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MCTM Future Primary Math

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This document outlines a verbal counting model proposed to help children develop a deeper understanding of numbers and math concepts rather than relying on rote memorization. The model uses letters to represent numbers and demonstrates counting, addition, subtraction, and other operations by building word problems using the letters. It also shows how this model could be applied to calendar math and comparing it to state math standards. The goal is to promote a more conceptual approach to early number sense over a procedural focus on counting and calculations.

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The Future of Primary Math: More Understanding/Less Counting by Joan A. Cotter, Ph.D. JoanCotter@RightStartMath.com MCTM Saturday, May 5, 2012 Duluth, Minnesota 1000 100 10 1 30 7 30 7 PowerPoint Presentation RightStartMath.com >Resources © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model 2 © Joan A. Cotter, Ph.D., 2012
Verbal 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 3 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective F +E 4 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective F +E A 5 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective F +E A B 6 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective F +E A B C 7 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective F +E A B C D E F 8 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective F +E A B C D E F A 9 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective F +E A B C D E F A B 10 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective F +E A B C D E F A B C D E 11 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective F +E A B C D E F A B C D E What is the sum? (It must be a letter.) 12 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective F +E K A B C D E F G H I J K 13 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective Now memorize the facts!! G +D 14 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective Now memorize the facts!! H + G F +D 15 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective Now memorize the facts!! H + G F +D D +C 16 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective Now memorize the facts!! H + G F +D D C +C +G 17 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective Now memorize the facts!! H E + G I F + +D D C +C +G 18 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective H –E Subtract with your fingers by counting backward. 19 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective J –F Subtract without using your fingers. 20 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective Try skip counting by B’s to T: B, D, . . . T. 21 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective Try skip counting by B’s to T: B, D, . . . T. What is D × E? 22 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective L is written AB because it is A J and B A’s 23 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective L is written AB because it is A J and B A’s huh? 24 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective L (twelve) is written AB because it is A J and B A’s 25 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective L (twelve) is written AB (12) because it is A J and B A’s 26 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective L (twelve) is written AB (12) because it is A J (one 10) and B A’s 27 © Joan A. Cotter, Ph.D., 2012
Verbal Counting Model From a child's perspective L (twelve) is written AB (12) because it is A J (one 10) and B A’s (two 1s). 28 © Joan A. Cotter, Ph.D., 2012
Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 29 © Joan A. Cotter, Ph.D., 2012
Calendar Math Calendar Counting August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 30 © Joan A. Cotter, Ph.D., 2012
Calendar Math Calendar Counting August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 31 © Joan A. Cotter, Ph.D., 2012
Calendar Math Calendar Counting August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 © Joan A. Cotter, Ph.D., 2012
Calendar Math Septemb Calendar Counting 1234567 August 89101214 1 2 113 11921 15112628 8 122820 67527 9 3 4 5 6 10 11 12 13 14 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31 33 © Joan A. Cotter, Ph.D., 2012
Calendar Math Septemb Calendar Counting 1234567 August 89101214 1 113 11921 2 15112628 122820 8 67527 9 3 4 5 6 10 11 12 13 14 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31 This is ordinal counting, not cardinal counting. 34 © Joan A. Cotter, Ph.D., 2012
Calendar Math Partial Calendar August 1 2 3 4 5 6 7 8 9 10 35 © Joan A. Cotter, Ph.D., 2012
Calendar Math Partial Calendar August 1 2 3 4 5 6 7 8 9 10 Children need the whole month to plan ahead. 36 © Joan A. Cotter, Ph.D., 2012
Calendar Math Septemb Calendar Patterning 1234567 August 89101214 1 2 113 11921 15112628 8 122820 67527 9 3 4 5 6 10 11 12 13 14 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31 Patterns are rarely based on 7s or proceed row by row. Patterns go on forever; they don’t stop at 31. 37 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Number Sense K: Represent quantities using whole numbers and understand relationships among whole numbers. 1–2: Understand place value and relationships among whole numbers. With the counting model, how difficult are the associated benchmarks for children to master? 38 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. 39 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. 40 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. 41 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. 42 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. 43 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting. 44 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting. • Add and subtract whole numbers up to 6, using objects. 45 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. 46 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. 47 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. 48 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. 49 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12. 50 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12. • Represent whole numbers up to 20 in various ways. 51 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. 52 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. 53 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. 54 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. 55 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12. 56 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12. • Represent whole numbers up to 20 in various ways. 57 © Joan A. Cotter, Ph.D., 2012
Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
Research on Counting Karen Wynn’s research 60 © Joan A. Cotter, Ph.D., 2012
Research on Counting Karen Wynn’s research 61 © Joan A. Cotter, Ph.D., 2012
Research on Counting Karen Wynn’s research 62 © Joan A. Cotter, Ph.D., 2012
Research on Counting Karen Wynn’s research 63 © Joan A. Cotter, Ph.D., 2012
Research on Counting Karen Wynn’s research 64 © Joan A. Cotter, Ph.D., 2012
Research on Counting Karen Wynn’s research 65 © Joan A. Cotter, Ph.D., 2012
Research on Counting Other research 66 © Joan A. Cotter, Ph.D., 2012
Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. 67 © Joan A. Cotter, Ph.D., 2012
Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. • Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. 68 © Joan A. Cotter, Ph.D., 2012
Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. • Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. • Adults, ages 18-50, from Boston. Edward Gibson and Michael Frank, MIT, 2008. 69 © Joan A. Cotter, Ph.D., 2012
Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. • Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. • Adults, ages 18-50, from Boston. Edward Gibson and Michael Frank, MIT, 2008. • Baby chicks from Italy. Lucia Regolin, University of Padova, 2009. 70 © Joan A. Cotter, Ph.D., 2012
Research on Counting In Japanese schools: • Children are discouraged from using counting for adding. 71 © Joan A. Cotter, Ph.D., 2012
Research on Counting In Japanese schools: • Children are discouraged from using counting for adding. • They consistently group in 5s. 72 © Joan A. Cotter, Ph.D., 2012
Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. 73 © Joan A. Cotter, Ph.D., 2012
Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. 74 © Joan A. Cotter, Ph.D., 2012
Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics long term.—Butterworth 75 © Joan A. Cotter, Ph.D., 2012
Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics long term.—Butterworth • Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit 76 © Joan A. Cotter, Ph.D., 2012
Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics long term.—Butterworth • Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit • Subitizing seems to be a necessary skill for understanding what the counting process means.— Glasersfeld 77 © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities 78 © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities “Think in pictures, because the brain remembers images better than it does anything else.” Ben Pridmore, World Memory Champion, 2009 79 © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities “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 80 © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities Japanese criteria for manipulatives • Representative of structure of numbers. • Easily manipulated by children. • Imaginable mentally. Japanese Council of Mathematics Education © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities Visualizing also needed in: • Reading • Sports • Creativity • Geography • Engineering • Construction © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities Visualizing also needed in: • Reading • Architecture • Sports • Astronomy • Creativity • Archeology • Geography • Chemistry • Engineering • Physics • Construction • Surgery © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities Try to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities Try to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities Now try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities Now try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities Early Roman numerals 1 I 2 II 3 III 4 IIII 5 V 8 VIII © Joan A. Cotter, Ph.D., 2012
Visualizing Quantities : Who could read the music? 93 © Joan A. Cotter, Ph.D., 2012
AN ALTERNATIVE to learning place value: Subitizing (groups of five) Math Way (of number naming) Place Value Cards Trading (with 4-digit numbers) 94 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Using fingers © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Using fingers © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Using fingers 97 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Using fingers 98 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Using fingers 99 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Using fingers 100 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives 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 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Recognizing 5 5 has a middle; 4 does not. © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Tally sticks 106 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Tally sticks 107 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Tally sticks 108 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Tally sticks 109 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Tally sticks 110 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Entering quantities © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Entering quantities 3 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Entering quantities 5 113 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Entering quantities 7 114 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Entering quantities 10 115 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives The stairs 116 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Adding © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Adding 4+3=7 © Joan A. Cotter, Ph.D., 2012
Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
Go to the Dump Game Objective: To learn the facts that total 10: 1+9 2+8 3+7 4+6 5+5 124 © Joan A. Cotter, Ph.D., 2012
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. 125 © Joan A. Cotter, Ph.D., 2012
Go to the Dump Game 6+ = 10 126 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 127 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 11 = ten 1 128 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 129 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 130 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 131 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9 132 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9 133 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 14 = ten 4 .... 19 = ten 9 134 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 .... 19 = ten 9 135 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 23 = 2-ten 3 .... 19 = ten 9 136 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 23 = 2-ten 3 .... .... 19 = ten 9 .... 99 = 9-ten 9 137 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 138 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7 139 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. 140 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. 141 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. 142 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. 143 © Joan A. Cotter, Ph.D., 2012
“Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. 144 © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) 145 © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. 146 © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. 147 © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. • Mathematics is the science of patterns. The patterned math way of counting greatly helps children learn number sense. 148 © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Compared to reading: 149 © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Compared to reading: • Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic. 150 © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Compared to reading: • Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic. • Just as we first teach the sound of the letters, we must first teach the name of the quantity (math way). 151 © Joan A. Cotter, Ph.D., 2012
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 152 © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens. © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens. © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names 6-ten = sixty The “ty” means tens. © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names 3-ten = thirty “Thir” also used in 1/3, 13 and 30. © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names 5-ten = fifty “Fif” also used in 1/5, 15 and 50. © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names 2-ten = twenty Two used to be pronounced “twoo.” © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names A word game fireplace place-fire © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news box-mail mailbox © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names ten 4 “Teen” also means ten. © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names ten 4 teen 4 “Teen” also means ten. © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names ten 4 teen 4 fourtee n “Teen” also means ten. © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names a one left © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names a one left a left-one © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names a one left a left-one eleven © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names two left Two pronounced “twoo.” © Joan A. Cotter, Ph.D., 2012
Math Way of Naming Numbers Traditional names two left twelve Two pronounced “twoo.” © Joan A. Cotter, Ph.D., 2012
Composing Numbers 3-ten © Joan A. Cotter, Ph.D., 2012
Composing Numbers 3-ten © Joan A. Cotter, Ph.D., 2012
Composing Numbers 3-ten © Joan A. Cotter, Ph.D., 2012
Composing Numbers 3-ten 30 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 3-ten 30 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 3-ten 30 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 3-ten 7 30 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 3-ten 7 30 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 3-ten 7 30 7 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 3-ten 7 30 7 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 3-ten 7 30 7 Notice the way we say the number, represent the number, and write the number all correspond. © Joan A. Cotter, Ph.D., 2012
Composing Numbers 7-ten 70 Another example. © Joan A. Cotter, Ph.D., 2012
Composing Numbers 7-ten 8 70 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 7-ten 8 70 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 7-ten 8 70 8 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 7-ten 8 78 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 10-ten © Joan A. Cotter, Ph.D., 2012
Composing Numbers 10-ten 100 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 10-ten 100 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 10-ten 100 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 1 hundred © Joan A. Cotter, Ph.D., 2012
Composing Numbers 1 hundred 100 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 1 hundred 100 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 1 hundred 100 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 1 hundred 100 © Joan A. Cotter, Ph.D., 2012
Composing Numbers 2 hundred © Joan A. Cotter, Ph.D., 2012
