korean formal [math way]Korean informal [not explicit]ChineseU.S.AverageHighestNumberCounted
    164. 164. © Joan A. Cotter, Ph.D., 2013164―Math‖ Way of Number Naming01020304050607080901004 5 6Age (yrs.)Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on youngchildrens counting: A natural experiment in numerical bilingualism. International Journalof Psychology, 23, 319-332.Korean formal [math way]Korean informal [not explicit]ChineseU.S.AverageHighestNumberCounted
    165. 165. © Joan A. Cotter, Ph.D., 2013165―Math‖ Way of Number Naming01020304050607080901004 5 6Age (yrs.)Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on youngchildrens counting: A natural experiment in numerical bilingualism. International Journalof Psychology, 23, 319-332.Korean formal [math way]Korean informal [not explicit]ChineseU.S.AverageHighestNumberCounted
    166. 166. © Joan A. Cotter, Ph.D., 2013166Math Way of Number Naming• Only 11 words are needed to count to 100the math way, 28 in English. (All Indo-European languages are non-standard innumber naming.)
    167. 167. © Joan A. Cotter, Ph.D., 2013167Math Way of Number Naming• Only 11 words are needed to count to 100the math way, 28 in English. (All Indo-European languages are non-standard innumber naming.)• Asian children learn mathematics using themath way of counting.
    168. 168. © Joan A. Cotter, Ph.D., 2013168Math Way of Number Naming• Only 11 words are needed to count to 100the math way, 28 in English. (All Indo-European languages are non-standard innumber naming.)• Asian children learn mathematics using themath way of counting.• They understand place value in first grade;only half of U.S. children understand placevalue at the end of fourth grade.
    169. 169. © Joan A. Cotter, Ph.D., 2013169Math Way of Number Naming• Only 11 words are needed to count to 100the math way, 28 in English. (All Indo-European languages are non-standard innumber naming.)• Asian children learn mathematics using themath way of counting.• They understand place value in first grade;only half of U.S. children understand placevalue at the end of fourth grade.• Mathematics is the science of patterns. Thepatterned math way of counting greatlyhelps children learn number sense.
    170. 170. © Joan A. Cotter, Ph.D., 2013170Math Way of Number NamingCompared to reading
    171. 171. © Joan A. Cotter, Ph.D., 2013171Math Way of Number Naming• Just as reciting the alphabet doesn‘t teachreading, counting doesn‘t teach arithmetic.Compared to reading
    172. 172. © Joan A. Cotter, Ph.D., 2013172Math Way of Number Naming• Just as reciting the alphabet doesn‘t teachreading, counting doesn‘t teach arithmetic.• Just as we first teach the sound of the letters,we must first teach the name of the quantity(math way).Compared to reading
    173. 173. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names4-ten = fortyThe ―ty‖meanstens.
    174. 174. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names4-ten = fortyThe ―ty‖meanstens.
    175. 175. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names6-ten = sixtyThe ―ty‖meanstens.
    176. 176. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names3-ten = thirty―Thir‖ alsoused in 1/3,13 and 30.
    177. 177. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names5-ten = fifty―Fif‖ alsoused in 1/5,15 and 50.
    178. 178. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names2-ten = twentyTwo used tobepronounced―twoo.‖
    179. 179. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesA word gamefireplace place-fire
    180. 180. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesA word gamefireplace place-firepaper-newsnewspaper
    181. 181. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesA word gamefireplace place-firepaper-newsbox-mail mailboxnewspaper
    182. 182. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesten 4Prefix -teenmeans ten.
    183. 183. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesten 4 teen 4Prefix -teenmeans ten.
    184. 184. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesten 4 teen 4 fourteenPrefix -teenmeans ten.
