NJMAC Visualization

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  • Montessori math materials ingeniously introduces children to our decimal system, but current research suggests that mathematical mastery can be better facilitated with simple enhancements in teaching techniques and material extensions. In this workshop, learn about research-based math discoveries, and explore ideas for Montessori math refinements, such as grouping the materials in fives to reduce counting and help the child in forming abstract images.
  • Show the baby 2 bears.
  • Show the baby 2 bears.
  • Show the baby 2 bears.
  • Show the baby 2 bears.
  • Show the baby 2 bears.
  • Stairs
  • Montessori math materials ingeniously introduces children to our decimal system, but current research suggests that mathematical mastery can be better facilitated with simple enhancements in teaching techniques and material extensions. In this workshop, learn about research-based math discoveries, and explore ideas for Montessori math refinements, such as grouping the materials in fives to reduce counting and help the child in forming abstract images.
  • NJMAC Visualization

    1. 1. Enriching Montessori Mathematics with Visualization by Joan A. Cotter, Ph.D. JoanCotter@rightstartmath.com 1000 3 2 5 5 100 10 7 x7 1NJMAC Conference March 2, 2012 Edison, New Jersey Presentations available: rightstartmath.com © Joan A. Cotter, Ph.D., 2012
    2. 2. Verbal Counting Model2 © Joan A. Cotter, Ph.D., 2012
    3. 3. Verbal Counting Model From a childs perspective Because we’re so familiar with 1, 2, 3, we’ll use letters. A=1 B=2 C=3 D=4 E = 5, and so forth3 © Joan A. Cotter, Ph.D., 2012
    4. 4. Verbal Counting Model From a childs perspective F +E4 © Joan A. Cotter, Ph.D., 2012
    5. 5. Verbal Counting Model From a childs perspective F +E A5 © Joan A. Cotter, Ph.D., 2012
    6. 6. Verbal Counting Model From a childs perspective F +E A B6 © Joan A. Cotter, Ph.D., 2012
    7. 7. Verbal Counting Model From a childs perspective F +E A B C7 © Joan A. Cotter, Ph.D., 2012
    8. 8. Verbal Counting Model From a childs perspective F +E A B C D E F8 © Joan A. Cotter, Ph.D., 2012
    9. 9. Verbal Counting Model From a childs perspective F +E A B C D E F A9 © Joan A. Cotter, Ph.D., 2012
    10. 10. Verbal Counting Model From a childs perspective F +E A B C D E F A B10 © Joan A. Cotter, Ph.D., 2012
    11. 11. Verbal Counting Model From a childs perspective F +E A B C D E F A B C D E11 © Joan A. Cotter, Ph.D., 2012
    12. 12. Verbal Counting Model From a childs perspective F +E A B C D E F A B C D E What is the sum? (It must be a letter.)12 © Joan A. Cotter, Ph.D., 2012
    13. 13. Verbal Counting Model From a childs perspective F +E K A B C D E F G H I J K13 © Joan A. Cotter, Ph.D., 2012
    14. 14. Verbal Counting Model From a childs perspective Now memorize the facts!! G +D14 © Joan A. Cotter, Ph.D., 2012
    15. 15. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D15 © Joan A. Cotter, Ph.D., 2012
    16. 16. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D D +C16 © Joan A. Cotter, Ph.D., 2012
    17. 17. Verbal Counting Model From a childs perspective Now memorize the facts!! H + G F +D D C +C +G17 © Joan A. Cotter, Ph.D., 2012
    18. 18. Verbal Counting Model From a childs perspective Now memorize the facts!! H E + G I F + +D D C +C +G18 © Joan A. Cotter, Ph.D., 2012
    19. 19. Verbal Counting Model From a childs perspective Try subtracting H by “taking away” –E19 © Joan A. Cotter, Ph.D., 2012
    20. 20. Verbal Counting Model From a childs perspective Try skip counting by B’s to T: B, D, . . . T.20 © Joan A. Cotter, Ph.D., 2012
    21. 21. Verbal Counting Model From a childs perspective Try skip counting by B’s to T: B, D, . . . T. What is D × E?21 © Joan A. Cotter, Ph.D., 2012
    22. 22. Verbal Counting Model From a childs perspective L is written AB because it is A J and B A’s22 © Joan A. Cotter, Ph.D., 2012
    23. 23. Verbal Counting Model From a childs perspective L is written AB because it is A J and B A’s huh?23 © Joan A. Cotter, Ph.D., 2012
    24. 24. Verbal Counting Model From a childs perspective L (twelve) is written AB because it is A J and B A’s24 © Joan A. Cotter, Ph.D., 2012
    25. 25. Verbal Counting Model From a childs perspective L (twelve) is written AB (12) because it is A J and B A’s25 © Joan A. Cotter, Ph.D., 2012
    26. 26. Verbal Counting Model From a childs perspective L (twelve) is written AB (12) because it is A J (one 10) and B A’s26 © Joan A. Cotter, Ph.D., 2012
    27. 27. Verbal Counting Model From a childs perspective L (twelve) is written AB (12) because it is A J (one 10) and B A’s (two 1s).27 © Joan A. Cotter, Ph.D., 2012
