This PowerPoint helps students to consider the concept of infinity.
AMS: Counting-Necessary or Detrimental? March 2011
1. Counting: Necessary or Detrimental? AMS Conference March 25, 2011 Chicago, Illinois by Joan A. Cotter, Ph.D. [email_address] 7 5 2 Presentation available: ALabacus.com 7 x 7 VII
61. 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.
62. 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 4 doesn’t include 1, 2 and 3.
63. 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.
64. 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.
91. 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)
92. Visualizing Mathematics “ Think in pictures, because the brain remembers images better than it does anything else.” Ben Pridmore, World Memory Champion, 2009
93. Visualizing Mathematics “ Mathematics is the activity of creating relationships, many of which are based in visual imagery. ” Wheatley and Cobb
94. Visualizing Mathematics “ The process of connecting symbols to imagery is at the heart of mathematics learning.” Dienes
95. 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
120. 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.
130. Naming Quantities What is 4 apples plus 3 more apples? Solving a problem without counting How would you find the answer without counting?
131. 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 “ Grouped in fives so the child does not need to count.” Black and White Bead Stairs A. M. Joosten This was the inspiration to group in 5s.
178. Problem Solving Solving a problem 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.
179. Problem Solving Solving a problem 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?
180. Problem Solving Solving a problem 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
181. Problem Solving Solving a problem 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?
182. Problem Solving Solving a problem 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
183. Problem Solving Solving a problem 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?
184. Problem Solving Solving a problem 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
185. Problem Solving Solving a problem 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.
186. Problem Solving Solving a problem 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.
187. Problem Solving Solving a problem 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.
188. Problem Solving Solving a problem 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?
190. 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.
191. Go to the Dump Game Objective: To learn the ways to partition 10: 1 + 9 2 + 8 3 + 7 4 + 6 5 + 5 Object of the game: To collect the most pairs that equal ten. It is similar to “Go Fish.” Children use the abacus while playing this “Go Fish” type game.
192. Go to the Dump Game The ways to partition 10.
195. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2
196. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3
197. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4
198. “ Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9
199. “ 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.
200. “ 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.
201. “ 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.
202. “ 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.
203. “ 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
204. “ Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 Only numbers under 100 need to be said the “math” way.
205. “ 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.
206. “ 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.
207. “ 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.
208. “ 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.
209. “ 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.
210. “ 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.
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215. Math Way of Naming Numbers Compared to reading:
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219. 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
220. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task:
221. Math Way of Naming Numbers Using 10s and 1s, ask the child to construct 48. Research task: Then ask the child to subtract 14.
222. 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 count 14.
223. 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.
224. 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.
225. 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.
226. 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.
227. 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.
228. 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.
229. 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.
230. 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.
231. 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.
232. 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.
233. Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens.
234. 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.
235. Math Way of Naming Numbers Traditional names 6-ten = sixty The “ty” means tens.
236. Math Way of Naming Numbers Traditional names 3-ten = thirty “ Thir” also used in 1/3, 13 and 30.
237. Math Way of Naming Numbers Traditional names 5-ten = fifty “ Fif” also used in 1/5, 15 and 50.
238. Math Way of Naming Numbers Traditional names 2-ten = twenty Two used to be pronounced “twoo.”
239. 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.
240. 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.
241. 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.
242. Math Way of Naming Numbers Traditional names ten 4 “ Teen” also means ten.
243. Math Way of Naming Numbers Traditional names ten 4 teen 4 “ Teen” also means ten.
244. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourteen “ Teen” also means ten.
245. Math Way of Naming Numbers Traditional names a one left
246. Math Way of Naming Numbers Traditional names a one left a left-one
247. Math Way of Naming Numbers Traditional names a one left a left-one eleven
248. Math Way of Naming Numbers Traditional names two left Two pronounced “twoo.”
249. Math Way of Naming Numbers Traditional names two left twelve Two pronounced “twoo.”
267. Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one hun-dred.
268. Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one hun-dred. 1 0 1 0
269. Composing Numbers 1 hundred 1 0 0 Of course, we can also read it as one hun-dred.
270. 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:
271. 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
272. 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
273. 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
274. 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 The Decimal Cards encourage reading numbers in the normal order.
376. Trading Side Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right. 1000 10 1 100
377. Trading Side Adding 4-digit numbers 3 658 + 2738 Enter the first number from left to right. 1000 10 1 100
378. Trading Side Adding 4-digit numbers 3 658 + 2738 Enter the first number from left to right. 1000 10 1 100
379. Trading Side Adding 4-digit numbers 3 6 58 + 2738 Enter the first number from left to right. 1000 10 1 100
380. Trading Side Adding 4-digit numbers 36 5 8 + 2738 Enter the first number from left to right. 1000 10 1 100
381. Trading Side Adding 4-digit numbers 365 8 + 2738 Enter the first number from left to right. 1000 10 1 100
382. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100
383. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100
384. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100
385. Trading Side Adding 4-digit numbers 3658 + 273 8 Add starting at the right. Write results after each step. 1000 10 1 100
386. 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
387. 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
388. 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
389. 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
390. 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
391. 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
392. 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
393. 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
394. 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
395. 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
396. 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
397. 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
398. 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
399. 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
400. 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
421. Multiplication on the AL Abacus 7 8 = This method was used in the Middle Ages, rather than memorize the facts > 5 5. For facts > 5 5
422. Multiplication on the AL Abacus 7 8 = This method was used in the Middle Ages, rather than memorize the facts > 5 5. For facts > 5 5
423. Multiplication on the AL Abacus 7 8 = This method was used in the Middle Ages, rather than memorize the facts > 5 5. For facts > 5 5 Tens:
424. Multiplication on the AL Abacus 7 8 = This method was used in the Middle Ages, rather than memorize the facts > 5 5. For facts > 5 5 Tens:
425. Multiplication on the AL Abacus 7 8 = This method was used in the Middle Ages, rather than memorize the facts > 5 5. For facts > 5 5 Tens: 20 + 30
426. Multiplication on the AL Abacus 7 8 = 50 + This method was used in the Middle Ages, rather than memorize the facts > 5 5. For facts > 5 5 Tens: 20 + 30 50
427. Multiplication on the AL Abacus 7 8 = 50 + This method was used in the Middle Ages, rather than memorize the facts > 5 5. For facts > 5 5 Tens: Ones: 20 + 30 50
428. Multiplication on the AL Abacus 7 8 = 50 + This method was used in the Middle Ages, rather than memorize the facts > 5 5. For facts > 5 5 Tens: Ones: 20 + 30 50
429. Multiplication on the AL Abacus 7 8 = 50 + This method was used in the Middle Ages, rather than memorize the facts > 5 5. For facts > 5 5 Tens: Ones: 3 2 20 + 30 50
505. “ Pie” Model Experts in visual literacy say that comparing quantities in pie charts is difficult because most people think linearly. It is easier to compare along a straight line than compare pie slices. askoxford.com
506. “ Pie” Model Experts in visual literacy say that comparing quantities in pie charts is difficult because most people think linearly. It is easier to compare along a straight line than compare pie slices. askoxford.com Specialists also suggest refraining from using more than one pie chart for comparison. statcan.ca
541. Counting: Necessary or Detrimental? AMS Conference March 25, 2011 Chicago, Illinois by Joan A. Cotter, Ph.D. [email_address] 7 5 2 Presentation available: ALabacus.com 7 x 7 VII