    185. 185. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesa one left
    186. 186. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesa one left a left-one
    187. 187. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesa one left a left-one eleven
    188. 188. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namestwo leftTwo saidas―twoo.‖
    189. 189. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namestwo left twelveTwo saidas―twoo.‖
    190. 190. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten
    191. 191. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten
    192. 192. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten3 0
    193. 193. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten3 0
    194. 194. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten3 0
    195. 195. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten73 0
    196. 196. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten73 0
    197. 197. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten73 07
    198. 198. © Joan A. Cotter, Ph.D., 20133 0Composing Numbers3-ten77
    199. 199. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten7Note the congruence in how we say thenumber, represent the number, and writethe number.3 07
    200. 200. © Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten1 0Another example.
    201. 201. © Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten 81 0
    202. 202. © Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten81 0
    203. 203. © Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten81 08
    204. 204. © Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten81 88
    205. 205. © Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten
    206. 206. © Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten1 0 0
    207. 207. © Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten1 0 0
    208. 208. © Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten1 0 0
    209. 209. © Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred
    210. 210. © Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 0 0
    211. 211. © Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 0 0
    212. 212. © Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 01 01 0 0
    213. 213. © Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 0 0
    214. 214. © Joan A. Cotter, Ph.D., 2013Composing Numbers2hundred
    215. 215. © Joan A. Cotter, Ph.D., 2013Composing Numbers2hundred
    216. 216. © Joan A. Cotter, Ph.D., 2013Composing Numbers2hundred2 0 0
    217. 217. © Joan A. Cotter, Ph.D., 2013217Learning the Facts
    218. 218. © Joan A. Cotter, Ph.D., 2013218Learning the FactsLimited success, especially for strugglingchildren, when learning is:
    219. 219. © Joan A. Cotter, Ph.D., 2013219Learning the Facts• Based on counting: whether dots,fingers, number lines, or countingwords.Limited success, especially for strugglingchildren, when learning is:
    220. 220. © Joan A. Cotter, Ph.D., 2013220Learning the Facts• Based on counting: whether dots,fingers, number lines, or countingwords.Limited success, especially for strugglingchildren, when learning is:• Based on rote memory: whether flashcards, timed tests, or computer games.
    221. 221. © Joan A. Cotter, Ph.D., 2013221Learning the Facts• Based on counting: whether dots,fingers, number lines, or countingwords.Limited success, especially for strugglingchildren, when learning is:• Based on rote memory: whether flashcards, timed tests, or computer games.• Based on skip counting: whether fingers or songs
    222. 222. © Joan A. Cotter, Ph.D., 2013222Fact Strategies
    223. 223. © Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =
    224. 224. © Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =
    225. 225. © Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =
    226. 226. © Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =Take 1 fromthe 5 and giveit to the 9.
    227. 227. © Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =Take 1 fromthe 5 and giveit to the 9.
    228. 228. © Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =Take 1 fromthe 5 and giveit to the 9.
    229. 229. © Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 = 14Take 1 fromthe 5 and giveit to the 9.
    230. 230. © Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =
    231. 231. © Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =
    232. 232. © Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =
    233. 233. © Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =
    234. 234. © Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =10 + 4 = 14
    235. 235. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 =
    236. 236. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 =
    237. 237. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 =Subtract 5;then 4.
    238. 238. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 =Subtract 5;then 4.
    239. 239. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 =Subtract 5;then 4.
    240. 240. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 = 6Subtract 5;then 4.
    241. 241. © Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 – 9 =
    242. 242. © Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 – 9 =Subtract 9from 10.
    243. 243. © Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 – 9 =Subtract 9from 10.
    244. 244. © Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 – 9 =Subtract 9from 10.
    245. 245. © Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 – 9 = 6Subtract 9from 10.
    246. 246. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =
    247. 247. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =Start with 9;go up to 15.
    248. 248. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =Start with 9;go up to 15.
    249. 249. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =Start with 9;go up to 15.
    250. 250. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =Start with 9;go up to 15.
    251. 251. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =1 + 5 = 6Start with 9;go up to 15.
