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NDMA Presentation

  1. 1. RightStart™ Mathematics in a
 Montessori Environment by Joan A. Cotter, Ph.D.
 JoanCotter@RightStartMath.com" 7 x 7 1000 3 2 5 5 100 10 1 New Discoveries !Montessori Academy! August 31, 2012
Hutchinson, Minnesota Other presentations available: rightstartmath.com© Joan A. Cotter, Ph.D., 2012
  2. 2. National Math Crisis © Joan A. Cotter, Ph.D., 2012
  3. 3. National Math Crisis•  25% of college freshmen take remedial math. © Joan A. Cotter, Ph.D., 2012
  4. 4. National Math Crisis•  25% of college freshmen take remedial math. •  In 2009, of the 1.5 million students who took theACT test, only 42% are ready for college algebra. © Joan A. Cotter, Ph.D., 2012
  5. 5. National Math Crisis•  25% of college freshmen take remedial math. •  In 2009, of the 1.5 million students who took theACT test, only 42% are ready for college algebra. •  A generation ago, the US produced 30% of theworld’s college grads; today it’s 14%. (CSM 2006) © Joan A. Cotter, Ph.D., 2012
  6. 6. National Math Crisis•  25% of college freshmen take remedial math. •  In 2009, of the 1.5 million students who took theACT test, only 42% are ready for college algebra. •  A generation ago, the US produced 30% of theworld’s college grads; today it’s 14%. (CSM 2006) •  Two-thirds of 4-year degrees in Japan and Chinaare in science and engineering; one-third in the U.S. © Joan A. Cotter, Ph.D., 2012
  7. 7. National Math Crisis•  25% of college freshmen take remedial math. •  In 2009, of the 1.5 million students who took theACT test, only 42% are ready for college algebra. •  A generation ago, the US produced 30% of theworld’s college grads; today it’s 14%. (CSM 2006) •  Two-thirds of 4-year degrees in Japan and Chinaare in science and engineering; one-third in the U.S. •  U.S. students, compared to the world, score high at4th grade, average at 8th, and near bottom at 12th. © Joan A. Cotter, Ph.D., 2012
  8. 8. National Math Crisis•  25% of college freshmen take remedial math. •  In 2009, of the 1.5 million students who took theACT test, only 42% are ready for college algebra. •  A generation ago, the US produced 30% of theworld’s college grads; today it’s 14%. (CSM 2006) •  Two-thirds of 4-year degrees in Japan and Chinaare in science and engineering; one-third in the U.S. •  U.S. students, compared to the world, score high at4th grade, average at 8th, and near bottom at 12th. •  Ready, Willing, and Unable to Serve says that 75% of17 to 24 year-olds are unfit for military service. (2010) © Joan A. Cotter, Ph.D., 2012
  9. 9. Math Education is Changing © Joan A. Cotter, Ph.D., 2012
  10. 10. Math Education is Changing•  The field of mathematics is doubling every 7 years. © Joan A. Cotter, Ph.D., 2012
  11. 11. Math Education is Changing•  The field of mathematics is doubling every 7 years. •  Math is used in many new ways. The workplaceneeds analytical thinkers and problem solvers. © Joan A. Cotter, Ph.D., 2012
  12. 12. Math Education is Changing•  The field of mathematics is doubling every 7 years. •  Math is used in many new ways. The workplaceneeds analytical thinkers and problem solvers. •  State exams require more than arithmetic:including geometry, algebra, probability, andstatistics. © Joan A. Cotter, Ph.D., 2012
  13. 13. Math Education is Changing•  The field of mathematics is doubling every 7 years. •  Math is used in many new ways. The workplaceneeds analytical thinkers and problem solvers. •  State exams require more than arithmetic:including geometry, algebra, probability, andstatistics. •  Brain research is providing clues on how to betterfacilitate learning, including math. © Joan A. Cotter, Ph.D., 2012
  14. 14. Math Education is Changing•  The field of mathematics is doubling every 7 years. •  Math is used in many new ways. The workplaceneeds analytical thinkers and problem solvers. •  State exams require more than arithmetic:including geometry, algebra, probability, andstatistics. •  Brain research is providing clues on how to betterfacilitate learning, including math. •  Calculators and computers have made computationwith many digits an unneeded skill. © Joan A. Cotter, Ph.D., 2012
  15. 15. Math Education is Changing•  The field of mathematics is doubling every 7 years. •  Math is used in many new ways. The workplaceneeds analytical thinkers and problem solvers. •  State exams require more than arithmetic:including geometry, algebra, probability, andstatistics. •  Brain research is providing clues on how to betterfacilitate learning, including math. •  Calculators and computers have made computationwith many digits an unneeded skill. •  There is a greater emphasis on STEM subjects. © Joan A. Cotter, Ph.D., 2012
  16. 16. Counting Model © Joan A. Cotter, Ph.D., 2012
