1. Overcoming Obstacles Learning Arithmetic through Visualizing with the AL Abacus IDA-UMB Conference March 12, 2011 Saint Paul, Minnesota by Joan A. Cotter, Ph.D. [email_address] 7 5 2 VII

2. Children with MD (Math Difficulties) Often experience difficulties with:

8. Children with MD (Math Difficulties) Often learn best when:

15. Learning Arithmetic Traditionally Counting

16. Learning Arithmetic Traditionally Counting Memorizing 390 Facts

17. Learning Arithmetic Traditionally Counting Memorizing 390 Facts Learning Procedures

18. Learning Arithmetic Traditionally Counting Memorizing 390 Facts Learning Procedures Solving Problems

19. Learning Arithmetic Traditionally Counting Memorizing 390 Facts Learning Procedures Solving Problems Place Value

20. Learning Arithmetic Visually Place Value Place value is the single most important topic in arithmetic.

21. Learning Arithmetic Visually Place Value Place value is the single most important topic in arithmetic. Naming Quantities

22. Learning Arithmetic Visually Place Value Place value is the single most important topic in arithmetic. Naming Quantities Visualizing 390 Facts

23. Learning Arithmetic Visually Place Value Place value is the single most important topic in arithmetic. Naming Quantities Visualizing 390 Facts Learning Procedures

24. Learning Arithmetic Visually Place Value Place value is the single most important topic in arithmetic. Naming Quantities Visualizing 390 Facts Learning Procedures Solving Problems

25. Counting Based-Arithmetic Arithmetic is deemed to be based on counting.

33. Counting Model From a child's 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 forth

34. Counting Model From a child's perspective F + E

35. Counting Model From a child's perspective F + E A

36. Counting Model From a child's perspective F + E A B

37. Counting Model From a child's perspective F + E A C B

38. Counting Model From a child's perspective F + E A F C D E B

39. Counting Model From a child's perspective F + E A A F C D E B

40. Counting Model From a child's perspective F + E A B A F C D E B

41. Counting Model From a child's perspective F + E A C D E B A F C D E B

42. Counting Model From a child's perspective F + E What is the sum? (It must be a letter.) A C D E B A F C D E B

43. Counting Model From a child's perspective K G I J K H A F C D E B F + E

44. Counting Model From a child's perspective Now memorize the facts!! G + D

45. Counting Model From a child's perspective Now memorize the facts!! G + D H + F

46. Counting Model From a child's perspective Now memorize the facts!! G + D H + F D + C

47. Counting Model From a child's perspective Now memorize the facts!! G + D H + F C + G D + C

48. Counting Model From a child's perspective Now memorize the facts!! E + I G + D H + F C + G D + C

49. Counting Model From a child's perspective Try subtracting by “taking away” H – E

50. Counting Model From a child's perspective Try skip counting by B’s to T : B , D , . . . T .

51. Counting Model From a child's perspective Try skip counting by B’s to T : B , D , . . . T . What is D  E ?

52. Counting Model From a child's perspective L is written AB because it is A J and B A’s

53. Counting Model From a child's perspective L is written AB because it is A J and B A’s huh?

54. Counting Model From a child's perspective L is written AB because it is A J and B A’s (twelve)

55. Counting Model From a child's perspective L is written AB because it is A J and B A’s (12) (twelve)

56. Counting Model From a child's perspective L is written AB because it is A J and B A’s (12) (one 10) (twelve)

57. Counting Model From a child's perspective L is written AB because it is A J and B A’s (12) (one 10) (two 1s). (twelve)

58. Counting Model Counting:

65. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 Sometimes calendars are used for counting.

66. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 Sometimes calendars are used for counting.

67. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31

68. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 This is ordinal, not cardinal counting. The 3 doesn’t include the 1 and the 2.

69. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 A calendar is NOT like a ruler. On a ruler the numbers are not in the spaces.

70. Calendar Math August 29 22 15 8 1 30 23 16 9 2 24 17 10 3 25 18 11 4 26 19 12 5 27 20 13 6 28 21 14 7 31 A calendar is NOT like a ruler. On a ruler the numbers are not in the spaces.

71. Calendar Math August 8 1 9 2 10 3 4 5 6 7 Always show the whole calendar. A child needs to see the whole before the parts. Children also need to learn to plan ahead.

75. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall

76. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall

77. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall

78. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall

79. Memorizing Math 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall Even worse if you have MD.

80. Memorizing Math Math needs to be taught so 95% is understood and only 5% memorized. Richard Skemp 58 69 69 Concept 8 23 32 Rote After 4 wks After 1 day Immediately Percentage Recall Even worse if you have MD.

81. Memorizing Math Flash Cards 9 + 7

87. Research on Counting Karen Wynn’s research Show the baby two teddy bears.

88. Research on Counting Karen Wynn’s research Then hide them with a screen.

89. Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.

90. Research on Counting Karen Wynn’s research Show the baby a third teddy bear and put it behind the screen.

91. Research on Counting Karen Wynn’s research Raise screen. Baby seeing 3 won’t look long because it is expected.

92. Research on Counting Karen Wynn’s research Researcher can change the number of teddy bears behind the screen.

93. Research on Counting Karen Wynn’s research A baby seeing 1 teddy bear will look much longer, because it’s unexpected.

100. Visualizing Mathematics Visualizing is an alternative to copious counting and mind-numbing memorization.

101. Visualizing Mathematics “ Think in pictures, because the brain remembers images better than it does anything else.” Ben Pridmore, World Memory Champion, 2009

102. 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)

103. Visualizing Mathematics “ The process of connecting symbols to imagery is at the heart of mathematics learning.” Dienes

104. Visualizing Mathematics “ Mathematics is the activity of creating relationships, many of which are based in visual imagery. ” Wheatley and Cobb

105. 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 others

109. Visualizing Mathematics Ready: How many?

110. Visualizing Mathematics Ready: How many?

111. Visualizing Mathematics Try again: How many?

112. Visualizing Mathematics Try again: How many?

113. Visualizing Mathematics Try again: How many?

114. Visualizing Mathematics Ready: How many?

115. Visualizing Mathematics Try again: How many?

116. Visualizing Mathematics Try to visualize 8 identical apples without grouping.

117. Visualizing Mathematics Try to visualize 8 identical apples without grouping.

118. Visualizing Mathematics Now try to visualize 5 as red and 3 as green.

119. Visualizing Mathematics Now try to visualize 5 as red and 3 as green.

120. Visualizing Mathematics Early Roman numerals I II III IIII V VIII 1 2 3 4 5 8 Romans grouped in fives. Notice 8 is 5 and 3.

121. Visualizing Mathematics Who could read the music? : Music needs 10 lines, two groups of five.

122. Naming Quantities Using fingers

123. Naming Quantities Using fingers Use left hand for 1-5 because we read from left to right.

124. Naming Quantities Using fingers

125. Naming Quantities Using fingers

126. Naming Quantities Using fingers Always show 7 as 5 and 2, not for example, as 4 and 3.

127. Naming Quantities Using fingers

128. 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.

129. Naming Quantities Recognizing 5

130. Naming Quantities Recognizing 5

131. 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.

132. Naming Quantities Tally sticks Lay the sticks flat on a surface, about 1 inch (2.5 cm) apart.

133. Naming Quantities Tally sticks

134. Naming Quantities Tally sticks

135. Naming Quantities Tally sticks Stick is horizontal, because it won’t fit diagonally and young children have problems with diagonals.

136. Naming Quantities Tally sticks

137. Naming Quantities Tally sticks Start a new row for every ten.

138. Naming Quantities What is 4 apples plus 3 more apples? Solving a problem without counting How would you find the answer without counting?

139. Naming Quantities What is 4 apples plus 3 more apples? Solving a problem without counting 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.

140. Naming Quantities A typical worksheet The child counts all the horsies and forgets the fact before turning the page.

141. Naming Quantities Number Chart 1 2 3 4 5

142. Naming Quantities Number Chart 1 2 3 4 5 To help the child learn the symbols

143. Naming Quantities Number Chart 6 1 7 2 8 3 9 4 10 5 To help the child learn the symbols

144. AL Abacus Double-sided AL abacus. Side 1 is grouped in 5s. Trading Side introduces algorithms with trading. 1000 10 1 100

145. AL Abacus Cleared

146. 3 Entering quantities AL Abacus Quantities are entered all at once, not counted.

147. 3 Entering quantities AL Abacus Quantities are entered all at once, not counted.

148. 5 AL Abacus Entering quantities Relate quantities to hands.