Composing Numbers 2 hundred © Joan A. Cotter, Ph.D., 2012
Composing Numbers 2 hundred 200 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 2s © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 2s 2 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 2s 2 4 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 2s 2 4 6 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 2s 2 4 6 8 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 2s 2 4 6 8 10 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 2s 2 4 6 8 10 12 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 2s 2 4 6 8 10 12 14 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 2s 2 4 6 8 10 12 14 16 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 2s 2 4 6 8 10 12 14 16 18 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 2s 2 4 6 8 10 12 14 16 18 20 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 5s © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 5s 5 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 5s 5 10 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 5s 5 10 15 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 5s 5 10 15 20 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 5s 5 10 15 20 25 © Joan A. Cotter, Ph.D., 2012
Counting by 2s and 5s Counting by 5s 5 10 15 20 25 30 © Joan A. Cotter, Ph.D., 2012
Evens and Odds Evens © Joan A. Cotter, Ph.D., 2012
Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
Evens and Odds Evens Use two fingers and touch each pair in succession. EVEN! © Joan A. Cotter, Ph.D., 2012
Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
Evens and Odds Odds Use two fingers and touch each pair in succession. ODD! © Joan A. Cotter, Ph.D., 2012
Fact Strategies 227 © Joan A. Cotter, Ph.D., 2012
Fact Strategies Complete the Ten 9+5= © Joan A. Cotter, Ph.D., 2012
Fact Strategies Complete the Ten 9+5= © Joan A. Cotter, Ph.D., 2012
Fact Strategies Complete the Ten 9+5= © Joan A. Cotter, Ph.D., 2012
Fact Strategies Complete the Ten 9+5= Take 1 from the 5 and give it to the 9. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Complete the Ten 9+5= Take 1 from the 5 and give it to the 9. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Complete the Ten 9+5= Take 1 from the 5 and give it to the 9. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Complete the Ten 9 + 5 = 14 Take 1 from the 5 and give it to the 9. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Two Fives 8+6= © Joan A. Cotter, Ph.D., 2012
Fact Strategies Two Fives 8+6= © Joan A. Cotter, Ph.D., 2012
Fact Strategies Two Fives 8+6= © Joan A. Cotter, Ph.D., 2012
Fact Strategies Two Fives 8+6= © Joan A. Cotter, Ph.D., 2012
Fact Strategies Two Fives 8+6= 10 + 4 = 14 © Joan A. Cotter, Ph.D., 2012
Fact Strategies Going Down 15 – 9 = © Joan A. Cotter, Ph.D., 2012
Fact Strategies Going Down 15 – 9 = © Joan A. Cotter, Ph.D., 2012
Fact Strategies Going Down 15 – 9 = Subtract 5; then 4. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Going Down 15 – 9 = Subtract 5; then 4. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Going Down 15 – 9 = Subtract 5; then 4. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Going Down 15 – 9 = 6 Subtract 5; then 4. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Subtract from 10 15 – 9 = © Joan A. Cotter, Ph.D., 2012
Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Subtract from 10 15 – 9 = 6 Subtract 9 from 10. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Going Up 15 – 9 = © Joan A. Cotter, Ph.D., 2012
Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Going Up 15 – 9 = 1+5=6 Start with 9; go up to 15. © Joan A. Cotter, Ph.D., 2012
Fact Strategies Multiplication 6× 4= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
Fact Strategies Multiplication 6× 4= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
Place Value Two aspects © Joan A. Cotter, Ph.D., 2012
Place Value Two aspects Static © Joan A. Cotter, Ph.D., 2012
Place Value Two aspects Static • Value of a digit is determined by position © Joan A. Cotter, Ph.D., 2012
Place Value Two aspects Static • Value of a digit is determined by position. • No position may have more than nine. © Joan A. Cotter, Ph.D., 2012
Place Value Two aspects Static • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position. © Joan A. Cotter, Ph.D., 2012
Place Value Two aspects Static • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position. • Place value cards show this aspect. © Joan A. Cotter, Ph.D., 2012
Place Value Two aspects Static • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position. • Place value cards show this aspect. Dynamic © Joan A. Cotter, Ph.D., 2012
Place Value Two aspects Static • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position. • Place value cards show this aspect. Dynamic • Ten ones = 1 ten; ten tens = 1 hundred; ten hundreds = 1 thousand, …. © Joan A. Cotter, Ph.D., 2012
Trading 1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
Trading Thousands 1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
Trading Hundreds 1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
Trading Tens 1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
Trading Ones 1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
Trading Adding 1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
Trading Adding 1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
Trading Adding 1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
Trading Adding 1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
Trading Adding 1000 100 10 1 8 +6 14 © Joan A. Cotter, Ph.D., 2012
Trading Adding 1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
Trading Adding 1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
Trading Adding 1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
Trading Adding 1000 100 10 1 8 +6 14 Same answer before and after trading. © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 7 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 7 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 6 Trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 3 © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity 1000 100 10 1 3 © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity • In the Bead Trading activity trading 10 ones for 1 ten occurs frequently; © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity • In the Bead Trading activity trading 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity • In the Bead Trading activity trading 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely. © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity • In the Bead Trading activity trading 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely. • Bead trading helps the child experience the greater value of each column from left to right. © Joan A. Cotter, Ph.D., 2012
Trading Bead Trading Activity • In the Bead Trading activity trading 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely. • Bead trading helps the child experience the greater value of each column from left to right. • To detect a pattern, there must be at least three examples in the sequence. Place value is a pattern. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 6 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 6 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 6 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 6 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 1 1 3658 + 2738 6396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Trading Adding 4-digit numbers 1000 100 10 1 1 1 3658 + 2738 6396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Number Sense K: Represent quantities using whole numbers and understand relationships among whole numbers. 1–2: Understand place value and relationships among whole numbers. With this alternate model, how difficult are the associated benchmarks for children to master? 328 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. 329 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. 330 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. 331 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. 332 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting. 333 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting. • Add and subtract whole numbers up to 6, using objects. 334 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. 335 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. 336 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. 337 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12. 338 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12. • Represent whole numbers up to 20 in various ways. 339 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. 340 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. 341 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. 342 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12. 343 © Joan A. Cotter, Ph.D., 2012
Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12. • Represent whole numbers up to 20 in various ways. 344 © Joan A. Cotter, Ph.D., 2012