    252. 252. © Joan A. Cotter, Ph.D., 2013MoneyPenny
    253. 253. © Joan A. Cotter, Ph.D., 2013MoneyNickel
    254. 254. © Joan A. Cotter, Ph.D., 2013MoneyDime
    255. 255. © Joan A. Cotter, Ph.D., 2013MoneyQuarter
    256. 256. © Joan A. Cotter, Ph.D., 2013MoneyQuarter
    257. 257. © Joan A. Cotter, Ph.D., 2013MoneyQuarter
    258. 258. © Joan A. Cotter, Ph.D., 2013MoneyQuarter
    259. 259. © Joan A. Cotter, Ph.D., 2013Trading1000 10 1100
    260. 260. © Joan A. Cotter, Ph.D., 2013TradingThousands1000 10 1100
    261. 261. © Joan A. Cotter, Ph.D., 2013TradingHundreds1000 10 1100
    262. 262. © Joan A. Cotter, Ph.D., 2013TradingTens1000 10 1100
    263. 263. © Joan A. Cotter, Ph.D., 2013TradingOnes1000 10 1100
    264. 264. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6
    265. 265. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6
    266. 266. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6
    267. 267. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6
    268. 268. © Joan A. Cotter, Ph.D., 2013TradingAdding8+ 6141000 10 1100
    269. 269. © Joan A. Cotter, Ph.D., 2013TradingAdding8+ 614Too manyones; trade 10ones for 1 ten.1000 10 1100
    270. 270. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 614Too manyones; trade 10ones for 1 ten.
    271. 271. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 614Too manyones; trade 10ones for 1 ten.
    272. 272. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 614Same answerbefore andafter trading.
    273. 273. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738
    274. 274. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter the firstnumber fromleft to right.
    275. 275. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right.
    276. 276. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right.
    277. 277. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right.
    278. 278. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right.
    279. 279. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right.
    280. 280. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults aftereach step.
    281. 281. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults after eachstep.
    282. 282. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults after eachstep.
    283. 283. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults after eachstep.
    284. 284. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults aftereach step.
    285. 285. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults aftereach step.1
    286. 286. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults aftereach step.1
    287. 287. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults aftereach step.1
    288. 288. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults aftereach step.1
    289. 289. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults aftereach step.1
    290. 290. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults aftereach step.1
    291. 291. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults aftereach step.1
    292. 292. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults aftereach step.1
    293. 293. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults aftereach step.1
    294. 294. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults aftereach step.1 1
    295. 295. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults aftereach step.1 1
    296. 296. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults after eachstep.1 1
    297. 297. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386396Add starting atthe right. Writeresults after eachstep.1 1
    298. 298. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386396Add starting atthe right. Writeresults after eachstep.1 1
    299. 299. © Joan A. Cotter, Ph.D., 2013299Meeting the Standards
    300. 300. © Joan A. Cotter, Ph.D., 2013300Meeting the StandardsPage 5―These Standards do not dictate curriculum orteaching methods. For example, just becausetopic A appears before topic B in the standardsfor a given grade, it does not necessarily meanthat topic A must be taught before topic B. Ateacher might prefer to teach topic B beforetopic A, or might choose to highlightconnections by teaching topic A and topic B atthe same time. Or, a teacher might prefer toteach a topic of his or her own choosing thatleads, as a byproduct, to students reaching thestandards for topics A and B.‖ —CCSS
    301. 301. © Joan A. Cotter, Ph.D., 2013301Meeting the StandardsPage 5 summary• Standards do not dictate curriculum orteaching methods.
    302. 302. © Joan A. Cotter, Ph.D., 2013302Meeting the StandardsPage 5 summary• Standards do not dictate curriculum orteaching methods.• Within a grade, topics may be taught inany order or taught indirectly.