  17. 17. Counting Model From a childs perspectiveBecause we’re so familiar with 1, 2, 3, we’ll use letters. A = 1 B = 2 C = 3 D = 4 E = 5, and so forth © Joan A. Cotter, Ph.D., 2012
  18. 18. Counting ModelFrom a childs perspective F +E! © Joan A. Cotter, Ph.D., 2012
  19. 19. Counting Model From a childs perspective F +E!A © Joan A. Cotter, Ph.D., 2012
  20. 20. Counting Model From a childs perspective F +E!A B © Joan A. Cotter, Ph.D., 2012
  21. 21. Counting Model From a childs perspective F +E!A B C © Joan A. Cotter, Ph.D., 2012
  22. 22. Counting Model From a childs perspective F +E!A B C D E F © Joan A. Cotter, Ph.D., 2012
  23. 23. Counting Model From a childs perspective F +E!A B C D E F A © Joan A. Cotter, Ph.D., 2012
  24. 24. Counting Model From a childs perspective F +E!A B C D E F A B © Joan A. Cotter, Ph.D., 2012
  25. 25. Counting Model From a childs perspective F +E!A B C D E F A B C D E © Joan A. Cotter, Ph.D., 2012
  26. 26. 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.)! © Joan A. Cotter, Ph.D., 2012
  27. 27. Counting Model From a childs perspective F +E KA B C D E F G H I J K © Joan A. Cotter, Ph.D., 2012
  28. 28. Counting Model From a childs perspective Now memorize the facts!! G! +D! © Joan A. Cotter, Ph.D., 2012
  29. 29. Counting Model From a childs perspective Now memorize the facts!! G! +D! © Joan A. Cotter, Ph.D., 2012
  30. 30. Counting Model From a childs perspective Now memorize the facts!! G! +D! D!+C! © Joan A. Cotter, Ph.D., 2012
  31. 31. Counting Model From a childs perspective Now memorize the facts!! G! +D! D! C!+C! +G! © Joan A. Cotter, Ph.D., 2012
  32. 32. Counting Model From a childs perspective Now memorize the facts!! G! +D! D! C!+C! +G! © Joan A. Cotter, Ph.D., 2012
  33. 33. Counting Model From a childs perspective Try subtracting Hby “taking away” – E © Joan A. Cotter, Ph.D., 2012
  34. 34. Counting Model From a childs perspective Try skip counting by B’s to T: B, D, . . . T. © Joan A. Cotter, Ph.D., 2012
  35. 35. Counting Model From a childs perspective Try skip counting by B’s to T: B, D, . . . T. What is D  E? © Joan A. Cotter, Ph.D., 2012
  36. 36. Counting Model From a childs perspective L is written AB!because it is A J !and B A’s ! © Joan A. Cotter, Ph.D., 2012
  37. 37. Counting Model From a childs perspective L is written AB!because it is A J !and B A’s ! huh? © Joan A. Cotter, Ph.D., 2012
  38. 38. Counting Model From a childs perspective L (twelve) is written AB!because it is A J !and B A’s ! © Joan A. Cotter, Ph.D., 2012
  39. 39. Counting Model From a childs perspective L (twelve) is written AB! (12) because it is A J !and B A’s ! © Joan A. Cotter, Ph.D., 2012
  40. 40. Counting Model From a childs perspective L (twelve) is written AB! (12) because it is A J !(one 10) and B A’s ! © Joan A. Cotter, Ph.D., 2012
  41. 41. 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). ! © Joan A. Cotter, Ph.D., 2012
  42. 42. Counting ModelIn Montessori, counting is pervasive: •  Number Rods •  Spindle Boxes •  Decimal materials •  Snake Game •  Dot Game •  Stamp Game •  Multiplication Board •  Bead Frame © Joan A. Cotter, Ph.D., 2012
  43. 43. Counting Model Summary © Joan A. Cotter, Ph.D., 2012
  44. 44. Counting Model Summary •  Is not natural; it takes years of practice. © Joan A. Cotter, Ph.D., 2012
  45. 45. Counting Model Summary •  Is not natural; it takes years of practice. •  Provides poor concept of quantity. © Joan A. Cotter, Ph.D., 2012
  46. 46. Counting Model Summary •  Is not natural; it takes years of practice. •  Provides poor concept of quantity. •  Ignores place value. © Joan A. Cotter, Ph.D., 2012
  47. 47. Counting Model Summary •  Is not natural; it takes years of practice. •  Provides poor concept of quantity. •  Ignores place value. •  Is very error prone. © Joan A. Cotter, Ph.D., 2012
  48. 48. 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. © Joan A. Cotter, Ph.D., 2012
  49. 49. 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 wayto master the facts. © Joan A. Cotter, Ph.D., 2012
  50. 50. 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! 31!Sometimes calendars are used for counting.! © Joan A. Cotter, Ph.D., 2012
  51. 51. 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! 31!Sometimes calendars are used for counting.! © Joan A. Cotter, Ph.D., 2012
  52. 52. 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! 31! © Joan A. Cotter, Ph.D., 2012
  53. 53. 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! 31!This is ordinal, not cardinal counting. The 3 doesn’t include the 1 and the 2.! 2012 © Joan A. Cotter, Ph.D.,
  54. 54. 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! 31!This is ordinal, not cardinal counting. The 4 doesn’t include 1, 2 and Joan A. Cotter, Ph.D., 2012 © 3.!