Research Highlights 345 © Joan A. Cotter, Ph.D., 2012
Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. 346 © Joan A. Cotter, Ph.D., 2012
Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. 347 © Joan A. Cotter, Ph.D., 2012
Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. 348 © Joan A. Cotter, Ph.D., 2012
Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. 349 © Joan A. Cotter, Ph.D., 2012
Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 350 © Joan A. Cotter, Ph.D., 2012
Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 351 © Joan A. Cotter, Ph.D., 2012
Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 352 © Joan A. Cotter, Ph.D., 2012
Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 353 © Joan A. Cotter, Ph.D., 2012
Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 354 © Joan A. Cotter, Ph.D., 2012
Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 355 © Joan A. Cotter, Ph.D., 2012
Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 356 © Joan A. Cotter, Ph.D., 2012
Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 357 © Joan A. Cotter, Ph.D., 2012
Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones. 358 © Joan A. Cotter, Ph.D., 2012
Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones. 359 © Joan A. Cotter, Ph.D., 2012
Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones. 360 © Joan A. Cotter, Ph.D., 2012
Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% © Joan A. Cotter, Ph.D., 2012
Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% Teens 10 + 3 94% 47% © Joan A. Cotter, Ph.D., 2012
Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% Teens 10 + 3 94% 47% 6 + 10 88% 33% © Joan A. Cotter, Ph.D., 2012
Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% Teens 10 + 3 94% 47% 6 + 10 88% 33% Circle Tens 78 75% 67% Place © Joan A. Cotter, Ph.D., 2012
Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% Teens 10 + 3 94% 47% 6 + 10 88% 33% Circle Tens 78 75% 67% Place 3924 44% 7% © Joan A. Cotter, Ph.D., 2012
Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% Teens 10 + 3 94% 47% 6 + 10 88% 33% Circle Tens 78 75% 67% Place 3924 44% 7% Mental 85 – 70 31% 0% Computation © Joan A. Cotter, Ph.D., 2012
Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% Teens 10 + 3 94% 47% 6 + 10 88% 33% Circle Tens 78 75% 67% Place 3924 44% 7% Mental 85 – 70 31% 0% Computation 2nd Graders in US (Reys): 9% © Joan A. Cotter, Ph.D., 2012
Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% Teens 10 + 3 94% 47% 6 + 10 88% 33% Circle Tens 78 75% 67% Place 3924 44% 7% Mental 85 – 70 31% 0% Computation 2nd Graders in US (Reys): 9% 38 + 24 = 512 or 0% 40% 57 + 35 = 812 © Joan A. Cotter, Ph.D., 2012
Framing the Future of Mathematics in Minnesota Place value, not counting, is the key to under- standing numbers. Place value is best taught by: • Subitizing (with groups of fives), • Initially using Math Way of number naming, • Incorporating Place Value Cards, • Patterning (trading with 4-digit numbers). © Joan A. Cotter, Ph.D., 2012
The Future of Primary Math: More Understanding/Less Counting by Joan A. Cotter, Ph.D. JoanCotter@RightStartMath.com MCTM Saturday, May 5, 2012 Duluth, Minnesota 1000 100 10 1 30 7 30 7 PowerPoint Presentation RightStartMath.com >Resources © Joan A. Cotter, Ph.D., 2012

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MCTM Future Primary Math

  • 1. The Future of Primary Math: More Understanding/Less Counting by Joan A. Cotter, Ph.D. JoanCotter@RightStartMath.com MCTM Saturday, May 5, 2012 Duluth, Minnesota 1000 100 10 1 30 7 30 7 PowerPoint Presentation RightStartMath.com >Resources © Joan A. Cotter, Ph.D., 2012
  • 2. Verbal Counting Model 2 © Joan A. Cotter, Ph.D., 2012
  • 3. Verbal 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 3 © Joan A. Cotter, Ph.D., 2012
  • 4. Verbal Counting Model From a child's perspective F +E 4 © Joan A. Cotter, Ph.D., 2012
  • 5. Verbal Counting Model From a child's perspective F +E A 5 © Joan A. Cotter, Ph.D., 2012
  • 6. Verbal Counting Model From a child's perspective F +E A B 6 © Joan A. Cotter, Ph.D., 2012
  • 7. Verbal Counting Model From a child's perspective F +E A B C 7 © Joan A. Cotter, Ph.D., 2012
  • 8. Verbal Counting Model From a child's perspective F +E A B C D E F 8 © Joan A. Cotter, Ph.D., 2012
  • 9. Verbal Counting Model From a child's perspective F +E A B C D E F A 9 © Joan A. Cotter, Ph.D., 2012
  • 10. Verbal Counting Model From a child's perspective F +E A B C D E F A B 10 © Joan A. Cotter, Ph.D., 2012
  • 11. Verbal Counting Model From a child's perspective F +E A B C D E F A B C D E 11 © Joan A. Cotter, Ph.D., 2012
  • 12. Verbal Counting Model From a child's perspective F +E A B C D E F A B C D E What is the sum? (It must be a letter.) 12 © Joan A. Cotter, Ph.D., 2012
  • 13. Verbal Counting Model From a child's perspective F +E K A B C D E F G H I J K 13 © Joan A. Cotter, Ph.D., 2012
  • 14. Verbal Counting Model From a child's perspective Now memorize the facts!! G +D 14 © Joan A. Cotter, Ph.D., 2012
  • 15. Verbal Counting Model From a child's perspective Now memorize the facts!! H + G F +D 15 © Joan A. Cotter, Ph.D., 2012
  • 16. Verbal Counting Model From a child's perspective Now memorize the facts!! H + G F +D D +C 16 © Joan A. Cotter, Ph.D., 2012
  • 17. Verbal Counting Model From a child's perspective Now memorize the facts!! H + G F +D D C +C +G 17 © Joan A. Cotter, Ph.D., 2012
  • 18. Verbal Counting Model From a child's perspective Now memorize the facts!! H E + G I F + +D D C +C +G 18 © Joan A. Cotter, Ph.D., 2012
  • 19. Verbal Counting Model From a child's perspective H –E Subtract with your fingers by counting backward. 19 © Joan A. Cotter, Ph.D., 2012
  • 20. Verbal Counting Model From a child's perspective J –F Subtract without using your fingers. 20 © Joan A. Cotter, Ph.D., 2012
  • 21. Verbal Counting Model From a child's perspective Try skip counting by B’s to T: B, D, . . . T. 21 © Joan A. Cotter, Ph.D., 2012
  • 22. Verbal Counting Model From a child's perspective Try skip counting by B’s to T: B, D, . . . T. What is D × E? 22 © Joan A. Cotter, Ph.D., 2012
  • 23. Verbal Counting Model From a child's perspective L is written AB because it is A J and B A’s 23 © Joan A. Cotter, Ph.D., 2012
  • 24. Verbal Counting Model From a child's perspective L is written AB because it is A J and B A’s huh? 24 © Joan A. Cotter, Ph.D., 2012
  • 25. Verbal Counting Model From a child's perspective L (twelve) is written AB because it is A J and B A’s 25 © Joan A. Cotter, Ph.D., 2012
  • 26. Verbal Counting Model From a child's perspective L (twelve) is written AB (12) because it is A J and B A’s 26 © Joan A. Cotter, Ph.D., 2012
  • 27. Verbal Counting Model From a child's perspective L (twelve) is written AB (12) because it is A J (one 10) and B A’s 27 © Joan A. Cotter, Ph.D., 2012
  • 28. Verbal Counting Model From a child's perspective L (twelve) is written AB (12) because it is A J (one 10) and B A’s (two 1s). 28 © Joan A. Cotter, Ph.D., 2012
  • 29. Calendar Math August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 29 © Joan A. Cotter, Ph.D., 2012
  • 30. Calendar Math Calendar Counting August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 30 © Joan A. Cotter, Ph.D., 2012
  • 31. Calendar Math Calendar Counting August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 31 © Joan A. Cotter, Ph.D., 2012
  • 32. Calendar Math Calendar Counting August 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 © Joan A. Cotter, Ph.D., 2012
  • 33. Calendar Math Septemb Calendar Counting 1234567 August 89101214 1 2 113 11921 15112628 8 122820 67527 9 3 4 5 6 10 11 12 13 14 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31 33 © Joan A. Cotter, Ph.D., 2012
  • 34. Calendar Math Septemb Calendar Counting 1234567 August 89101214 1 113 11921 2 15112628 122820 8 67527 9 3 4 5 6 10 11 12 13 14 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31 This is ordinal counting, not cardinal counting. 34 © Joan A. Cotter, Ph.D., 2012
  • 35. Calendar Math Partial Calendar August 1 2 3 4 5 6 7 8 9 10 35 © Joan A. Cotter, Ph.D., 2012
  • 36. Calendar Math Partial Calendar August 1 2 3 4 5 6 7 8 9 10 Children need the whole month to plan ahead. 36 © Joan A. Cotter, Ph.D., 2012
  • 37. Calendar Math Septemb Calendar Patterning 1234567 August 89101214 1 2 113 11921 15112628 8 122820 67527 9 3 4 5 6 10 11 12 13 14 7 2234 20 15 16 17 18 19 20 21 29 3 22 23 24 25 26 27 28 29 30 31 Patterns are rarely based on 7s or proceed row by row. Patterns go on forever; they don’t stop at 31. 37 © Joan A. Cotter, Ph.D., 2012
  • 38. Minnesota Standards Number Sense K: Represent quantities using whole numbers and understand relationships among whole numbers. 1–2: Understand place value and relationships among whole numbers. With the counting model, how difficult are the associated benchmarks for children to master? 38 © Joan A. Cotter, Ph.D., 2012