149. 5 AL Abacus Entering quantities Relate quantities to hands.

150. 7 AL Abacus Entering quantities

151. 7 AL Abacus Entering quantities

152. AL Abacus 10 Entering quantities

153. AL Abacus 10 Entering quantities

154. AL Abacus The stairs Can use to “count” 1 to 10. Also read quantities on the right side.

155. AL Abacus Adding

156. AL Abacus Adding 4 + 3 =

157. AL Abacus Adding 4 + 3 =

158. AL Abacus Adding 4 + 3 =

159. AL Abacus Adding 4 + 3 =

160. AL Abacus Adding 4 + 3 = 7 Answer is seen immediately, no counting needed.

161. Go to the Dump Game Objective: 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.

162. Go to the Dump Game Objective: 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.

163. Go to the Dump Game Children use the abacus while playing this “Go Fish” type game.

164. Go to the Dump Game A game viewed from above. Starting

165. Go to the Dump Game Each player takes 5 cards. 7 2 7 9 5 7 4 2 6 1 3 8 3 4 9 Starting

166. Go to the Dump Game Does YellowCap have any pairs? [no] 7 2 7 9 5 7 2 4 6 1 3 8 3 4 9 Finding pairs

167. Go to the Dump Game Does BlueCap have any pairs? [yes, 1] 7 2 7 9 5 7 2 4 6 1 3 8 3 4 9 Finding pairs

168. Go to the Dump Game Does BlueCap have any pairs? [yes, 1] 7 2 7 9 5 7 2 1 3 8 Finding pairs 4 6 3 4 9

169. Go to the Dump Game Does BlueCap have any pairs? [yes, 1] 4 6 7 2 7 9 5 7 2 1 3 8 3 4 9 Finding pairs

170. Go to the Dump Game Does PinkCap have any pairs? [yes, 2] 4 6 7 2 7 9 5 7 2 1 3 8 3 4 9 Finding pairs

171. Go to the Dump Game Does PinkCap have any pairs? [yes, 2] 4 6 7 2 7 9 5 3 4 9 Finding pairs 7 2 1 3 8

172. Go to the Dump Game Does PinkCap have any pairs? [yes, 2] 4 6 7 2 7 9 5 2 1 8 3 4 9 Finding pairs 7 3

173. Go to the Dump Game Does PinkCap have any pairs? [yes, 2] 4 6 7 2 7 9 5 1 3 4 9 Finding pairs 7 3 2 8

174. Go to the Dump Game The player asks the player on her left. 2 4 6 7 2 7 9 5 1 3 4 9 2 8 Playing

175. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? The player asks the player on her left. 2 4 6 7 2 7 9 5 1 3 4 9 2 8 Playing

176. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? The player asks the player on her left. 7 4 6 2 7 9 5 1 4 9 2 8 Playing

177. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? The player asks the player on her left. 7 4 6 2 7 9 5 1 4 9 2 8 Playing 3

178. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 3? 4 6 2 7 9 5 1 4 9 2 8 Playing 7 3

179. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? YellowCap gets another turn. 4 6 2 7 9 5 1 4 9 2 8 Playing 7 3

180. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? Go to the dump. YellowCap gets another turn. 4 6 2 7 9 5 1 4 9 2 8 Playing 7 3

181. Go to the Dump Game BlueCap, do you have a 3? BlueCap, do you have an 8? Go to the dump. 2 4 6 7 3 2 7 9 5 1 4 9 2 8 Playing

182. Go to the Dump Game 2 8 4 6 7 3 2 2 7 9 5 1 4 9 Playing 1

183. Go to the Dump Game PinkCap, do you have a 6? 2 8 4 6 7 3 2 2 7 9 5 1 4 9 Playing 1

184. Go to the Dump Game PinkCap, do you have a 6? Go to the dump. 2 8 4 6 7 3 2 2 7 9 5 1 4 9 Playing 1

185. Go to the Dump Game 5 2 8 4 6 7 3 2 2 7 9 5 1 4 9 Playing 1

186. Go to the Dump Game 2 8 5 4 6 7 3 2 2 7 9 5 1 4 9 Playing 1

187. Go to the Dump Game YellowCap, do you have a 9? 1 9 2 8 5 4 6 7 3 2 2 7 9 5 4 9 Playing 1

188. Go to the Dump Game YellowCap, do you have a 9? 1 9 2 8 5 4 6 7 3 2 2 7 5 4 9 Playing 1

189. Go to the Dump Game YellowCap, do you have a 9? 1 9 2 8 5 4 6 7 3 2 2 7 5 4 9 Playing 1 9

190. Go to the Dump Game 1 9 1 9 5 4 6 7 3 2 2 7 5 4 9 Playing

191. Go to the Dump Game PinkCap is not out of the game. Her turn ends, but she takes 5 more cards. 1 9 5 4 6 7 3 2 2 7 5 4 9 Playing 2 9 1 7 7

