1. Teaching Math to Childrenwith Special NeedsMassHOPE - TEACHWorcester, MASaturday, April 279:45am 10:45amJoan A. Cotter, Ph.D.JoanCotter@RightStartMath.com

2. Characteristics of LDChild usually: 3. Characteristics of LDChild usually: Is often more creative. 4. Characteristics of LDChild usually: Is often more creative. Cannot learn by rote. 5. Characteristics of LDChild usually: Is often more creative. Cannot learn by rote. Must understand and make sense ofconceptin order to remember. 6. Characteristics of LDChild usually: Is often more creative. Cannot learn by rote. Must understand and make sense ofconceptin order to remember. Is more visual and often hands-on. 7. Characteristics of LDChild usually: Is often more creative. Cannot learn by rote. Must understand and make sense ofconceptin order to remember. Is more visual and often hands-on. Dislikes worksheets. 8. Myths about LD 9. Myths about LD1. People with LD have lower intelligence. (33%are gifted.) 10. Myths about LD1. People with LD have lower intelligence. (33%are gifted.)2. They are lazy or stubborn. 11. Myths about LD1. People with LD have lower intelligence. (33%are gifted.)2. They are lazy or stubborn.3. Children with LD can be cured or will outgrowit. 12. Myths about LD1. People with LD have lower intelligence. (33%are gifted.)2. They are lazy or stubborn.3. Children with LD can be cured or will outgrowit.4. Boys are more likely to be affected. 13. Myths about LD1. People with LD have lower intelligence. (33%are gifted.)2. They are lazy or stubborn.3. Children with LD can be cured or will outgrowit.4. Boys are more likely to be affected.5. Dyslexia and learning disability are the samething. 14. Myths about LD1. People with LD have lower intelligence. (33%are gifted.)2. They are lazy or stubborn.3. Children with LD can be cured or will outgrowit.4. Boys are more likely to be affected.5. Dyslexia and learning disability are the samething.6. Adults with LD and ADD cannot succeed inhigher education. 15. Problems Occurring in Math(Dyscalculia) 16. Problems Occurring in Math(Dyscalculia) Reversals in writing numbers 17. Problems Occurring in Math(Dyscalculia) Reversals in writing numbers Poor number sense 18. Problems Occurring in Math(Dyscalculia) Reversals in writing numbers Poor number sense Slow fact retrieval 19. Problems Occurring in Math(Dyscalculia) Reversals in writing numbers Poor number sense Slow fact retrieval Errors in computation 20. Problems Occurring in Math(Dyscalculia) Reversals in writing numbers Poor number sense Slow fact retrieval Errors in computation Difficulty in solving word problems 21. Problems Occurring in Math(Dyscalculia) Reversals in writing numbers Poor number sense Slow fact retrieval Errors in computation Difficulty in solving word problemsDyscalculia only affects arithmetic, not otherbranches of math. 22. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13 23. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13 24. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13 25. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13 26. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13 27. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13 28. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13 29. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13 30. Time Needed to MemorizeAccording to a study with college students, it took them 31. Time Needed to Memorize 93 minutes to learn 200 nonsensesyllables.According to a study with college students, it took them 32. Time Needed to Memorize 93 minutes to learn 200 nonsensesyllables. 24 minutes to learn 200 words of prose.According to a study with college students, it took them 33. Time Needed to Memorize 93 minutes to learn 200 nonsensesyllables. 24 minutes to learn 200 words of prose. 10 minutes to learn 200 words of poetry.According to a study with college students, it took them 34. Memorizing MathPercentage RecallImmediatelyAfter 1 day After 4wks.Rote 32 23 8Concept69 69 58 35. Memorizing MathPercentage RecallImmediatelyAfter 1 day After 4wks.Rote 32 23 8Concept69 69 58 36. Memorizing MathPercentage RecallImmediatelyAfter 1 day After 4wks.Rote 32 23 8Concept69 69 58 37. Memorizing MathPercentage RecallImmediatelyAfter 1 day After 4wks.Rote 32 23 8Concept69 69 58 38. Memorizing MathMath needs to be taught so 95%is understood and only 5%memorized.Richard SkempPercentage RecallImmediatelyAfter 1 day After 4wks.Rote 32 23 8Concept69 69 58 39. Flash Cards & Times Tests 40. Flash Cards & Times Tests Often used to teach rote. 41. Flash Cards & Times Tests Often used to teach rote. Liked only by those who dont needthem. 42. Flash Cards & Times Tests Often used to teach rote. Liked only by those who dont needthem. Give the false impression that mathisntabout thinking. 43. Flash Cards & Times Tests Often used to teach rote. Liked only by those who dont needthem. Give the false impression that mathisntabout thinking. Often produce stress children understress stop learning. 