  • 39. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. 39 © Joan A. Cotter, Ph.D., 2012
  • 40. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. 40 © Joan A. Cotter, Ph.D., 2012
  • 41. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. 41 © Joan A. Cotter, Ph.D., 2012
  • 42. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. 42 © Joan A. Cotter, Ph.D., 2012
  • 43. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. 43 © Joan A. Cotter, Ph.D., 2012
  • 44. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting. 44 © Joan A. Cotter, Ph.D., 2012
  • 45. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting. • Add and subtract whole numbers up to 6, using objects. 45 © Joan A. Cotter, Ph.D., 2012
  • 46. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. 46 © Joan A. Cotter, Ph.D., 2012
  • 47. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. 47 © Joan A. Cotter, Ph.D., 2012
  • 48. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. 48 © Joan A. Cotter, Ph.D., 2012
  • 49. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. 49 © Joan A. Cotter, Ph.D., 2012
  • 50. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12. 50 © Joan A. Cotter, Ph.D., 2012
  • 51. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12. • Represent whole numbers up to 20 in various ways. 51 © Joan A. Cotter, Ph.D., 2012
  • 52. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. 52 © Joan A. Cotter, Ph.D., 2012
  • 53. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. 53 © Joan A. Cotter, Ph.D., 2012
  • 54. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. 54 © Joan A. Cotter, Ph.D., 2012
  • 55. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. 55 © Joan A. Cotter, Ph.D., 2012
  • 56. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12. 56 © Joan A. Cotter, Ph.D., 2012
  • 57. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12. • Represent whole numbers up to 20 in various ways. 57 © Joan A. Cotter, Ph.D., 2012
  • 58. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
  • 59. Research on Counting Karen Wynn’s research © Joan A. Cotter, Ph.D., 2012
  • 60. Research on Counting Karen Wynn’s research 60 © Joan A. Cotter, Ph.D., 2012
  • 61. Research on Counting Karen Wynn’s research 61 © Joan A. Cotter, Ph.D., 2012
  • 62. Research on Counting Karen Wynn’s research 62 © Joan A. Cotter, Ph.D., 2012
  • 63. Research on Counting Karen Wynn’s research 63 © Joan A. Cotter, Ph.D., 2012
  • 64. Research on Counting Karen Wynn’s research 64 © Joan A. Cotter, Ph.D., 2012
  • 65. Research on Counting Karen Wynn’s research 65 © Joan A. Cotter, Ph.D., 2012
  • 66. Research on Counting Other research 66 © Joan A. Cotter, Ph.D., 2012
  • 67. Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. 67 © Joan A. Cotter, Ph.D., 2012
  • 68. Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. • Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. 68 © Joan A. Cotter, Ph.D., 2012
  • 69. Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. • Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. • Adults, ages 18-50, from Boston. Edward Gibson and Michael Frank, MIT, 2008. 69 © Joan A. Cotter, Ph.D., 2012
  • 70. Research on Counting Other research • Australian Aboriginal children from two tribes. Brian Butterworth, University College London, 2008. • Adult Pirahã from Amazon region. Edward Gibson and Michael Frank, MIT, 2008. • Adults, ages 18-50, from Boston. Edward Gibson and Michael Frank, MIT, 2008. • Baby chicks from Italy. Lucia Regolin, University of Padova, 2009. 70 © Joan A. Cotter, Ph.D., 2012
  • 71. Research on Counting In Japanese schools: • Children are discouraged from using counting for adding. 71 © Joan A. Cotter, Ph.D., 2012
  • 72. Research on Counting In Japanese schools: • Children are discouraged from using counting for adding. • They consistently group in 5s. 72 © Joan A. Cotter, Ph.D., 2012
  • 73. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. 73 © Joan A. Cotter, Ph.D., 2012
  • 74. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. 74 © Joan A. Cotter, Ph.D., 2012
  • 75. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics long term.—Butterworth 75 © Joan A. Cotter, Ph.D., 2012
  • 76. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics long term.—Butterworth • Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit 76 © Joan A. Cotter, Ph.D., 2012
  • 77. Research on Counting Subitizing • Subitizing is quick recognition of quantity without counting. • Human babies and some animals can subitize small quantities at birth. • Children who can subitize perform better in mathematics long term.—Butterworth • Subitizing “allows the child to grasp the whole and the elements at the same time.”—Benoit • Subitizing seems to be a necessary skill for understanding what the counting process means.— Glasersfeld 77 © Joan A. Cotter, Ph.D., 2012
  • 78. Visualizing Quantities 78 © Joan A. Cotter, Ph.D., 2012
  • 79. Visualizing Quantities “Think in pictures, because the brain remembers images better than it does anything else.” Ben Pridmore, World Memory Champion, 2009 79 © Joan A. Cotter, Ph.D., 2012
  • 80. Visualizing Quantities “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 80 © Joan A. Cotter, Ph.D., 2012
  • 81. Visualizing Quantities Japanese criteria for manipulatives • Representative of structure of numbers. • Easily manipulated by children. • Imaginable mentally. Japanese Council of Mathematics Education © Joan A. Cotter, Ph.D., 2012
  • 82. Visualizing Quantities Visualizing also needed in: • Reading • Sports • Creativity • Geography • Engineering • Construction © Joan A. Cotter, Ph.D., 2012
  • 83. Visualizing Quantities Visualizing also needed in: • Reading • Architecture • Sports • Astronomy • Creativity • Archeology • Geography • Chemistry • Engineering • Physics • Construction • Surgery © Joan A. Cotter, Ph.D., 2012
  • 84. Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
  • 85. Visualizing Quantities Ready: How many? © Joan A. Cotter, Ph.D., 2012
  • 86. Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
  • 87. Visualizing Quantities Try again: How many? © Joan A. Cotter, Ph.D., 2012
  • 88. Visualizing Quantities Try to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  • 89. Visualizing Quantities Try to visualize 8 identical apples without grouping. © Joan A. Cotter, Ph.D., 2012
  • 90. Visualizing Quantities Now try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  • 91. Visualizing Quantities Now try to visualize 5 as red and 3 as green. © Joan A. Cotter, Ph.D., 2012
  • 92. Visualizing Quantities Early Roman numerals 1 I 2 II 3 III 4 IIII 5 V 8 VIII © Joan A. Cotter, Ph.D., 2012
  • 93. Visualizing Quantities : Who could read the music? 93 © Joan A. Cotter, Ph.D., 2012
  • 94. AN ALTERNATIVE to learning place value: Subitizing (groups of five) Math Way (of number naming) Place Value Cards Trading (with 4-digit numbers) 94 © Joan A. Cotter, Ph.D., 2012
  • 95. Grouping in Fives Using fingers © Joan A. Cotter, Ph.D., 2012
  • 96. Grouping in Fives Using fingers © Joan A. Cotter, Ph.D., 2012
  • 97. Grouping in Fives Using fingers 97 © Joan A. Cotter, Ph.D., 2012
  • 98. Grouping in Fives Using fingers 98 © Joan A. Cotter, Ph.D., 2012
  • 99. Grouping in Fives Using fingers 99 © Joan A. Cotter, Ph.D., 2012
  • 100. Grouping in Fives Using fingers 100 © Joan A. Cotter, Ph.D., 2012
  • 101. Grouping in Fives 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 © Joan A. Cotter, Ph.D., 2012
  • 102. Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
  • 103. Grouping in Fives Recognizing 5 © Joan A. Cotter, Ph.D., 2012
  • 104. Grouping in Fives Recognizing 5 5 has a middle; 4 does not. © Joan A. Cotter, Ph.D., 2012
  • 105. Grouping in Fives Tally sticks © Joan A. Cotter, Ph.D., 2012
  • 106. Grouping in Fives Tally sticks 106 © Joan A. Cotter, Ph.D., 2012
  • 107. Grouping in Fives Tally sticks 107 © Joan A. Cotter, Ph.D., 2012
  • 108. Grouping in Fives Tally sticks 108 © Joan A. Cotter, Ph.D., 2012
  • 109. Grouping in Fives Tally sticks 109 © Joan A. Cotter, Ph.D., 2012
  • 110. Grouping in Fives Tally sticks 110 © Joan A. Cotter, Ph.D., 2012
  • 111. Grouping in Fives Entering quantities © Joan A. Cotter, Ph.D., 2012
  • 112. Grouping in Fives Entering quantities 3 © Joan A. Cotter, Ph.D., 2012
  • 113. Grouping in Fives Entering quantities 5 113 © Joan A. Cotter, Ph.D., 2012
  • 114. Grouping in Fives Entering quantities 7 114 © Joan A. Cotter, Ph.D., 2012
  • 115. Grouping in Fives Entering quantities 10 115 © Joan A. Cotter, Ph.D., 2012
  • 116. Grouping in Fives The stairs 116 © Joan A. Cotter, Ph.D., 2012
  • 117. Grouping in Fives Adding © Joan A. Cotter, Ph.D., 2012
  • 118. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
  • 119. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
  • 120. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
  • 121. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
  • 122. Grouping in Fives Adding 4+3=7 © Joan A. Cotter, Ph.D., 2012
