Teaching Math to Childrenwith Special NeedsMassHOPE - TEACHWorcester, MASaturday, April 279:45am — 10:45amJoan A. Cotter, ...
Characteristics of LDChild usually:
Characteristics of LDChild usually:• Is often more creative.
Characteristics of LDChild usually:• Is often more creative.• Cannot learn by rote.
Characteristics of LDChild usually:• Is often more creative.• Cannot learn by rote.• Must understand and make sense ofconc...
Characteristics of LDChild usually:• Is often more creative.• Cannot learn by rote.• Must understand and make sense ofconc...
Characteristics of LDChild usually:• Is often more creative.• Cannot learn by rote.• Must understand and make sense ofconc...
Myths about LD
Myths about LD1. People with LD have lower intelligence. (33%are gifted.)
Myths about LD1. People with LD have lower intelligence. (33%are gifted.)2. They are lazy or stubborn.
Myths about LD1. People with LD have lower intelligence. (33%are gifted.)2. They are lazy or stubborn.3. Children with LD ...
Myths about LD1. People with LD have lower intelligence. (33%are gifted.)2. They are lazy or stubborn.3. Children with LD ...
Myths about LD1. People with LD have lower intelligence. (33%are gifted.)2. They are lazy or stubborn.3. Children with LD ...
Myths about LD1. People with LD have lower intelligence. (33%are gifted.)2. They are lazy or stubborn.3. Children with LD ...
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 c...
Problems Occurring in Math(Dyscalculia)• Reversals in writing numbers• Poor number sense• Slow fact retrieval• Errors in c...
Problems Occurring in Math(Dyscalculia)• Reversals in writing numbers• Poor number sense• Slow fact retrieval• Errors in c...
Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicatio...
Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicatio...
Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicatio...
Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicatio...
Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicatio...
Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicatio...
Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicatio...
Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicatio...
Time Needed to MemorizeAccording to a study with college students, it took them
Time Needed to Memorize• 93 minutes to learn 200 nonsensesyllables.According to a study with college students, it took them
Time Needed to Memorize• 93 minutes to learn 200 nonsensesyllables.• 24 minutes to learn 200 words of prose.According to a...
Time Needed to Memorize• 93 minutes to learn 200 nonsensesyllables.• 24 minutes to learn 200 words of prose.• 10 minutes t...
Memorizing MathPercentage RecallImmediatelyAfter 1 day After 4wks.Rote 32 23 8Concept69 69 58
Memorizing MathPercentage RecallImmediatelyAfter 1 day After 4wks.Rote 32 23 8Concept69 69 58
Memorizing MathPercentage RecallImmediatelyAfter 1 day After 4wks.Rote 32 23 8Concept69 69 58
Memorizing MathPercentage RecallImmediatelyAfter 1 day After 4wks.Rote 32 23 8Concept69 69 58
Memorizing MathMath needs to be taught so 95%is understood and only 5%memorized.Richard SkempPercentage RecallImmediatelyA...
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 needthem.
Flash Cards & Times Tests• Often used to teach rote.• Liked only by those who don‟t needthem.• Give the false impression t...
Flash Cards & Times Tests• Often used to teach rote.• Liked only by those who don‟t needthem.• Give the false impression t...
Flash Cards & Times Tests• Often used to teach rote.• Liked only by those who don‟t needthem.• Give the false impression t...
Flash Cards & Times Tests• Often used to teach rote.• Liked only by those who don‟t needthem.• Give the false impression t...
Flash Cards & Times Tests• Often used to teach rote.• Liked only by those who don‟t needthem.• Give the false impression t...
Flash Cards & Times Tests• Often used to teach rote.• Liked only by those who don‟t needthem.• Give the false impression t...
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspective
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveBecause were so familiar with 1, 2, 3, we‟ll use...
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveF + E =
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveAF + E =
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA BF + E =
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA CBF + E =
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA FC D EBF + E =
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveAA FC D EBF + E =
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA BA FC D EBF + E =
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA C D EBA FC D EBF + E =
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA C D EBA FC D EBWhat is the sum?(It must be a l...
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveG I J KHA FC D EBF + E =K
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D
© 2010 Joan A.Traditional CountingFrom a childs perspectiveTry subtractingby „takingaway‟H– E
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveTry skip counting by Bs to T:B, D, . . . , T.
© Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveTry skip counting by Bs to T:B, D, . . . , T.Wha...
Adding on a Number LineD + C =
Adding on a Number LineD + C =* A B C D E F G H I J K L M
Adding on a Number LineD + C =* A B C D E F G H I J K L M
Adding on a Number LineD + C =* A B C D E F G H I J K L MWe‟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.• Ha...
DominoesMost 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.• P...
© Joan A. Cotter, Ph.D., 2013VisualizingJapanese criteria for manipulatives
© Joan A. Cotter, Ph.D., 2013Visualizing• Visual is related to seeing.
© Joan A. Cotter, Ph.D., 2013Visualizing• Visual is related to seeing.• Visualize is to form a mental image.
© Joan A. Cotter, Ph.D., 2013VisualizingTry to visualize 8 identical apples withoutgrouping.
© Joan A. Cotter, Ph.D., 2013VisualizingTry to visualize 8 identical apples withoutgrouping.
© Joan A. Cotter, Ph.D., 2013VisualizingNow try to visualize 8 apples: 5 red and 3 green.
© Joan A. Cotter, Ph.D., 2013VisualizingNow try to visualize 8 apples: 5 red and 3 green.
© Joan A. Cotter, Ph.D., 2013VisualizingCan you visualize this rod?
© Joan A. Cotter, Ph.D., 2013Grouping in Fives
© Joan A. Cotter, Ph.D., 2013Grouping in FivesIIIIIIIIIIVVIII123458Early Roman numerals
© Joan A. Cotter, Ph.D., 2013Grouping in FivesMusical staff
© Joan A. Cotter, Ph.D., 2013Clocks and nickelsGrouping in Fives
© Joan A. Cotter, Ph.D., 2013Grouping in FivesClocks and nickels
© Joan A. Cotter, Ph.D., 2013Grouping in FivesTally marks
© Joan A. Cotter, Ph.D., 2011Grouping in FivesSubitizing• Instant recognition of quantity is called subitizing.
© Joan A. Cotter, Ph.D., 2011Grouping in FivesSubitizing• Instant recognition of quantity is called subitizing.• Grouping ...
© Joan A. Cotter, Ph.D., 2011Subitizing• Five-month-old infants can subitize to 1–3.
© Joan A. Cotter, Ph.D., 2011Subitizing• Three-year-olds can subitize to 1–5.• Five-month-old infants can subitize to 1–3.
© Joan A. Cotter, Ph.D., 2011Subitizing• Three-year-olds can subitize to 1–5.• Four-year-olds can subitize 1–10 bygrouping...
© Joan A. Cotter, Ph.D., 2011Subitizing• Three-year-olds can subitize to 1–5.• Four-year-olds can subitize 1–10 bygrouping...
© Joan A. Cotter, Ph.D., 2011Research on Subitizing
© Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
© Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
© Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
© Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
© Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
© Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
© Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
© Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
© Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns researchYou could say subitizing ismuch more “natural” than...