  55. 55. 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! 31! 1 2 3 4 5 6A calendar is NOT a ruler. On a ruler the numbers are not in the spaces.!Cotter, Ph.D., 2012 © Joan A.
  56. 56. Calendar Math August 1! 2! 3! 4! 5! 6! 7! 8! 9! 10!Always 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., 2012
  57. 57. Calendar Math The calendar is not a number line. •  No quantity is involved. •  Numbers are in spaces, not at lines like a ruler. © Joan A. Cotter, Ph.D., 2012
  58. 58. 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. © Joan A. Cotter, Ph.D., 2012
  59. 59. 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. © Joan A. Cotter, Ph.D., 2012
  60. 60. Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58 © Joan A. Cotter, Ph.D., 2012
  61. 61. Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58 © Joan A. Cotter, Ph.D., 2012
  62. 62. Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58 © Joan A. Cotter, Ph.D., 2012
  63. 63. Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58 © Joan A. Cotter, Ph.D., 2012
  64. 64. Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58 © Joan A. Cotter, Ph.D., 2012
  65. 65. Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58 © Joan A. Cotter, Ph.D., 2012
  66. 66. Memorizing Math Percentage Recall Immediately After 1 day After 4 wks Rote 32 23 8 Concept 69 69 58 Math needs to be taught so 95% is understood and only 5% memorized. Richard Skemp © Joan A. Cotter, Ph.D., 2012
  67. 67. Memorizing Math 9 ! +7 ! Flash cards: © Joan A. Cotter, Ph.D., 2012
  68. 68. Memorizing Math 9 ! +7 ! Flash cards: •  Are often used to teach rote. © Joan A. Cotter, Ph.D., 2012
  69. 69. Memorizing Math 9 ! +7 ! Flash cards: •  Are often used to teach rote. •  Are liked only by those who don’t need them. © Joan A. Cotter, Ph.D., 2012
  70. 70. 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. © Joan A. Cotter, Ph.D., 2012
  71. 71. 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. © Joan A. Cotter, Ph.D., 2012
  72. 72. 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 learningdisabilities. •  Give the false impression that math isn’t aboutthinking. •  Often produce stress – children under stressstop learning. © Joan A. Cotter, Ph.D., 2012
  73. 73. 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 learningdisabilities. •  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
  74. 74. Research on Counting Karen Wynn’s research Show the baby two teddy bears. ! © Joan A. Cotter, Ph.D., 2012
  75. 75. Research on Counting Karen Wynn’s research Then hide them with a screen.! © Joan A. Cotter, Ph.D., 2012
  76. 76. Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.! © Joan A. Cotter, Ph.D., 2012
  77. 77. Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.! © Joan A. Cotter, Ph.D., 2012
  78. 78. Research on Counting Karen Wynn’s research Raise screen. Baby seeing 3 won’t look long because it is expected.! Joan A. Cotter, Ph.D., 2012 ©
  79. 79. Research on Counting Karen Wynn’s research Researcher can change the number of teddy bears behind the screen.! A. Cotter, Ph.D., 2012 © Joan
  80. 80. Research on Counting Karen Wynn’s research A baby seeing 1 teddy bear will look much longer, because it’s unexpected.! 2012 © Joan A. Cotter, Ph.D.,
  81. 81. Research on Counting Other research © Joan A. Cotter, Ph.D., 2012
  82. 82. 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., 2012
  83. 83. 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., 2012
  84. 84. 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., 2012
  85. 85. 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., 2012
  86. 86. Research on Counting In Japanese schools: •  Children are discouraged from usingcounting for adding. © Joan A. Cotter, Ph.D., 2012
  87. 87. Research on Counting In Japanese schools: •  Children are discouraged from usingcounting for adding. •  They consistently group in 5s. © Joan A. Cotter, Ph.D., 2012
  88. 88. Research on Counting Subitizing •  Subitizing is quick recognition of quantitywithout counting. © Joan A. Cotter, Ph.D., 2012
  89. 89. Research on Counting Subitizing •  Subitizing is quick recognition of quantitywithout counting. •  Human babies and some animals can subitizesmall quantities at birth. © Joan A. Cotter, Ph.D., 2012