44. Flash Cards & Times Tests Often used to teach rote. Liked only by those who dont needthem. Give the false impression that mathisntabout thinking. Often produce stress children understress stop learning. Not concrete use abstract symbols. 45. Flash Cards & Times Tests Often used to teach rote. Liked only by those who dont needthem. Give the false impression that mathisntabout thinking. Often produce stress children understress stop learning. Not concrete use abstract symbols. Cause stress (may become physicallyill). 46. Flash Cards & Times Tests Often used to teach rote. Liked only by those who dont needthem. Give the false impression that mathisntabout thinking. Often produce stress children understress stop learning. Not concrete use abstract symbols. Cause stress (may become physicallyill). 47. Flash Cards & Times Tests Often used to teach rote. Liked only by those who dont needthem. Give the false impression that mathisntabout thinking. Often produce stress children understress stop learning. Not concrete use abstract symbols. Cause stress (may become physicallyill). 48. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspective 49. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveBecause were so familiar with 1, 2, 3, well useletters.A = 1B = 2C = 3D = 4E = 5, and so forth 50. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveF + E = 51. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveAF + E = 52. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA BF + E = 53. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA CBF + E = 54. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA FC D EBF + E = 55. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveAA FC D EBF + E = 56. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA BA FC D EBF + E = 57. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA C D EBA FC D EBF + E = 58. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA C D EBA FC D EBWhat is the sum?(It must be a letter.)F + E = 59. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveG I J KHA FC D EBF + E =K 60. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D 61. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D 62. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D 63. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D 64. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D 65. 2010 Joan A.Traditional CountingFrom a childs perspectiveTry subtractingby takingawayH E 66. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveTry skip counting by Bs to T:B, D, . . . , T. 67. Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveTry skip counting by Bs to T:B, D, . . . , T.What is D x E? 68. Adding on a Number LineD + C = 69. Adding on a Number LineD + C =* A B C D E F G H I J K L M 70. Adding on a Number LineD + C =* A B C D E F G H I J K L M 71. Adding on a Number LineD + C =* A B C D E F G H I J K L MWere counting spaces, not lines. 72. Number Line Drawbacks 73. Number Line Drawbacks Quantity of a number not obvious. 74. Number Line Drawbacks Quantity of a number not obvious. Requires counting in order to compute. 75. Number Line Drawbacks Quantity of a number not obvious. Requires counting in order to compute. Ignores place value. 76. Number Line Drawbacks Quantity of a number not obvious. Requires counting in order to compute. Ignores place value. Hard to visualize. 77. DominoesMost domino patterns cannot be added.3 + 1 4 78. Domino Pattern Drawbacks 79. Domino Pattern Drawbacks Patterns generally cannot be added. 80. Domino Pattern Drawbacks Patterns generally cannot be added. They are not used outside of games. 81. Domino Pattern Drawbacks Patterns generally cannot be added. They are not used outside of games. Emphasize only 16. 82. Domino Pattern Drawbacks Patterns generally cannot be added. They are not used outside of games. Emphasize only 16. Precious little mathematics. 83. Joan A. Cotter, Ph.D., 2013VisualizingJapanese criteria for manipulatives 84. Joan A. Cotter, Ph.D., 2013Visualizing Visual is related to seeing. 85. Joan A. Cotter, Ph.D., 2013Visualizing Visual is related to seeing. Visualize is to form a mental image. 86. Joan A. Cotter, Ph.D., 2013VisualizingTry to visualize 8 identical apples withoutgrouping. 87. Joan A. Cotter, Ph.D., 2013VisualizingTry to visualize 8 identical apples withoutgrouping. 88. Joan A. Cotter, Ph.D., 2013VisualizingNow try to visualize 8 apples: 5 red and 3 green. 89. Joan A. Cotter, Ph.D., 2013VisualizingNow try to visualize 8 apples: 5 red and 3 green. 90. Joan A. Cotter, Ph.D., 2013VisualizingCan you visualize this rod? 91. Joan A. Cotter, Ph.D., 2013Grouping in Fives 92. Joan A. Cotter, Ph.D., 2013Grouping in FivesIIIIIIIIIIVVIII123458Early Roman numerals 93. Joan A. Cotter, Ph.D., 2013Grouping in FivesMusical staff 94. Joan A. Cotter, Ph.D., 2013Clocks and nickelsGrouping in Fives 95. Joan A. Cotter, Ph.D., 2013Grouping in FivesClocks and nickels 96. Joan A. Cotter, Ph.D., 2013Grouping in FivesTally marks 97. Joan A. Cotter, Ph.D., 2011Grouping in FivesSubitizing Instant recognition of quantity is called subitizing. 