© Joan A. Cotter, Ph.D., 2011Learning 1–10Using fingers
© Joan A. Cotter, Ph.D., 2011Learning 1–10Using fingers
© Joan A. Cotter, Ph.D., 2011Learning 1–10Using fingers
© Joan A. Cotter, Ph.D., 2011Learning 1–10Using fingers
© Joan A. Cotter, Ph.D., 2011Learning 1–10Using fingers
© Joan A. Cotter, Ph.D., 2011Learning 1–10Using fingers
© Joan A. Cotter, Ph.D., 2011Learning 1–10Subitizing 5
© Joan A. Cotter, Ph.D., 2011Learning 1–10Subitizing 5
© Joan A. Cotter, Ph.D., 2011Learning 1–10Subitizing 55 has a middle; 4 does not.
© Joan A. Cotter, Ph.D., 2011Learning 1–10Tally sticks
© Joan A. Cotter, Ph.D., 2011Learning 1–10Tally sticks
© Joan A. Cotter, Ph.D., 2011Learning 1–10Tally sticks
© Joan A. Cotter, Ph.D., 2011Learning 1–10Tally sticksFive as a group.
© Joan A. Cotter, Ph.D., 2011Learning 1–10Tally sticks
© Joan A. Cotter, Ph.D., 2011Learning 1–10Tally sticks
© Joan A. Cotter, Ph.D., 2013Learning 1–10Entering quantities
© Joan A. Cotter, Ph.D., 20133Learning 1–10Entering quantities
© Joan A. Cotter, Ph.D., 20135Learning 1–10Entering quantities
© Joan A. Cotter, Ph.D., 20137Learning 1–10Entering quantities
© Joan A. Cotter, Ph.D., 2013Learning 1–1010Entering quantities
© Joan A. Cotter, Ph.D., 2013Learning 1–10The stairs
© Joan A. Cotter, Ph.D., 2013Learning 1–10Adding
© Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 =Adding
© Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 =Adding
© Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 =Adding
© Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 =Adding
© Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 = 7Adding
© Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 = 7AddingJapanese children learn to do this mentally.
© Joan A. Cotter, Ph.D., 2012Games
© Joan A. Cotter, Ph.D., 2012GamesGamesMath
© Joan A. Cotter, Ph.D., 2012GamesGamesMath=
© Joan A. Cotter, Ph.D., 2012GamesGamesMathBooksReading=
© Joan A. Cotter, Ph.D., 2012GamesGamesMathBooksReadingGames provide interesting repetition neededfor automatic responses.=
© Joan A. Cotter, Ph.D., 2012GamesGamesMathBooksReadingGames provide interesting repetition neededfor automatic responses....
© Joan A. Cotter, Ph.D., 2012Go to the Dump
© Joan A. Cotter, Ph.D., 2012Go to the DumpObjective: To learn and master the factsthat total 10:
© Joan A. Cotter, Ph.D., 2012Go to the DumpObjective: To learn and master the factsthat total 10:1 + 92 + 83 + 74 + 65 + 5
© Joan A. Cotter, Ph.D., 2012Go to the DumpObjective: To learn and master the factsthat total 10:1 + 92 + 83 + 74 + 65 + 5
© Joan A. Cotter, Ph.D., 2012Go to the DumpObjective: To learn and master the factsthat total 10:1 + 92 + 83 + 74 + 65 + 5...
© Joan A. Cotter, Ph.D., 2013Writing Numbers61728394105Sandpaper 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 Naming11 = ten 1
© Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 2
© Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 3
© Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4
© Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 9
© Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten
© Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-te...
© Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-te...
© Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-te...
© Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-te...
© Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming137 = 1 hundred 3-ten 7
© Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming137 = 1 hundred 3-ten 7or137 = 1 hundred and 3-ten 7
© Joan A. Cotter, Ph.D., 2013Math Way of Number Naming• Only 11 words are needed to count to 100the math way, 28 in Englis...
© Joan A. Cotter, Ph.D., 2013Math Way of Number Naming• Only 11 words are needed to count to 100the math way, 28 in Englis...
© Joan A. Cotter, Ph.D., 2013Math Way of Number Naming• Only 11 words are needed to count to 100the math way, 28 in Englis...
© Joan A. Cotter, Ph.D., 2013Math Way of Number Naming• Only 11 words are needed to count to 100the math way, 28 in Englis...
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingCompared to reading
© Joan A. Cotter, Ph.D., 2013Math Way of Number Naming• Just as reciting the alphabet doesn‟t teachreading, counting doesn...
© Joan A. Cotter, Ph.D., 2013Math Way of Number Naming• Just as reciting the alphabet doesn‟t teachreading, counting doesn...
© Joan A. Cotter, Ph.D., 2013Place-Value Cards3 03 0 0 03 th- ou-sand3 hun-dred3- ten3 0 0
© Joan A. Cotter, Ph.D., 2013Place-Value Cards5 084 0 0 02 0 0
© Joan A. Cotter, Ph.D., 2013Place-Value Cards5 084 0 0 0 4 0 0 02 0 05 082 0 0
© Joan A. Cotter, Ph.D., 2013Place-Value Cards5 084 0 0 0 4 0 0 02 0 04 0 0 02 0 05 085 082 0 0
© Joan A. Cotter, Ph.D., 2013Place-Value Cards3 0 0 08
© Joan A. Cotter, Ph.D., 2013Place-Value Cards3 0 0 0 3 0 0 088
© Joan A. Cotter, Ph.D., 2013Place-Value Cards3 0 0 0 3 0 0 3 0 00 0888
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names4-ten = fortyThe “ty”meanstens.
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names4-ten = fortyThe “ty”meanstens.
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names6-ten = sixtyThe “ty”meanstens.
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names3-ten = thirty“Thir” alsoused in 1/3,13 and 30.
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names5-ten = fifty“Fif” alsoused in 1/5,15 and 50.
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names2-ten = twentyTwo used tobepronounced“twoo.”
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesA word gamefireplace place-fire
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesA word gamefireplace place-firepaper-newsnewspaper
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesA word gamefireplace place-firepaper-newsbox-mail mailb...
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesten 4Prefix -teenmeans ten.
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesten 4 teen 4Prefix -teenmeans ten.
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesten 4 teen 4 fourteenPrefix -teenmeans ten.
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesa one left
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesa one left a left-one
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesa one left a left-one eleven
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namestwo leftTwo saidas“twoo.”
© Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namestwo left twelveTwo saidas“twoo.”
© Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten
© Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten
© Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten3 0
© Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten3 0
© Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten3 0
© Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten73 0
© Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten73 0
© Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten73 07
© Joan A. Cotter, Ph.D., 20133 0Composing Numbers3-ten77
© Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten7Note the congruence in how we say thenumber, represent the number, and...
© Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten1 0Another example.
© Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten 81 0
© Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten81 0
© Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten81 08
© Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten81 88
© Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten
© Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten1 0 0
© Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten1 0 0
© Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten1 0 0
© Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred
© Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 0 0
© Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 0 0
© Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 01 01 0 0
© Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 0 0
© Joan A. Cotter, Ph.D., 2013Composing Numbers2hundred
© Joan A. Cotter, Ph.D., 2013Composing Numbers2hundred
© Joan A. Cotter, Ph.D., 2013Composing Numbers2hundred2 0 0
© Joan A. Cotter, Ph.D., 2013Learning the Facts
© Joan A. Cotter, Ph.D., 2013Learning the FactsLimited success, especially for strugglingchildren, when learning is:
© Joan A. Cotter, Ph.D., 2013Learning the Facts• Based on counting: whether dots,fingers, number lines, or countingwords.L...
© Joan A. Cotter, Ph.D., 2013Learning the Facts• Based on counting: whether dots,fingers, number lines, or countingwords.L...