  • 123. Grouping in Fives Adding 4+3= © Joan A. Cotter, Ph.D., 2012
  • 124. Go to the Dump Game Objective: To learn the facts that total 10: 1+9 2+8 3+7 4+6 5+5 124 © Joan A. Cotter, Ph.D., 2012
  • 125. 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. 125 © Joan A. Cotter, Ph.D., 2012
  • 126. Go to the Dump Game 6+ = 10 126 © Joan A. Cotter, Ph.D., 2012
  • 127. “Math” Way of Naming Numbers 127 © Joan A. Cotter, Ph.D., 2012
  • 128. “Math” Way of Naming Numbers 11 = ten 1 128 © Joan A. Cotter, Ph.D., 2012
  • 129. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 129 © Joan A. Cotter, Ph.D., 2012
  • 130. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 130 © Joan A. Cotter, Ph.D., 2012
  • 131. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 131 © Joan A. Cotter, Ph.D., 2012
  • 132. “Math” Way of Naming Numbers 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9 132 © Joan A. Cotter, Ph.D., 2012
  • 133. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 13 = ten 3 14 = ten 4 .... 19 = ten 9 133 © Joan A. Cotter, Ph.D., 2012
  • 134. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 14 = ten 4 .... 19 = ten 9 134 © Joan A. Cotter, Ph.D., 2012
  • 135. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 .... 19 = ten 9 135 © Joan A. Cotter, Ph.D., 2012
  • 136. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 23 = 2-ten 3 .... 19 = ten 9 136 © Joan A. Cotter, Ph.D., 2012
  • 137. “Math” Way of Naming Numbers 11 = ten 1 20 = 2-ten 12 = ten 2 21 = 2-ten 1 13 = ten 3 22 = 2-ten 2 14 = ten 4 23 = 2-ten 3 .... .... 19 = ten 9 .... 99 = 9-ten 9 137 © Joan A. Cotter, Ph.D., 2012
  • 138. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 138 © Joan A. Cotter, Ph.D., 2012
  • 139. “Math” Way of Naming Numbers 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7 139 © Joan A. Cotter, Ph.D., 2012
  • 140. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. 140 © Joan A. Cotter, Ph.D., 2012
  • 141. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. 141 © Joan A. Cotter, Ph.D., 2012
  • 142. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. 142 © Joan A. Cotter, Ph.D., 2012
  • 143. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. 143 © Joan A. Cotter, Ph.D., 2012
  • 144. “Math” Way of Naming Numbers 100 Chinese Average Highest Number Counted U.S. 90 Korean formal [math way] Korean informal [not explicit] 80 70 60 50 40 30 20 10 0 4 5 6 Ages (yrs.) Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. 144 © Joan A. Cotter, Ph.D., 2012
  • 145. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) 145 © Joan A. Cotter, Ph.D., 2012
  • 146. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. 146 © Joan A. Cotter, Ph.D., 2012
  • 147. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. 147 © Joan A. Cotter, Ph.D., 2012
  • 148. Math Way of Naming Numbers • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo-European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. • Mathematics is the science of patterns. The patterned math way of counting greatly helps children learn number sense. 148 © Joan A. Cotter, Ph.D., 2012
  • 149. Math Way of Naming Numbers Compared to reading: 149 © Joan A. Cotter, Ph.D., 2012
  • 150. Math Way of Naming Numbers Compared to reading: • Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic. 150 © Joan A. Cotter, Ph.D., 2012
  • 151. Math Way of Naming Numbers Compared to reading: • Just as reciting the alphabet doesn’t teach reading, counting doesn’t teach arithmetic. • Just as we first teach the sound of the letters, we must first teach the name of the quantity (math way). 151 © Joan A. Cotter, Ph.D., 2012
  • 152. 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 152 © Joan A. Cotter, Ph.D., 2012
  • 153. Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens. © Joan A. Cotter, Ph.D., 2012
  • 154. Math Way of Naming Numbers Traditional names 4-ten = forty The “ty” means tens. © Joan A. Cotter, Ph.D., 2012
  • 155. Math Way of Naming Numbers Traditional names 6-ten = sixty The “ty” means tens. © Joan A. Cotter, Ph.D., 2012
  • 156. Math Way of Naming Numbers Traditional names 3-ten = thirty “Thir” also used in 1/3, 13 and 30. © Joan A. Cotter, Ph.D., 2012
  • 157. Math Way of Naming Numbers Traditional names 5-ten = fifty “Fif” also used in 1/5, 15 and 50. © Joan A. Cotter, Ph.D., 2012
  • 158. Math Way of Naming Numbers Traditional names 2-ten = twenty Two used to be pronounced “twoo.” © Joan A. Cotter, Ph.D., 2012
  • 159. Math Way of Naming Numbers Traditional names A word game fireplace place-fire © Joan A. Cotter, Ph.D., 2012
  • 160. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news © Joan A. Cotter, Ph.D., 2012
  • 161. Math Way of Naming Numbers Traditional names A word game fireplace place-fire newspaper paper-news box-mail mailbox © Joan A. Cotter, Ph.D., 2012
  • 162. Math Way of Naming Numbers Traditional names ten 4 “Teen” also means ten. © Joan A. Cotter, Ph.D., 2012
  • 163. Math Way of Naming Numbers Traditional names ten 4 teen 4 “Teen” also means ten. © Joan A. Cotter, Ph.D., 2012
  • 164. Math Way of Naming Numbers Traditional names ten 4 teen 4 fourtee n “Teen” also means ten. © Joan A. Cotter, Ph.D., 2012
  • 165. Math Way of Naming Numbers Traditional names a one left © Joan A. Cotter, Ph.D., 2012
  • 166. Math Way of Naming Numbers Traditional names a one left a left-one © Joan A. Cotter, Ph.D., 2012
  • 167. Math Way of Naming Numbers Traditional names a one left a left-one eleven © Joan A. Cotter, Ph.D., 2012
  • 168. Math Way of Naming Numbers Traditional names two left Two pronounced “twoo.” © Joan A. Cotter, Ph.D., 2012
  • 169. Math Way of Naming Numbers Traditional names two left twelve Two pronounced “twoo.” © Joan A. Cotter, Ph.D., 2012
  • 170. Composing Numbers 3-ten © Joan A. Cotter, Ph.D., 2012
  • 171. Composing Numbers 3-ten © Joan A. Cotter, Ph.D., 2012
  • 172. Composing Numbers 3-ten © Joan A. Cotter, Ph.D., 2012
  • 173. Composing Numbers 3-ten 30 © Joan A. Cotter, Ph.D., 2012
  • 174. Composing Numbers 3-ten 30 © Joan A. Cotter, Ph.D., 2012
  • 175. Composing Numbers 3-ten 30 © Joan A. Cotter, Ph.D., 2012
  • 176. Composing Numbers 3-ten 7 30 © Joan A. Cotter, Ph.D., 2012
  • 177. Composing Numbers 3-ten 7 30 © Joan A. Cotter, Ph.D., 2012
  • 178. Composing Numbers 3-ten 7 30 7 © Joan A. Cotter, Ph.D., 2012
  • 179. Composing Numbers 3-ten 7 30 7 © Joan A. Cotter, Ph.D., 2012
  • 180. Composing Numbers 3-ten 7 30 7 Notice the way we say the number, represent the number, and write the number all correspond. © Joan A. Cotter, Ph.D., 2012
  • 181. Composing Numbers 7-ten 70 Another example. © Joan A. Cotter, Ph.D., 2012
  • 182. Composing Numbers 7-ten 8 70 © Joan A. Cotter, Ph.D., 2012
  • 183. Composing Numbers 7-ten 8 70 © Joan A. Cotter, Ph.D., 2012
  • 184. Composing Numbers 7-ten 8 70 8 © Joan A. Cotter, Ph.D., 2012
  • 185. Composing Numbers 7-ten 8 78 © Joan A. Cotter, Ph.D., 2012
  • 186. Composing Numbers 10-ten © Joan A. Cotter, Ph.D., 2012
  • 187. Composing Numbers 10-ten 100 © Joan A. Cotter, Ph.D., 2012
  • 188. Composing Numbers 10-ten 100 © Joan A. Cotter, Ph.D., 2012
  • 189. Composing Numbers 10-ten 100 © Joan A. Cotter, Ph.D., 2012
  • 190. Composing Numbers 1 hundred © Joan A. Cotter, Ph.D., 2012
  • 191. Composing Numbers 1 hundred 100 © Joan A. Cotter, Ph.D., 2012
  • 192. Composing Numbers 1 hundred 100 © Joan A. Cotter, Ph.D., 2012
  • 193. Composing Numbers 1 hundred 100 © Joan A. Cotter, Ph.D., 2012
  • 194. Composing Numbers 1 hundred 100 © Joan A. Cotter, Ph.D., 2012
  • 195. Composing Numbers 2 hundred © Joan A. Cotter, Ph.D., 2012
  • 196. Composing Numbers 2 hundred © Joan A. Cotter, Ph.D., 2012
  • 197. Composing Numbers 2 hundred 200 © Joan A. Cotter, Ph.D., 2012
  • 198. Counting by 2s and 5s © Joan A. Cotter, Ph.D., 2012
  • 199. Counting by 2s and 5s Counting by 2s © Joan A. Cotter, Ph.D., 2012
  • 200. Counting by 2s and 5s Counting by 2s 2 © Joan A. Cotter, Ph.D., 2012
  • 201. Counting by 2s and 5s Counting by 2s 2 4 © Joan A. Cotter, Ph.D., 2012
  • 202. Counting by 2s and 5s Counting by 2s 2 4 6 © Joan A. Cotter, Ph.D., 2012
  • 203. Counting by 2s and 5s Counting by 2s 2 4 6 8 © Joan A. Cotter, Ph.D., 2012
  • 204. Counting by 2s and 5s Counting by 2s 2 4 6 8 10 © Joan A. Cotter, Ph.D., 2012
  • 205. Counting by 2s and 5s Counting by 2s 2 4 6 8 10 12 © Joan A. Cotter, Ph.D., 2012
  • 206. Counting by 2s and 5s Counting by 2s 2 4 6 8 10 12 14 © Joan A. Cotter, Ph.D., 2012
  • 207. Counting by 2s and 5s Counting by 2s 2 4 6 8 10 12 14 16 © Joan A. Cotter, Ph.D., 2012
  • 208. Counting by 2s and 5s Counting by 2s 2 4 6 8 10 12 14 16 18 © Joan A. Cotter, Ph.D., 2012
  • 209. Counting by 2s and 5s Counting by 2s 2 4 6 8 10 12 14 16 18 20 © Joan A. Cotter, Ph.D., 2012
  • 210. Counting by 2s and 5s Counting by 5s © Joan A. Cotter, Ph.D., 2012
  • 211. Counting by 2s and 5s Counting by 5s 5 © Joan A. Cotter, Ph.D., 2012
  • 212. Counting by 2s and 5s Counting by 5s 5 10 © Joan A. Cotter, Ph.D., 2012