  90. 90. Research on Counting Subitizing •  Subitizing is quick recognition of quantitywithout counting. •  Human babies and some animals can subitizesmall quantities at birth. •  Children who can subitize perform better inmathematics.—Butterworth © Joan A. Cotter, Ph.D., 2012
  91. 91. Research on Counting Subitizing •  Subitizing is quick recognition of quantitywithout counting. •  Human babies and some animals can subitizesmall quantities at birth. •  Children who can subitize perform better inmathematics.—Butterworth •  Subitizing “allows the child to grasp the wholeand the elements at the same time.”—Benoit © Joan A. Cotter, Ph.D., 2012
  92. 92. Research on Counting Subitizing •  Subitizing is quick recognition of quantitywithout counting. •  Human babies and some animals can subitizesmall quantities at birth. •  Children who can subitize perform better inmathematics.—Butterworth •  Subitizing “allows the child to grasp the wholeand the elements at the same time.”—Benoit •  Subitizing seems to be a necessary skill forunderstanding what the counting process means.—Glasersfeld © Joan A. Cotter, Ph.D., 2012
  93. 93. Research on Counting Finger gnosia •  Finger gnosia is the ability to know which fingerscan been lightly touched without looking. © Joan A. Cotter, Ph.D., 2012
  94. 94. Research on Counting Finger gnosia •  Finger gnosia is the ability to know which fingerscan been lightly touched without looking. •  Part of the brain controlling fingers is adjacent tomath part of the brain. © Joan A. Cotter, Ph.D., 2012
  95. 95. Research on Counting Finger gnosia •  Finger gnosia is the ability to know which fingerscan been lightly touched without looking. •  Part of the brain controlling fingers is adjacent tomath part of the brain. •  Children who use their fingers as representationaltools perform better in mathematics—Butterworth © Joan A. Cotter, Ph.D., 2012
  96. 96. Visualizing Mathematics © Joan A. Cotter, Ph.D., 2012
  97. 97. Visualizing Mathematics“In our concern about the memorization ofmath facts or solving problems, we must notforget that the root of mathematical study isthe creation of mental pictures in theimagination and manipulating those imagesand relationships using the power of reasonand logic.” Mindy Holte (E1) © Joan A. Cotter, Ph.D., 2012
  98. 98. Visualizing Mathematics “Think in pictures, because thebrain remembers images betterthan it does anything else.”   Ben Pridmore, World Memory Champion, 2009 © Joan A. Cotter, Ph.D., 2012
  99. 99. Visualizing Mathematics“Mathematics is the activity ofcreating relationships, many of whichare based in visual imagery.” Wheatley and Cobb © Joan A. Cotter, Ph.D., 2012
  100. 100. Visualizing Mathematics“The process of connecting symbols toimagery is at the heart of mathematicslearning.” Dienes © Joan A. Cotter, Ph.D., 2012
  101. 101. Visualizing Mathematics“The role of physical manipulativeswas to help the child form thosevisual images and thus to eliminatethe need for the physicalmanipulatives.” Ginsberg and others © Joan A. Cotter, Ph.D., 2012
  102. 102. 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
  103. 103. Visualizing Mathematics Visualizing also needed in: •  Reading •  Sports •  Creativity •  Geography •  Engineering •  Construction © Joan A. Cotter, Ph.D., 2012
  104. 104. Visualizing Mathematics Visualizing also needed in: •  Reading •  Architecture •  Sports •  Astronomy •  Creativity •  Archeology •  Geography •  Chemistry •  Engineering •  Physics •  Construction •  Surgery © Joan A. Cotter, Ph.D., 2012
  105. 105. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
  106. 106. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
  107. 107. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
  108. 108. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
  109. 109. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
  110. 110. Visualizing Mathematics Ready: How many? © Joan A. Cotter, Ph.D., 2012
  111. 111. Visualizing Mathematics Try again: How many? © Joan A. Cotter, Ph.D., 2012