    303. 303. © Joan A. Cotter, Ph.D., 2013303Meeting the StandardsKindergarten (K.NBT)Know number names and the countsequence.1. Count to 100 by ones and by tens.2. Count forward beginning from a givennumber within the known sequence(instead of having to begin at 1).3. Write numbers from 0 to 20. Represent anumber of objects with a written numeral0-20 (with 0 representing a count of noobjects).
    304. 304. © Joan A. Cotter, Ph.D., 2013304Meeting the StandardsKindergarten (K.CC)1. Count to 100 by ones and by tens.2. Count forward beginning from a givennumber.
    305. 305. © Joan A. Cotter, Ph.D., 2013305Meeting the StandardsKindergarten (K.CC)1. Count to 100 by ones and by tens.2. Count forward beginning from a givennumber.
    306. 306. © Joan A. Cotter, Ph.D., 2013306Meeting the StandardsKindergarten (K.CC)1. Count to 100 by ones and by tens.2. Count forward beginning from a givennumber.
    307. 307. © Joan A. Cotter, Ph.D., 2013Meeting the Standards61728394105Kindergarten (K.CC)3. Write numbers from 0 to 20.Number Chart
    308. 308. © Joan A. Cotter, Ph.D., 2013308Meeting the StandardsKindergarten (K.NBT)Work with numbers 11–19.1. Compose and partition numbers from 11 to19 into ten ones and some further ones.
    309. 309. © Joan A. Cotter, Ph.D., 2013309Meeting the StandardsKindergarten (K.NBT)Work with numbers 11–19.1. Compose and partition numbers from 11 to19 into ten ones and some further ones.
    310. 310. © Joan A. Cotter, Ph.D., 2013310Meeting the StandardsKindergarten (K.NBT)Work with numbers 11–19.1. Compose and partition numbers from 11 to19 into ten ones and some further ones.1 861 06
    311. 311. © Joan A. Cotter, Ph.D., 2013311Meeting the StandardsKindergarten (K.OA)Understand addition and subtraction.1. Represent addition and subtraction withobjects, fingers, . . . equations.2. Solve addition and subtraction wordproblems, and add and subtract within 10.3. Partition numbers less than or equal to 10into pairs.4. For any number from 1 to 9, find thenumber that makes 10.5. Fluently add and subtract within 5.
    312. 312. © Joan A. Cotter, Ph.D., 2013312Meeting the StandardsKindergarten (K.OA)2. Solve addition and subtraction wordproblems, and add and subtract within 10.WholePart PartPart-wholecircles
    313. 313. © Joan A. Cotter, Ph.D., 2013313Meeting the StandardsUsing part-whole circles to solve problemsLee received 3 goldfish as a gift. NowLee has 5. How many did Lee have tostart with?
    314. 314. © Joan A. Cotter, Ph.D., 2013314Meeting the StandardsUsing part-whole circles to solve problemsLee received 3 goldfish as a gift. NowLee has 5. How many did Lee have tostart with?Is 3 a part or whole?
    315. 315. © Joan A. Cotter, Ph.D., 2013315Meeting the StandardsUsing part-whole circles to solve problemsLee received 3 goldfish as a gift. NowLee has 5. How many did Lee have tostart with?Is 3 a part or whole?3
    316. 316. © Joan A. Cotter, Ph.D., 2013316Meeting the StandardsUsing part-whole circles to solve problemsLee received 3 goldfish as a gift. NowLee has 5. How many did Lee have tostart with?Is 5 a part or whole?3
    317. 317. © Joan A. Cotter, Ph.D., 2013317Meeting the StandardsUsing part-whole circles to solve problemsLee received 3 goldfish as a gift. NowLee has 5. How many did Lee have tostart with?Is 5 a part or whole?35
    318. 318. © Joan A. Cotter, Ph.D., 2013318Meeting the StandardsUsing part-whole circles to solve problemsLee received 3 goldfish as a gift. NowLee has 5. How many did Lee have tostart with?What is the missing part?35
    319. 319. © Joan A. Cotter, Ph.D., 2013319Meeting the StandardsUsing part-whole circles to solve problemsLee received 3 goldfish as a gift. NowLee has 5. How many did Lee have tostart with?What is the missing part?352
    320. 320. © Joan A. Cotter, Ph.D., 2013320Meeting the StandardsKindergarten (K.OA)4. For any number from 1 to 9, find thenumber that makes 10.107 3
    321. 321. © Joan A. Cotter, Ph.D., 2013321Meeting the StandardsGrade 1 (1.OA)Understand and apply properties ofoperations and the relationship betweenaddition and subtraction.1. Apply properties of operations asstrategies to add and subtract,commutative property and associativeproperty of addition.2. Understand subtraction as an unknown-addend problem. [Subtract by going up.]