    28. 28. Verbal Counting Model Summary28 © Joan A. Cotter, Ph.D., 2012
    29. 29. Verbal Counting Model Summary • Is not natural; it takes years of practice.29 © Joan A. Cotter, Ph.D., 2012
    30. 30. Verbal Counting Model Summary • Is not natural; it takes years of practice. • Provides poor concept of quantity.30 © Joan A. Cotter, Ph.D., 2012
    31. 31. Verbal Counting Model Summary • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value.31 © Joan A. Cotter, Ph.D., 2012
    32. 32. Verbal Counting Model Summary • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value. • Is very error prone.32 © Joan A. Cotter, Ph.D., 2012
    33. 33. Verbal Counting Model Summary • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value. • Is very error prone. • Is tedious and time-consuming.33 © Joan A. Cotter, Ph.D., 2012
    34. 34. Verbal Counting Model Summary • Is not natural; it takes years of practice. • Provides poor concept of quantity. • Ignores place value. • Is very error prone. • Is tedious and time-consuming. • Does not provide an efficient way to master the facts.34 © Joan A. Cotter, Ph.D., 2012
    35. 35. Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31Sometimes calendars are used for counting. © Joan A. Cotter, Ph.D., 201235
    36. 36. Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31Sometimes calendars are used for counting. © Joan A. Cotter, Ph.D., 201236
    37. 37. Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 3137 © Joan A. Cotter, Ph.D., 2012
    38. 38. Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31This is ordinal, not cardinal counting. The 3 doesn’t include the 1 and the 2.Joan A. Cotter, Ph.D., 2012 ©38
    39. 39. Calendar Math Septemb 1234567 August 89101214 1 113 11921 2 15112628 122820 8 67527 9 3 4 10 11 12 13 14 5 6 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31This is ordinal, not cardinal counting. The 4 doesn’t include 1, 2 and 3. © Joan A. Cotter, Ph.D., 201239
    40. 40. Calendar Math Septemb 1234567 August 89101214 1 113 11921 2 15112628 122820 8 67527 9 3 4 5 10 11 12 13 14 6 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31 1 2 3 4 5 6A calendar is NOT a ruler. On a ruler the numbers are not in the spaces. © Joan A. Cotter, Ph.D., 201240
    41. 41. Calendar Math August 1 2 3 4 5 6 7 8 9 10Always show the whole calendar. A child needs to see the wholebefore the parts. Children also need to learn to plan ahead. © Joan A. Cotter, Ph.D., 201241
    42. 42. Calendar Math The calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler.42 © Joan A. Cotter, Ph.D., 2012
    43. 43. Calendar Math The calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler. Children need to see the whole month, not just part. • Purpose of calendar is to plan ahead. • Many ways to show the current date.43 © Joan A. Cotter, Ph.D., 2012
    44. 44. Calendar Math The calendar is not a number line. • No quantity is involved. • Numbers are in spaces, not at lines like a ruler. Children need to see the whole month, not just part. • Purpose of calendar is to plan ahead. • Many ways to show the current date. Calendars give a narrow view of patterning. • Patterns do not necessarily involve numbers. • Patterns rarely proceed row by row. • Patterns go on forever; they don’t stop at 31.44 © Joan A. Cotter, Ph.D., 2012
    45. 45. Memorizing Math 9 +7 Flash cards:• Are often used to teach rote.• Are liked only by those who don’t need them.• Don’t work for those with learning disabilities.• Give the false impression that math isn’t aboutthinking.• Often produce stress – children under stressstop learning.• Are not concrete – use abstract symbols. © Joan A. Cotter, Ph.D., 2012
    46. 46. Learning Arithmetic Compared to reading: • A child learns to read. • Later a child uses reading to learn. • A child learns to do arithmetic. • Later a child uses arithmetic to solve problems.Show the baby two teddy bears. © Joan A. Cotter, Ph.D., 2012
    47. 47. Research on Counting Karen Wynn’s researchShow the baby two teddy bears. © Joan A. Cotter, Ph.D., 2012
    48. 48. Research on Counting Karen Wynn’s researchShow the baby two teddy bears. © Joan A. Cotter, Ph.D., 2012
    49. 49. Research on Counting Karen Wynn’s researchThen hide them with a screen. © Joan A. Cotter, Ph.D., 201249
    50. 50. Research on Counting Karen Wynn’s researchShow the baby a third teddy bear and put it behind the screen. © Joan A. Cotter, Ph.D., 201250
    51. 51. Research on Counting Karen Wynn’s researchShow the baby a third teddy bear and put it behind the screen. © Joan A. Cotter, Ph.D., 201251
    52. 52. Research on Counting Karen Wynn’s researchRaise screen. Baby seeing 3 won’t look long because it is expected. © Joan A. Cotter, Ph.D., 201252