  112. 112. Visualizing Mathematics Try to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  113. 113. Visualizing Mathematics Try to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  114. 114. Visualizing Mathematics Now try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  115. 115. Visualizing Mathematics Now try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  116. 116. Visualizing Mathematics Early Roman numerals 1 I 2 II 3 III 4 IIII 5 V 8 VIII Romans grouped in fives. Notice 8 is 5 and 3.! © Joan A. Cotter, Ph.D., 2012
  117. 117. Visualizing Mathematics : Who could read the music? Music needs 10 lines, two groups of five.! © Joan A. Cotter, Ph.D., 2012
  118. 118. Research on Counting Teach Counting •  Finger gnosia is the ability to know which fingerscan been lightly touched without looking. •  Part of the brain controlling fingers is adjacent tomath part of the brain. •  Children who use their fingers as representationaltools perform better in mathematics—Butterworth © Joan A. Cotter, Ph.D., 2012
  119. 119. Very Early Computation Numerals In English there are two ways of writing numbers: 3578 Three thousand five hundred seventy eight © Joan A. Cotter, Ph.D., 2012
  120. 120. Very Early Computation Numerals In English there are two ways of writing numbers: 3578 Three thousand five hundred seventy eightIn Chinese there is only one way of writing numbers: 3 Th 5 H 7 T 8 U (8 characters) © Joan A. Cotter, Ph.D., 2012
  121. 121. Very Early Computation Calculating rods Because their characters are cumbersometo use for computing, the Chinese usedcalculating rods, beginning in the 4thcentury BC. © Joan A. Cotter, Ph.D., 2012
  122. 122. Very Early Computation Calculating rods © Joan A. Cotter, Ph.D., 2012
  123. 123. Very Early Computation Calculating rods Numerals for Ones and Hundreds (Even Powers of Ten) © Joan A. Cotter, Ph.D., 2012
  124. 124. Very Early Computation Calculating rods Numerals for Ones and Hundreds (Even Powers of Ten) © Joan A. Cotter, Ph.D., 2012
  125. 125. Very Early Computation Calculating rods Numerals for Ones and Hundreds (Odd Powers of Ten) Numerals for Tens and Thousands (Odd Powers of Ten) © Joan A. Cotter, Ph.D., 2012
  126. 126. Very Early Computation Calculating rods 3578 © Joan A. Cotter, Ph.D., 2012
  127. 127. Very Early Computation Calculating rods 3578 3578,3578They grouped, not in thousands, but ten-thousands! © Joan A. Cotter, Ph.D., 2012
  128. 128. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
  129. 129. Naming Quantities Using fingers Naming quantities is a three-period lesson. © Joan A. Cotter, Ph.D., 2012
  130. 130. Naming Quantities Using fingers Use left hand for 1-5 because we read from left to right.! © Joan A. Cotter, Ph.D., 2012
  131. 131. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
  132. 132. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
  133. 133. Naming Quantities Using fingers Always show 7 as 5 and 2, not for example, as 4 and 3.! © Joan A. Cotter, Ph.D., 2012
  134. 134. Naming Quantities Using fingers © Joan A. Cotter, Ph.D., 2012
  135. 135. 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. Cotter Also set to music. Listen and download sheet music from Web site.! Joan A. Cotter, Ph.D., 2012 ©
  136. 136. Naming Quantities Recognizing 5 © Joan A. Cotter, Ph.D., 2012
  137. 137. Naming Quantities Recognizing 5 © Joan A. Cotter, Ph.D., 2012
  138. 138. 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.! 2012 Joan A. Cotter, Ph.D.,
  139. 139. Naming Quantities Tally sticks Lay the sticks flat on a surface, about 1 inch (2.5 cm) apart.! © Joan A. Cotter, Ph.D., 2012
  140. 140. Naming Quantities Tally sticks © Joan A. Cotter, Ph.D., 2012
  141. 141. Naming Quantities Tally sticks © Joan A. Cotter, Ph.D., 2012
  142. 142. Naming Quantities Tally sticks Stick is horizontal, because it won’t fit diagonally and young childrenhave problems with diagonals.! © Joan A. Cotter, Ph.D., 2012
  143. 143. Naming Quantities Tally sticks © Joan A. Cotter, Ph.D., 2012
  144. 144. Naming Quantities Tally sticks Start a new row for every ten.! © Joan A. Cotter, Ph.D., 2012
  145. 145. 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., 2012
  146. 146. 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.Then take 1 from the 3 and give it to the 4 to make 5 and 2.! © Joan A. Cotter, Ph.D., 2012
  147. 147. Naming QuantitiesNumber Chart 1" 2" 3" 4" 5! © Joan A. Cotter, Ph.D., 2012
