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Learning Disabilities Mass HOPE April 2013

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This document summarizes a presentation on teaching math to children with special needs. It discusses the characteristics of children with learning disabilities, myths about learning disabilities, problems occurring in math like dyscalculia, and effective teaching strategies like teaching for understanding versus rules and procedures. It also covers topics like memorization, flash cards, counting strategies, and visualizing mathematics concepts.

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Teaching Math to Children with Special Needs MassHOPE - TEACH Worcester, MA Saturday, April 27 9:45am — 10:45am Joan A. Cotter, Ph.D. JoanCotter@RightStartMath.com
Characteristics of LD Child usually:
Characteristics of LD Child usually: • Is often more creative.
Characteristics of LD Child usually: • Is often more creative. • Cannot learn by rote.
Characteristics of LD Child usually: • Is often more creative. • Cannot learn by rote. • Must understand and make sense of concept in order to remember.
Characteristics of LD Child usually: • Is often more creative. • Cannot learn by rote. • Must understand and make sense of concept in order to remember. • Is more visual and often hands-on.
Characteristics of LD Child usually: • Is often more creative. • Cannot learn by rote. • Must understand and make sense of concept in order to remember. • Is more visual and often hands-on. • Dislikes worksheets.
Myths about LD
Myths about LD 1. People with LD have lower intelligence. (33% are gifted.)
Myths about LD 1. People with LD have lower intelligence. (33% are gifted.) 2. They are lazy or stubborn.
Myths about LD 1. People with LD have lower intelligence. (33% are gifted.) 2. They are lazy or stubborn. 3. Children with LD can be cured or will outgrow it.
Myths about LD 1. People with LD have lower intelligence. (33% are gifted.) 2. They are lazy or stubborn. 3. Children with LD can be cured or will outgrow it. 4. Boys are more likely to be affected.
Myths about LD 1. People with LD have lower intelligence. (33% are gifted.) 2. They are lazy or stubborn. 3. Children with LD can be cured or will outgrow it. 4. Boys are more likely to be affected. 5. Dyslexia and learning disability are the same thing.
Myths about LD 1. People with LD have lower intelligence. (33% are gifted.) 2. They are lazy or stubborn. 3. Children with LD can be cured or will outgrow it. 4. Boys are more likely to be affected. 5. Dyslexia and learning disability are the same thing. 6. Adults with LD and ADD cannot succeed in higher education.
Problems Occurring in Math (Dyscalculia)
Problems Occurring in Math (Dyscalculia) • Reversals in writing numbers
Problems Occurring in Math (Dyscalculia) • Reversals in writing numbers • Poor number sense
Problems Occurring in Math (Dyscalculia) • Reversals in writing numbers • Poor number sense • Slow fact retrieval
Problems Occurring in Math (Dyscalculia) • Reversals in writing numbers • Poor number sense • Slow fact retrieval • Errors in computation
Problems Occurring in Math (Dyscalculia) • Reversals in writing numbers • Poor number sense • Slow fact retrieval • Errors in computation • Difficulty in solving word problems
Problems Occurring in Math (Dyscalculia) • Reversals in writing numbers • Poor number sense • Slow fact retrieval • Errors in computation • Difficulty in solving word problems Dyscalculia only affects arithmetic, not other branches of math.
Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
Time Needed to Memorize According to a study with college students, it took them
Time Needed to Memorize • 93 minutes to learn 200 nonsense syllables. According to a study with college students, it took them
Time Needed to Memorize • 93 minutes to learn 200 nonsense syllables. • 24 minutes to learn 200 words of prose. According to a study with college students, it took them
Time Needed to Memorize • 93 minutes to learn 200 nonsense syllables. • 24 minutes to learn 200 words of prose. • 10 minutes to learn 200 words of poetry. According to a study with college students, it took them
Memorizing Math Percentage Recall Immediatel y After 1 day After 4 wks. Rote 32 23 8 Concept 69 69 58
Memorizing Math Percentage Recall Immediatel y After 1 day After 4 wks. Rote 32 23 8 Concept 69 69 58
Memorizing Math Percentage Recall Immediatel y After 1 day After 4 wks. Rote 32 23 8 Concept 69 69 58
Memorizing Math Percentage Recall Immediatel y After 1 day After 4 wks. Rote 32 23 8 Concept 69 69 58
Memorizing Math Math needs to be taught so 95% is understood and only 5% memorized. Richard Skemp Percentage Recall Immediatel y After 1 day After 4 wks. Rote 32 23 8 Concept 69 69 58
Flash Cards & Times Tests
Flash Cards & Times Tests • Often used to teach rote.
Flash Cards & Times Tests • Often used to teach rote. • Liked only by those who don‟t need them.
Flash Cards & Times Tests • Often used to teach rote. • Liked only by those who don‟t need them. • Give the false impression that math isn‟t about thinking.
Flash Cards & Times Tests • Often used to teach rote. • Liked only by those who don‟t need them. • Give the false impression that math isn‟t about thinking. • Often produce stress – children under stress stop learning.
Flash Cards & Times Tests • Often used to teach rote. • Liked only by those who don‟t need them. • Give the false impression that math isn‟t about thinking. • Often produce stress – children under stress stop learning. • Not concrete – use abstract symbols.
Flash Cards & Times Tests • Often used to teach rote. • Liked only by those who don‟t need them. • Give the false impression that math isn‟t about thinking. • Often produce stress – children under stress stop learning. • Not concrete – use abstract symbols. • Cause stress (may become physically ill).
Flash Cards & Times Tests • Often used to teach rote. • Liked only by those who don‟t need them. • Give the false impression that math isn‟t about thinking. • Often produce stress – children under stress stop learning. • Not concrete – use abstract symbols. • Cause stress (may become physically ill).
Flash Cards & Times Tests • Often used to teach rote. • Liked only by those who don‟t need them. • Give the false impression that math isn‟t about thinking. • Often produce stress – children under stress stop learning. • Not concrete – use abstract symbols. • Cause stress (may become physically ill).
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective
© Joan A. Cotter, Ph.D., 2013 Traditional Counting 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
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective F + E =
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective A F + E =
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective A B F + E =
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective A CB F + E =
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective A FC D EB F + E =
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective AA FC D EB F + E =
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective A BA FC D EB F + E =
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective A C D EBA FC D EB F + E =
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective A C D EBA FC D EB What is the sum? (It must be a letter.) F + E =
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective G I J KHA FC D EB F + E = K
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective Now memorize the facts!! G + D
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective Now memorize the facts!! G + D
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective Now memorize the facts!! G + D
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective Now memorize the facts!! G + D
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective Now memorize the facts!! G + D
© 2010 Joan A. Traditional Counting From a child's perspective Try subtracting by „taking away‟ H – E
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective Try skip counting by B's to T: B, D, . . . , T.
© Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective Try skip counting by B's to T: B, D, . . . , T. What is D x E?
Adding on a Number Line D + C =
Adding on a Number Line D + C = * A B C D E F G H I J K L M
Adding on a Number Line D + C = * A B C D E F G H I J K L M
Adding on a Number Line D + C = * A B C D E F G H I J K L M We‟re counting spaces, not lines.
Number Line Drawbacks
Number Line Drawbacks • Quantity of a number not obvious.
Number Line Drawbacks • Quantity of a number not obvious. • Requires counting in order to compute.
Number Line Drawbacks • Quantity of a number not obvious. • Requires counting in order to compute. • Ignores place value.
Number Line Drawbacks • Quantity of a number not obvious. • Requires counting in order to compute. • Ignores place value. • Hard to visualize.
Dominoes Most domino patterns cannot be added. 3 + 1 ≠ 4
Domino Pattern Drawbacks
Domino Pattern Drawbacks • Patterns generally cannot be added.