© Joan A. Cotter, Ph.D., 2013Learning the Facts• Based on counting: whether dots,fingers, number lines, or countingwords.L...
© Joan A. Cotter, Ph.D., 2013Fact Strategies
© Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =
© Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =
© Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =
© Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =Take 1 fromthe 5 and giveit to the 9.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =Take 1 fromthe 5 and giveit to the 9.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =Take 1 fromthe 5 and giveit to the 9.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 = 14Take 1 fromthe 5 and giveit to the 9.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =
© Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =
© Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =
© Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =
© Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =10 + 4 = 14
© Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 =
© Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 =
© Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 =Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 =Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 =Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 = 6Subtract 5;then 4.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 – 9 =
© Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 – 9 =Subtract 9from 10.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 – 9 =Subtract 9from 10.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 – 9 =Subtract 9from 10.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 – 9 = 6Subtract 9from 10.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =
© Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =Start with 9;go up to 15.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =Start with 9;go up to 15.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =Start with 9;go up to 15.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =Start with 9;go up to 15.
© Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =1 + 5 = 6Start with 9;go up to 15.
© Joan A. Cotter, Ph.D., 2013MoneyPenny
© Joan A. Cotter, Ph.D., 2013MoneyNickel
© Joan A. Cotter, Ph.D., 2013MoneyDime
© Joan A. Cotter, Ph.D., 2013MoneyQuarter
© Joan A. Cotter, Ph.D., 2013MoneyQuarter
© Joan A. Cotter, Ph.D., 2013MoneyQuarter
© Joan A. Cotter, Ph.D., 2013MoneyQuarter
© Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts6 x 4 =(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts6 x 4 =(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts6 x 4 =(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts6 x 4 =(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts6 x 4 =(6 taken 4 times)
© Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts9 x 3 =
© Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts9 x 3 =
© Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts9 x 3 = 30
© Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts9 x 3 =30 – 3 = 27
© Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts7 x 7 =
© Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts7 x 7 =
© Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts7 x 7 =25 +
© Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts7 x 7 =25 + 10 + 10
© Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts7 x 7 =25 + 10 + 10+ 4 = 49
© Joan A. Cotter, Ph.D., 2013Trading1000 10 1100
© Joan A. Cotter, Ph.D., 2013TradingThousands1000 10 1100
© Joan A. Cotter, Ph.D., 2013TradingHundreds1000 10 1100
© Joan A. Cotter, Ph.D., 2013TradingTens1000 10 1100
© Joan A. Cotter, Ph.D., 2013TradingOnes1000 10 1100
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6
© Joan A. Cotter, Ph.D., 2013TradingAdding8+ 6141000 10 1100
© Joan A. Cotter, Ph.D., 2013TradingAdding8+ 614Too manyones; trade 10ones for 1 ten.1000 10 1100
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 614Too manyones; trade 10ones for 1 ten.
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 614Too manyones; trade 10ones for 1 ten.
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 614Same answerbefore andafter trading.
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter the firstnumber fromleft to right.
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right.
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right.
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right.
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right.
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right.
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults aft...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults aft...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults aft...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults aft...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults af...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults af...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults af...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults af...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults a...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults a...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults a...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults a...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults a...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults ...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults ...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults ...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults ...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386396Add starting atthe right. Writeresults...
© Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386396Add starting atthe right. Writeresults...
© Joan A. Cotter, Ph.D., 2013Part-Whole Circles
© Joan A. Cotter, Ph.D., 2013Part-Whole CirclesWholePart Part
© Joan A. Cotter, Ph.D., 2013Part-Whole CirclesWhat is the whole?3 2
© Joan A. Cotter, Ph.D., 2013Part-Whole CirclesWhat is the whole?352
© Joan A. Cotter, Ph.D., 2013Part-Whole CirclesWhat is the other part?107
© Joan A. Cotter, Ph.D., 2013Part-Whole CirclesWhat is the other part?3107
© Joan A. Cotter, Ph.D., 2011Short Division
© Joan A. Cotter, Ph.D., 2011Short Division• Means we don‟t write the stuffunderneath.
© Joan A. Cotter, Ph.D., 2011Short Division• Means we don‟t write the stuffunderneath.• Should be used for single-digit di...
© Joan A. Cotter, Ph.D., 2011Short Division• Means we don‟t write the stuffunderneath.• Should be used for single-digit di...
© Joan A. Cotter, Ph.D., 2011Short Division• Means we don‟t write the stuffunderneath.• Should be used for single-digit di...
© Joan A. Cotter, Ph.D., 2011Short Division• Means we don‟t write the stuffunderneath.• Should be used for single-digit di...
© Joan A. Cotter, Ph.D., 2011Short Division• Means we don‟t write the stuffunderneath.• Should be used for single-digit di...
© Joan A. Cotter, Ph.D., 2011Short Division3 4 7 1)
© Joan A. Cotter, Ph.D., 2011Short Division)The little lines help to keep track of placevalue.3 4 7 1
© Joan A. Cotter, Ph.D., 2011Short Division)400 ÷ 3 = ? [100] Write the 1 on the line.3 4 7 1
© Joan A. Cotter, Ph.D., 2011Short Division1)400 ÷ 3 = ? [100] Write the 1 on the line.3 4 7 1
© Joan A. Cotter, Ph.D., 2011Short Division1)400 ÷ 3 = ? [100] Write the 1 on the line.What is the remainder? [100] How ma...
© Joan A. Cotter, Ph.D., 20113 4 7 1Short Division1400 ÷ 3 = ? [100] Write the 1 on the line.What is the remainder? [100] ...
© Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionLet‟s show the 17 tens by writing a 1 beforethe 7.11)
© Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionLet‟s show the 17 tens by writing a 1 beforethe 7. Divide the tens: 17 t...
© Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionLet‟s show the 17 tens by writing a 1 beforethe 7. Divide the tens: 17 t...
© Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionTo find the remainder go up. 3 x 5 = 15; howfar is 15 from 17. [2]1 51)
© Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionTo find the remainder go up. 3 x 5 = 15; howfar is 15 from 17. [2]How ma...
© Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionTo find the remainder go up. 3 x 5 = 15; howfar is 15 from 17. [2]How ma...
© Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionTo find the remainder go up. 3 x 5 = 15; howfar is 15 from 17. [2]How ma...
© Joan A. Cotter, Ph.D., 20113 4 7 1Short Division21 ÷ 3 = ? [7]1 521)
© Joan A. Cotter, Ph.D., 20113 4 7 1Short Division)1 5 72121 ÷ 3 = ? [7]
© Joan A. Cotter, Ph.D., 20113 4 7 1Short Division)1 5 721
© Joan A. Cotter, Ph.D., 2011Short DivisionAnother example
© Joan A. Cotter, Ph.D., 2011Short Division)9 8 0 5 3
© Joan A. Cotter, Ph.D., 2011Short Division)9 8 0 5 380 hundred ÷ 9 = ? [8 hundred]
© Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division8)
© Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division88)
© Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division8) 8
© Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division) 88 9
© Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division4) 88 9
© Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division4) 88 9
© Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division8 9 4) 48
© Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division)8 9 4 r748
© Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division8 9 4)r748
Teaching Math
Teaching Math• Talk positively about math.
Teaching Math• Talk positively about math.• Teach for understanding; concretemodels should lead to mental models.
Teaching Math• Talk positively about math.• Teach for understanding; concretemodels should lead to mental models.• Encoura...