    322. 322. © Joan A. Cotter, Ph.D., 2013322Meeting the StandardsGrade 1 (1.OA)1. Apply properties of operations asstrategies to add and subtract,commutative property and associativeproperty of addition.6 + 3 = 93 + 6 = 9
    323. 323. © Joan A. Cotter, Ph.D., 2013323Meeting the StandardsGrade 1 (1.OA)Work with addition and subtractionequations.7. Understand the meaning of the equal sign.8. Determine the unknown whole number inan addition or subtraction equation.
    324. 324. © Joan A. Cotter, Ph.D., 2013324Meeting the StandardsGrade 1 (1.OA)7. Understand the meaning of the equal sign.10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10Math balance
    325. 325. © Joan A. Cotter, Ph.D., 2013325Meeting the StandardsGrade 1 (1.OA)7. Understand the meaning of the equal sign.10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 107 = 7
    326. 326. © Joan A. Cotter, Ph.D., 2013326Meeting the StandardsGrade 1 (1.OA)7. Understand the meaning of the equal sign.10 = 3 + 710 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
    327. 327. © Joan A. Cotter, Ph.D., 2013327Meeting the StandardsGrade 1 (1.OA)7. Understand the meaning of the equal sign.10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 108 + 2 = 10
    328. 328. © Joan A. Cotter, Ph.D., 2013328Meeting the StandardsGrade 1 (1.OA)7. Understand the meaning of the equal sign.10 9 8 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 107 + 7 = 147
    329. 329. © Joan A. Cotter, Ph.D., 2013329Meeting the StandardsGrade 1 (1.OA)8 + _ = 118. Determine the unknown whole number inan addition or subtraction equation.
    330. 330. © Joan A. Cotter, Ph.D., 2013330Meeting the StandardsGrade 1 (1.OA)10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 108 + 3 = 118. Determine the unknown whole number inan addition or subtraction equation.
    331. 331. © Joan A. Cotter, Ph.D., 2013331Meeting the StandardsGrade 1 (1.OA)Extend the counting sequence.1. Count to 120, starting at any number lessthan 120.
    332. 332. © Joan A. Cotter, Ph.D., 2013332Meeting the StandardsGrade 1 (1.OA)Extend the counting sequence.1. Count to 120, starting at any number lessthan 120.1 0 01 091 0 01 09
    333. 333. © Joan A. Cotter, Ph.D., 2013333Meeting the StandardsGrade 1 (1.NBT)Understanding place value.3. Compare two two-digit numbers, recordingthe results of comparisons with symbols >,=, <.4. Add a two-digit number and a multiple of10.5. Mentally find 10 more or 10 less than thenumber, without having to count.6. Subtract multiples of 10 in the range 10-90from multiples of 10.
    334. 334. © Joan A. Cotter, Ph.D., 2013334Meeting the StandardsGrade 1 (1.NBT)3. Compare two two-digit numbers, recordingthe results of comparisons with symbols >,=, <.4 06 6 04
    335. 335. © Joan A. Cotter, Ph.D., 2013335Meeting the StandardsGrade 1 (1.NBT)3. Compare two two-digit numbers, recordingthe results of comparisons with symbols >,=, <.4 06 6 0446 64Put two dots by greaternumber.