    53. 53. Research on Counting Karen Wynn’s researchResearcher can change the number of teddy bears behind the screen. © Joan A. Cotter, Ph.D., 201253
    54. 54. Research on Counting Karen Wynn’s researchA baby seeing 1 teddy bear will look much longer, because it’s unexpected.Joan A. Cotter, Ph.D., 2012 ©54
    55. 55. Research on Counting Other research55 © Joan A. Cotter, Ph.D., 2012
    56. 56. Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008.These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 201256
    57. 57. Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. • Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008.These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 201257
    58. 58. Research on Counting Other research • 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.These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 201258
    59. 59. Research on Counting Other research • 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.These groups matched quantities without using counting words. © Joan A. Cotter, Ph.D., 201259
    60. 60. Research on Counting In Japanese schools: • Children are discouraged from using counting for adding.60 © Joan A. Cotter, Ph.D., 2012
    61. 61. Research on Counting In Japanese schools: • Children are discouraged from using counting for adding. • They consistently group in 5s.61 © Joan A. Cotter, Ph.D., 2012
    62. 62. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting.62 © Joan A. Cotter, Ph.D., 2012
    63. 63. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth.63 © Joan A. Cotter, Ph.D., 2012
    64. 64. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics long term.—Butterworth64 © Joan A. Cotter, Ph.D., 2012
    65. 65. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics long term.—Butterworth • Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit65 © Joan A. Cotter, Ph.D., 2012
    66. 66. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics long term.—Butterworth • Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit • Subitizing seems to be a necessary skill for understanding what the counting process means.— Glasersfeld66 © Joan A. Cotter, Ph.D., 2012
    67. 67. Visualizing Mathematics67 © Joan A. Cotter, Ph.D., 2012
    68. 68. Visualizing Mathematics “In our concern about the memorization of math facts or solving problems, we must not forget that the root of mathematical study is the creation of mental pictures in the imagination and manipulating those images and relationships using the power of reason and logic.” Mindy Holte (E I)68 © Joan A. Cotter, Ph.D., 2012
    69. 69. Visualizing Mathematics “Think in pictures, because the brain remembers images better than it does anything else.” Ben Pridmore, World Memory Champion, 200969 © Joan A. Cotter, Ph.D., 2012
    70. 70. Visualizing Mathematics “The role of physical manipulatives was to help the child form those visual images and thus to eliminate the need for the physical manipulatives.” Ginsberg and others70 © Joan A. Cotter, Ph.D., 2012
    71. 71. Visualizing Mathematics Japanese criteria for manipulatives• Representative of structure of numbers.• Easily manipulated by children.• Imaginable mentally. Japanese Council of Mathematics Education © Joan A. Cotter, Ph.D., 2012
    72. 72. Visualizing Mathematics Visualizing also needed in:• Reading• Sports• Creativity• Geography• Engineering• Construction © Joan A. Cotter, Ph.D., 2012
    73. 73. Visualizing Mathematics Visualizing also needed in:• Reading • Architecture• Sports • Astronomy• Creativity • Archeology• Geography • Chemistry• Engineering • Physics• Construction • Surgery © Joan A. Cotter, Ph.D., 2012
    74. 74. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
    75. 75. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
    76. 76. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
    77. 77. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
    78. 78. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
    79. 79. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
    80. 80. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
    81. 81. Visualizing MathematicsTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
    82. 82. Visualizing MathematicsTry to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
    83. 83. Visualizing MathematicsNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
    84. 84. Visualizing MathematicsNow try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
    85. 85. Visualizing Mathematics Early Roman numerals 1 I 2 II 3 III 4 IIII 5 V 8 VIIIRomans grouped in fives. Notice 8 is 5 and 3. © Joan A. Cotter, Ph.D., 2012