  148. 148. Naming Quantities Number Chart 1" 2"To help the 3"child learnthe symbols 4" 5! © Joan A. Cotter, Ph.D., 2012
  149. 149. Naming Quantities Number Chart 1" 6! 2" 7!To help the 3" 8!child learnthe symbols 4" 9! 5! 10! © Joan A. Cotter, Ph.D., 2012
  150. 150. Naming Quantities Pairing Finger Cards Use two sets of finger cards and match them.! © Joan A. Cotter, Ph.D., 2012
  151. 151. Naming Quantities Ordering Finger Cards Putting the finger cards in order.! © Joan A. Cotter, Ph.D., 2012
  152. 152. Naming Quantities Matching Numbers to Finger Cards 5! 1! 10!Match the number to the finger card.! © Joan A. Cotter, Ph.D., 2012
  153. 153. Naming Quantities Matching Fingers to Number Cards 9! 1! 10! 4! 6! 2! 3! 7! 8! 5!Match the finger card to the number.! © Joan A. Cotter, Ph.D., 2012
  154. 154. Naming Quantities Finger Card Memory game Use two sets of finger cards and play Memory.! © Joan A. Cotter, Ph.D., 2012
  155. 155. Naming Quantities Number Rods © Joan A. Cotter, Ph.D., 2012
  156. 156. Naming Quantities Number Rods © Joan A. Cotter, Ph.D., 2012
  157. 157. Naming Quantities Number Rods Using different colors.! © Joan A. Cotter, Ph.D., 2012
  158. 158. Naming Quantities Spindle Box 45 dark-colored and 10 light-colored spindles. Could be in separate©containers.! Joan A. Cotter, Ph.D., 2012
  159. 159. Naming Quantities Spindle Box 45 dark-colored and 10 light-colored spindles in two containers.! © Joan A. Cotter, Ph.D., 2012
  160. 160. 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., 2012
  161. 161. 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., 2012
  162. 162. 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., 2012
  163. 163. 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., 2012
  164. 164. 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., 2012
  165. 165. 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., 2012
  166. 166. 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., 2012
  167. 167. Naming Quantities Black and White Bead Stairs “Grouped in fives so the child does not need to count.” A. M. Joosten This was the inspiration to group in 5s.! © Joan A. Cotter, Ph.D., 2012
  168. 168. AL Abacus 1000 100 10 1Double-sided AL abacus. Side 1 is grouped in 5s.!Trading Side introduces algorithms with trading. ! © Joan A. Cotter, Ph.D., 2012
  169. 169. AL Abacus Cleared © Joan A. Cotter, Ph.D., 2012
  170. 170. AL Abacus Entering quantities 3 Quantities are entered all at once, not counted.! © Joan A. Cotter, Ph.D., 2012
  171. 171. AL Abacus Entering quantities 5 Relate quantities to hands.! © Joan A. Cotter, Ph.D., 2012
  172. 172. AL Abacus Entering quantities 7 © Joan A. Cotter, Ph.D., 2012
  173. 173. AL Abacus Entering quantities 10 © Joan A. Cotter, Ph.D., 2012
  174. 174. AL Abacus The stairs Can use to “count” 1 to 10. Also read quantities on the right side.!© Joan A. Cotter, Ph.D., 2012
  175. 175. AL Abacus Adding © Joan A. Cotter, Ph.D., 2012
  176. 176. AL Abacus Adding 4 + 3 = © Joan A. Cotter, Ph.D., 2012
  177. 177. AL Abacus Adding 4 + 3 = © Joan A. Cotter, Ph.D., 2012
  178. 178. AL Abacus Adding 4 + 3 = © Joan A. Cotter, Ph.D., 2012
  179. 179. AL Abacus Adding 4 + 3 = © Joan A. Cotter, Ph.D., 2012
  180. 180. AL Abacus Adding 4 + 3 = 7 Answer is seen immediately, no counting needed.! © Joan A. Cotter, Ph.D., 2012
  181. 181. Go to the Dump Game Aim: To learn the facts that total 10: 1 + 9! 2 + 8! 3 + 7! 4 + 6! 5 + 5!Children use the abacus while playing this “Go Fish” type game.! © Joan A. Cotter, Ph.D., 2012
  182. 182. 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., 2012
  183. 183. Go to the Dump GameThe ways to partition 10.! © Joan A. Cotter, Ph.D., 2012
  184. 184. Go to the Dump Game StartingA game viewed from above.! © Joan A. Cotter, Ph.D., 2012
  185. 185. 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., 2012
  186. 186. 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., 2012
  187. 187. 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., 2012
  188. 188. 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., 2012
  189. 189. 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., 2012
  190. 190. 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., 2012
  191. 191. 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., 2012