192. Go to the Dump Game 6 5 1 Winner? 4 5 9 5

193. Go to the Dump Game No counting. Combine both stacks. Winner? 4 5 9 6 5 1

194. Go to the Dump Game Whose pile is the highest? Winner? 4 6 5 5 9 1

195. Go to the Dump Game No shuffling needed for next game. Next game

196. Part-Whole Circles Part-whole circles help children see relationships and solve problems.

197. Part-Whole Circles Whole Part-whole circles help children see relationships and solve problems.

198. Part-Whole Circles Whole Part-whole circles help children see relationships and solve problems. Part Part

199. Part-Whole Circles 10 If 10 is the whole

200. Part-Whole Circles 10 4 and 4 is one part,

201. Part-Whole Circles 10 4 What is the other part?

202. Part-Whole Circles 10 4 6 What is the other part?

203. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? A missing addend problem, considered very difficult for first graders. They can do it with Part-Whole Circles.

204. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 3 a part or whole?

205. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 3 a part or whole? 3

206. Part-Whole Circles 3 Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 5 a part or whole?

207. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Is 5 a part or whole ? 5 3

208. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 5 3 What is the missing part?

209. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? What is the missing part? 5 3 2

210. Part-Whole Circles 5 3 2 Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? Write the equation.

211. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 2 + 3 = 5 5 3 2 Write the equation.

212. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 2 + 3 = 5 5 3 2 3 + 2 = 5 Write the equation.

213. Part-Whole Circles Lee received 3 goldfish as a gift. Now Lee has 5. How many did Lee have to start with? 2 + 3 = 5 5 3 2 3 + 2 = 5 5 – 3 = 2 Write the equation. Is this an addition or subtraction problem?

214. Part-Whole Circles Part-whole circles help young children solve problems. Writing equations do not.

215. Part-Whole Circles Do not try to help children solve story problems by teaching “key” words.

216. “ Math” Way of Naming Numbers

217. “ Math” Way of Naming Numbers 11 = ten 1

218. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2

219. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3

220. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4

221. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9

222. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.

223. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.

224. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.

225. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3 Don’t say “2-ten s .” We don’t say 3 hundred s eleven for 311.

226. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3 . . . . . . . . 99 = 9-ten 9

227. “ Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 Only numbers under 100 need to be said the “math” way.

228. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Shows how far children from 3 countries can count at ages 4, 5, and 6.

229. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Purple is Chinese. Note jump between ages 5 and 6.

230. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Dark green is Korean “math” way.

231. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Average Highest Number Counted Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Dotted green is everyday Korean; notice smaller jump between ages 5 and 6.

232. “ Math” Way of Naming Numbers 0 10 20 30 40 50 60 70 80 90 100 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. Korean formal [math way] Korean informal [not explicit] Chinese U.S. Average Highest Number Counted Red is English speakers. They learn same amount between ages 4-5 and 5-6.

237. Math Way of Naming Numbers Compared to reading:

240. 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 Researchers

241. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task:

242. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14.

243. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.

244. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.

245. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.

246. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.

247. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.

248. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.

249. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.

250. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14.

251. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.

252. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.

253. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones.

254. Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens.

255. 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.

256. Math Way of Naming Numbers Traditional names 6-ten = sixty The “ty” means tens.

257. Math Way of Naming Numbers Traditional names 3-ten = thirty “ Thir” also used in 1/3, 13 and 30.

258. Math Way of Naming Numbers Traditional names 5-ten = fifty “ Fif” also used in 1/5, 15 and 50.

259. Math Way of Naming Numbers Traditional names 2-ten = twenty Two used to be pronounced “twoo.”

260. 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.

261. Math Way of Naming Numbers Traditional names A word game fireplace place-fire paper-news newspaper Say the syllables backward. This is how we say the teen numbers.