Domino Pattern Drawbacks • Patterns generally cannot be added. • They are not used outside of games.
Domino Pattern Drawbacks • Patterns generally cannot be added. • They are not used outside of games. • Emphasize only 1–6.
Domino Pattern Drawbacks • Patterns generally cannot be added. • They are not used outside of games. • Emphasize only 1–6. • Precious little mathematics.
© Joan A. Cotter, Ph.D., 2013 Visualizing Japanese criteria for manipulatives
© Joan A. Cotter, Ph.D., 2013 Visualizing • Visual is related to seeing.
© Joan A. Cotter, Ph.D., 2013 Visualizing • Visual is related to seeing. • Visualize is to form a mental image.
© Joan A. Cotter, Ph.D., 2013 Visualizing Try to visualize 8 identical apples without grouping.
© Joan A. Cotter, Ph.D., 2013 Visualizing Try to visualize 8 identical apples without grouping.
© Joan A. Cotter, Ph.D., 2013 Visualizing Now try to visualize 8 apples: 5 red and 3 green.
© Joan A. Cotter, Ph.D., 2013 Visualizing Now try to visualize 8 apples: 5 red and 3 green.
© Joan A. Cotter, Ph.D., 2013 Visualizing Can you visualize this rod?
© Joan A. Cotter, Ph.D., 2013 Grouping in Fives
© Joan A. Cotter, Ph.D., 2013 Grouping in Fives I II III IIII V VIII 1 2 3 4 5 8 Early Roman numerals
© Joan A. Cotter, Ph.D., 2013 Grouping in Fives Musical staff
© Joan A. Cotter, Ph.D., 2013 Clocks and nickels Grouping in Fives
© Joan A. Cotter, Ph.D., 2013 Grouping in Fives Clocks and nickels
© Joan A. Cotter, Ph.D., 2013 Grouping in Fives Tally marks
© Joan A. Cotter, Ph.D., 2011 Grouping in Fives Subitizing • Instant recognition of quantity is called subitizing.
© Joan A. Cotter, Ph.D., 2011 Grouping in Fives Subitizing • Instant recognition of quantity is called subitizing. • Grouping in fives extends subitizing beyond five.
© Joan A. Cotter, Ph.D., 2011 Subitizing • Five-month-old infants can subitize to 1–3.
© Joan A. Cotter, Ph.D., 2011 Subitizing • Three-year-olds can subitize to 1– 5. • Five-month-old infants can subitize to 1–3.
© Joan A. Cotter, Ph.D., 2011 Subitizing • Three-year-olds can subitize to 1– 5. • Four-year-olds can subitize 1–10 by grouping with five. • Five-month-old infants can subitize to 1–3.
© Joan A. Cotter, Ph.D., 2011 Subitizing • Three-year-olds can subitize to 1– 5. • Four-year-olds can subitize 1–10 by grouping with five. • Five-month-old infants can subitize to 1–3. • Counting is analogous to sounding out a word; subitizing, to recognizing the word.
© Joan A. Cotter, Ph.D., 2011 Research on Subitizing
© Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
© Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
© Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
© Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
© Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
© Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
© Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
© Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
© Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research You could say subitizing is much more “natural” than counting.
© Joan A. Cotter, Ph.D., 2011 Learning 1–10 Using fingers
© Joan A. Cotter, Ph.D., 2011 Learning 1–10 Using fingers
© Joan A. Cotter, Ph.D., 2011 Learning 1–10 Using fingers
© Joan A. Cotter, Ph.D., 2011 Learning 1–10 Using fingers
© Joan A. Cotter, Ph.D., 2011 Learning 1–10 Using fingers
© Joan A. Cotter, Ph.D., 2011 Learning 1–10 Using fingers
© Joan A. Cotter, Ph.D., 2011 Learning 1–10 Subitizing 5
© Joan A. Cotter, Ph.D., 2011 Learning 1–10 Subitizing 5
© Joan A. Cotter, Ph.D., 2011 Learning 1–10 Subitizing 5 5 has a middle; 4 does not.
© Joan A. Cotter, Ph.D., 2011 Learning 1–10 Tally sticks
© Joan A. Cotter, Ph.D., 2011 Learning 1–10 Tally sticks
© Joan A. Cotter, Ph.D., 2011 Learning 1–10 Tally sticks
© Joan A. Cotter, Ph.D., 2011 Learning 1–10 Tally sticks Five as a group.
© Joan A. Cotter, Ph.D., 2011 Learning 1–10 Tally sticks
© Joan A. Cotter, Ph.D., 2011 Learning 1–10 Tally sticks
© Joan A. Cotter, Ph.D., 2013 Learning 1–10 Entering quantities
© Joan A. Cotter, Ph.D., 2013 3 Learning 1–10 Entering quantities
© Joan A. Cotter, Ph.D., 2013 5 Learning 1–10 Entering quantities
© Joan A. Cotter, Ph.D., 2013 7 Learning 1–10 Entering quantities
© Joan A. Cotter, Ph.D., 2013 Learning 1–10 10 Entering quantities
© Joan A. Cotter, Ph.D., 2013 Learning 1–10 The stairs
© Joan A. Cotter, Ph.D., 2013 Learning 1–10 Adding
© Joan A. Cotter, Ph.D., 2013 Learning 1–10 4 + 3 = Adding
© Joan A. Cotter, Ph.D., 2013 Learning 1–10 4 + 3 = Adding
© Joan A. Cotter, Ph.D., 2013 Learning 1–10 4 + 3 = Adding
© Joan A. Cotter, Ph.D., 2013 Learning 1–10 4 + 3 = Adding
© Joan A. Cotter, Ph.D., 2013 Learning 1–10 4 + 3 = 7 Adding
© Joan A. Cotter, Ph.D., 2013 Learning 1–10 4 + 3 = 7 Adding Japanese children learn to do this mentally.
© Joan A. Cotter, Ph.D., 2012 Games
© Joan A. Cotter, Ph.D., 2012 Games Games Math
© Joan A. Cotter, Ph.D., 2012 Games Games Math =
© Joan A. Cotter, Ph.D., 2012 Games Games Math Books Reading =
© Joan A. Cotter, Ph.D., 2012 Games Games Math Books Reading Games provide interesting repetition needed for automatic responses. =
© Joan A. Cotter, Ph.D., 2012 Games Games Math Books Reading Games provide interesting repetition needed for automatic responses. More importantly, games provide an application for the new information! =
© Joan A. Cotter, Ph.D., 2012 Go to the Dump
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Objective: To learn and master the facts that total 10:
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Objective: To learn and master the facts that total 10: 1 + 9 2 + 8 3 + 7 4 + 6 5 + 5
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Objective: To learn and master the facts that total 10: 1 + 9 2 + 8 3 + 7 4 + 6 5 + 5
© Joan A. Cotter, Ph.D., 2012 Go to the Dump Objective: To learn and master the facts that total 10: 1 + 9 2 + 8 3 + 7 4 + 6 5 + 5 It is played similar to Go Fish.
© Joan A. Cotter, Ph.D., 2013 Writing Numbers 61 72 83 94 105 Sandpaper numbers also help. Number Chart
© Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming
© Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1
© Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2
© Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3
© Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4
© Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9
© Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten
© Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1
© Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2
© Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3
© Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3 . . . . . . . . 99 = 9-ten 9
© Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 137 = 1 hundred 3-ten 7
© Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo- European languages are non-standard in number naming.)
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo- European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting.
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo- European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade.
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming • Only 11 words are needed to count to 100 the math way, 28 in English. (All Indo- European languages are non-standard in number naming.) • Asian children learn mathematics using the math way of counting. • They understand place value in first grade; only half of U.S. children understand place value at the end of fourth grade. • Mathematics is the science of patterns. The patterned math way of counting greatly helps children learn number sense.