Teaching Math• Talk positively about math.• Teach for understanding; concretemodels should lead to mental models.• Encoura...
Teaching Math• Talk positively about math.• Teach for understanding; concretemodels should lead to mental models.• Encoura...
Teaching Math• Talk positively about math.• Teach for understanding; concretemodels should lead to mental models.• Encoura...
Teaching Math to Childrenwith Special NeedsMassHOPE - TEACHWorcester, MASaturday, April 279:45am — 10:45amJoan A. Cotter, ...
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Learning Disabilities Mass HOPE April 2013

  1. 1. Teaching Math to Childrenwith Special NeedsMassHOPE - TEACHWorcester, MASaturday, April 279:45am — 10:45amJoan A. Cotter, Ph.D.JoanCotter@RightStartMath.com
  2. 2. Characteristics of LDChild usually:
  3. 3. Characteristics of LDChild usually:• Is often more creative.
  4. 4. Characteristics of LDChild usually:• Is often more creative.• Cannot learn by rote.
  5. 5. Characteristics of LDChild usually:• Is often more creative.• Cannot learn by rote.• Must understand and make sense ofconceptin order to remember.
  6. 6. Characteristics of LDChild usually:• Is often more creative.• Cannot learn by rote.• Must understand and make sense ofconceptin order to remember.• Is more visual and often hands-on.
  7. 7. Characteristics of LDChild usually:• Is often more creative.• Cannot learn by rote.• Must understand and make sense ofconceptin order to remember.• Is more visual and often hands-on.• Dislikes worksheets.
  8. 8. Myths about LD
  9. 9. Myths about LD1. People with LD have lower intelligence. (33%are gifted.)
  10. 10. Myths about LD1. People with LD have lower intelligence. (33%are gifted.)2. They are lazy or stubborn.
  11. 11. Myths about LD1. People with LD have lower intelligence. (33%are gifted.)2. They are lazy or stubborn.3. Children with LD can be cured or will outgrowit.
  12. 12. Myths about LD1. People with LD have lower intelligence. (33%are gifted.)2. They are lazy or stubborn.3. Children with LD can be cured or will outgrowit.4. Boys are more likely to be affected.
  13. 13. Myths about LD1. People with LD have lower intelligence. (33%are gifted.)2. They are lazy or stubborn.3. Children with LD can be cured or will outgrowit.4. Boys are more likely to be affected.5. Dyslexia and learning disability are the samething.
  14. 14. Myths about LD1. People with LD have lower intelligence. (33%are gifted.)2. They are lazy or stubborn.3. Children with LD can be cured or will outgrowit.4. Boys are more likely to be affected.5. Dyslexia and learning disability are the samething.6. Adults with LD and ADD cannot succeed inhigher education.
  15. 15. Problems Occurring in Math(Dyscalculia)
  16. 16. Problems Occurring in Math(Dyscalculia)• Reversals in writing numbers
  17. 17. Problems Occurring in Math(Dyscalculia)• Reversals in writing numbers• Poor number sense
  18. 18. Problems Occurring in Math(Dyscalculia)• Reversals in writing numbers• Poor number sense• Slow fact retrieval
  19. 19. Problems Occurring in Math(Dyscalculia)• Reversals in writing numbers• Poor number sense• Slow fact retrieval• Errors in computation
  20. 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. 21. Problems Occurring in Math(Dyscalculia)• Reversals in writing numbers• Poor number sense• Slow fact retrieval• Errors in computation• Difficulty in solving word problemsDyscalculia only affects arithmetic, not otherbranches of math.
  22. 22. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13
  23. 23. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13
  24. 24. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13
  25. 25. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13
  26. 26. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13
  27. 27. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13
  28. 28. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13
  29. 29. Teaching forUnderstandingEmphasizesunderstandingTeaches a fewgeneralizationsIdentifies globalrelationshipsBroad applicationTakes longer to learn,but is retained moreeasilyDifficult to teachDifficult to testTeaching Rules andProceduresEmphasizes recallTeaches many rulesIdentifies sequentialstepsLimited contextIs learned more quickly,but is quicklyforgottenIs easy to teachCalifornia Mathematics Framework, 1985, p. 13
  30. 30. Time Needed to MemorizeAccording to a study with college students, it took them
  31. 31. Time Needed to Memorize• 93 minutes to learn 200 nonsensesyllables.According to a study with college students, it took them
  32. 32. Time Needed to Memorize• 93 minutes to learn 200 nonsensesyllables.• 24 minutes to learn 200 words of prose.According to a study with college students, it took them
  33. 33. Time Needed to Memorize• 93 minutes to learn 200 nonsensesyllables.• 24 minutes to learn 200 words of prose.• 10 minutes to learn 200 words of poetry.According to a study with college students, it took them
  34. 34. Memorizing MathPercentage RecallImmediatelyAfter 1 day After 4wks.Rote 32 23 8Concept69 69 58
  35. 35. Memorizing MathPercentage RecallImmediatelyAfter 1 day After 4wks.Rote 32 23 8Concept69 69 58
  36. 36. Memorizing MathPercentage RecallImmediatelyAfter 1 day After 4wks.Rote 32 23 8Concept69 69 58
  37. 37. Memorizing MathPercentage RecallImmediatelyAfter 1 day After 4wks.Rote 32 23 8Concept69 69 58
  38. 38. Memorizing MathMath needs to be taught so 95%is understood and only 5%memorized.Richard SkempPercentage RecallImmediatelyAfter 1 day After 4wks.Rote 32 23 8Concept69 69 58
  39. 39. Flash Cards & Times Tests
  40. 40. Flash Cards & Times Tests• Often used to teach rote.
  41. 41. Flash Cards & Times Tests• Often used to teach rote.• Liked only by those who don‟t needthem.
  42. 42. Flash Cards & Times Tests• Often used to teach rote.• Liked only by those who don‟t needthem.• Give the false impression that mathisn‟tabout thinking.
  43. 43. Flash Cards & Times Tests• Often used to teach rote.• Liked only by those who don‟t needthem.• Give the false impression that mathisn‟tabout thinking.• Often produce stress – children understress stop learning.
  44. 44. Flash Cards & Times Tests• Often used to teach rote.• Liked only by those who don‟t needthem.• Give the false impression that mathisn‟tabout thinking.• Often produce stress – children understress stop learning.• Not concrete – use abstract symbols.
  45. 45. Flash Cards & Times Tests• Often used to teach rote.• Liked only by those who don‟t needthem.• Give the false impression that mathisn‟tabout thinking.• Often produce stress – children understress stop learning.• Not concrete – use abstract symbols.• Cause stress (may become physicallyill).
  46. 46. Flash Cards & Times Tests• Often used to teach rote.• Liked only by those who don‟t needthem.• Give the false impression that mathisn‟tabout thinking.• Often produce stress – children understress stop learning.• Not concrete – use abstract symbols.• Cause stress (may become physicallyill).
  47. 47. Flash Cards & Times Tests• Often used to teach rote.• Liked only by those who don‟t needthem.• Give the false impression that mathisn‟tabout thinking.• Often produce stress – children understress stop learning.• Not concrete – use abstract symbols.• Cause stress (may become physicallyill).