    86. 86. Visualizing Mathematics : Who could read the music?Music needs 10 lines, two groups of five. © Joan A. Cotter, Ph.D., 201286
    87. 87. Very Early Computation Numerals In English there are two ways of writing numbers: Numerals: 357887 © Joan A. Cotter, Ph.D., 2012
    88. 88. Very Early Computation Numerals In English there are two ways of writing numbers: Numerals: 3578 Words: Three thousand five hundred seventy-eight88 © Joan A. Cotter, Ph.D., 2012
    89. 89. Very Early Computation Numerals In English there are two ways of writing numbers: Numerals: 3578 Words: Three thousand five hundred seventy-eight In ancient Chinese there was only one way of writing numbers: 3 Th 5 H 7 T 8 U (8 characters)89 © Joan A. Cotter, Ph.D., 2012
    90. 90. Very Early Computation Calculating rods Because their characters are cumbersome to use for computing, the Chinese used calculating rods, beginning in the 4th century BC.90 © Joan A. Cotter, Ph.D., 2012
    91. 91. Very Early Computation Calculating rods91 © Joan A. Cotter, Ph.D., 2012
    92. 92. Very Early Computation Calculating rods Numerals for Ones and Hundreds (Even Powers of Ten)92 © Joan A. Cotter, Ph.D., 2012
    93. 93. Very Early Computation Calculating rods Numerals for Ones and Hundreds (Even Powers of Ten)93 © Joan A. Cotter, Ph.D., 2012
    94. 94. Very Early Computation Calculating rods Numerals for Ones and Hundreds (Even Powers of Ten) Numerals for Tens and Thousands (Odd Powers of Ten)94 © Joan A. Cotter, Ph.D., 2012
    95. 95. Very Early Computation Calculating rods 357895 © Joan A. Cotter, Ph.D., 2012
    96. 96. Very Early Computation Calculating rods 3578 3578,3578 They grouped, not in thousands, but ten-thousands!96 © Joan A. Cotter, Ph.D., 2012
    97. 97. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
    98. 98. Naming Quantities Using fingersNaming quantities is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
    99. 99. Naming Quantities Using fingersUse left hand for 1-5 because we read from left to right. © Joan A. Cotter, Ph.D., 2012
    100. 100. Naming Quantities Using fingers100 © Joan A. Cotter, Ph.D., 2012
    101. 101. Naming Quantities Using fingers101 © Joan A. Cotter, Ph.D., 2012
    102. 102. Naming Quantities Using fingersAlways show 7 as 5 and 2, not for example, as 4 and 3. © Joan A. Cotter, Ph.D., 2012102
    103. 103. Naming Quantities Using fingers103 © Joan A. Cotter, Ph.D., 2012
    104. 104. Naming Quantities Yellow is the Sun Yellow is the sun. Six is five and one. Why is the sky so blue? Seven is five and two. Salty is the sea. Eight is five and three. Hear the thunder roar. Nine is five and four. Ducks will swim and dive. Ten is five and five. –Joan A. CotterAlso set to music. Listen and download sheet music from Web site. © Joan A. Cotter, Ph.D., 2012
    105. 105. Naming Quantities Recognizing 5 © Joan A. Cotter, Ph.D., 2012
    106. 106. Naming Quantities Recognizing 5 © Joan A. Cotter, Ph.D., 2012
    107. 107. Naming Quantities Recognizing 5 5 has a middle; 4 does not.Look at your hand; your middle finger is longer to remind you 5 has a middle. A. Cotter, Ph.D., 2012 © Joan
    108. 108. Naming Quantities Tally sticksLay the sticks flat on a surface, about 1 inch (2.5 cm) apart. © Joan A. Cotter, Ph.D., 2012
    109. 109. Naming Quantities Tally sticks109 © Joan A. Cotter, Ph.D., 2012
    110. 110. Naming Quantities Tally sticks110 © Joan A. Cotter, Ph.D., 2012
    111. 111. Naming Quantities Tally sticksStick is horizontal, because it won’t fit diagonally and young children haveproblems with diagonals.111 © Joan A. Cotter, Ph.D., 2012
    112. 112. Naming Quantities Tally sticks112 © Joan A. Cotter, Ph.D., 2012
    113. 113. Naming Quantities Tally sticksStart a new row for every ten. © Joan A. Cotter, Ph.D., 2012113
    114. 114. Naming Quantities Solving a problem without counting What is 4 apples plus 3 more apples?How would you find the answer without counting? © Joan A. Cotter, Ph.D., 2012114
    115. 115. Naming Quantities Solving a problem without counting What is 4 apples plus 3 more apples?To remember 4 + 3, the Japanese child is taught to visualize 4 and 3. Thentake 1 from the 3 and give it to the 4 to make 5 and 2. © Joan A. Cotter, Ph.D., 2012115
    116. 116. Naming QuantitiesNumberChart 1 2 3 4 5 © Joan A. Cotter, Ph.D., 2012
    117. 117. Naming Quantities Number Chart 1 2To help the 3child learnthe symbols 4 5 © Joan A. Cotter, Ph.D., 2012
    118. 118. Naming Quantities Number Chart 1 6 2 7To help the 3 8child learnthe symbols 4 9 5 10 © Joan A. Cotter, Ph.D., 2012