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Compared to reading
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming • Just as reciting the alphabet doesn‟t teach reading, counting doesn‟t teach arithmetic. Compared to reading
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming • 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). Compared to reading
© Joan A. Cotter, Ph.D., 2013 Place-Value Cards 3 0 3 0 0 0 3 th- ou-sand 3 hun-dred 3- ten 3 0 0
© Joan A. Cotter, Ph.D., 2013 Place-Value Cards 5 0 8 4 0 0 0 2 0 0
© Joan A. Cotter, Ph.D., 2013 Place-Value Cards 5 0 8 4 0 0 0 4 0 0 0 2 0 0 5 0 82 0 0
© Joan A. Cotter, Ph.D., 2013 Place-Value Cards 5 0 8 4 0 0 0 4 0 0 0 2 0 0 4 0 0 02 0 05 08 5 0 82 0 0
© Joan A. Cotter, Ph.D., 2013 Place-Value Cards 3 0 0 0 8
© Joan A. Cotter, Ph.D., 2013 Place-Value Cards 3 0 0 0 3 0 0 0 8 8
© Joan A. Cotter, Ph.D., 2013 Place-Value Cards 3 0 0 0 3 0 0 3 0 00 08 8 8
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names 4-ten = forty The “ty” means tens.
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names 4-ten = forty The “ty” means tens.
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names 6-ten = sixty The “ty” means tens.
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names 3-ten = thirty “Thir” also used in 1/3, 13 and 30.
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names 5-ten = fifty “Fif” also used in 1/5, 15 and 50.
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names 2-ten = twenty Two used to be pronounced “twoo.”
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names A word game fireplace place-fire
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names A word game fireplace place-fire paper-newsnewspaper
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names A word game fireplace place-fire paper-news box-mail mailbox newspaper
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names ten 4 Prefix -teen means ten.
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names ten 4 teen 4 Prefix -teen means ten.
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names ten 4 teen 4 fourteen Prefix -teen means ten.
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names a one left
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names a one left a left-one
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names a one left a left-one eleven
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names two left Two said as “twoo.”
© Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names two left twelve Two said as “twoo.”
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten 3 0
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten 3 0
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten 3 0
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten 7 3 0
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten 7 3 0
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten 7 3 0 7
© Joan A. Cotter, Ph.D., 2013 3 0 Composing Numbers 3-ten 7 7
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten 7 Note the congruence in how we say the number, represent the number, and write the number. 3 07
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 1-ten 1 0 Another example.
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 1-ten 8 1 0
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 1-ten 8 1 0
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 1-ten 8 1 0 8
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 1-ten 8 1 88
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 10-ten
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 10-ten 1 0 0
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 10-ten 1 0 0
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 10-ten 1 0 0
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 1 hundred
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 1 hundred 1 0 0
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 1 hundred 1 0 0
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 1 hundred 1 01 01 0 0
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 1 hundred 1 0 0
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 2 hundred
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 2 hundred
© Joan A. Cotter, Ph.D., 2013 Composing Numbers 2 hundred 2 0 0
© Joan A. Cotter, Ph.D., 2013 Learning the Facts
© Joan A. Cotter, Ph.D., 2013 Learning the Facts Limited success, especially for struggling children, when learning is:
© Joan A. Cotter, Ph.D., 2013 Learning the Facts • Based on counting: whether dots, fingers, number lines, or counting words. Limited success, especially for struggling children, when learning is:
© Joan A. Cotter, Ph.D., 2013 Learning the Facts • Based on counting: whether dots, fingers, number lines, or counting words. Limited success, especially for struggling children, when learning is: • Based on rote memory: whether flash cards, timed tests, or computer games.
© Joan A. Cotter, Ph.D., 2013 Learning the Facts • Based on counting: whether dots, fingers, number lines, or counting words. Limited success, especially for struggling children, when learning is: • Based on rote memory: whether flash cards, timed tests, or computer games. • Based on skip counting: whether fingers or songs
© Joan A. Cotter, Ph.D., 2013 Fact Strategies
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Complete the Ten 9 + 5 =
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Complete the Ten 9 + 5 =
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Complete the Ten 9 + 5 =
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Complete the Ten 9 + 5 = 14 Take 1 from the 5 and give it to the 9.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Two Fives 8 + 6 =
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Two Fives 8 + 6 =
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Two Fives 8 + 6 =
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Two Fives 8 + 6 =
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Two Fives 8 + 6 = 10 + 4 = 14
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Down 15 – 9 =
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Down 15 – 9 =
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Down 15 – 9 = 6 Subtract 5; then 4.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Subtract from 10 15 – 9 =
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Subtract from 10 15 – 9 = 6 Subtract 9 from 10.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Up 15 – 9 =
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15.
© Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Up 15 – 9 = 1 + 5 = 6 Start with 9; go up to 15.
© Joan A. Cotter, Ph.D., 2013 Money Penny
© Joan A. Cotter, Ph.D., 2013 Money Nickel
© Joan A. Cotter, Ph.D., 2013 Money Dime
© Joan A. Cotter, Ph.D., 2013 Money Quarter
© Joan A. Cotter, Ph.D., 2013 Money Quarter
© Joan A. Cotter, Ph.D., 2013 Money Quarter
© Joan A. Cotter, Ph.D., 2013 Money Quarter
© Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 6 x 4 = (6 taken 4 times)
© Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 6 x 4 = (6 taken 4 times)
© Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 6 x 4 = (6 taken 4 times)
© Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 6 x 4 = (6 taken 4 times)
© Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 6 x 4 = (6 taken 4 times)
© Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 9 x 3 =
© Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 9 x 3 =
© Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 9 x 3 = 30
© Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 9 x 3 = 30 – 3 = 27
© Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 7 x 7 =
© Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 7 x 7 =
© Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 7 x 7 = 25 +
© Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 7 x 7 = 25 + 10 + 10
© Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 7 x 7 = 25 + 10 + 10 + 4 = 49
© Joan A. Cotter, Ph.D., 2013 Trading 1000 10 1100
© Joan A. Cotter, Ph.D., 2013 Trading Thousands 1000 10 1100
© Joan A. Cotter, Ph.D., 2013 Trading Hundreds 1000 10 1100
© Joan A. Cotter, Ph.D., 2013 Trading Tens 1000 10 1100
© Joan A. Cotter, Ph.D., 2013 Trading Ones 1000 10 1100
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 8 + 6
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 8 + 6
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 8 + 6
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 8 + 6
© Joan A. Cotter, Ph.D., 2013 Trading Adding 8 + 6 14 1000 10 1100
© Joan A. Cotter, Ph.D., 2013 Trading Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. 1000 10 1100
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten.
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten.
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 8 + 6 14 Same answer before and after trading.
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right.
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Enter numbers from left to right.
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Enter numbers from left to right.
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Enter numbers from left to right.
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Enter numbers from left to right.
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Enter numbers from left to right.
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step.
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 1
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 1 1
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 1 1
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 1 1
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 6396 Add starting at the right. Write results after each step. 1 1
© Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 6396 Add starting at the right. Write results after each step. 1 1
© Joan A. Cotter, Ph.D., 2013 Part-Whole Circles
© Joan A. Cotter, Ph.D., 2013 Part-Whole Circles Whole Part Part
© Joan A. Cotter, Ph.D., 2013 Part-Whole Circles What is the whole? 3 2
© Joan A. Cotter, Ph.D., 2013 Part-Whole Circles What is the whole? 3 5 2
© Joan A. Cotter, Ph.D., 2013 Part-Whole Circles What is the other part?10 7
© Joan A. Cotter, Ph.D., 2013 Part-Whole Circles What is the other part? 3 10 7
© Joan A. Cotter, Ph.D., 2011 Short Division
© Joan A. Cotter, Ph.D., 2011 Short Division • Means we don‟t write the stuff underneath.