262. Math Way of Naming Numbers Traditional names A word game fireplace place-fire paper-news box-mail mailbox newspaper Say the syllables backward. This is how we say the teen numbers.

263. Math Way of Naming Numbers Traditional names ten 4 “ Teen” also means ten.

264. Math Way of Naming Numbers Traditional names ten 4 teen 4 “ Teen” also means ten.

265. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourteen “ Teen” also means ten.

266. Math Way of Naming Numbers Traditional names a one left

267. Math Way of Naming Numbers Traditional names a one left a left-one

268. Math Way of Naming Numbers Traditional names a one left a left-one eleven

269. Math Way of Naming Numbers Traditional names two left Two pronounced “twoo.”

270. Math Way of Naming Numbers Traditional names two left twelve Two pronounced “twoo.”

271. Composing Numbers 3-ten

272. Composing Numbers 3-ten

273. Composing Numbers 3-ten 3 0

274. Composing Numbers 3-ten 3 0 Point to the 3 and say 3.

275. Composing Numbers 3-ten 3 0 Point to 0 and say 10. The 0 makes 3 a ten.

276. Composing Numbers 3-ten 7 3 0

277. Composing Numbers 3-ten 7 3 0

278. Composing Numbers 3-ten 7 3 0 7

279. Composing Numbers 3-ten 7 3 0 7 Place the 7 on top of the 0 of the 30.

280. Composing Numbers 3-ten 7 Notice the way we say the number, represent the number, and write the number all correspond. 3 0 7

281. Composing Numbers 7-ten

282. Composing Numbers 7-ten 70 is visualizable—again because of the fives’ grouping.

283. Composing Numbers 7-ten 7 0 70 is visualizable—again because of the fives’ grouping.

284. Composing Numbers 7-ten 8 7 0 70 is visualizable—again because of the fives’ grouping.

285. Composing Numbers 7-ten 8 7 0 70 is visualizable—again because of the fives’ grouping.

286. Composing Numbers 7-ten 8 7 0 8 70 is visualizable—again because of the fives grouping.

287. Composing Numbers 7-ten 8 7 8 8 Place the 8 on top of the 0 of the 70.

288. Composing Numbers 10-ten

289. Composing Numbers 10-ten 1 0 0

290. Composing Numbers 10-ten 1 0 0

291. Composing Numbers 10-ten 1 0 0

292. Composing Numbers 1 hundred

293. Composing Numbers 1 hundred 1 0 0

294. Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one hun-dred.

295. Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one hun-dred. 1 0 1 0

296. Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one hun-dred.

297. Composing Numbers 2 hundred

298. Composing Numbers 2 hundred 2 0 0 Just the edges of the abacuses are shown.

299. Composing Numbers 2 hundred 2 0 0 Just the edges of the abacuses are shown.

300. Composing Numbers 6 hundred 6 0 0 Maintaining the fives’ grouping.

301. Composing Numbers 10 hundred 1 0 0 0

302. Composing Numbers 10 hundred 1 0 0 0

303. Composing Numbers 1 thousand 1 0 0 0 Of course, we can also read it as one th-ou-sand.

304. Composing Numbers 1 thousand 1 0 0 0 Of course, we can also read it as one th-ou-sand.

305. Composing Numbers 1 thousand 1 0 0 0 Of course, we can also read it as one th-ou-sand.

306. Composing Numbers 1 thousand 1 0 0 0 Of course, we can also read it as one th-ou-sand.

307. Composing Numbers Reading numbers backward 2 5 8 4 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text:

308. Composing Numbers 2 5 8 4 5 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward

309. Composing Numbers 2 5 8 4 2 5 8 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward

310. Composing Numbers 2 5 8 4 2 5 8 4 To read a number, students are often instructed to start at the right (ones column), contrary to normal reading of numbers and text: Reading numbers backward

311. Fact Strategies

314. Fact Strategies Complete the Ten 9 + 5 =

315. Fact Strategies Complete the Ten 9 + 5 =

316. Fact Strategies Complete the Ten 9 + 5 =

317. Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.

318. Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9. Use two hands and move the bead simultaneously.

319. Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.