    119. 119. Naming Quantities Pairing Finger Cards QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª andand a TIFFQuickTimeª and aa QuickTimeª and a QuickTimeª areTIFF (LZW) decompressor TIFF (LZW) decompressor (LZW) decompressor areTIFF (LZW)to see this picture. needed(LZW)seeathis picture. see decompressor are neededto see this picture. to to see this picture. QuickTimeª this picture. are needed to and needed decompressor TIFF are neededUse two sets of finger cards and match them. © Joan A. Cotter, Ph.D., 2012119
    120. 120. Naming Quantities Ordering Finger Cards QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor QuickTimeª and a are needed to see this picture. TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture.Putting the finger cards in order. © Joan A. Cotter, Ph.D., 2012120
    121. 121. Naming Quantities Matching Numbers to Finger Cards QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. 5 1 QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. 10Match the number to the finger card. © Joan A. Cotter, Ph.D., 2012121
    122. 122. Naming Quantities Matching Fingers to Number Cards 9 1 10 4 6 QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. 2 3 7 8 5 QuickTimeª and a TIFF (LZW) decompressor are needed to see this picture. QuickTimeª and aa QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor QuickTimeª and a TIFF (LZW) decompressor QuickTimeª and a TIFF (LZW) decompressor are needed QuickTimeªpicture. and are neededtotoseedecompressor TIFF (LZW) this picture. see TIFF (LZW) decompressor are neededtotoseedecompressor TIFF (LZW) this picture. are needed toseethis picture. are needed toseethis picture. are needed seethis picture. thisMatch the finger card to the number. © Joan A. Cotter, Ph.D., 2012122
    123. 123. Naming Quantities Finger Card Memory game QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. needed to see this picture. are TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. are needed to see this picture. QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. needed to see this picture. are QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. needed to see this picture. are QuickTimeª and a QuickTimeª and a QuickTimeª and a QuickTimeª and a TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor TIFF (LZW) decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. needed to see this picture. areUse two sets of finger cards and play Memory. © Joan A. Cotter, Ph.D., 2012123
    124. 124. Naming Quantities Number Rods124 © Joan A. Cotter, Ph.D., 2012
    125. 125. Naming Quantities Number Rods125 © Joan A. Cotter, Ph.D., 2012
    126. 126. Naming Quantities Number RodsUsing different colors. © Joan A. Cotter, Ph.D., 2012126
    127. 127. Naming Quantities Spindle Box45 dark-colored and 10 light-colored spindles. Could be in separate containers. Cotter, Ph.D., 2012 © Joan A.127
    128. 128. Naming Quantities Spindle Box45 dark-colored and 10 light-colored spindles in two containers. © Joan A. Cotter, Ph.D., 2012128
    129. 129. Naming Quantities Spindle Box 0 1 2 3 4The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012129
    130. 130. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012130
    131. 131. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012131
    132. 132. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012132
    133. 133. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012133
    134. 134. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012134
    135. 135. Naming Quantities Spindle Box 5 6 7 8 9The child takes blue spindles with left hand and yellow with right. © Joan A. Cotter, Ph.D., 2012135
    136. 136. Naming Quantities Black and White Bead Stairs “Grouped in fives so the child does not need to count.” A. M. JoostenThis was the inspiration to group in 5s. © Joan A. Cotter, Ph.D., 2012136
    137. 137. AL Abacus Cleared © Joan A. Cotter, Ph.D., 2012
    138. 138. AL Abacus Entering quantities 3Quantities are entered all at once, not counted. © Joan A. Cotter, Ph.D., 2012
    139. 139. AL Abacus Entering quantities 5Relate quantities to hands. © Joan A. Cotter, Ph.D., 2012139
    140. 140. AL Abacus Entering quantities 7140 © Joan A. Cotter, Ph.D., 2012
    141. 141. AL Abacus Entering quantities 10141 © Joan A. Cotter, Ph.D., 2012
    142. 142. AL Abacus The stairsCan use to “count” 1 to 10. Also read quantities on the right side. © Joan A. Cotter, Ph.D., 2012142
    143. 143. AL Abacus Adding © Joan A. Cotter, Ph.D., 2012
    144. 144. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
    145. 145. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
    146. 146. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
    147. 147. AL Abacus Adding4+3= © Joan A. Cotter, Ph.D., 2012