© Joan A. Cotter, Ph.D., 2011 Short Division • Means we don‟t write the stuff underneath. • Should be used for single-digit divisors.
© Joan A. Cotter, Ph.D., 2011 Short Division • Means we don‟t write the stuff underneath. • Should be used for single-digit divisors. • Is easier to understand than long division.
© Joan A. Cotter, Ph.D., 2011 Short Division • Means we don‟t write the stuff underneath. • Should be used for single-digit divisors. • Is easier to understand than long division. • Best taught before long division.
© Joan A. Cotter, Ph.D., 2011 Short Division • Means we don‟t write the stuff underneath. • Should be used for single-digit divisors. • Is easier to understand than long division. • Best taught before long division. • Is much more useful in real life.
© Joan A. Cotter, Ph.D., 2011 Short Division • Means we don‟t write the stuff underneath. • Should be used for single-digit divisors. • Is easier to understand than long division. • Best taught before long division. • Is much more useful in real life.
© Joan A. Cotter, Ph.D., 2011 Short Division 3 4 7 1)
© Joan A. Cotter, Ph.D., 2011 Short Division ) The little lines help to keep track of place value. 3 4 7 1
© Joan A. Cotter, Ph.D., 2011 Short Division ) 400 ÷ 3 = ? [100] Write the 1 on the line. 3 4 7 1
© Joan A. Cotter, Ph.D., 2011 Short Division 1 ) 400 ÷ 3 = ? [100] Write the 1 on the line. 3 4 7 1
© Joan A. Cotter, Ph.D., 2011 Short Division 1 ) 400 ÷ 3 = ? [100] Write the 1 on the line. What is the remainder? [100] How many tens is that? [10] 3 4 7 1
© Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division 1 400 ÷ 3 = ? [100] Write the 1 on the line. What is the remainder? [100] How many tens is that? [10] How many tens do we have? [10 + 7 = 17] )
© Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division Let‟s show the 17 tens by writing a 1 before the 7. 1 1 )
© Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division Let‟s show the 17 tens by writing a 1 before the 7. Divide the tens: 17 tens ÷ 3 = ? [5] 1 1 )
© Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division Let‟s show the 17 tens by writing a 1 before the 7. Divide the tens: 17 tens ÷ 3 = ? [5] 1 ) 1 5
© Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division To find the remainder go up. 3 x 5 = 15; how far is 15 from 17. [2] 1 5 1 )
© Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division To find the remainder go up. 3 x 5 = 15; how far is 15 from 17. [2] How many ones do we have? [20 + 1] 1 5 1 )
© Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division To find the remainder go up. 3 x 5 = 15; how far is 15 from 17. [2] How many ones do we have? [20 + 1] 1 5 1 )
© Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division To find the remainder go up. 3 x 5 = 15; how far is 15 from 17. [2] How many ones do we have? [20 + 1] 1 5 1 ) 2
© Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division 21 ÷ 3 = ? [7] 1 5 21 )
© Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division ) 1 5 7 21 21 ÷ 3 = ? [7]
© Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division ) 1 5 7 21
© Joan A. Cotter, Ph.D., 2011 Short Division Another example
© Joan A. Cotter, Ph.D., 2011 Short Division )9 8 0 5 3
© Joan A. Cotter, Ph.D., 2011 Short Division )9 8 0 5 3 80 hundred ÷ 9 = ? [8 hundred]
© Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division 8 )
© Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division 8 8 )
© Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division 8 ) 8
© Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division ) 8 8 9
© Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division 4 ) 8 8 9
© Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division 4 ) 8 8 9
© Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division 8 9 4 ) 48
© Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division ) 8 9 4 r7 48
© Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division 8 9 4 ) r7 48
Teaching Math
Teaching Math • Talk positively about math.
Teaching Math • Talk positively about math. • Teach for understanding; concrete models should lead to mental models.
Teaching Math • Talk positively about math. • Teach for understanding; concrete models should lead to mental models. • Encourage children to ask questions. “I don‟t get it” is not a question.
Teaching Math • Talk positively about math. • Teach for understanding; concrete models should lead to mental models. • Encourage children to ask questions. “I don‟t get it” is not a question. • Use correct vocabulary.
Teaching Math • Talk positively about math. • Teach for understanding; concrete models should lead to mental models. • Encourage children to ask questions. “I don‟t get it” is not a question. • Use correct vocabulary. • Ask a question once; allow 3 seconds for a response. Resist rephrasing.
Teaching Math • Talk positively about math. • Teach for understanding; concrete models should lead to mental models. • Encourage children to ask questions. “I don‟t get it” is not a question. • Use correct vocabulary. • Ask a question once; allow 3 seconds for a response. Resist rephrasing. • Give a child 2-3 seconds to say a fact.
Teaching Math to Children with Special Needs MassHOPE - TEACH Worcester, MA Saturday, April 27 9:45am — 10:45am Joan A. Cotter, Ph.D. JoanCotter@RightStartMath.com

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Learning Disabilities Mass HOPE April 2013

  • 1. Teaching Math to Children with Special Needs MassHOPE - TEACH Worcester, MA Saturday, April 27 9:45am — 10:45am Joan A. Cotter, Ph.D. JoanCotter@RightStartMath.com
  • 3. Characteristics of LD Child usually: • Is often more creative.
  • 4. Characteristics of LD Child usually: • Is often more creative. • Cannot learn by rote.
  • 5. Characteristics of LD Child usually: • Is often more creative. • Cannot learn by rote. • Must understand and make sense of concept in order to remember.
  • 6. Characteristics of LD Child usually: • Is often more creative. • Cannot learn by rote. • Must understand and make sense of concept in order to remember. • Is more visual and often hands-on.
  • 7. Characteristics of LD Child usually: • Is often more creative. • Cannot learn by rote. • Must understand and make sense of concept in order to remember. • Is more visual and often hands-on. • Dislikes worksheets.
  • 9. Myths about LD 1. People with LD have lower intelligence. (33% are gifted.)
  • 10. Myths about LD 1. People with LD have lower intelligence. (33% are gifted.) 2. They are lazy or stubborn.
  • 11. Myths about LD 1. People with LD have lower intelligence. (33% are gifted.) 2. They are lazy or stubborn. 3. Children with LD can be cured or will outgrow it.
  • 12. Myths about LD 1. People with LD have lower intelligence. (33% are gifted.) 2. They are lazy or stubborn. 3. Children with LD can be cured or will outgrow it. 4. Boys are more likely to be affected.
  • 13. Myths about LD 1. People with LD have lower intelligence. (33% are gifted.) 2. They are lazy or stubborn. 3. Children with LD can be cured or will outgrow it. 4. Boys are more likely to be affected. 5. Dyslexia and learning disability are the same thing.
  • 14. Myths about LD 1. People with LD have lower intelligence. (33% are gifted.) 2. They are lazy or stubborn. 3. Children with LD can be cured or will outgrow it. 4. Boys are more likely to be affected. 5. Dyslexia and learning disability are the same thing. 6. Adults with LD and ADD cannot succeed in higher education.
  • 15. Problems Occurring in Math (Dyscalculia)
  • 16. Problems Occurring in Math (Dyscalculia) • Reversals in writing numbers
  • 17. Problems Occurring in Math (Dyscalculia) • Reversals in writing numbers • Poor number sense
  • 18. Problems Occurring in Math (Dyscalculia) • Reversals in writing numbers • Poor number sense • Slow fact retrieval
  • 19. Problems Occurring in Math (Dyscalculia) • Reversals in writing numbers • Poor number sense • Slow fact retrieval • Errors in computation
  • 20. Problems Occurring in Math (Dyscalculia) • Reversals in writing numbers • Poor number sense • Slow fact retrieval • Errors in computation • Difficulty in solving word problems
  • 21. Problems Occurring in Math (Dyscalculia) • Reversals in writing numbers • Poor number sense • Slow fact retrieval • Errors in computation • Difficulty in solving word problems Dyscalculia only affects arithmetic, not other branches of math.