  192. 192. 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., 2012
  193. 193. 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., 2012
  194. 194. 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., 2012
  195. 195. 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., 2012
  196. 196. 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., 2012
  197. 197. 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 Playing © Joan A. Cotter, Ph.D., 2012
  198. 198. 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., 2012
  199. 199. 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., 2012
  200. 200. 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. Playing © Joan A. Cotter, Ph.D., 2012
  201. 201. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 4 9 Playing © Joan A. Cotter, Ph.D., 2012
  202. 202. 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? © Joan A. Cotter, Ph.D., 2012
  203. 203. 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? © Joan A. Cotter, Ph.D., 2012
  204. 204. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 5 4 9 Playing © Joan A. Cotter, Ph.D., 2012
  205. 205. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 5 4 9 Playing © Joan A. Cotter, Ph.D., 2012
  206. 206. Go to the Dump Game 7 3 2 2 7 9 5 2 8 4 6 1 5 4 9YellowCap, do you have a 9? Playing © Joan A. Cotter, Ph.D., 2012
  207. 207. Go to the Dump Game 7 3 2 2 7 5 2 8 4 6 1 5 4 9YellowCap, do you have a 9? Playing © Joan A. Cotter, Ph.D., 2012
  208. 208. Go to the Dump Game 7 3 2 2 7 5 2 8 4 6 19 5 4 9YellowCap, do you have a 9? Playing © Joan A. Cotter, Ph.D., 2012
  209. 209. Go to the Dump Game 7 3 2 2 7 5 2 1 8 9 4 6 5 4 9 Playing © Joan A. Cotter, Ph.D., 2012
  210. 210. 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 moreJoan A. Cotter, Ph.D., 2012 © cards.!
  211. 211. Go to the Dump Game 9 1 4 6 5 5 Winner? © Joan A. Cotter, Ph.D., 2012
  212. 212. Go to the Dump Game 9 1 4 6 5 Winner?No counting. Combine both stacks.! © Joan A. Cotter, Ph.D., 2012
  213. 213. Go to the Dump Game 9 1 4 6 5 Winner?Whose stack is the highest?! © Joan A. Cotter, Ph.D., 2012
  214. 214. Go to the Dump Game Next gameNo shuffling needed for next game.! © Joan A. Cotter, Ph.D., 2012
  215. 215. “Math” Way of Naming Numbers © Joan A. Cotter, Ph.D., 2012
  216. 216. “Math” Way of Naming Numbers11 = ten 1 © Joan A. Cotter, Ph.D., 2012
  217. 217. “Math” Way of Naming Numbers11 = ten 1 12 = ten 2 © Joan A. Cotter, Ph.D., 2012
  218. 218. “Math” Way of Naming Numbers11 = ten 1 12 = ten 2 13 = ten 3 © Joan A. Cotter, Ph.D., 2012
  219. 219. “Math” Way of Naming Numbers11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 © Joan A. Cotter, Ph.D., 2012
  220. 220. “Math” Way of Naming Numbers11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 © Joan A. Cotter, Ph.D., 2012
  221. 221. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.! © Joan A. Cotter, Ph.D., 2012
  222. 222. “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 9 Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.! © Joan A. Cotter, Ph.D., 2012
  223. 223. “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 9 Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.! © Joan A. Cotter, Ph.D., 2012
  224. 224. “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 Don’t say “2-tens.” We don’t say 3 hundreds eleven for 311.! © Joan A. Cotter, Ph.D., 2012
  225. 225. “Math” Way of Naming Numbers11 = 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 9 © Joan A. Cotter, Ph.D., 2012
  226. 226. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 Only numbers under 100 need to be said the “math” way. ! © Joan A. Cotter, Ph.D., 2012
  227. 227. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7 Only numbers under 100 need to be said the “math” way. ! © Joan A. Cotter, Ph.D., 2012
  228. 228. “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, andJoan A.!Cotter, Ph.D., 2012 © 6.
  229. 229. “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., 2012
  230. 230. “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., 2012
  231. 231. “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 Joan A. Cotter, 6.! 2012 © 5 and Ph.D.,
  232. 232. “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 Joan A. Cotter, Ph.D., 2012 © and 5-6.!