  48. 48. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspective
  49. 49. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveBecause were so familiar with 1, 2, 3, we‟ll useletters.A = 1B = 2C = 3D = 4E = 5, and so forth
  50. 50. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveF + E =
  51. 51. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveAF + E =
  52. 52. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA BF + E =
  53. 53. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA CBF + E =
  54. 54. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA FC D EBF + E =
  55. 55. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveAA FC D EBF + E =
  56. 56. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA BA FC D EBF + E =
  57. 57. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA C D EBA FC D EBF + E =
  58. 58. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveA C D EBA FC D EBWhat is the sum?(It must be a letter.)F + E =
  59. 59. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveG I J KHA FC D EBF + E =K
  60. 60. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D
  61. 61. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D
  62. 62. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D
  63. 63. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D
  64. 64. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveNow memorize the facts!!G+ D
  65. 65. © 2010 Joan A.Traditional CountingFrom a childs perspectiveTry subtractingby „takingaway‟H– E
  66. 66. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveTry skip counting by Bs to T:B, D, . . . , T.
  67. 67. © Joan A. Cotter, Ph.D., 2013Traditional CountingFrom a childs perspectiveTry skip counting by Bs to T:B, D, . . . , T.What is D x E?
  68. 68. Adding on a Number LineD + C =
  69. 69. Adding on a Number LineD + C =* A B C D E F G H I J K L M
  70. 70. Adding on a Number LineD + C =* A B C D E F G H I J K L M
  71. 71. Adding on a Number LineD + C =* A B C D E F G H I J K L MWe‟re counting spaces, not lines.
  72. 72. Number Line Drawbacks
  73. 73. Number Line Drawbacks• Quantity of a number not obvious.
  74. 74. Number Line Drawbacks• Quantity of a number not obvious.• Requires counting in order to compute.
  75. 75. Number Line Drawbacks• Quantity of a number not obvious.• Requires counting in order to compute.• Ignores place value.
  76. 76. Number Line Drawbacks• Quantity of a number not obvious.• Requires counting in order to compute.• Ignores place value.• Hard to visualize.
  77. 77. DominoesMost domino patterns cannot be added.3 + 1 ≠4
  78. 78. Domino Pattern Drawbacks
  79. 79. Domino Pattern Drawbacks• Patterns generally cannot be added.
  80. 80. Domino Pattern Drawbacks• Patterns generally cannot be added.• They are not used outside of games.
  81. 81. Domino Pattern Drawbacks• Patterns generally cannot be added.• They are not used outside of games.• Emphasize only 1–6.
  82. 82. Domino Pattern Drawbacks• Patterns generally cannot be added.• They are not used outside of games.• Emphasize only 1–6.• Precious little mathematics.
  83. 83. © Joan A. Cotter, Ph.D., 2013VisualizingJapanese criteria for manipulatives
  84. 84. © Joan A. Cotter, Ph.D., 2013Visualizing• Visual is related to seeing.
  85. 85. © Joan A. Cotter, Ph.D., 2013Visualizing• Visual is related to seeing.• Visualize is to form a mental image.
  86. 86. © Joan A. Cotter, Ph.D., 2013VisualizingTry to visualize 8 identical apples withoutgrouping.
  87. 87. © Joan A. Cotter, Ph.D., 2013VisualizingTry to visualize 8 identical apples withoutgrouping.
  88. 88. © Joan A. Cotter, Ph.D., 2013VisualizingNow try to visualize 8 apples: 5 red and 3 green.
  89. 89. © Joan A. Cotter, Ph.D., 2013VisualizingNow try to visualize 8 apples: 5 red and 3 green.
  90. 90. © Joan A. Cotter, Ph.D., 2013VisualizingCan you visualize this rod?
  91. 91. © Joan A. Cotter, Ph.D., 2013Grouping in Fives
  92. 92. © Joan A. Cotter, Ph.D., 2013Grouping in FivesIIIIIIIIIIVVIII123458Early Roman numerals
  93. 93. © Joan A. Cotter, Ph.D., 2013Grouping in FivesMusical staff
  94. 94. © Joan A. Cotter, Ph.D., 2013Clocks and nickelsGrouping in Fives
  95. 95. © Joan A. Cotter, Ph.D., 2013Grouping in FivesClocks and nickels
  96. 96. © Joan A. Cotter, Ph.D., 2013Grouping in FivesTally marks
  97. 97. © Joan A. Cotter, Ph.D., 2011Grouping in FivesSubitizing• Instant recognition of quantity is called subitizing.
  98. 98. © Joan A. Cotter, Ph.D., 2011Grouping in FivesSubitizing• Instant recognition of quantity is called subitizing.• Grouping in fives extends subitizing beyond five.
  99. 99. © Joan A. Cotter, Ph.D., 2011Subitizing• Five-month-old infants can subitize to 1–3.
  100. 100. © Joan A. Cotter, Ph.D., 2011Subitizing• Three-year-olds can subitize to 1–5.• Five-month-old infants can subitize to 1–3.
  101. 101. © Joan A. Cotter, Ph.D., 2011Subitizing• Three-year-olds can subitize to 1–5.• Four-year-olds can subitize 1–10 bygrouping with five.• Five-month-old infants can subitize to 1–3.
  102. 102. © Joan A. Cotter, Ph.D., 2011Subitizing• Three-year-olds can subitize to 1–5.• Four-year-olds can subitize 1–10 bygrouping with five.• Five-month-old infants can subitize to 1–3.• Counting is analogous to sounding out aword; subitizing, to recognizing the word.
  103. 103. © Joan A. Cotter, Ph.D., 2011Research on Subitizing
  104. 104. © Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
  105. 105. © Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
  106. 106. © Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
  107. 107. © Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
  108. 108. © Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
  109. 109. © Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
  110. 110. © Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
  111. 111. © Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns research
  112. 112. © Joan A. Cotter, Ph.D., 2011Research on SubitizingKaren Wynns researchYou could say subitizing ismuch more “natural” thancounting.
  113. 113. © Joan A. Cotter, Ph.D., 2011Learning 1–10Using fingers
  114. 114. © Joan A. Cotter, Ph.D., 2011Learning 1–10Using fingers
  115. 115. © Joan A. Cotter, Ph.D., 2011Learning 1–10Using fingers
  116. 116. © Joan A. Cotter, Ph.D., 2011Learning 1–10Using fingers
  117. 117. © Joan A. Cotter, Ph.D., 2011Learning 1–10Using fingers
  118. 118. © Joan A. Cotter, Ph.D., 2011Learning 1–10Using fingers
  119. 119. © Joan A. Cotter, Ph.D., 2011Learning 1–10Subitizing 5
  120. 120. © Joan A. Cotter, Ph.D., 2011Learning 1–10Subitizing 5
  121. 121. © Joan A. Cotter, Ph.D., 2011Learning 1–10Subitizing 55 has a middle; 4 does not.
  122. 122. © Joan A. Cotter, Ph.D., 2011Learning 1–10Tally sticks
  123. 123. © Joan A. Cotter, Ph.D., 2011Learning 1–10Tally sticks
  124. 124. © Joan A. Cotter, Ph.D., 2011Learning 1–10Tally sticks
  125. 125. © Joan A. Cotter, Ph.D., 2011Learning 1–10Tally sticksFive as a group.