320. Fact Strategies Complete the Ten 9 + 5 = 14 Take 1 from the 5 and give it to the 9.

321. Fact Strategies Two Fives 8 + 6 =

322. Fact Strategies Two Fives 8 + 6 =

323. Fact Strategies Two Fives 8 + 6 = Two fives make 10.

324. Fact Strategies Two Fives 8 + 6 = Just add the “leftovers.”

325. Fact Strategies Two Fives 8 + 6 = 10 + 4 = 14 Just add the “leftovers.”

326. Fact Strategies Two Fives 7 + 5 = Another example.

327. Fact Strategies Two Fives 7 + 5 =

328. Fact Strategies Two Fives 7 + 5 = 12

329. Fact Strategies Going Down 15 – 9 =

330. Fact Strategies Going Down 15 – 9 =

331. Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.

332. Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.

333. Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.

334. Fact Strategies Going Down 15 – 9 = 6 Subtract 5; then 4.

335. Fact Strategies Subtract from 10 15 – 9 =

336. Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.

337. Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.

338. Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.

339. Fact Strategies Subtract from 10 15 – 9 = 6 Subtract 9 from 10.

340. Fact Strategies Going Up 13 – 9 =

341. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.

342. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.

343. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.

344. Fact Strategies Going Up 13 – 9 = Start with 9; go up to 13.

345. Fact Strategies Going Up 13 – 9 = 1 + 3 = 4 Start with 9; go up to 13.

346. Money Penny

347. Money Nickel

348. Money Dime

349. Money Quarter

350. Money Quarter

351. Money Quarter

352. Money Quarter

353. Trading Side Cleared 1000 10 1 100

354. Trading Side Thousands 1000 10 1 100

355. Trading Side Hundreds 1000 10 1 100 The third wire from each end is not used.

356. Trading Side Tens 1000 10 1 100 The third wire from each end is not used.

357. Trading Side Ones 1000 10 1 100 The third wire from each end is not used.

358. Trading Side Adding 8 + 6 1000 10 1 100

359. Trading Side Adding 8 + 6 1000 10 1 100

360. Trading Side Adding 8 + 6 1000 10 1 100

361. Trading Side Adding 8 + 6 1000 10 1 100

362. Trading Side Adding 8 + 6 14 1000 10 1 100

363. Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. You can see the 10 ones (yellow). 1000 10 1 100

364. Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. 1000 10 1 100

365. Trading Side Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. 1000 10 1 100

366. Trading Side Adding 8 + 6 14 Same answer before and after trading. 1000 10 1 100

367. Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 1000 10 1 100

368. Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 7 1000 10 1 100

369. Trading Side Bead Trading game Object: To get a high score by adding numbers on the green cards. 7 1000 10 1 100

370. Trading Side Bead Trading game 6 Turn over another card. Enter 6 beads. Do we need to trade? 1000 10 1 100

371. Trading Side Bead Trading game 6 Turn over another card. Enter 6 beads. Do we need to trade? 1000 10 1 100

372. Trading Side Bead Trading game 6 Turn over another card. Enter 6 beads. Do we need to trade? 1000 10 1 100

373. Trading Side Bead Trading game 6 Trade 10 ones for 1 ten. 1000 10 1 100

374. Trading Side Bead Trading game 6 1000 10 1 100

375. Trading Side Bead Trading game 6 1000 10 1 100

376. Trading Side Bead Trading game 9 1000 10 1 100

377. Trading Side Bead Trading game 9 1000 10 1 100

378. Trading Side Bead Trading game 9 Another trade. 1000 10 1 100

379. Trading Side Bead Trading game 9 Another trade. 1000 10 1 100

380. Trading Side Bead Trading game 3 1000 10 1 100

381. Trading Side Bead Trading game 3 1000 10 1 100

387. Trading Side Adding 4-digit numbers 3658 + 2738 1000 10 1 100

388. Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right. 1000 10 1 100

389. Trading Side Adding 4-digit numbers 3 658 + 2738 Enter the first number from left to right. 1000 10 1 100

390. Trading Side Adding 4-digit numbers 3 658 + 2738 Enter the first number from left to right. 1000 10 1 100

391. Trading Side Adding 4-digit numbers 3 6 58 + 2738 Enter the first number from left to right. 1000 10 1 100

392. Trading Side Adding 4-digit numbers 36 5 8 + 2738 Enter the first number from left to right. 1000 10 1 100

393. Trading Side Adding 4-digit numbers 365 8 + 2738 Enter the first number from left to right. 1000 10 1 100

394. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100

395. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100

396. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100

397. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100

398. Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. . . . 6 ones. Did anything else happen? 1000 10 1 100