  233. 233. 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.) © Joan A. Cotter, Ph.D., 2012
  234. 234. 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. © Joan A. Cotter, Ph.D., 2012
  235. 235. 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. © Joan A. Cotter, Ph.D., 2012
  236. 236. 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. © Joan A. Cotter, Ph.D., 2012
  237. 237. Math Way of Naming Numbers Compared to reading: © Joan A. Cotter, Ph.D., 2012
  238. 238. Math Way of Naming Numbers Compared to reading: •  Just as reciting the alphabet doesn’t teach reading,counting doesn’t teach arithmetic. © Joan A. Cotter, Ph.D., 2012
  239. 239. 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 mustfirst teach the name of the quantity (math way). © Joan A. Cotter, Ph.D., 2012
  240. 240. 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 mustfirst teach the name of the quantity (math way). •  Montessorians do use the math way of namingnumbers but are too quick to switch to traditionalnames. Use the math way for a longer period of time. © Joan A. Cotter, Ph.D., 2012
  241. 241. Math Way of Naming Numbers“Rather, the increased gap between Chinese andU.S. students and that of Chinese Americans andCaucasian Americans may be due primarily to thenature of their initial gap prior to formal schooling,such as counting efficiency and base-ten numbersense.” Jian Wang and Emily Lin, 2005 Researchers © Joan A. Cotter, Ph.D., 2012
  242. 242. Math Way of Naming NumbersResearch task: Using 10s and 1s, ask thechild to construct 48. © Joan A. Cotter, Ph.D., 2012
  243. 243. Math Way of Naming NumbersResearch task: Using 10s and 1s, ask thechild to construct 48. Then ask the child tosubtract 14. © Joan A. Cotter, Ph.D., 2012
  244. 244. 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. © Joan A. Cotter, Ph.D., 2012
  245. 245. 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. © Joan A. Cotter, Ph.D., 2012
  246. 246. 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. © Joan A. Cotter, Ph.D., 2012
  247. 247. 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. © Joan A. Cotter, Ph.D., 2012
  248. 248. 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. © Joan A. Cotter, Ph.D., 2012
  249. 249. 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. © Joan A. Cotter, Ph.D., 2012
  250. 250. 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. © Joan A. Cotter, Ph.D., 2012
  251. 251. 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. © Joan A. Cotter, Ph.D., 2012
  252. 252. 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. © Joan A. Cotter, Ph.D., 2012
  253. 253. 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. © Joan A. Cotter, Ph.D., 2012
  254. 254. 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. © Joan A. Cotter, Ph.D., 2012
  255. 255. Math Way of Naming Numbers Traditional names 4-ten = fortyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
  256. 256. 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
  257. 257. Math Way of Naming Numbers Traditional names 6-ten = sixtyThe “ty”means tens. © Joan A. Cotter, Ph.D., 2012
  258. 258. Math Way of Naming Numbers Traditional names 3-ten = thirty“Thir” alsoused in 1/3,13 and 30. © Joan A. Cotter, Ph.D., 2012
  259. 259. Math Way of Naming Numbers Traditional names 5-ten = fifty“Fif” alsoused in 1/5,15 and 50. © Joan A. Cotter, Ph.D., 2012
  260. 260. Math Way of Naming Numbers Traditional names 2-ten = twentyTwo used to bepronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  261. 261. Math Way of Naming Numbers Traditional names A word game fireplace place-fire Say the syllables backward. This is how we say the teen numbers.! © Joan A. Cotter, Ph.D., 2012
  262. 262. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news Say the syllables backward. This is how we say the teen numbers.! © Joan A. Cotter, Ph.D., 2012
  263. 263. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news box-mail mailbox Say the syllables backward. This is how we say the teen numbers.! © Joan A. Cotter, Ph.D., 2012
  264. 264. Math Way of Naming Numbers Traditional names ten 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  265. 265. Math Way of Naming Numbers Traditional names ten 4 teen 4“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  266. 266. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourteen“Teen” alsomeans ten. © Joan A. Cotter, Ph.D., 2012
  267. 267. Math Way of Naming Numbers Traditional names a one left © Joan A. Cotter, Ph.D., 2012
  268. 268. Math Way of Naming Numbers Traditional names a one left a left-one © Joan A. Cotter, Ph.D., 2012
  269. 269. Math Way of Naming Numbers Traditional names a one left a left-one eleven © Joan A. Cotter, Ph.D., 2012
  270. 270. Math Way of Naming Numbers Traditional names two leftTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  271. 271. Math Way of Naming Numbers Traditional names two left twelveTwopronounced“twoo.” © Joan A. Cotter, Ph.D., 2012
  272. 272. Composing Numbers 3-ten © Joan A. Cotter, Ph.D., 2012
  273. 273. Composing Numbers 3-ten © Joan A. Cotter, Ph.D., 2012
  274. 274. Composing Numbers 3-ten 30 © Joan A. Cotter, Ph.D., 2012

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