  126. 126. © Joan A. Cotter, Ph.D., 2011Learning 1–10Tally sticks
  127. 127. © Joan A. Cotter, Ph.D., 2011Learning 1–10Tally sticks
  128. 128. © Joan A. Cotter, Ph.D., 2013Learning 1–10Entering quantities
  129. 129. © Joan A. Cotter, Ph.D., 20133Learning 1–10Entering quantities
  130. 130. © Joan A. Cotter, Ph.D., 20135Learning 1–10Entering quantities
  131. 131. © Joan A. Cotter, Ph.D., 20137Learning 1–10Entering quantities
  132. 132. © Joan A. Cotter, Ph.D., 2013Learning 1–1010Entering quantities
  133. 133. © Joan A. Cotter, Ph.D., 2013Learning 1–10The stairs
  134. 134. © Joan A. Cotter, Ph.D., 2013Learning 1–10Adding
  135. 135. © Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 =Adding
  136. 136. © Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 =Adding
  137. 137. © Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 =Adding
  138. 138. © Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 =Adding
  139. 139. © Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 = 7Adding
  140. 140. © Joan A. Cotter, Ph.D., 2013Learning 1–104 + 3 = 7AddingJapanese children learn to do this mentally.
  141. 141. © Joan A. Cotter, Ph.D., 2012Games
  142. 142. © Joan A. Cotter, Ph.D., 2012GamesGamesMath
  143. 143. © Joan A. Cotter, Ph.D., 2012GamesGamesMath=
  144. 144. © Joan A. Cotter, Ph.D., 2012GamesGamesMathBooksReading=
  145. 145. © Joan A. Cotter, Ph.D., 2012GamesGamesMathBooksReadingGames provide interesting repetition neededfor automatic responses.=
  146. 146. © Joan A. Cotter, Ph.D., 2012GamesGamesMathBooksReadingGames provide interesting repetition neededfor automatic responses.More importantly, games provide anapplicationfor the new information!=
  147. 147. © Joan A. Cotter, Ph.D., 2012Go to the Dump
  148. 148. © Joan A. Cotter, Ph.D., 2012Go to the DumpObjective: To learn and master the factsthat total 10:
  149. 149. © Joan A. Cotter, Ph.D., 2012Go to the DumpObjective: To learn and master the factsthat total 10:1 + 92 + 83 + 74 + 65 + 5
  150. 150. © Joan A. Cotter, Ph.D., 2012Go to the DumpObjective: To learn and master the factsthat total 10:1 + 92 + 83 + 74 + 65 + 5
  151. 151. © Joan A. Cotter, Ph.D., 2012Go to the DumpObjective: To learn and master the factsthat total 10:1 + 92 + 83 + 74 + 65 + 5It is played similar to Go Fish.
  152. 152. © Joan A. Cotter, Ph.D., 2013Writing Numbers61728394105Sandpaper numbers also help.Number Chart
  153. 153. © Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming
  154. 154. © Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 1
  155. 155. © Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 2
  156. 156. © Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 3
  157. 157. © Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4
  158. 158. © Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 9
  159. 159. © Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten
  160. 160. © Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten21 = 2-ten 1
  161. 161. © Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten21 = 2-ten 122 = 2-ten 2
  162. 162. © Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten21 = 2-ten 122 = 2-ten 223 = 2-ten 3
  163. 163. © Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming11 = ten 112 = ten 213 = ten 314 = ten 4. . . .19 = ten 920 = 2-ten21 = 2-ten 122 = 2-ten 223 = 2-ten 3. . . .. . . .99 = 9-ten 9
  164. 164. © Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming137 = 1 hundred 3-ten 7
  165. 165. © Joan A. Cotter, Ph.D., 2013“Math” Way of Number Naming137 = 1 hundred 3-ten 7or137 = 1 hundred and 3-ten 7
  166. 166. © Joan A. Cotter, Ph.D., 2013Math Way of Number Naming• Only 11 words are needed to count to 100the math way, 28 in English. (All Indo-European languages are non-standard innumber naming.)
  167. 167. © Joan A. Cotter, Ph.D., 2013Math Way of Number Naming• Only 11 words are needed to count to 100the math way, 28 in English. (All Indo-European languages are non-standard innumber naming.)• Asian children learn mathematics using themath way of counting.
  168. 168. © Joan A. Cotter, Ph.D., 2013Math Way of Number Naming• Only 11 words are needed to count to 100the math way, 28 in English. (All Indo-European languages are non-standard innumber naming.)• Asian children learn mathematics using themath way of counting.• They understand place value in first grade;only half of U.S. children understand placevalue at the end of fourth grade.
  169. 169. © Joan A. Cotter, Ph.D., 2013Math Way of Number Naming• Only 11 words are needed to count to 100the math way, 28 in English. (All Indo-European languages are non-standard innumber naming.)• Asian children learn mathematics using themath way of counting.• They understand place value in first grade;only half of U.S. children understand placevalue at the end of fourth grade.• Mathematics is the science of patterns. Thepatterned math way of counting greatlyhelps children learn number sense.
  170. 170. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingCompared to reading
  171. 171. © Joan A. Cotter, Ph.D., 2013Math Way of Number Naming• Just as reciting the alphabet doesn‟t teachreading, counting doesn‟t teach arithmetic.Compared to reading
  172. 172. © Joan A. Cotter, Ph.D., 2013Math Way of Number Naming• Just as reciting the alphabet doesn‟t teachreading, 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. 173. © Joan A. Cotter, Ph.D., 2013Place-Value Cards3 03 0 0 03 th- ou-sand3 hun-dred3- ten3 0 0
  174. 174. © Joan A. Cotter, Ph.D., 2013Place-Value Cards5 084 0 0 02 0 0
  175. 175. © Joan A. Cotter, Ph.D., 2013Place-Value Cards5 084 0 0 0 4 0 0 02 0 05 082 0 0
  176. 176. © Joan A. Cotter, Ph.D., 2013Place-Value Cards5 084 0 0 0 4 0 0 02 0 04 0 0 02 0 05 085 082 0 0
  177. 177. © Joan A. Cotter, Ph.D., 2013Place-Value Cards3 0 0 08
  178. 178. © Joan A. Cotter, Ph.D., 2013Place-Value Cards3 0 0 0 3 0 0 088
  179. 179. © Joan A. Cotter, Ph.D., 2013Place-Value Cards3 0 0 0 3 0 0 3 0 00 0888
  180. 180. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names4-ten = fortyThe “ty”meanstens.
  181. 181. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names4-ten = fortyThe “ty”meanstens.
  182. 182. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names6-ten = sixtyThe “ty”meanstens.
  183. 183. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names3-ten = thirty“Thir” alsoused in 1/3,13 and 30.
  184. 184. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names5-ten = fifty“Fif” alsoused in 1/5,15 and 50.
  185. 185. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular names2-ten = twentyTwo used tobepronounced“twoo.”
  186. 186. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesA word gamefireplace place-fire
  187. 187. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesA word gamefireplace place-firepaper-newsnewspaper
  188. 188. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesA word gamefireplace place-firepaper-newsbox-mail mailboxnewspaper
  189. 189. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesten 4Prefix -teenmeans ten.
  190. 190. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesten 4 teen 4Prefix -teenmeans ten.
  191. 191. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesten 4 teen 4 fourteenPrefix -teenmeans ten.
  192. 192. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesa one left
  193. 193. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesa one left a left-one
  194. 194. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namesa one left a left-one eleven
  195. 195. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namestwo leftTwo saidas“twoo.”
  196. 196. © Joan A. Cotter, Ph.D., 2013Math Way of Number NamingRegular namestwo left twelveTwo saidas“twoo.”