399. Trading Side Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1 Is it okay to show the extra ten by writing a 1 above the tens column? 1000 10 1 100

400. Trading Side Adding 4-digit numbers 3658 + 27 3 8 6 Add starting at the right. Write results after each step. 1 1000 10 1 100

401. Trading Side Adding 4-digit numbers 3658 + 27 3 8 6 Add starting at the right. Write results after each step. 1 Do we need to trade? [no] 1000 10 1 100

402. Trading Side Adding 4-digit numbers 3658 + 2738 9 6 Add starting at the right. Write results after each step. 1 1000 10 1 100

403. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 1000 10 1 100

404. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 Do we need to trade? [yes] 1000 10 1 100

405. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 Notice the number of yellow beads. [3] Notice the number of blue beads left. [3] Coincidence? No, because 13 – 10 = 3. 1000 10 1 100

406. Trading Side Adding 4-digit numbers 3658 + 2 7 38 96 Add starting at the right. Write results after each step. 1 1000 10 1 100

407. Trading Side Adding 4-digit numbers 3658 + 2738 3 96 Add starting at the right. Write results after each step. 1 1000 10 1 100

408. Trading Side Adding 4-digit numbers 3658 + 2738 3 96 Add starting at the right. Write results after each step. 1 1 1000 10 1 100

409. Trading Side Adding 4-digit numbers 3658 + 2 738 396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100

410. Trading Side Adding 4-digit numbers 3658 + 2 738 396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100

411. Trading Side Adding 4-digit numbers 3658 + 2738 6 396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100

412. Trading Side Adding 4-digit numbers 3658 + 2738 6396 Add starting at the right. Write results after each step. 1 1 1000 10 1 100

413. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)

414. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)

415. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)

416. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)

417. Multiplication on the AL Abacus Basic facts 6  4 = (6 taken 4 times)

418. Multiplication on the AL Abacus Basic facts 9  3 =

419. Multiplication on the AL Abacus Basic facts 9  3 =

420. Multiplication on the AL Abacus Basic facts 9  3 = 30

421. Multiplication on the AL Abacus Basic facts 9  3 = 30 – 3 = 27

422. Multiplication on the AL Abacus Basic facts 4  8 =

423. Multiplication on the AL Abacus Basic facts 4  8 =

424. Multiplication on the AL Abacus Basic facts 4  8 =

425. Multiplication on the AL Abacus Basic facts 4  8 = 20 + 12 = 32

426. Multiplication on the AL Abacus Basic facts 7  7 =

427. Multiplication on the AL Abacus Basic facts 7  7 =

428. Multiplication on the AL Abacus Basic facts 7  7 = 25 + 10 + 10 + 4 = 49

429. Multiplication on the AL Abacus Commutative property 5  6 =

430. Multiplication on the AL Abacus Commutative property 5  6 =

431. Multiplication on the AL Abacus Commutative property 5  6 = 6  5

432. Multiples Patterns Twos 2 4 6 8 10 12 14 16 18 20 Recognizing multiples needed for fractions and algebra.

433. Multiples Patterns Twos 2 4 6 8 10 1 2 14 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.

434. Multiples Patterns Twos 2 4 6 8 10 12 1 4 16 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.

435. Multiples Patterns Twos 2 4 6 8 10 12 14 1 6 18 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.

436. Multiples Patterns Twos 2 4 6 8 10 12 14 16 1 8 20 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.

437. Multiples Patterns Twos 2 4 6 8 1 0 12 14 16 18 2 0 The ones repeat in the second row. Recognizing multiples needed for fractions and algebra.

438. Multiples Patterns Fours 4 8 12 16 20 24 28 32 36 40

439. Multiples Patterns Fours 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0 The ones repeat in the second row.

440. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80

441. Multiples Patterns Sixes and Eights 6 1 2 1 8 2 4 3 0 3 6 4 2 4 8 5 4 6 0 8 16 24 32 40 48 56 64 72 80

442. Multiples Patterns Sixes and Eights 6 1 2 1 8 2 4 3 0 3 6 4 2 4 8 5 4 6 0 8 1 6 2 4 3 2 4 0 4 8 5 6 6 4 7 2 8 0 Again the ones repeat in the second row.

443. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 4 0 48 56 64 72 8 0 The ones in the 8s show the multiples of 2.

444. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 3 2 4 0 48 56 64 7 2 8 0 The ones in the 8s show the multiples of 2.

445. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 2 4 3 2 4 0 48 56 6 4 7 2 8 0 The ones in the 8s show the multiples of 2.

446. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 1 6 2 4 3 2 4 0 48 5 6 6 4 7 2 8 0 The ones in the 8s show the multiples of 2.

447. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 1 6 2 4 3 2 4 0 4 8 5 6 6 4 7 2 8 0 The ones in the 8s show the multiples of 2.

448. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 6  4 6  4 is the fourth number (multiple).

449. Multiples Patterns Sixes and Eights 6 12 18 24 30 36 42 48 54 60 8 16 24 32 40 48 56 64 72 80 8  7 8  7 is the seventh number (multiple).

450. Multiples Patterns Nines 9 18 27 36 45 90 81 72 63 54 The second row is written in reverse order. Also the digits in each number add to 9.

454. Multiples Patterns Threes 3 6 9 1 2 15 18 2 1 24 27 3 0 The 3s have several patterns: Observe the ones.

455. Multiples Patterns Threes 3 6 9 1 2 15 18 2 1 24 27 3 0 The 3s have several patterns: Observe the ones.

456. Multiples Patterns Threes 3 6 9 1 2 15 18 2 1 2 4 27 3 0 The 3s have several patterns: Observe the ones.

457. Multiples Patterns Threes 3 6 9 1 2 1 5 18 2 1 2 4 27 3 0 The 3s have several patterns: Observe the ones.

458. Multiples Patterns Threes 3 6 9 1 2 1 5 18 2 1 2 4 27 3 0 The 3s have several patterns: Observe the ones.

459. Multiples Patterns Threes 3 6 9 1 2 1 5 18 2 1 2 4 2 7 3 0 The 3s have several patterns: Observe the ones.

460. Multiples Patterns Threes 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 The 3s have several patterns: Observe the ones.

461. Multiples Patterns Threes 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 0 The 3s have several patterns: Observe the ones.

462. Multiples Patterns Threes 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 30 The 3s have several patterns: The tens are the same in each row.

463. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.

464. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.

465. Multiples Patterns Threes 3 6 9 12 15 18 21 24 27 30 The 3s have several patterns: Add the digits in the columns.

466. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.

467. Multiples Patterns Sevens 7 14 2 1 28 35 4 2 49 56 6 3 70 The 7s have the 1, 2, 3… pattern.

468. Multiples Patterns Sevens 7 1 4 2 1 28 3 5 4 2 49 5 6 6 3 70 The 7s have the 1, 2, 3… pattern.

469. Multiples Patterns Sevens 7 1 4 2 1 2 8 3 5 4 2 4 9 5 6 6 3 7 0 The 7s have the 1, 2, 3… pattern.

470. Multiples Patterns Sevens 7 14 21 28 35 42 49 56 63 70 The 7s have the 1, 2, 3… pattern.

471. Multiples Patterns Sevens 7 14 21 2 8 35 42 4 9 56 63 7 0 The 7s have the 1, 2, 3… pattern.

472. Research Highlights TASK EXPER CTRL TEENS 10 + 3 94% 47% 6 + 10 88% 33% CIRCLE TENS 78 75% 67% 3924 44% 7% 14 as 10 & 4 48 – 14 81% 33%

473. Research Highlights TASK EXPER CTRL 26-TASK (tens) 6 (ones) 94% 100% Other research questions asked. 2 (tens) 63% 13% MENTAL COMP: 85 – 70 31% 0% 2nd Graders in U.S. (Reys): 9% 38 + 24 = 512 or 0% 40% 57 + 35 = 812

474. Some Important Conclusions

480. Overcoming Obstacles Learning Arithmetic through Visualizing with the AL Abacus IDA-UMB Conference March 12, 2011 Saint Paul, Minnesota by Joan A. Cotter, Ph.D. [email_address] 7 5 2 VII

481. Overcoming Obstacles Learning Arithmetic through Visualizing with the AL Abacus IDA-UMB Conference March 12, 2011 Saint Paul, Minnesota by Joan A. Cotter, Ph.D. [email_address] 7 5 2 (PowerPoint is available on Alabacus.com under Resources.) VII

482. ABACUS WORK PAGE 1000 10 1 100