  • 213. Counting by 2s and 5s Counting by 5s 5 10 15 © Joan A. Cotter, Ph.D., 2012
  • 214. Counting by 2s and 5s Counting by 5s 5 10 15 20 © Joan A. Cotter, Ph.D., 2012
  • 215. Counting by 2s and 5s Counting by 5s 5 10 15 20 25 © Joan A. Cotter, Ph.D., 2012
  • 216. Counting by 2s and 5s Counting by 5s 5 10 15 20 25 30 © Joan A. Cotter, Ph.D., 2012
  • 217. Evens and Odds Evens © Joan A. Cotter, Ph.D., 2012
  • 218. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  • 219. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  • 220. Evens and Odds Evens Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  • 221. Evens and Odds Evens Use two fingers and touch each pair in succession. EVEN! © Joan A. Cotter, Ph.D., 2012
  • 222. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  • 223. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  • 224. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  • 225. Evens and Odds Odds Use two fingers and touch each pair in succession. © Joan A. Cotter, Ph.D., 2012
  • 226. Evens and Odds Odds Use two fingers and touch each pair in succession. ODD! © Joan A. Cotter, Ph.D., 2012
  • 227. Fact Strategies 227 © Joan A. Cotter, Ph.D., 2012
  • 228. Fact Strategies Complete the Ten 9+5= © Joan A. Cotter, Ph.D., 2012
  • 229. Fact Strategies Complete the Ten 9+5= © Joan A. Cotter, Ph.D., 2012
  • 230. Fact Strategies Complete the Ten 9+5= © Joan A. Cotter, Ph.D., 2012
  • 231. Fact Strategies Complete the Ten 9+5= Take 1 from the 5 and give it to the 9. © Joan A. Cotter, Ph.D., 2012
  • 232. Fact Strategies Complete the Ten 9+5= Take 1 from the 5 and give it to the 9. © Joan A. Cotter, Ph.D., 2012
  • 233. Fact Strategies Complete the Ten 9+5= Take 1 from the 5 and give it to the 9. © Joan A. Cotter, Ph.D., 2012
  • 234. Fact Strategies Complete the Ten 9 + 5 = 14 Take 1 from the 5 and give it to the 9. © Joan A. Cotter, Ph.D., 2012
  • 235. Fact Strategies Two Fives 8+6= © Joan A. Cotter, Ph.D., 2012
  • 236. Fact Strategies Two Fives 8+6= © Joan A. Cotter, Ph.D., 2012
  • 237. Fact Strategies Two Fives 8+6= © Joan A. Cotter, Ph.D., 2012
  • 238. Fact Strategies Two Fives 8+6= © Joan A. Cotter, Ph.D., 2012
  • 239. Fact Strategies Two Fives 8+6= 10 + 4 = 14 © Joan A. Cotter, Ph.D., 2012
  • 240. Fact Strategies Going Down 15 – 9 = © Joan A. Cotter, Ph.D., 2012
  • 241. Fact Strategies Going Down 15 – 9 = © Joan A. Cotter, Ph.D., 2012
  • 242. Fact Strategies Going Down 15 – 9 = Subtract 5; then 4. © Joan A. Cotter, Ph.D., 2012
  • 243. Fact Strategies Going Down 15 – 9 = Subtract 5; then 4. © Joan A. Cotter, Ph.D., 2012
  • 244. Fact Strategies Going Down 15 – 9 = Subtract 5; then 4. © Joan A. Cotter, Ph.D., 2012
  • 245. Fact Strategies Going Down 15 – 9 = 6 Subtract 5; then 4. © Joan A. Cotter, Ph.D., 2012
  • 246. Fact Strategies Subtract from 10 15 – 9 = © Joan A. Cotter, Ph.D., 2012
  • 247. Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10. © Joan A. Cotter, Ph.D., 2012
  • 248. Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10. © Joan A. Cotter, Ph.D., 2012
  • 249. Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10. © Joan A. Cotter, Ph.D., 2012
  • 250. Fact Strategies Subtract from 10 15 – 9 = 6 Subtract 9 from 10. © Joan A. Cotter, Ph.D., 2012
  • 251. Fact Strategies Going Up 15 – 9 = © Joan A. Cotter, Ph.D., 2012
  • 252. Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15. © Joan A. Cotter, Ph.D., 2012
  • 253. Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15. © Joan A. Cotter, Ph.D., 2012
  • 254. Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15. © Joan A. Cotter, Ph.D., 2012
  • 255. Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15. © Joan A. Cotter, Ph.D., 2012
  • 256. Fact Strategies Going Up 15 – 9 = 1+5=6 Start with 9; go up to 15. © Joan A. Cotter, Ph.D., 2012
  • 257. Fact Strategies Multiplication 6× 4= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
  • 258. Fact Strategies Multiplication 6× 4= (6 taken 4 times) © Joan A. Cotter, Ph.D., 2012
  • 259. Place Value Two aspects © Joan A. Cotter, Ph.D., 2012
  • 260. Place Value Two aspects Static © Joan A. Cotter, Ph.D., 2012
  • 261. Place Value Two aspects Static • Value of a digit is determined by position © Joan A. Cotter, Ph.D., 2012
  • 262. Place Value Two aspects Static • Value of a digit is determined by position. • No position may have more than nine. © Joan A. Cotter, Ph.D., 2012
  • 263. Place Value Two aspects Static • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position. © Joan A. Cotter, Ph.D., 2012
  • 264. Place Value Two aspects Static • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position. • Place value cards show this aspect. © Joan A. Cotter, Ph.D., 2012
  • 265. Place Value Two aspects Static • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position. • Place value cards show this aspect. Dynamic © Joan A. Cotter, Ph.D., 2012
  • 266. Place Value Two aspects Static • Value of a digit is determined by position. • No position may have more than nine. • As you progress to the left, value at each position is ten times greater than previous position. • Place value cards show this aspect. Dynamic • Ten ones = 1 ten; ten tens = 1 hundred; ten hundreds = 1 thousand, …. © Joan A. Cotter, Ph.D., 2012
  • 267. Trading 1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • 268. Trading Thousands 1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • 269. Trading Hundreds 1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • 270. Trading Tens 1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • 271. Trading Ones 1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • 272. Trading Adding 1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • 273. Trading Adding 1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • 274. Trading Adding 1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • 275. Trading Adding 1000 100 10 1 8 +6 © Joan A. Cotter, Ph.D., 2012
  • 276. Trading Adding 1000 100 10 1 8 +6 14 © Joan A. Cotter, Ph.D., 2012
  • 277. Trading Adding 1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  • 278. Trading Adding 1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  • 279. Trading Adding 1000 100 10 1 8 +6 14 Too many ones; trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  • 280. Trading Adding 1000 100 10 1 8 +6 14 Same answer before and after trading. © Joan A. Cotter, Ph.D., 2012
  • 281. Trading Bead Trading Activity 1000 100 10 1 © Joan A. Cotter, Ph.D., 2012
  • 282. Trading Bead Trading Activity 1000 100 10 1 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
  • 283. Trading Bead Trading Activity 1000 100 10 1 7 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
  • 284. Trading Bead Trading Activity 1000 100 10 1 7 Object: To get a high score by adding numbers on the green cards. © Joan A. Cotter, Ph.D., 2012
  • 285. Trading Bead Trading Activity 1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  • 286. Trading Bead Trading Activity 1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  • 287. Trading Bead Trading Activity 1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  • 288. Trading Bead Trading Activity 1000 100 10 1 6 Trade 10 ones for 1 ten. © Joan A. Cotter, Ph.D., 2012
  • 289. Trading Bead Trading Activity 1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  • 290. Trading Bead Trading Activity 1000 100 10 1 6 © Joan A. Cotter, Ph.D., 2012
  • 291. Trading Bead Trading Activity 1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
  • 292. Trading Bead Trading Activity 1000 100 10 1 9 © Joan A. Cotter, Ph.D., 2012
  • 293. Trading Bead Trading Activity 1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
  • 294. Trading Bead Trading Activity 1000 100 10 1 9 Another trade. © Joan A. Cotter, Ph.D., 2012
  • 295. Trading Bead Trading Activity 1000 100 10 1 3 © Joan A. Cotter, Ph.D., 2012
  • 296. Trading Bead Trading Activity 1000 100 10 1 3 © Joan A. Cotter, Ph.D., 2012
  • 297. Trading Bead Trading Activity • In the Bead Trading activity trading 10 ones for 1 ten occurs frequently; © Joan A. Cotter, Ph.D., 2012
  • 298. Trading Bead Trading Activity • In the Bead Trading activity trading 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; © Joan A. Cotter, Ph.D., 2012
  • 299. Trading Bead Trading Activity • In the Bead Trading activity trading 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely. © Joan A. Cotter, Ph.D., 2012
  • 300. Trading Bead Trading Activity • In the Bead Trading activity trading 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely. • Bead trading helps the child experience the greater value of each column from left to right. © Joan A. Cotter, Ph.D., 2012
  • 301. Trading Bead Trading Activity • In the Bead Trading activity trading 10 ones for 1 ten occurs frequently; 10 tens for 1 hundred, less often; 10 hundreds for 1 thousand, rarely. • Bead trading helps the child experience the greater value of each column from left to right. • To detect a pattern, there must be at least three examples in the sequence. Place value is a pattern. © Joan A. Cotter, Ph.D., 2012