  197. 197. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten
  198. 198. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten
  199. 199. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten3 0
  200. 200. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten3 0
  201. 201. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten3 0
  202. 202. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten73 0
  203. 203. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten73 0
  204. 204. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten73 07
  205. 205. © Joan A. Cotter, Ph.D., 20133 0Composing Numbers3-ten77
  206. 206. © Joan A. Cotter, Ph.D., 2013Composing Numbers3-ten7Note the congruence in how we say thenumber, represent the number, and writethe number.3 07
  207. 207. © Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten1 0Another example.
  208. 208. © Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten 81 0
  209. 209. © Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten81 0
  210. 210. © Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten81 08
  211. 211. © Joan A. Cotter, Ph.D., 2013Composing Numbers1-ten81 88
  212. 212. © Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten
  213. 213. © Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten1 0 0
  214. 214. © Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten1 0 0
  215. 215. © Joan A. Cotter, Ph.D., 2013Composing Numbers10-ten1 0 0
  216. 216. © Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred
  217. 217. © Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 0 0
  218. 218. © Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 0 0
  219. 219. © Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 01 01 0 0
  220. 220. © Joan A. Cotter, Ph.D., 2013Composing Numbers1hundred1 0 0
  221. 221. © Joan A. Cotter, Ph.D., 2013Composing Numbers2hundred
  222. 222. © Joan A. Cotter, Ph.D., 2013Composing Numbers2hundred
  223. 223. © Joan A. Cotter, Ph.D., 2013Composing Numbers2hundred2 0 0
  224. 224. © Joan A. Cotter, Ph.D., 2013Learning the Facts
  225. 225. © Joan A. Cotter, Ph.D., 2013Learning the FactsLimited success, especially for strugglingchildren, when learning is:
  226. 226. © Joan A. Cotter, Ph.D., 2013Learning the Facts• Based on counting: whether dots,fingers, number lines, or countingwords.Limited success, especially for strugglingchildren, when learning is:
  227. 227. © Joan A. Cotter, Ph.D., 2013Learning the Facts• Based on counting: whether dots,fingers, number lines, or countingwords.Limited success, especially for strugglingchildren, when learning is:• Based on rote memory: whether flashcards, timed tests, or computer games.
  228. 228. © Joan A. Cotter, Ph.D., 2013Learning the Facts• Based on counting: whether dots,fingers, number lines, or countingwords.Limited success, especially for strugglingchildren, when learning is:• Based on rote memory: whether flashcards, timed tests, or computer games.• Based on skip counting: whether fingers or songs
  229. 229. © Joan A. Cotter, Ph.D., 2013Fact Strategies
  230. 230. © Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =
  231. 231. © Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =
  232. 232. © Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =
  233. 233. © Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =Take 1 fromthe 5 and giveit to the 9.
  234. 234. © Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =Take 1 fromthe 5 and giveit to the 9.
  235. 235. © Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 =Take 1 fromthe 5 and giveit to the 9.
  236. 236. © Joan A. Cotter, Ph.D., 2013Fact StrategiesComplete the Ten9 + 5 = 14Take 1 fromthe 5 and giveit to the 9.
  237. 237. © Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =
  238. 238. © Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =
  239. 239. © Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =
  240. 240. © Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =
  241. 241. © Joan A. Cotter, Ph.D., 2013Fact StrategiesTwo Fives8 + 6 =10 + 4 = 14
  242. 242. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 =
  243. 243. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 =
  244. 244. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 =Subtract 5;then 4.
  245. 245. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 =Subtract 5;then 4.
  246. 246. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 =Subtract 5;then 4.
  247. 247. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Down15 – 9 = 6Subtract 5;then 4.
  248. 248. © Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 – 9 =
  249. 249. © Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 – 9 =Subtract 9from 10.
  250. 250. © Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 – 9 =Subtract 9from 10.
  251. 251. © Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 – 9 =Subtract 9from 10.
  252. 252. © Joan A. Cotter, Ph.D., 2013Fact StrategiesSubtract from 1015 – 9 = 6Subtract 9from 10.
  253. 253. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =
  254. 254. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =Start with 9;go up to 15.
  255. 255. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =Start with 9;go up to 15.
  256. 256. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =Start with 9;go up to 15.
  257. 257. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =Start with 9;go up to 15.
  258. 258. © Joan A. Cotter, Ph.D., 2013Fact StrategiesGoing Up15 – 9 =1 + 5 = 6Start with 9;go up to 15.
  259. 259. © Joan A. Cotter, Ph.D., 2013MoneyPenny
  260. 260. © Joan A. Cotter, Ph.D., 2013MoneyNickel
  261. 261. © Joan A. Cotter, Ph.D., 2013MoneyDime
  262. 262. © Joan A. Cotter, Ph.D., 2013MoneyQuarter
  263. 263. © Joan A. Cotter, Ph.D., 2013MoneyQuarter
  264. 264. © Joan A. Cotter, Ph.D., 2013MoneyQuarter
  265. 265. © Joan A. Cotter, Ph.D., 2013MoneyQuarter
  266. 266. © Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts6 x 4 =(6 taken 4 times)
  267. 267. © Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts6 x 4 =(6 taken 4 times)
  268. 268. © Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts6 x 4 =(6 taken 4 times)
  269. 269. © Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts6 x 4 =(6 taken 4 times)
  270. 270. © Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts6 x 4 =(6 taken 4 times)
  271. 271. © Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts9 x 3 =
  272. 272. © Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts9 x 3 =
  273. 273. © Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts9 x 3 = 30
  274. 274. © Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts9 x 3 =30 – 3 = 27
  275. 275. © Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts7 x 7 =
  276. 276. © Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts7 x 7 =
  277. 277. © Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts7 x 7 =25 +
  278. 278. © Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts7 x 7 =25 + 10 + 10
  279. 279. © Joan A. Cotter, Ph.D., 2013Multiplication StrategiesBasic facts7 x 7 =25 + 10 + 10+ 4 = 49
  280. 280. © Joan A. Cotter, Ph.D., 2013Trading1000 10 1100
  281. 281. © Joan A. Cotter, Ph.D., 2013TradingThousands1000 10 1100
  282. 282. © Joan A. Cotter, Ph.D., 2013TradingHundreds1000 10 1100
  283. 283. © Joan A. Cotter, Ph.D., 2013TradingTens1000 10 1100
  284. 284. © Joan A. Cotter, Ph.D., 2013TradingOnes1000 10 1100
  285. 285. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6
  286. 286. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6
  287. 287. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6
  288. 288. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 6
  289. 289. © Joan A. Cotter, Ph.D., 2013TradingAdding8+ 6141000 10 1100
  290. 290. © Joan A. Cotter, Ph.D., 2013TradingAdding8+ 614Too manyones; trade 10ones for 1 ten.1000 10 1100
  291. 291. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 614Too manyones; trade 10ones for 1 ten.
  292. 292. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 614Too manyones; trade 10ones for 1 ten.
  293. 293. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding8+ 614Same answerbefore andafter trading.
  294. 294. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738
  295. 295. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter the firstnumber fromleft to right.
  296. 296. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right.
  297. 297. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right.
  298. 298. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right.
  299. 299. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right.
  300. 300. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Enter numbersfrom left to right.
  301. 301. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults aftereach step.
  302. 302. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults after eachstep.
  303. 303. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults after eachstep.
  304. 304. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738Add starting atthe right. Writeresults after eachstep.
  305. 305. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults aftereach step.