  • 22. Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
  • 23. Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
  • 24. Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
  • 25. Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
  • 26. Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
  • 27. Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
  • 28. Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
  • 29. Teaching for Understanding Emphasizes understanding Teaches a few generalizations Identifies global relationships Broad application Takes longer to learn, but is retained more easily Difficult to teach Difficult to test Teaching Rules and Procedures Emphasizes recall Teaches many rules Identifies sequential steps Limited context Is learned more quickly, but is quickly forgotten Is easy to teachCalifornia Mathematics Framework, 1985, p. 13
  • 30. Time Needed to Memorize According to a study with college students, it took them
  • 31. Time Needed to Memorize • 93 minutes to learn 200 nonsense syllables. According to a study with college students, it took them
  • 32. Time Needed to Memorize • 93 minutes to learn 200 nonsense syllables. • 24 minutes to learn 200 words of prose. According to a study with college students, it took them
  • 33. Time Needed to Memorize • 93 minutes to learn 200 nonsense syllables. • 24 minutes to learn 200 words of prose. • 10 minutes to learn 200 words of poetry. According to a study with college students, it took them
  • 34. Memorizing Math Percentage Recall Immediatel y After 1 day After 4 wks. Rote 32 23 8 Concept 69 69 58
  • 35. Memorizing Math Percentage Recall Immediatel y After 1 day After 4 wks. Rote 32 23 8 Concept 69 69 58
  • 36. Memorizing Math Percentage Recall Immediatel y After 1 day After 4 wks. Rote 32 23 8 Concept 69 69 58
  • 37. Memorizing Math Percentage Recall Immediatel y After 1 day After 4 wks. Rote 32 23 8 Concept 69 69 58
  • 38. Memorizing Math Math needs to be taught so 95% is understood and only 5% memorized. Richard Skemp Percentage Recall Immediatel y After 1 day After 4 wks. Rote 32 23 8 Concept 69 69 58
  • 39. Flash Cards & Times Tests
  • 40. Flash Cards & Times Tests • Often used to teach rote.
  • 41. Flash Cards & Times Tests • Often used to teach rote. • Liked only by those who don‟t need them.
  • 42. Flash Cards & Times Tests • Often used to teach rote. • Liked only by those who don‟t need them. • Give the false impression that math isn‟t about thinking.
  • 43. Flash Cards & Times Tests • Often used to teach rote. • Liked only by those who don‟t need them. • Give the false impression that math isn‟t about thinking. • Often produce stress – children under stress stop learning.
  • 44. Flash Cards & Times Tests • Often used to teach rote. • Liked only by those who don‟t need them. • Give the false impression that math isn‟t about thinking. • Often produce stress – children under stress stop learning. • Not concrete – use abstract symbols.
  • 45. Flash Cards & Times Tests • Often used to teach rote. • Liked only by those who don‟t need them. • Give the false impression that math isn‟t about thinking. • Often produce stress – children under stress stop learning. • Not concrete – use abstract symbols. • Cause stress (may become physically ill).
  • 46. Flash Cards & Times Tests • Often used to teach rote. • Liked only by those who don‟t need them. • Give the false impression that math isn‟t about thinking. • Often produce stress – children under stress stop learning. • Not concrete – use abstract symbols. • Cause stress (may become physically ill).
  • 47. Flash Cards & Times Tests • Often used to teach rote. • Liked only by those who don‟t need them. • Give the false impression that math isn‟t about thinking. • Often produce stress – children under stress stop learning. • Not concrete – use abstract symbols. • Cause stress (may become physically ill).
  • 48. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective
  • 49. © Joan A. Cotter, Ph.D., 2013 Traditional Counting 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
  • 50. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective F + E =
  • 51. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective A F + E =
  • 52. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective A B F + E =
  • 53. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective A CB F + E =
  • 54. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective A FC D EB F + E =
  • 55. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective AA FC D EB F + E =
  • 56. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective A BA FC D EB F + E =
  • 57. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective A C D EBA FC D EB F + E =
  • 58. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective A C D EBA FC D EB What is the sum? (It must be a letter.) F + E =
  • 59. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective G I J KHA FC D EB F + E = K
  • 60. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective Now memorize the facts!! G + D
  • 61. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective Now memorize the facts!! G + D
  • 62. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective Now memorize the facts!! G + D
  • 63. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective Now memorize the facts!! G + D
  • 64. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective Now memorize the facts!! G + D
  • 65. © 2010 Joan A. Traditional Counting From a child's perspective Try subtracting by „taking away‟ H – E
  • 66. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective Try skip counting by B's to T: B, D, . . . , T.
  • 67. © Joan A. Cotter, Ph.D., 2013 Traditional Counting From a child's perspective Try skip counting by B's to T: B, D, . . . , T. What is D x E?
  • 68. Adding on a Number Line D + C =
  • 69. Adding on a Number Line D + C = * A B C D E F G H I J K L M
  • 70. Adding on a Number Line D + C = * A B C D E F G H I J K L M
  • 71. Adding on a Number Line D + C = * A B C D E F G H I J K L M We‟re counting spaces, not lines.
  • 73. Number Line Drawbacks • Quantity of a number not obvious.
  • 74. Number Line Drawbacks • Quantity of a number not obvious. • Requires counting in order to compute.
  • 75. Number Line Drawbacks • Quantity of a number not obvious. • Requires counting in order to compute. • Ignores place value.
  • 76. Number Line Drawbacks • Quantity of a number not obvious. • Requires counting in order to compute. • Ignores place value. • Hard to visualize.
  • 77. Dominoes Most domino patterns cannot be added. 3 + 1 ≠ 4
  • 79. Domino Pattern Drawbacks • Patterns generally cannot be added.
  • 80. Domino Pattern Drawbacks • Patterns generally cannot be added. • They are not used outside of games.
  • 81. Domino Pattern Drawbacks • Patterns generally cannot be added. • They are not used outside of games. • Emphasize only 1–6.
  • 82. Domino Pattern Drawbacks • Patterns generally cannot be added. • They are not used outside of games. • Emphasize only 1–6. • Precious little mathematics.
  • 83. © Joan A. Cotter, Ph.D., 2013 Visualizing Japanese criteria for manipulatives
  • 84. © Joan A. Cotter, Ph.D., 2013 Visualizing • Visual is related to seeing.
  • 85. © Joan A. Cotter, Ph.D., 2013 Visualizing • Visual is related to seeing. • Visualize is to form a mental image.
  • 86. © Joan A. Cotter, Ph.D., 2013 Visualizing Try to visualize 8 identical apples without grouping.
  • 87. © Joan A. Cotter, Ph.D., 2013 Visualizing Try to visualize 8 identical apples without grouping.
  • 88. © Joan A. Cotter, Ph.D., 2013 Visualizing Now try to visualize 8 apples: 5 red and 3 green.
  • 89. © Joan A. Cotter, Ph.D., 2013 Visualizing Now try to visualize 8 apples: 5 red and 3 green.
  • 90. © Joan A. Cotter, Ph.D., 2013 Visualizing Can you visualize this rod?
  • 91. © Joan A. Cotter, Ph.D., 2013 Grouping in Fives
  • 92. © Joan A. Cotter, Ph.D., 2013 Grouping in Fives I II III IIII V VIII 1 2 3 4 5 8 Early Roman numerals
  • 93. © Joan A. Cotter, Ph.D., 2013 Grouping in Fives Musical staff
  • 94. © Joan A. Cotter, Ph.D., 2013 Clocks and nickels Grouping in Fives
  • 95. © Joan A. Cotter, Ph.D., 2013 Grouping in Fives Clocks and nickels
  • 96. © Joan A. Cotter, Ph.D., 2013 Grouping in Fives Tally marks
  • 97. © Joan A. Cotter, Ph.D., 2011 Grouping in Fives Subitizing • Instant recognition of quantity is called subitizing.