98. Joan A. Cotter, Ph.D., 2011Grouping in FivesSubitizing Instant recognition of quantity is called subitizing. Grouping in fives extends subitizing beyond five. 99. Joan A. Cotter, Ph.D., 2011Subitizing Five-month-old infants can subitize to 13. 100. Joan A. Cotter, Ph.D., 2011Subitizing Three-year-olds can subitize to 15. Five-month-old infants can subitize to 13. 101. Joan A. Cotter, Ph.D., 2011Subitizing Three-year-olds can subitize to 15. Four-year-olds can subitize 110 bygrouping with five. Five-month-old infants can subitize to 13. 102. Joan A. Cotter, Ph.D., 2011Subitizing Three-year-olds can subitize to 15. Four-year-olds can subitize 110 bygrouping with five. Five-month-old infants can subitize to 13. Counting is analogous to sounding out aword; subitizing, to recognizing the word. 103. Joan A. Cotter, Ph.D., 2011Research on Subitizing 104. Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research 105. Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research 106. Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research 107. Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research 108. Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research 109. Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research 110. Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research 111. Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research 112. Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns researchYou could say subitizing ismuch more natural thancounting. 113. Joan A. Cotter, Ph.D., 2011Learning 110Using fingers 114. Joan A. Cotter, Ph.D., 2011Learning 110Using fingers 115. Joan A. Cotter, Ph.D., 2011Learning 110Using fingers 116. Joan A. Cotter, Ph.D., 2011Learning 110Using fingers 117. Joan A. Cotter, Ph.D., 2011Learning 110Using fingers 118. Joan A. Cotter, Ph.D., 2011Learning 110Using fingers 119. Joan A. Cotter, Ph.D., 2011Learning 110Subitizing 5 120. Joan A. Cotter, Ph.D., 2011Learning 110Subitizing 5 121. Joan A. Cotter, Ph.D., 2011Learning 110Subitizing 55 has a middle; 4 does not. 122. Joan A. Cotter, Ph.D., 2011Learning 110Tally sticks 123. Joan A. Cotter, Ph.D., 2011Learning 110Tally sticks 124. Joan A. Cotter, Ph.D., 2011Learning 110Tally sticks 125. Joan A. Cotter, Ph.D., 2011Learning 110Tally sticksFive as a group. 126. Joan A. Cotter, Ph.D., 2011Learning 110Tally sticks 127. Joan A. Cotter, Ph.D., 2011Learning 110Tally sticks 128. Joan A. Cotter, Ph.D., 2013Learning 110Entering quantities 129. Joan A. Cotter, Ph.D., 20133Learning 110Entering quantities 130. Joan A. Cotter, Ph.D., 20135Learning 110Entering quantities 131. Joan A. Cotter, Ph.D., 20137Learning 110Entering quantities 132. Joan A. Cotter, Ph.D., 2013Learning 11010Entering quantities 133. Joan A. Cotter, Ph.D., 2013Learning 110The stairs 134. Joan A. Cotter, Ph.D., 2013Learning 110Adding 135. Joan A. Cotter, Ph.D., 2013Learning 1104 + 3 =Adding 136. Joan A. Cotter, Ph.D., 2013Learning 1104 + 3 =Adding 137. Joan A. Cotter, Ph.D., 2013Learning 1104 + 3 =Adding 138. Joan A. Cotter, Ph.D., 2013Learning 1104 + 3 =Adding 139. Joan A. Cotter, Ph.D., 2013Learning 1104 + 3 = 7Adding 140. Joan A. Cotter, Ph.D., 2013Learning 1104 + 3 = 7AddingJapanese children learn to do this mentally. 141. Joan A. Cotter, Ph.D., 2012Games 142. Joan A. Cotter, Ph.D., 2012GamesGamesMath 143. Joan A. Cotter, Ph.D., 2012GamesGamesMath= 144. Joan A. Cotter, Ph.D., 2012GamesGamesMathBooksReading= 145. Joan A. Cotter, Ph.D., 2012GamesGamesMathBooksReadingGames provide interesting repetition neededfor automatic responses.= 146. Joan A. Cotter, Ph.D., 2012GamesGamesMathBooksReadingGames provide interesting repetition neededfor automatic responses.More importantly, games provide anapplicationfor the new information!= 147. Joan A. Cotter, Ph.D., 2012Go to the Dump 148. Joan A. Cotter, Ph.D., 2012Go to the DumpObjective: To learn and master the factsthat total 10: 149. Joan A. Cotter, Ph.D., 2012Go to the DumpObjective: To learn and master the factsthat total 10:1 + 92 + 83 + 74 + 65 + 5 150. Joan A. Cotter, Ph.D., 2012Go to the DumpObjective: To learn and master the factsthat total 10:1 + 92 + 83 + 74 + 65 + 5 151. Joan A. Cotter, Ph.D., 2012Go to the DumpObjective: To learn and master the factsthat total 10:1 + 92 + 83 + 74 + 65 + 5It is played similar to Go Fish. 152. Joan A. Cotter, Ph.D., 2013Writing Numbers61728394105Sandpaper numbers also help.Number Chart 153. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming 154. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming11 = ten 1 155. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming11 = ten 112 = ten 2 156. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming11 = ten 112 = ten 213 = ten 3 157. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4 158. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 9 159. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten 160. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten21 = 2-ten 1 161. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten21 = 2-ten 122 = 2-ten 2 162. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten21 = 2-ten 122 = 2-ten 223 = 2-ten 3 163. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten21 = 2-ten 122 = 2-ten 223 = 2-ten 3. . . .. . . .99 = 9-ten 9 164. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming137 = 1 hundred 3-ten 7 165. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming137 = 1 hundred 3-ten 7or137 = 1 hundred and 3-ten 7 166. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming Only 11 words are needed to count to 100the math way, 28 in English. (All Indo-European languages are non-standard innumber naming.) 167. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming Only 11 words are needed to count to 100the math way, 28 in English. (All Indo-European languages are non-standard innumber naming.) Asian children learn mathematics using themath way of counting. 168. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming Only 11 words are needed to count to 100the math way, 28 in English. (All Indo-European languages are non-standard innumber naming.) Asian children learn mathematics using themath way of counting. They understand place value in first grade;only half of U.S. children understand placevalue at the end of fourth grade. 169. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming Only 11 words are needed to count to 100the math way, 28 in English. (All Indo-European languages are non-standard innumber naming.) Asian children learn mathematics using themath way of counting. They understand place value in first grade;only half of U.S. children understand placevalue at the end of fourth grade. Mathematics is the science of patterns. Thepatterned math way of counting greatlyhelps children learn number sense. 170. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingCompared to reading 171. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming Just as reciting the alphabet doesnt teachreading, counting doesnt teach arithmetic.Compared to reading 172. Joan A. Cotter, Ph.D., 2013Math Way of Number Naming Just as reciting the alphabet doesnt teachreading, counting doesnt teach arithmetic. Just as we first teach the sound of the letters,we must first teach the name of the quantity(math way).Compared to reading 173. Joan A. Cotter, Ph.D., 2013Place-Value Cards3 03 0 0 03 th- ou-sand3 hun-dred3- ten3 0 0 174. Joan A. Cotter, Ph.D., 2013Place-Value Cards5 084 0 0 02 0 0 175. Joan A. Cotter, Ph.D., 2013Place-Value Cards5 084 0 0 0 4 0 0 02 0 05 082 0 0 176. Joan A. Cotter, Ph.D., 2013Place-Value Cards5 084 0 0 0 4 0 0 02 0 04 0 0 02 0 05 085 082 0 0 177. Joan A. Cotter, Ph.D., 2013Place-Value Cards3 0 0 08 178. Joan A. Cotter, Ph.D., 2013Place-Value Cards3 0 0 0 3 0 0 088 179. Joan A. Cotter, Ph.D., 2013Place-Value Cards3 0 0 0 3 0 0 3 0 00 0888 180. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names4-ten = fortyThe tymeanstens. 181. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names4-ten = fortyThe tymeanstens. 182. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names6-ten = sixtyThe tymeanstens. 183. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names3-ten = thirtyThir alsoused in 1/3,13 and 30. 184. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names5-ten = fiftyFif alsoused in 1/5,15 and 50. 185. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names2-ten = twentyTwo used tobepronouncedtwoo. 186. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesA word gamefireplace place-fire 187. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesA word gamefireplace place-firepaper-newsnewspaper 188. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesA word gamefireplace place-firepaper-newsbox-mail mailboxnewspaper 189. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesten 4Prefix -teenmeans ten. 190. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesten 4 teen 4Prefix -teenmeans ten. 191. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesten 4 teen 4 fourteenPrefix -teenmeans ten. 192. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesa one left 193. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesa one left a left-one 194. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesa one left a left-one eleven 195. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namestwo leftTwo saidastwoo. 196. Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namestwo left twelveTwo saidastwoo. 197. Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten 198. Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten 199. Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten3 0 200. Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten3 0 201. Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten3 0 202. Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten73 0 203. Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten73 0 204. Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten73 07 205. Joan A. Cotter, Ph.D., 20133 0Composing Numbers3-ten77 206. Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten7Note the congruence in how we say thenumber, represent the number, and writethe number.3 07 207. Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten1 0Another example. 208. Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten 81 0 209. Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten81 0 210. Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten81 08 211. Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten81 88 212. Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten 213. Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten1 0 0 214. Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten1 0 0 215. Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten1 0 0 216. Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred 217. Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 0 0 218. Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 0 0 219. Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 01 01 0 0 220. Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 0 0 221. Joan A. Cotter, Ph.D., 2013Composing Numbers2hundred 222. Joan A. Cotter, Ph.D., 2013Composing Numbers2hundred 223. Joan A. Cotter, Ph.D., 2013Composing Numbers2hundred2 0 0 224. Joan A. Cotter, Ph.D., 2013Learning the Facts 225. Joan A. Cotter, Ph.D., 2013Learning the FactsLimited success, especially for strugglingchildren, when learning is: 226. Joan A. Cotter, Ph.D., 2013Learning the Facts Based on counting: whether dots,fingers, number lines, or countingwords.Limited success, especially for strugglingchildren, when learning is: 227. Joan A. Cotter, Ph.D., 2013Learning the Facts Based on counting: whether dots,fingers, number lines, or countingwords.Limited success, especially for strugglingchildren, when learning is: Based on rote memory: whether flashcards, timed tests, or computer games. 228. Joan A. Cotter, Ph.D., 2013Learning the Facts Based on counting: whether dots,fingers, number lines, or countingwords.Limited success, especially for strugglingchildren, when learning is: Based on rote memory: whether flashcards, timed tests, or computer games. Based on skip counting: whether fingers or songs 229. Joan A. Cotter, Ph.D., 2013Fact Strategies 230. Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 = 231. Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 = 232. Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 = 233. Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =Take 1 fromthe 5 and giveit to the 9. 234. Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =Take 1 fromthe 5 and giveit to the 9. 235. Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =Take 1 fromthe 5 and giveit to the 9. 236. Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 = 14Take 1 fromthe 5 and giveit to the 9. 237. Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 = 238. Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 = 239. Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 = 240. Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 = 241. Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =10 + 4 = 14 242. Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 9 = 243. Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 9 = 244. Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 9 =Subtract 5;then 4. 245. Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 9 =Subtract 5;then 4. 246. Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 9 =Subtract 5;then 4. 247. Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 9 = 6Subtract 5;then 4. 248. Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 9 = 249. Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 9 =Subtract 9from 10. 250. Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 9 =Subtract 9from 10. 251. Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 9 =Subtract 9from 10. 252. Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 9 = 6Subtract 9from 10. 253. Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 9 = 254. Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 9 =Start with 9;go up to 15. 255. Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 9 =Start with 9;go up to 15. 256. Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 9 =Start with 9;go up to 15. 257. Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 9 =Start with 9;go up to 15. 258. Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 9 =1 + 5 = 6Start with 9;go up to 15. 259. Joan A. Cotter, Ph.D., 2013MoneyPenny 260. Joan A. Cotter, Ph.D., 2013MoneyNickel 261. Joan A. Cotter, Ph.D., 2013MoneyDime 262. Joan A. Cotter, Ph.D., 2013MoneyQuarter 263. Joan A. Cotter, Ph.D., 2013MoneyQuarter 264. Joan A. Cotter, Ph.D., 2013MoneyQuarter 265. Joan A. Cotter, Ph.D., 2013MoneyQuarter 266. Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts6 x 4 =(6 taken 4 times) 267. Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts6 x 4 =(6 taken 4 times) 268. Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts6 x 4 =(6 taken 4 times) 269. Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts6 x 4 =(6 taken 4 times) 270. Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts6 x 4 =(6 taken 4 times) 271. Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts9 x 3 = 272. Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts9 x 3 = 273. Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts9 x 3 = 30 274. Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts9 x 3 =30 3 = 27 275. Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts7 x 7 = 276. Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts7 x 7 = 277. Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts7 x 7 =25 + 278. Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts7 x 7 =25 + 10 + 10 279. Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts7 x 7 =25 + 10 + 10+ 4 = 49 280. Joan A. Cotter, Ph.D., 2013Trading1000 10 1100 281. Joan A. Cotter, Ph.D., 2013TradingThousands1000 10 1100 282. Joan A. Cotter, Ph.D., 2013TradingHundreds1000 10 1100 283. Joan A. Cotter, Ph.D., 2013TradingTens1000 10 1100 284. Joan A. Cotter, Ph.D., 2013TradingOnes1000 10 1100 285. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6 286. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6 287. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6 288. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6 289. Joan A. Cotter, Ph.D., 2013TradingAdding8+ 6141000 10 1100 290. Joan A. Cotter, Ph.D., 2013TradingAdding8+ 614Too manyones; trade 10ones for 1 ten.1000 10 1100 291. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 614Too manyones; trade 10ones for 1 ten. 292. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 614Too manyones; trade 10ones for 1 ten. 293. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 614Same answerbefore andafter trading. 294. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738 295. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter the firstnumber fromleft to right. 296. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right. 297. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right. 298. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right. 299. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right. 300. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right. 301. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults aftereach step. 302. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults after eachstep. 303. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults after eachstep. 304. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults after eachstep. 305. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults aftereach step. 306. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults aftereach step.1 307. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults aftereach step.1 308. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults aftereach step.1 309. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults aftereach step.1 310. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults aftereach step.1 311. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults aftereach step.1 312. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults aftereach step.1 313. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults aftereach step.1 314. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults aftereach step.1 315. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults aftereach step.1 1 316. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults aftereach step.1 1 317. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults after eachstep.1 1 318. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386396Add starting atthe right. Writeresults after eachstep.1 1 319. Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386396Add starting atthe right. Writeresults after eachstep.1 1 320. Joan A. Cotter, Ph.D., 2013Part-Whole Circles 321. Joan A. Cotter, Ph.D., 2013Part-Whole CirclesWholePart Part 322. Joan A. Cotter, Ph.D., 2013Part-Whole CirclesWhat is the whole?3 2 323. Joan A. Cotter, Ph.D., 2013Part-Whole CirclesWhat is the whole?352 324. Joan A. Cotter, Ph.D., 2013Part-Whole CirclesWhat is the other part?107 325. Joan A. Cotter, Ph.D., 2013Part-Whole CirclesWhat is the other part?3107 326. Joan A. Cotter, Ph.D., 2011Short Division 327. Joan A. Cotter, Ph.D., 2011Short Division Means we dont write the stuffunderneath. 328. Joan A. Cotter, Ph.D., 2011Short Division Means we dont write the stuffunderneath. Should be used for single-digit divisors. 