  • 302. Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 © Joan A. Cotter, Ph.D., 2012
  • 303. Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • 304. Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • 305. Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • 306. Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • 307. Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • 308. Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Enter the first number from left to right. © Joan A. Cotter, Ph.D., 2012
  • 309. Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 310. Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 311. Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 312. Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 313. Trading Adding 4-digit numbers 1000 100 10 1 3658 + 2738 6 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 314. Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 6 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 315. Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 6 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 316. Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 6 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 317. Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 318. Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 319. Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 320. Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 321. Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 96 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 322. Trading Adding 4-digit numbers 1000 100 10 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 323. Trading Adding 4-digit numbers 1000 100 10 1 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 324. Trading Adding 4-digit numbers 1000 100 10 1 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 325. Trading Adding 4-digit numbers 1000 100 10 1 1 1 3658 + 2738 396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 326. Trading Adding 4-digit numbers 1000 100 10 1 1 1 3658 + 2738 6396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 327. Trading Adding 4-digit numbers 1000 100 10 1 1 1 3658 + 2738 6396 Add starting at the right. Write results after each step. © Joan A. Cotter, Ph.D., 2012
  • 328. Minnesota Standards Number Sense K: Represent quantities using whole numbers and understand relationships among whole numbers. 1–2: Understand place value and relationships among whole numbers. With this alternate model, how difficult are the associated benchmarks for children to master? 328 © Joan A. Cotter, Ph.D., 2012
  • 329. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. 329 © Joan A. Cotter, Ph.D., 2012
  • 330. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. 330 © Joan A. Cotter, Ph.D., 2012
  • 331. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. 331 © Joan A. Cotter, Ph.D., 2012
  • 332. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. 332 © Joan A. Cotter, Ph.D., 2012
  • 333. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting. 333 © Joan A. Cotter, Ph.D., 2012
  • 334. Minnesota Standards Kindergarten Represent quantities using whole numbers and understand relationships among whole numbers. • Count forward to 31, backward from 10. • Count number of objects and identify the quantity. • Compare the number of objects in two or more sets. • Given a number, identify one more or one less. • Recognize number of objects up to 6, without counting. • Add and subtract whole numbers up to 6, using objects. 334 © Joan A. Cotter, Ph.D., 2012
  • 335. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. 335 © Joan A. Cotter, Ph.D., 2012
  • 336. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. 336 © Joan A. Cotter, Ph.D., 2012
  • 337. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. 337 © Joan A. Cotter, Ph.D., 2012
  • 338. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12. 338 © Joan A. Cotter, Ph.D., 2012
  • 339. Minnesota Standards Grade 1 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 120. • Count by 2s to 30 and by 5s to 120. • Count backwards from 30. • Demonstrate understanding of odd and even to 12. • Represent whole numbers up to 20 in various ways. 339 © Joan A. Cotter, Ph.D., 2012
  • 340. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. 340 © Joan A. Cotter, Ph.D., 2012
  • 341. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. 341 © Joan A. Cotter, Ph.D., 2012
  • 342. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. 342 © Joan A. Cotter, Ph.D., 2012
  • 343. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12. 343 © Joan A. Cotter, Ph.D., 2012
  • 344. Minnesota Standards Grade 2 Understand place value and relationships among whole numbers. • Read, write, compare and order numbers to 999. • Count by 2s, 5s, 10s from any given whole number. • Understand the significance of groups of ten. • Demonstrate understanding of odd and even up to 12. • Represent whole numbers up to 20 in various ways. 344 © Joan A. Cotter, Ph.D., 2012
  • 345. Research Highlights 345 © Joan A. Cotter, Ph.D., 2012
  • 346. Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. 346 © Joan A. Cotter, Ph.D., 2012
  • 347. Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. 347 © Joan A. Cotter, Ph.D., 2012
  • 348. Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. 348 © Joan A. Cotter, Ph.D., 2012
  • 349. Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. 349 © Joan A. Cotter, Ph.D., 2012
  • 350. Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 350 © Joan A. Cotter, Ph.D., 2012
  • 351. Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 351 © Joan A. Cotter, Ph.D., 2012
  • 352. Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 352 © Joan A. Cotter, Ph.D., 2012
  • 353. Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 353 © Joan A. Cotter, Ph.D., 2012
  • 354. Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 354 © Joan A. Cotter, Ph.D., 2012
  • 355. Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 355 © Joan A. Cotter, Ph.D., 2012
  • 356. Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 356 © Joan A. Cotter, Ph.D., 2012
  • 357. Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children thinking of 14 as 14 ones counted 14. 357 © Joan A. Cotter, Ph.D., 2012
  • 358. Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones. 358 © Joan A. Cotter, Ph.D., 2012
  • 359. Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones. 359 © Joan A. Cotter, Ph.D., 2012
  • 360. Research Highlights Research task: Using 10s and 1s, ask the child to construct 48. Then ask the child to subtract 14. Children who understand tens remove a ten and 4 ones. 360 © Joan A. Cotter, Ph.D., 2012
  • 361. Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% © Joan A. Cotter, Ph.D., 2012
  • 362. Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% Teens 10 + 3 94% 47% © Joan A. Cotter, Ph.D., 2012
  • 363. Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% Teens 10 + 3 94% 47% 6 + 10 88% 33% © Joan A. Cotter, Ph.D., 2012
  • 364. Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% Teens 10 + 3 94% 47% 6 + 10 88% 33% Circle Tens 78 75% 67% Place © Joan A. Cotter, Ph.D., 2012
  • 365. Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% Teens 10 + 3 94% 47% 6 + 10 88% 33% Circle Tens 78 75% 67% Place 3924 44% 7% © Joan A. Cotter, Ph.D., 2012
  • 366. Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% Teens 10 + 3 94% 47% 6 + 10 88% 33% Circle Tens 78 75% 67% Place 3924 44% 7% Mental 85 – 70 31% 0% Computation © Joan A. Cotter, Ph.D., 2012
  • 367. Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% Teens 10 + 3 94% 47% 6 + 10 88% 33% Circle Tens 78 75% 67% Place 3924 44% 7% Mental 85 – 70 31% 0% Computation 2nd Graders in US (Reys): 9% © Joan A. Cotter, Ph.D., 2012
  • 368. Research Highlights TASK EXPER CTRL 14 as 10 & 4 48 – 14 81% 33% Teens 10 + 3 94% 47% 6 + 10 88% 33% Circle Tens 78 75% 67% Place 3924 44% 7% Mental 85 – 70 31% 0% Computation 2nd Graders in US (Reys): 9% 38 + 24 = 512 or 0% 40% 57 + 35 = 812 © Joan A. Cotter, Ph.D., 2012
  • 369. Framing the Future of Mathematics in Minnesota Place value, not counting, is the key to under- standing numbers. Place value is best taught by: • Subitizing (with groups of fives), • Initially using Math Way of number naming, • Incorporating Place Value Cards, • Patterning (trading with 4-digit numbers). © Joan A. Cotter, Ph.D., 2012
  • 370. The Future of Primary Math: More Understanding/Less Counting by Joan A. Cotter, Ph.D. JoanCotter@RightStartMath.com MCTM Saturday, May 5, 2012 Duluth, Minnesota 1000 100 10 1 30 7 30 7 PowerPoint Presentation RightStartMath.com >Resources © Joan A. Cotter, Ph.D., 2012

Editor's Notes

  • #59: Show the baby 2 bears.
  • #60: Show the baby 2 bears.
  • #61: Show the baby 2 bears.
  • #62: Show the baby 2 bears.
  • #117: Stairs