  • 98. © Joan A. Cotter, Ph.D., 2011 Grouping in Fives Subitizing • Instant recognition of quantity is called subitizing. • Grouping in fives extends subitizing beyond five.
  • 99. © Joan A. Cotter, Ph.D., 2011 Subitizing • Five-month-old infants can subitize to 1–3.
  • 100. © Joan A. Cotter, Ph.D., 2011 Subitizing • Three-year-olds can subitize to 1– 5. • Five-month-old infants can subitize to 1–3.
  • 101. © Joan A. Cotter, Ph.D., 2011 Subitizing • Three-year-olds can subitize to 1– 5. • Four-year-olds can subitize 1–10 by grouping with five. • Five-month-old infants can subitize to 1–3.
  • 102. © Joan A. Cotter, Ph.D., 2011 Subitizing • Three-year-olds can subitize to 1– 5. • Four-year-olds can subitize 1–10 by grouping with five. • Five-month-old infants can subitize to 1–3. • Counting is analogous to sounding out a word; subitizing, to recognizing the word.
  • 103. © Joan A. Cotter, Ph.D., 2011 Research on Subitizing
  • 104. © Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
  • 105. © Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
  • 106. © Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
  • 107. © Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
  • 108. © Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
  • 109. © Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
  • 110. © Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
  • 111. © Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research
  • 112. © Joan A. Cotter, Ph.D., 2011 Research on Subitizing Karen Wynn's research You could say subitizing is much more “natural” than counting.
  • 113. © Joan A. Cotter, Ph.D., 2011 Learning 1–10 Using fingers
  • 114. © Joan A. Cotter, Ph.D., 2011 Learning 1–10 Using fingers
  • 115. © Joan A. Cotter, Ph.D., 2011 Learning 1–10 Using fingers
  • 116. © Joan A. Cotter, Ph.D., 2011 Learning 1–10 Using fingers
  • 117. © Joan A. Cotter, Ph.D., 2011 Learning 1–10 Using fingers
  • 118. © Joan A. Cotter, Ph.D., 2011 Learning 1–10 Using fingers
  • 119. © Joan A. Cotter, Ph.D., 2011 Learning 1–10 Subitizing 5
  • 120. © Joan A. Cotter, Ph.D., 2011 Learning 1–10 Subitizing 5
  • 121. © Joan A. Cotter, Ph.D., 2011 Learning 1–10 Subitizing 5 5 has a middle; 4 does not.
  • 122. © Joan A. Cotter, Ph.D., 2011 Learning 1–10 Tally sticks
  • 123. © Joan A. Cotter, Ph.D., 2011 Learning 1–10 Tally sticks
  • 124. © Joan A. Cotter, Ph.D., 2011 Learning 1–10 Tally sticks
  • 125. © Joan A. Cotter, Ph.D., 2011 Learning 1–10 Tally sticks Five as a group.
  • 126. © Joan A. Cotter, Ph.D., 2011 Learning 1–10 Tally sticks
  • 127. © Joan A. Cotter, Ph.D., 2011 Learning 1–10 Tally sticks
  • 128. © Joan A. Cotter, Ph.D., 2013 Learning 1–10 Entering quantities
  • 129. © Joan A. Cotter, Ph.D., 2013 3 Learning 1–10 Entering quantities
  • 130. © Joan A. Cotter, Ph.D., 2013 5 Learning 1–10 Entering quantities
  • 131. © Joan A. Cotter, Ph.D., 2013 7 Learning 1–10 Entering quantities
  • 132. © Joan A. Cotter, Ph.D., 2013 Learning 1–10 10 Entering quantities
  • 133. © Joan A. Cotter, Ph.D., 2013 Learning 1–10 The stairs
  • 134. © Joan A. Cotter, Ph.D., 2013 Learning 1–10 Adding
  • 135. © Joan A. Cotter, Ph.D., 2013 Learning 1–10 4 + 3 = Adding
  • 136. © Joan A. Cotter, Ph.D., 2013 Learning 1–10 4 + 3 = Adding
  • 137. © Joan A. Cotter, Ph.D., 2013 Learning 1–10 4 + 3 = Adding
  • 138. © Joan A. Cotter, Ph.D., 2013 Learning 1–10 4 + 3 = Adding
  • 139. © Joan A. Cotter, Ph.D., 2013 Learning 1–10 4 + 3 = 7 Adding
  • 140. © Joan A. Cotter, Ph.D., 2013 Learning 1–10 4 + 3 = 7 Adding Japanese children learn to do this mentally.
  • 141. © Joan A. Cotter, Ph.D., 2012 Games
  • 142. © Joan A. Cotter, Ph.D., 2012 Games Games Math
  • 143. © Joan A. Cotter, Ph.D., 2012 Games Games Math =
  • 144. © Joan A. Cotter, Ph.D., 2012 Games Games Math Books Reading =
  • 145. © Joan A. Cotter, Ph.D., 2012 Games Games Math Books Reading Games provide interesting repetition needed for automatic responses. =
  • 146. © Joan A. Cotter, Ph.D., 2012 Games Games Math Books Reading Games provide interesting repetition needed for automatic responses. More importantly, games provide an application for the new information! =
  • 147. © Joan A. Cotter, Ph.D., 2012 Go to the Dump
  • 148. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Objective: To learn and master the facts that total 10:
  • 149. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Objective: To learn and master the facts that total 10: 1 + 9 2 + 8 3 + 7 4 + 6 5 + 5
  • 150. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Objective: To learn and master the facts that total 10: 1 + 9 2 + 8 3 + 7 4 + 6 5 + 5
  • 151. © Joan A. Cotter, Ph.D., 2012 Go to the Dump Objective: To learn and master the facts that total 10: 1 + 9 2 + 8 3 + 7 4 + 6 5 + 5 It is played similar to Go Fish.
  • 152. © Joan A. Cotter, Ph.D., 2013 Writing Numbers 61 72 83 94 105 Sandpaper numbers also help. Number Chart
  • 153. © Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming
  • 154. © Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1
  • 155. © Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2
  • 156. © Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3
  • 157. © Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4
  • 158. © Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9
  • 159. © Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten
  • 160. © Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1
  • 161. © Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2
  • 162. © Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3
  • 163. © Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 11 = ten 1 12 = ten 2 13 = ten 3 14 = ten 4 . . . . 19 = ten 9 20 = 2-ten 21 = 2-ten 1 22 = 2-ten 2 23 = 2-ten 3 . . . . . . . . 99 = 9-ten 9
  • 164. © Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 137 = 1 hundred 3-ten 7
  • 165. © Joan A. Cotter, Ph.D., 2013 “Math” Way of Number Naming 137 = 1 hundred 3-ten 7 or 137 = 1 hundred and 3-ten 7
  • 166. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming • 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.)
  • 167. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming • 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.
  • 168. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming • 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.
  • 169. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming • 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.
  • 170. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Compared to reading
  • 171. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming • Just as reciting the alphabet doesn‟t teach reading, counting doesn‟t teach arithmetic. Compared to reading
  • 172. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming • 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). Compared to reading
  • 173. © Joan A. Cotter, Ph.D., 2013 Place-Value Cards 3 0 3 0 0 0 3 th- ou-sand 3 hun-dred 3- ten 3 0 0
  • 174. © Joan A. Cotter, Ph.D., 2013 Place-Value Cards 5 0 8 4 0 0 0 2 0 0
  • 175. © Joan A. Cotter, Ph.D., 2013 Place-Value Cards 5 0 8 4 0 0 0 4 0 0 0 2 0 0 5 0 82 0 0
  • 176. © Joan A. Cotter, Ph.D., 2013 Place-Value Cards 5 0 8 4 0 0 0 4 0 0 0 2 0 0 4 0 0 02 0 05 08 5 0 82 0 0
  • 177. © Joan A. Cotter, Ph.D., 2013 Place-Value Cards 3 0 0 0 8
  • 178. © Joan A. Cotter, Ph.D., 2013 Place-Value Cards 3 0 0 0 3 0 0 0 8 8
  • 179. © Joan A. Cotter, Ph.D., 2013 Place-Value Cards 3 0 0 0 3 0 0 3 0 00 08 8 8
  • 180. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names 4-ten = forty The “ty” means tens.