    336. 336. © Joan A. Cotter, Ph.D., 2013336Meeting the StandardsGrade 1 (1.NBT)3. Compare two two-digit numbers, recordingthe results of comparisons with symbols >,=, <.4 06 6 0446 64..Put two dots by greaternumber.
    337. 337. © Joan A. Cotter, Ph.D., 2013337Meeting the StandardsGrade 1 (1.NBT)3. Compare two two-digit numbers, recordingthe results of comparisons with symbols >,=, <.4 06 6 0446 64..Put two dots by greaternumber.Put one dot by lessernumber.
    338. 338. © Joan A. Cotter, Ph.D., 2013338Meeting the StandardsGrade 1 (1.NBT)3. Compare two two-digit numbers, recordingthe results of comparisons with symbols >,=, <.4 06 6 0446 64...Put two dots by greaternumber.Put one dot by lessernumber.
    339. 339. © Joan A. Cotter, Ph.D., 2013339Meeting the StandardsGrade 1 (1.NBT)3. Compare two two-digit numbers, recordingthe results of comparisons with symbols >,=, <.4 06 6 0446 64...Put two dots by greaternumber.Put one dot by lessernumber.
    340. 340. © Joan A. Cotter, Ph.D., 2013340Meeting the StandardsGrade 1 (1.NBT)3. Compare two two-digit numbers, recordingthe results of comparisons with symbols >,=, <.4 06 6 0446 64...Put two dots by greaternumber.Put one dot by lessernumber.
    341. 341. © Joan A. Cotter, Ph.D., 2013341Meeting the StandardsGrade 1 (1.NBT)4. Add a two-digit number and a multiple of10.5. Mentally find 10 more or 10 less than thenumber, without having to count.24 + 10 = __
    342. 342. © Joan A. Cotter, Ph.D., 2013342Meeting the StandardsGrade 1 (1.NBT)4. Add a two-digit number and a multiple of10.5. Mentally find 10 more or 10 less than thenumber, without having to count.24 + 10 = 34
    343. 343. © Joan A. Cotter, Ph.D., 2013343Meeting the StandardsGrade 1 (1.NBT)4. Add a two-digit number and a multiple of10.5. Mentally find 10 more or 10 less than thenumber, without having to count.24 – 10 = __
    344. 344. © Joan A. Cotter, Ph.D., 2013344Meeting the StandardsGrade 1 (1.NBT)4. Add a two-digit number and a multiple of10.5. Mentally find 10 more or 10 less than thenumber, without having to count.24 – 10 = 14
    345. 345. © Joan A. Cotter, Ph.D., 2013345Meeting the StandardsGrade 1 (1.NBT)6. Subtract multiples of 10 in the range 10-90from multiples of 10.90 – 30 = __
    346. 346. © Joan A. Cotter, Ph.D., 2013346Meeting the StandardsGrade 1 (1.NBT)6. Subtract multiples of 10 in the range 10-90from multiples of 10.90 – 30 = 60
    347. 347. © Joan A. Cotter, Ph.D., 2013347Meeting the StandardsGrade 2 (2.OA)Work with equal groups of objects to gainfoundations for multiplication.3. Determine whether a group of objects (up to20) has an odd or even number of members.4. Use addition to find the total number ofobjects arranged in rectangular arrays.
    348. 348. © Joan A. Cotter, Ph.D., 2013348Meeting the StandardsGrade 2 (2.OA)3. Determine whether a group of objects (up to20) has an odd or even number of members.Is 17 evenor odd?
    349. 349. © Joan A. Cotter, Ph.D., 2013349Meeting the StandardsGrade 2 (2.OA)3. Determine whether a group of objects (up to20) has an odd or even number of members.Is 17 evenor odd?