  306. 306. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults aftereach step.1
  307. 307. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults aftereach step.1
  308. 308. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386Add starting atthe right. Writeresults aftereach step.1
  309. 309. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults aftereach step.1
  310. 310. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults aftereach step.1
  311. 311. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults aftereach step.1
  312. 312. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults aftereach step.1
  313. 313. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 273896Add starting atthe right. Writeresults aftereach step.1
  314. 314. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults aftereach step.1
  315. 315. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults aftereach step.1 1
  316. 316. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults aftereach step.1 1
  317. 317. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 2738396Add starting atthe right. Writeresults after eachstep.1 1
  318. 318. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386396Add starting atthe right. Writeresults after eachstep.1 1
  319. 319. © Joan A. Cotter, Ph.D., 20131000 10 1100TradingAdding 4-digit numbers3658+ 27386396Add starting atthe right. Writeresults after eachstep.1 1
  320. 320. © Joan A. Cotter, Ph.D., 2013Part-Whole Circles
  321. 321. © Joan A. Cotter, Ph.D., 2013Part-Whole CirclesWholePart Part
  322. 322. © Joan A. Cotter, Ph.D., 2013Part-Whole CirclesWhat is the whole?3 2
  323. 323. © Joan A. Cotter, Ph.D., 2013Part-Whole CirclesWhat is the whole?352
  324. 324. © Joan A. Cotter, Ph.D., 2013Part-Whole CirclesWhat is the other part?107
  325. 325. © Joan A. Cotter, Ph.D., 2013Part-Whole CirclesWhat is the other part?3107
  326. 326. © Joan A. Cotter, Ph.D., 2011Short Division
  327. 327. © Joan A. Cotter, Ph.D., 2011Short Division• Means we don‟t write the stuffunderneath.
  328. 328. © Joan A. Cotter, Ph.D., 2011Short Division• Means we don‟t write the stuffunderneath.• Should be used for single-digit divisors.
  329. 329. © Joan A. Cotter, Ph.D., 2011Short Division• Means we don‟t write the stuffunderneath.• Should be used for single-digit divisors.• Is easier to understand than longdivision.
  330. 330. © Joan A. Cotter, Ph.D., 2011Short Division• Means we don‟t write the stuffunderneath.• Should be used for single-digit divisors.• Is easier to understand than longdivision.• Best taught before long division.
  331. 331. © Joan A. Cotter, Ph.D., 2011Short Division• Means we don‟t write the stuffunderneath.• Should be used for single-digit divisors.• Is easier to understand than longdivision.• Best taught before long division.• Is much more useful in real life.
  332. 332. © Joan A. Cotter, Ph.D., 2011Short Division• Means we don‟t write the stuffunderneath.• Should be used for single-digit divisors.• Is easier to understand than longdivision.• Best taught before long division.• Is much more useful in real life.
  333. 333. © Joan A. Cotter, Ph.D., 2011Short Division3 4 7 1)
  334. 334. © Joan A. Cotter, Ph.D., 2011Short Division)The little lines help to keep track of placevalue.3 4 7 1
  335. 335. © Joan A. Cotter, Ph.D., 2011Short Division)400 ÷ 3 = ? [100] Write the 1 on the line.3 4 7 1
  336. 336. © Joan A. Cotter, Ph.D., 2011Short Division1)400 ÷ 3 = ? [100] Write the 1 on the line.3 4 7 1
  337. 337. © Joan A. Cotter, Ph.D., 2011Short Division1)400 ÷ 3 = ? [100] Write the 1 on the line.What is the remainder? [100] How many tensis that? [10]3 4 7 1
  338. 338. © Joan A. Cotter, Ph.D., 20113 4 7 1Short Division1400 ÷ 3 = ? [100] Write the 1 on the line.What is the remainder? [100] How many tensis that? [10] How many tens do we have? [10+ 7 = 17])
  339. 339. © Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionLet‟s show the 17 tens by writing a 1 beforethe 7.11)
  340. 340. © Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionLet‟s show the 17 tens by writing a 1 beforethe 7. Divide the tens: 17 tens ÷ 3 = ? [5]11)
  341. 341. © Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionLet‟s show the 17 tens by writing a 1 beforethe 7. Divide the tens: 17 tens ÷ 3 = ? [5]1)1 5
  342. 342. © Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionTo find the remainder go up. 3 x 5 = 15; howfar is 15 from 17. [2]1 51)
  343. 343. © Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionTo find the remainder go up. 3 x 5 = 15; howfar is 15 from 17. [2]How many ones do we have? [20 + 1]1 51)
  344. 344. © Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionTo find the remainder go up. 3 x 5 = 15; howfar is 15 from 17. [2]How many ones do we have? [20 + 1]1 51)
  345. 345. © Joan A. Cotter, Ph.D., 20113 4 7 1Short DivisionTo find the remainder go up. 3 x 5 = 15; howfar is 15 from 17. [2]How many ones do we have? [20 + 1]1 51) 2
  346. 346. © Joan A. Cotter, Ph.D., 20113 4 7 1Short Division21 ÷ 3 = ? [7]1 521)
  347. 347. © Joan A. Cotter, Ph.D., 20113 4 7 1Short Division)1 5 72121 ÷ 3 = ? [7]
  348. 348. © Joan A. Cotter, Ph.D., 20113 4 7 1Short Division)1 5 721
  349. 349. © Joan A. Cotter, Ph.D., 2011Short DivisionAnother example
  350. 350. © Joan A. Cotter, Ph.D., 2011Short Division)9 8 0 5 3
  351. 351. © Joan A. Cotter, Ph.D., 2011Short Division)9 8 0 5 380 hundred ÷ 9 = ? [8 hundred]
  352. 352. © Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division8)
  353. 353. © Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division88)
  354. 354. © Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division8) 8
  355. 355. © Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division) 88 9
  356. 356. © Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division4) 88 9
  357. 357. © Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division4) 88 9
  358. 358. © Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division8 9 4) 48
  359. 359. © Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division)8 9 4 r748
  360. 360. © Joan A. Cotter, Ph.D., 20119 8 0 5 3Short Division8 9 4)r748
  361. 361. Teaching Math
  362. 362. Teaching Math• Talk positively about math.
  363. 363. Teaching Math• Talk positively about math.• Teach for understanding; concretemodels should lead to mental models.
  364. 364. Teaching Math• Talk positively about math.• Teach for understanding; concretemodels should lead to mental models.• Encourage children to ask questions. “Idon‟t get it” is not a question.
  365. 365. Teaching Math• Talk positively about math.• Teach for understanding; concretemodels should lead to mental models.• Encourage children to ask questions. “Idon‟t get it” is not a question.• Use correct vocabulary.
  366. 366. Teaching Math• Talk positively about math.• Teach for understanding; concretemodels should lead to mental models.• Encourage children to ask questions. “Idon‟t get it” is not a question.• Use correct vocabulary.• Ask a question once; allow 3 secondsfor a response. Resist rephrasing.
  367. 367. Teaching Math• Talk positively about math.• Teach for understanding; concretemodels should lead to mental models.• Encourage children to ask questions. “Idon‟t get it” is not a question.• Use correct vocabulary.• Ask a question once; allow 3 secondsfor a response. Resist rephrasing.• Give a child 2-3 seconds to say a fact.
  368. 368. Teaching Math to Childrenwith Special NeedsMassHOPE - TEACHWorcester, MASaturday, April 279:45am — 10:45amJoan A. Cotter, Ph.D.JoanCotter@RightStartMath.com
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