    148. 148. AL Abacus Adding 4+3=7Answer is seen immediately, no counting needed. © Joan A. Cotter, Ph.D., 2012
    149. 149. Go to the Dump Game Aim: To learn the facts that total 10: 1+9 2+8 3+7 4+6 5+5Children use the abacus while playing this “Go Fish” type game. © Joan A. Cotter, Ph.D., 2012149
    150. 150. Go to the Dump Game Aim: To learn the facts that total 10: 1+9 2+8 3+7 4+6 5+5 Object of the game: To collect the most pairs that equal ten.Children use the abacus while playing this “Go Fish” type game. © Joan A. Cotter, Ph.D., 2012150
    151. 151. Go to the Dump GameThe ways to partition 10. © Joan A. Cotter, Ph.D., 2012151
    152. 152. Go to the Dump Game StartingA game viewed from above. © Joan A. Cotter, Ph.D., 2012152
    153. 153. Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 StartingEach player takes 5 cards. © Joan A. Cotter, Ph.D., 2012153
    154. 154. Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 Finding pairsDoes YellowCap have any pairs? [no] © Joan A. Cotter, Ph.D., 2012154
    155. 155. Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 Finding pairsDoes BlueCap have any pairs? [yes, 1] © Joan A. Cotter, Ph.D., 2012155
    156. 156. Go to the Dump Game 72 7 9 5 72 1 3 8 4 6 34 9 Finding pairsDoes BlueCap have any pairs? [yes, 1] © Joan A. Cotter, Ph.D., 2012156
    157. 157. Go to the Dump Game 72 7 9 5 4 6 72 1 3 8 34 9 Finding pairsDoes BlueCap have any pairs? [yes, 1] © Joan A. Cotter, Ph.D., 2012157
    158. 158. Go to the Dump Game 72 7 9 5 4 6 72 1 3 8 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012158
    159. 159. Go to the Dump Game 72 7 9 5 4 6 72 1 3 8 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012159
    160. 160. Go to the Dump Game 72 7 9 5 7 3 4 6 2 1 8 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012160
    161. 161. Go to the Dump Game 72 7 9 5 2 8 4 6 1 34 9 Finding pairsDoes PinkCap have any pairs? [yes, 2] © Joan A. Cotter, Ph.D., 2012161
    162. 162. Go to the Dump Game 72 7 9 5 2 8 4 6 1 34 9 PlayingThe player asks the player on her left. © Joan A. Cotter, Ph.D., 2012162
    163. 163. Go to the Dump Game BlueCap, do you have an3? have a 3? 72 7 9 5 2 8 4 6 1 34 9 PlayingThe player asks the player on her left. © Joan A. Cotter, Ph.D., 2012163
    164. 164. Go to the Dump Game BlueCap, do you have an3? have a 3? 72 7 9 5 3 2 8 4 6 1 4 9 PlayingThe player asks the player on her left. © Joan A. Cotter, Ph.D., 2012164
    165. 165. Go to the Dump Game 7 3 BlueCap, do you have an3? have a 3? 2 7 9 5 2 8 4 6 1 4 9 Playing165 © Joan A. Cotter, Ph.D., 2012
    166. 166. Go to the Dump Game 7 3 BlueCap, do you have an3? have a 8? 2 7 9 5 2 8 4 6 1 4 9 PlayingYellowCap gets another turn. © Joan A. Cotter, Ph.D., 2012166
    167. 167. Go to the Dump Game 7 3 BlueCap, do you have an3? have a 8? 2 7 9 5 2 8 4 6 1 4 9 Go to the dump. PlayingYellowCap gets another turn. © Joan A. Cotter, Ph.D., 2012167
    168. 168. Go to the Dump Game 7 3 BlueCap, do you have an3? have a 8? 2 2 7 9 5 2 8 4 6 1 4 9 Go to the dump. Playing168 © Joan A. Cotter, Ph.D., 2012
    169. 169. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 4 9 Playing169 © Joan A. Cotter, Ph.D., 2012
    170. 170. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 4 9 PinkCap, do you Playing have a 6?170 © Joan A. Cotter, Ph.D., 2012
    171. 171. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 4 9 PinkCap, do you Go to the dump. Playing have a 6?171 © Joan A. Cotter, Ph.D., 2012
    172. 172. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 5 4 9 Playing172 © Joan A. Cotter, Ph.D., 2012
    173. 173. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 5 4 9 Playing173 © Joan A. Cotter, Ph.D., 2012
    174. 174. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 5 4 9 YellowCap, do you have a 9? Playing174 © Joan A. Cotter, Ph.D., 2012
    175. 175. Go to the Dump Game 7 3 2 2 7 5 2 8 4 6 1 5 4 9 YellowCap, do you have a 9? Playing175 © Joan A. Cotter, Ph.D., 2012
    176. 176. Go to the Dump Game 7 3 2 2 7 5 2 8 4 6 19 5 4 9 YellowCap, do you have a 9? Playing176 © Joan A. Cotter, Ph.D., 2012
    177. 177. Go to the Dump Game 7 3 2 2 7 5 2 1 8 9 4 6 5 4 9 Playing177 © Joan A. Cotter, Ph.D., 2012
    178. 178. Go to the Dump Game 7 3 2 2 7 5 2 1 8 9 4 6 2 9 1 7 7 5 4 9 PlayingPinkCap is not out of the game. Her turn ends, but she takes 5 more cards.Joan A. Cotter, Ph.D., 2012 ©178
    179. 179. Go to the Dump Game 9 1 4 6 5 5 Winner?179 © Joan A. Cotter, Ph.D., 2012
    180. 180. Go to the Dump Game 9 1 4 6 5 Winner?No counting. Combine both stacks. © Joan A. Cotter, Ph.D., 2012180
    181. 181. Go to the Dump Game 9 1 4 6 5 Winner?Whose stack is the highest? © Joan A. Cotter, Ph.D., 2012181