329. Joan A. Cotter, Ph.D., 2011Short Division Means we dont write the stuffunderneath. Should be used for single-digit divisors. Is easier to understand than longdivision. 330. Joan A. Cotter, Ph.D., 2011Short Division Means we dont write the stuffunderneath. Should be used for single-digit divisors. Is easier to understand than longdivision. Best taught before long division. 331. Joan A. Cotter, Ph.D., 2011Short Division Means we dont write the stuffunderneath. Should be used for single-digit divisors. Is easier to understand than longdivision. Best taught before long division. Is much more useful in real life. 332. Joan A. Cotter, Ph.D., 2011Short Division Means we dont write the stuffunderneath. Should be used for single-digit divisors. Is easier to understand than longdivision. Best taught before long division. Is much more useful in real life. 333. Joan A. Cotter, Ph.D., 2011Short Division3 4 7 1) 334. Joan A. Cotter, Ph.D., 2011Short Division)The little lines help to keep track of placevalue.3 4 7 1 335. Joan A. Cotter, Ph.D., 2011Short Division)400 3 = ? [100] Write the 1 on the line.3 4 7 1 336. Joan A. Cotter, Ph.D., 2011Short Division1)400 3 = ? [100] Write the 1 on the line.3 4 7 1 337. Joan A. Cotter, Ph.D., 2011Short Division1)400 3 = ? [100] Write the 1 on the line.What is the remainder? [100] How many tensis that? [10]3 4 7 1 338. Joan A. Cotter, Ph.D., 20113 4 7 1Short Division1400 3 = ? [100] Write the 1 on the line.What is the remainder? [100] How many tensis that? [10] How many tens do we have? [10+ 7 = 17]) 339. Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionLets show the 17 tens by writing a 1 beforethe 7.11) 340. Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionLets show the 17 tens by writing a 1 beforethe 7. Divide the tens: 17 tens 3 = ? [5]11) 341. Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionLets show the 17 tens by writing a 1 beforethe 7. Divide the tens: 17 tens 3 = ? [5]1)1 5 342. Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionTo find the remainder go up. 3 x 5 = 15; howfar is 15 from 17. [2]1 51) 343. Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionTo find the remainder go up. 3 x 5 = 15; howfar is 15 from 17. [2]How many ones do we have? [20 + 1]1 51) 344. Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionTo find the remainder go up. 3 x 5 = 15; howfar is 15 from 17. [2]How many ones do we have? [20 + 1]1 51) 345. Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionTo find the remainder go up. 3 x 5 = 15; howfar is 15 from 17. [2]How many ones do we have? [20 + 1]1 51) 2 346. Joan A. Cotter, Ph.D., 20113 4 7 1Short Division21 3 = ? [7]1 521) 347. Joan A. Cotter, Ph.D., 20113 4 7 1Short Division)1 5 72121 3 = ? [7] 348. Joan A. Cotter, Ph.D., 20113 4 7 1Short Division)1 5 721 349. Joan A. Cotter, Ph.D., 2011Short DivisionAnother example 350. Joan A. Cotter, Ph.D., 2011Short Division)9 8 0 5 3 351. Joan A. Cotter, Ph.D., 2011Short Division)9 8 0 5 380 hundred 9 = ? [8 hundred] 352. Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division8) 353. Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division88) 354. Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division8) 8 355. Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division) 88 9 356. Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division4) 88 9 357. Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division4) 88 9 358. Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division8 9 4) 48 359. Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division)8 9 4 r748 360. Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division8 9 4)r748 361. Teaching Math 362. Teaching Math Talk positively about math. 363. Teaching Math Talk positively about math. Teach for understanding; concretemodels should lead to mental models. 364. Teaching Math Talk positively about math. Teach for understanding; concretemodels should lead to mental models. Encourage children to ask questions. Idont get it is not a question. 365. Teaching Math Talk positively about math. Teach for understanding; concretemodels should lead to mental models. Encourage children to ask questions. Idont get it is not a question. Use correct vocabulary. 366. Teaching Math Talk positively about math. Teach for understanding; concretemodels should lead to mental models. Encourage children to ask questions. Idont get it is not a question. Use correct vocabulary. Ask a question once; allow 3 secondsfor a response. Resist rephrasing. 367. Teaching Math Talk positively about math. Teach for understanding; concretemodels should lead to mental models. Encourage children to ask questions. Idont get it is not a question. Use correct vocabulary. Ask a question once; allow 3 secondsfor a response. Resist rephrasing. Give a child 2-3 seconds to say a fact. 368. Teaching Math to Childrenwith Special NeedsMassHOPE - TEACHWorcester, MASaturday, April 279:45am 10:45amJoan A. Cotter, Ph.D.JoanCotter@RightStartMath.com