  • 181. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names 4-ten = forty The “ty” means tens.
  • 182. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names 6-ten = sixty The “ty” means tens.
  • 183. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names 3-ten = thirty “Thir” also used in 1/3, 13 and 30.
  • 184. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names 5-ten = fifty “Fif” also used in 1/5, 15 and 50.
  • 185. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names 2-ten = twenty Two used to be pronounced “twoo.”
  • 186. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names A word game fireplace place-fire
  • 187. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names A word game fireplace place-fire paper-newsnewspaper
  • 188. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names A word game fireplace place-fire paper-news box-mail mailbox newspaper
  • 189. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names ten 4 Prefix -teen means ten.
  • 190. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names ten 4 teen 4 Prefix -teen means ten.
  • 191. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names ten 4 teen 4 fourteen Prefix -teen means ten.
  • 192. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names a one left
  • 193. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names a one left a left-one
  • 194. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names a one left a left-one eleven
  • 195. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names two left Two said as “twoo.”
  • 196. © Joan A. Cotter, Ph.D., 2013 Math Way of Number Naming Regular names two left twelve Two said as “twoo.”
  • 197. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten
  • 198. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten
  • 199. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten 3 0
  • 200. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten 3 0
  • 201. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten 3 0
  • 202. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten 7 3 0
  • 203. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten 7 3 0
  • 204. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten 7 3 0 7
  • 205. © Joan A. Cotter, Ph.D., 2013 3 0 Composing Numbers 3-ten 7 7
  • 206. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 3-ten 7 Note the congruence in how we say the number, represent the number, and write the number. 3 07
  • 207. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 1-ten 1 0 Another example.
  • 208. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 1-ten 8 1 0
  • 209. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 1-ten 8 1 0
  • 210. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 1-ten 8 1 0 8
  • 211. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 1-ten 8 1 88
  • 212. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 10-ten
  • 213. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 10-ten 1 0 0
  • 214. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 10-ten 1 0 0
  • 215. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 10-ten 1 0 0
  • 216. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 1 hundred
  • 217. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 1 hundred 1 0 0
  • 218. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 1 hundred 1 0 0
  • 219. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 1 hundred 1 01 01 0 0
  • 220. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 1 hundred 1 0 0
  • 221. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 2 hundred
  • 222. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 2 hundred
  • 223. © Joan A. Cotter, Ph.D., 2013 Composing Numbers 2 hundred 2 0 0
  • 224. © Joan A. Cotter, Ph.D., 2013 Learning the Facts
  • 225. © Joan A. Cotter, Ph.D., 2013 Learning the Facts Limited success, especially for struggling children, when learning is:
  • 226. © Joan A. Cotter, Ph.D., 2013 Learning the Facts • Based on counting: whether dots, fingers, number lines, or counting words. Limited success, especially for struggling children, when learning is:
  • 227. © Joan A. Cotter, Ph.D., 2013 Learning the Facts • Based on counting: whether dots, fingers, number lines, or counting words. Limited success, especially for struggling children, when learning is: • Based on rote memory: whether flash cards, timed tests, or computer games.
  • 228. © Joan A. Cotter, Ph.D., 2013 Learning the Facts • Based on counting: whether dots, fingers, number lines, or counting words. Limited success, especially for struggling children, when learning is: • Based on rote memory: whether flash cards, timed tests, or computer games. • Based on skip counting: whether fingers or songs
  • 229. © Joan A. Cotter, Ph.D., 2013 Fact Strategies
  • 230. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Complete the Ten 9 + 5 =
  • 231. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Complete the Ten 9 + 5 =
  • 232. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Complete the Ten 9 + 5 =
  • 233. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.
  • 234. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.
  • 235. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Complete the Ten 9 + 5 = Take 1 from the 5 and give it to the 9.
  • 236. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Complete the Ten 9 + 5 = 14 Take 1 from the 5 and give it to the 9.
  • 237. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Two Fives 8 + 6 =
  • 238. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Two Fives 8 + 6 =
  • 239. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Two Fives 8 + 6 =
  • 240. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Two Fives 8 + 6 =
  • 241. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Two Fives 8 + 6 = 10 + 4 = 14
  • 242. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Down 15 – 9 =
  • 243. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Down 15 – 9 =
  • 244. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
  • 245. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
  • 246. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Down 15 – 9 = Subtract 5; then 4.
  • 247. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Down 15 – 9 = 6 Subtract 5; then 4.
  • 248. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Subtract from 10 15 – 9 =
  • 249. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
  • 250. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
  • 251. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Subtract from 10 15 – 9 = Subtract 9 from 10.
  • 252. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Subtract from 10 15 – 9 = 6 Subtract 9 from 10.
  • 253. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Up 15 – 9 =
  • 254. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15.
  • 255. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15.
  • 256. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15.
  • 257. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Up 15 – 9 = Start with 9; go up to 15.
  • 258. © Joan A. Cotter, Ph.D., 2013 Fact Strategies Going Up 15 – 9 = 1 + 5 = 6 Start with 9; go up to 15.
  • 259. © Joan A. Cotter, Ph.D., 2013 Money Penny
  • 260. © Joan A. Cotter, Ph.D., 2013 Money Nickel
  • 261. © Joan A. Cotter, Ph.D., 2013 Money Dime
  • 262. © Joan A. Cotter, Ph.D., 2013 Money Quarter
  • 263. © Joan A. Cotter, Ph.D., 2013 Money Quarter
  • 264. © Joan A. Cotter, Ph.D., 2013 Money Quarter
  • 265. © Joan A. Cotter, Ph.D., 2013 Money Quarter
  • 266. © Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 6 x 4 = (6 taken 4 times)
  • 267. © Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 6 x 4 = (6 taken 4 times)
  • 268. © Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 6 x 4 = (6 taken 4 times)
  • 269. © Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 6 x 4 = (6 taken 4 times)
  • 270. © Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 6 x 4 = (6 taken 4 times)
  • 271. © Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 9 x 3 =
  • 272. © Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 9 x 3 =
  • 273. © Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 9 x 3 = 30
  • 274. © Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 9 x 3 = 30 – 3 = 27
  • 275. © Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 7 x 7 =
  • 276. © Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 7 x 7 =
  • 277. © Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 7 x 7 = 25 +
  • 278. © Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 7 x 7 = 25 + 10 + 10
  • 279. © Joan A. Cotter, Ph.D., 2013 Multiplication Strategies Basic facts 7 x 7 = 25 + 10 + 10 + 4 = 49
  • 280. © Joan A. Cotter, Ph.D., 2013 Trading 1000 10 1100
  • 281. © Joan A. Cotter, Ph.D., 2013 Trading Thousands 1000 10 1100
  • 282. © Joan A. Cotter, Ph.D., 2013 Trading Hundreds 1000 10 1100
  • 283. © Joan A. Cotter, Ph.D., 2013 Trading Tens 1000 10 1100
  • 284. © Joan A. Cotter, Ph.D., 2013 Trading Ones 1000 10 1100
  • 285. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 8 + 6
  • 286. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 8 + 6
  • 287. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 8 + 6
  • 288. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 8 + 6
  • 289. © Joan A. Cotter, Ph.D., 2013 Trading Adding 8 + 6 14 1000 10 1100
  • 290. © Joan A. Cotter, Ph.D., 2013 Trading Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten. 1000 10 1100
  • 291. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten.