    182. 182. Go to the Dump Game Next gameNo shuffling needed for next game. © Joan A. Cotter, Ph.D., 2012182
    183. 183. “Math” Way of Naming Numbers183 © Joan A. Cotter, Ph.D., 2012
    184. 184. “Math” Way of Naming Numbers 11 = ten 1184 © Joan A. Cotter, Ph.D., 2012
    185. 185. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2185 © Joan A. Cotter, Ph.D., 2012
    186. 186. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3186 © Joan A. Cotter, Ph.D., 2012
    187. 187. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4187 © Joan A. Cotter, Ph.D., 2012
    188. 188. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9188 © Joan A. Cotter, Ph.D., 2012
    189. 189. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012189
    190. 190. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 14 = ten 4 .... 19 = ten 9Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012190
    191. 191. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 .... 19 = ten 9Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012191
    192. 192. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 23 = 2-ten 3 .... 19 = ten 9Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311. © Joan A. Cotter, Ph.D., 2012192
    193. 193. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 23 = 2-ten 3 .... .... 19 = ten 9 .... 99 = 9-ten 9193 © Joan A. Cotter, Ph.D., 2012
    194. 194. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7Only numbers under 100 need to be said the “math” way. © Joan A. Cotter, Ph.D., 2012194
    195. 195. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7Only numbers under 100 need to be said the “math” way. © Joan A. Cotter, Ph.D., 2012195
    196. 196. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.Shows how far children from 3 countries can count at ages 4, 5, and 6. © Joan A. Cotter, Ph.D., 2012196
    197. 197. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.Purple is Chinese. Note jump between ages 5 and 6. © Joan A. Cotter, Ph.D., 2012197
    198. 198. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.Dark green is Korean “math” way. © Joan A. Cotter, Ph.D., 2012198
    199. 199. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.Dotted green is everyday Korean; notice smaller jump between ages 5 and Joan A. Cotter, Ph.D., 2012 © 6.199
    200. 200. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young childrens counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332.Red is English speakers. They learn same amount between ages 4-5 and©5-6. Cotter, Ph.D., 2012 Joan A.200
    201. 201. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.)201 © Joan A. Cotter, Ph.D., 2012
    202. 202. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting.202 © Joan A. Cotter, Ph.D., 2012
    203. 203. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade.203 © Joan A. Cotter, Ph.D., 2012
    204. 204. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. • Mathematics is the science of patterns. The patterned math way of counting greatly helps children learn number sense.204 © Joan A. Cotter, Ph.D., 2012
    205. 205. Math Way of Naming Numbers Compared to reading:205 © Joan A. Cotter, Ph.D., 2012
    206. 206. Math Way of Naming Numbers Compared to reading: • Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic.206 © Joan A. Cotter, Ph.D., 2012
    207. 207. Math Way of Naming Numbers Compared to reading: • Just as reciting the alphabet doesn’t teach reading, 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).207 © Joan A. Cotter, Ph.D., 2012
    208. 208. Math Way of Naming Numbers Compared to reading: • Just as reciting the alphabet doesn’t teach reading, 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). • Montessorians do use the math way of naming numbers but are too quick to switch to traditional names. Use the math way for a longer period of time.208 © Joan A. Cotter, Ph.D., 2012
    209. 209. Math Way of Naming Numbers “Rather, the increased gap between Chinese and U.S. students and that of Chinese Americans and Caucasian Americans may be due primarily to the nature of their initial gap prior to formal schooling, such as counting efficiency and base-ten number sense.” Jian Wang and Emily Lin, 2005 Researchers209 © Joan A. Cotter, Ph.D., 2012
    210. 210. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48.210 © Joan A. Cotter, Ph.D., 2012
    211. 211. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14.211 © Joan A. Cotter, Ph.D., 2012
    212. 212. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones count 14.212 © Joan A. Cotter, Ph.D., 2012
    213. 213. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.213 © Joan A. Cotter, Ph.D., 2012