    350. 350. © Joan A. Cotter, Ph.D., 2013350Meeting the StandardsGrade 2 (2.OA)4. Use addition to find the total number ofobjects arranged in rectangular arrays.
    351. 351. © Joan A. Cotter, Ph.D., 2013351Meeting the StandardsGrade 2 (2.OA)4. Use addition to find the total number ofobjects arranged in rectangular arrays.5 + 5 + 5 + 5 = 20
    352. 352. © Joan A. Cotter, Ph.D., 2013352Meeting the StandardsGrade 2 (2.OA)4. Use addition to find the total number ofobjects arranged in rectangular arrays.10 9 8 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 105 + 5 + 5 + 5 = 207
    353. 353. © Joan A. Cotter, Ph.D., 2013353Meeting the StandardsGrade 2 (2.NBT)Number and Operations in Base Ten.2. Count within 1000; skip-count by 2s, 5s,10s, and 100s.3. Read and write numbers to 1000 usingbase-ten numerals, number names, andexpanded form.4. Compare two three-digit numbers basedon meanings of the hundreds, tens, andones digits, using >, =, and <.
    354. 354. © Joan A. Cotter, Ph.D., 2013354Meeting the StandardsGrade 2 (2.NBT)2. Skip-count by 2s, 5s, 10s, and 100s.
    355. 355. © Joan A. Cotter, Ph.D., 2013355Meeting the StandardsGrade 2 (2.NBT)2. Skip-count by 2s, 5s, 10s, and 100s.5,
    356. 356. © Joan A. Cotter, Ph.D., 2013356Meeting the StandardsGrade 2 (2.NBT)2. Skip-count by 2s, 5s, 10s, and 100s.5, 10,
    357. 357. © Joan A. Cotter, Ph.D., 2013357Meeting the StandardsGrade 2 (2.NBT)2. Skip-count by 2s, 5s, 10s, and 100s.5, 10, 15, . . .
    358. 358. © Joan A. Cotter, Ph.D., 2013358Meeting the StandardsGrade 2 (2.NBT)2. Skip-count by 2s, 5s, 10s, and 100s.1000 10 1100100, 200, 300, . . .
    359. 359. © Joan A. Cotter, Ph.D., 2013359Meeting the StandardsGrade 2 (2.NBT)2. Count within 1000.3. Read and write numbers to 1000 usingbase-ten numerals, number names, andexpanded form.3 0 07 08378,
    360. 360. © Joan A. Cotter, Ph.D., 2013360Meeting the StandardsGrade 2 (2.NBT)2. Count within 1000.3. Read and write numbers to 1000 usingbase-ten numerals, number names, andexpanded form.3 0 07 083 0 07 09378, 379,
    361. 361. © Joan A. Cotter, Ph.D., 2013361Meeting the StandardsGrade 2 (2.NBT)2. Count within 1000.3. Read and write numbers to 1000 usingbase-ten numerals, number names, andexpanded form.3 0 07 083 0 07 093 0 08 0378, 379, 380
    362. 362. © Joan A. Cotter, Ph.D., 2013362Meeting the StandardsGrade 2 (2.NBT)4. Compare two three-digit numbers basedon meanings of the hundreds, tens, andones digits, using >, =, and <.7 0 00 06 6 0 07 00706 > 670
    363. 363. © Joan A. Cotter, Ph.D., 2013363ObjectivesI. Review the traditional counting trajectory.II. Experience traditional counting like a child.III. Group in 5s and 10s: an alternative tocounting.IV. Meet CCSS without counting.
    364. 364. © Joan A. Cotter, Ph.D., 2013Teaching Primary Mathematics with MoreUnderstanding and Less CountingNational Council of Supervisors of MathematicsMonday, April 16, 2013Denver, ColoradoJoan A. Cotter, Ph.D.JoanCotter@RightStartMath.comandTracy Mittleider, MESdTracy@RightStartMath.com

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