  • 292. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 8 + 6 14 Too many ones; trade 10 ones for 1 ten.
  • 293. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 8 + 6 14 Same answer before and after trading.
  • 294. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738
  • 295. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Enter the first number from left to right.
  • 296. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Enter numbers from left to right.
  • 297. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Enter numbers from left to right.
  • 298. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Enter numbers from left to right.
  • 299. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Enter numbers from left to right.
  • 300. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Enter numbers from left to right.
  • 301. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
  • 302. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
  • 303. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
  • 304. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 Add starting at the right. Write results after each step.
  • 305. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step.
  • 306. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1
  • 307. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1
  • 308. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 6 Add starting at the right. Write results after each step. 1
  • 309. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
  • 310. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
  • 311. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
  • 312. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
  • 313. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 96 Add starting at the right. Write results after each step. 1
  • 314. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 1
  • 315. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 1 1
  • 316. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 1 1
  • 317. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 396 Add starting at the right. Write results after each step. 1 1
  • 318. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 6396 Add starting at the right. Write results after each step. 1 1
  • 319. © Joan A. Cotter, Ph.D., 2013 1000 10 1100 Trading Adding 4-digit numbers 3658 + 2738 6396 Add starting at the right. Write results after each step. 1 1
  • 320. © Joan A. Cotter, Ph.D., 2013 Part-Whole Circles
  • 321. © Joan A. Cotter, Ph.D., 2013 Part-Whole Circles Whole Part Part
  • 322. © Joan A. Cotter, Ph.D., 2013 Part-Whole Circles What is the whole? 3 2
  • 323. © Joan A. Cotter, Ph.D., 2013 Part-Whole Circles What is the whole? 3 5 2
  • 324. © Joan A. Cotter, Ph.D., 2013 Part-Whole Circles What is the other part?10 7
  • 325. © Joan A. Cotter, Ph.D., 2013 Part-Whole Circles What is the other part? 3 10 7
  • 326. © Joan A. Cotter, Ph.D., 2011 Short Division
  • 327. © Joan A. Cotter, Ph.D., 2011 Short Division • Means we don‟t write the stuff underneath.
  • 328. © Joan A. Cotter, Ph.D., 2011 Short Division • Means we don‟t write the stuff underneath. • Should be used for single-digit divisors.
  • 329. © Joan A. Cotter, Ph.D., 2011 Short Division • Means we don‟t write the stuff underneath. • Should be used for single-digit divisors. • Is easier to understand than long division.
  • 330. © Joan A. Cotter, Ph.D., 2011 Short Division • Means we don‟t write the stuff underneath. • Should be used for single-digit divisors. • Is easier to understand than long division. • Best taught before long division.
  • 331. © Joan A. Cotter, Ph.D., 2011 Short Division • Means we don‟t write the stuff underneath. • Should be used for single-digit divisors. • Is easier to understand than long division. • Best taught before long division. • Is much more useful in real life.
  • 332. © Joan A. Cotter, Ph.D., 2011 Short Division • Means we don‟t write the stuff underneath. • Should be used for single-digit divisors. • Is easier to understand than long division. • Best taught before long division. • Is much more useful in real life.
  • 333. © Joan A. Cotter, Ph.D., 2011 Short Division 3 4 7 1)
  • 334. © Joan A. Cotter, Ph.D., 2011 Short Division ) The little lines help to keep track of place value. 3 4 7 1
  • 335. © Joan A. Cotter, Ph.D., 2011 Short Division ) 400 ÷ 3 = ? [100] Write the 1 on the line. 3 4 7 1
  • 336. © Joan A. Cotter, Ph.D., 2011 Short Division 1 ) 400 ÷ 3 = ? [100] Write the 1 on the line. 3 4 7 1
  • 337. © Joan A. Cotter, Ph.D., 2011 Short Division 1 ) 400 ÷ 3 = ? [100] Write the 1 on the line. What is the remainder? [100] How many tens is that? [10] 3 4 7 1
  • 338. © Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division 1 400 ÷ 3 = ? [100] Write the 1 on the line. What is the remainder? [100] How many tens is that? [10] How many tens do we have? [10 + 7 = 17] )
  • 339. © Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division Let‟s show the 17 tens by writing a 1 before the 7. 1 1 )
  • 340. © Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division Let‟s show the 17 tens by writing a 1 before the 7. Divide the tens: 17 tens ÷ 3 = ? [5] 1 1 )
  • 341. © Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division Let‟s show the 17 tens by writing a 1 before the 7. Divide the tens: 17 tens ÷ 3 = ? [5] 1 ) 1 5
  • 342. © Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division To find the remainder go up. 3 x 5 = 15; how far is 15 from 17. [2] 1 5 1 )
  • 343. © Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division To find the remainder go up. 3 x 5 = 15; how far is 15 from 17. [2] How many ones do we have? [20 + 1] 1 5 1 )
  • 344. © Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division To find the remainder go up. 3 x 5 = 15; how far is 15 from 17. [2] How many ones do we have? [20 + 1] 1 5 1 )
  • 345. © Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division To find the remainder go up. 3 x 5 = 15; how far is 15 from 17. [2] How many ones do we have? [20 + 1] 1 5 1 ) 2
  • 346. © Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division 21 ÷ 3 = ? [7] 1 5 21 )
  • 347. © Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division ) 1 5 7 21 21 ÷ 3 = ? [7]
  • 348. © Joan A. Cotter, Ph.D., 2011 3 4 7 1 Short Division ) 1 5 7 21
  • 349. © Joan A. Cotter, Ph.D., 2011 Short Division Another example
  • 350. © Joan A. Cotter, Ph.D., 2011 Short Division )9 8 0 5 3
  • 351. © Joan A. Cotter, Ph.D., 2011 Short Division )9 8 0 5 3 80 hundred ÷ 9 = ? [8 hundred]
  • 352. © Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division 8 )
  • 353. © Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division 8 8 )
  • 354. © Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division 8 ) 8
  • 355. © Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division ) 8 8 9
  • 356. © Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division 4 ) 8 8 9
  • 357. © Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division 4 ) 8 8 9
  • 358. © Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division 8 9 4 ) 48
  • 359. © Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division ) 8 9 4 r7 48
  • 360. © Joan A. Cotter, Ph.D., 2011 9 8 0 5 3 Short Division 8 9 4 ) r7 48
  • 362. Teaching Math • Talk positively about math.
  • 363. Teaching Math • Talk positively about math. • Teach for understanding; concrete models should lead to mental models.
  • 364. Teaching Math • Talk positively about math. • Teach for understanding; concrete models should lead to mental models. • Encourage children to ask questions. “I don‟t get it” is not a question.
  • 365. Teaching Math • Talk positively about math. • Teach for understanding; concrete models should lead to mental models. • Encourage children to ask questions. “I don‟t get it” is not a question. • Use correct vocabulary.
  • 366. Teaching Math • Talk positively about math. • Teach for understanding; concrete models should lead to mental models. • Encourage children to ask questions. “I don‟t get it” is not a question. • Use correct vocabulary. • Ask a question once; allow 3 seconds for a response. Resist rephrasing.
  • 367. Teaching Math • Talk positively about math. • Teach for understanding; concrete models should lead to mental models. • Encourage children to ask questions. “I don‟t get it” is not a question. • Use correct vocabulary. • Ask a question once; allow 3 seconds for a response. Resist rephrasing. • Give a child 2-3 seconds to say a fact.
  • 368. Teaching Math to Children with Special Needs MassHOPE - TEACH Worcester, MA Saturday, April 27 9:45am — 10:45am Joan A. Cotter, Ph.D. JoanCotter@RightStartMath.com