    214. 214. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.214 © Joan A. Cotter, Ph.D., 2012
    215. 215. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.215 © Joan A. Cotter, Ph.D., 2012
    216. 216. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.216 © Joan A. Cotter, Ph.D., 2012
    217. 217. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.217 © Joan A. Cotter, Ph.D., 2012
    218. 218. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.218 © Joan A. Cotter, Ph.D., 2012
    219. 219. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.219 © Joan A. Cotter, Ph.D., 2012
    220. 220. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.220 © Joan A. Cotter, Ph.D., 2012
    221. 221. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.221 © Joan A. Cotter, Ph.D., 2012
    222. 222. Math Way of Naming Numbers Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.222 © Joan A. Cotter, Ph.D., 2012
    223. 223. Math Way of Naming Numbers Traditional names4-ten =fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
    224. 224. Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens.The traditional names for 40, 60, 70, 80, and 90 follow a pattern. © Joan A. Cotter, Ph.D., 2012
    225. 225. Math Way of Naming Numbers Traditional names6-ten = sixtyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
    226. 226. Math Way of Naming Numbers Traditional names3-ten = thirty“Thir” alsoused in 1/3,13 and 30. © Joan A. Cotter, Ph.D., 2012
    227. 227. Math Way of Naming Numbers Traditional names5-ten = fifty“Fif” alsoused in 1/5,15 and 50. © Joan A. Cotter, Ph.D., 2012
    228. 228. Math Way of Naming Numbers Traditional names2-ten = twentyTwo used to bepronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
    229. 229. Math Way of Naming Numbers Traditional names A word game fireplace place-fireSay the syllables backward. This is how we say the teen numbers. © Joan A. Cotter, Ph.D., 2012
    230. 230. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-newsSay the syllables backward. This is how we say the teen numbers. © Joan A. Cotter, Ph.D., 2012
    231. 231. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news box-mail mailboxSay the syllables backward. This is how we say the teen numbers. © Joan A. Cotter, Ph.D., 2012
    232. 232. Math Way of Naming Numbers Traditional names ten 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
    233. 233. Math Way of Naming Numbers Traditional names ten 4 teen 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
    234. 234. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourtee n“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
    235. 235. Math Way of Naming Numbers Traditional names a one left © Joan A. Cotter, Ph.D., 2012
    236. 236. Math Way of Naming Numbers Traditional names a one left a left-one © Joan A. Cotter, Ph.D., 2012
    237. 237. Math Way of Naming Numbers Traditional names a one left a left-one eleven © Joan A. Cotter, Ph.D., 2012
    238. 238. Math Way of Naming Numbers Traditional names two leftTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
    239. 239. Math Way of Naming Numbers Traditional names two left twelveTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
    240. 240. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
    241. 241. Composing Numbers3-ten © Joan A. Cotter, Ph.D., 2012
    242. 242. Composing Numbers3-ten30 © Joan A. Cotter, Ph.D., 2012
    243. 243. Composing Numbers 3-ten 30Point to the 3 and say 3. © Joan A. Cotter, Ph.D., 2012
    244. 244. Composing Numbers 3-ten 30Point to 0 and say 10. The 0 makes 3 a ten. © Joan A. Cotter, Ph.D., 2012
    245. 245. Composing Numbers3-ten 730 © Joan A. Cotter, Ph.D., 2012
    246. 246. Composing Numbers3-ten 730 © Joan A. Cotter, Ph.D., 2012
    247. 247. Composing Numbers3-ten 730 7 © Joan A. Cotter, Ph.D., 2012
    248. 248. Composing Numbers 3-ten 7 30 7Place the 7 on top of the 0 of the 30. © Joan A. Cotter, Ph.D., 2012
    249. 249. Composing Numbers 3-ten 7 30 7Notice the way we say the number, represent thenumber, and write the number all correspond. © Joan A. Cotter, Ph.D., 2012
    250. 250. Composing Numbers 7-ten 6 78 6Another example. © Joan A. Cotter, Ph.D., 2012
    251. 251. Composing Numbers 7-ten 6 78 6 In the UK, pupils are expected to know the amount remaining: 24, that is 100 – 76.Another example. © Joan A. Cotter, Ph.D., 2012
    252. 252. Composing Numbers10-ten © Joan A. Cotter, Ph.D., 2012
    253. 253. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012
    254. 254. Composing Numbers10-ten100 © Joan A. Cotter, Ph.D., 2012

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