teaching math to children with special needs · • rote counting to 100 in kindergarten. •...

© Joan A. Cotter, Ph.D., 2015

Joan A. Cotter, [email protected]

Sioux Empire Christian Home EducatorsHomeschool Conference

Sioux Falls, SDSaturday, May 2, 20153:00 p.m.– 4:00 p.m.

Teaching Math to Childrenwith Special Needs


•PeoplewithLDhavelowerintelligence.(33%aregifted.)

•Theyarelazyorstubborn.

•ChildrenwithLDcanbecuredorwilloutgrowit.

•Boysaremorelikelytobeaffected.(Girlsareequallyaffected.)

•Dyslexiaandlearningdisabilityarethesamething.

•StudentswithLDandADHDcannotsucceedinhighereducation.

Myths about LD


Childusually:

•Ismorecreative.

•Cannotlearnbyrote.

•Mustunderstandandmakesenseofaconceptinordertorememberit.

•Ismorevisualandhands-on.

•Dislikesworksheets.

•Findsitverydifficulttounlearn.

Characteristics of LD


•Reversalsinwritingnumbers

•Poornumbersense

•Slowfactretrieval

•Errorsincomputation

•Difficultyinsolvingwordproblems

Dyscalculiamainlyaffectsarithmetic,nototherbranchesofmath.

Problems Occurring with MathDyscalculia


•Counting•Learnsequence(numbernamesbyheart)

•One-to-onecorrespondence(onecountperobject)

•Cardinalityprinciple(lastnumbertellshowmany)

•Memorizingfacts•Flashcardsandtimedtests

•Rhymesandsongs

•Memorizingalgorithms(procedures)

•Usingkeywordstosolvestoryproblems

How Math is Traditionally Taught


Traditional Counting From a child’s perspective

Becausewe’resofamiliarwith1,2,3,we’lluselettersofthealphabet.

A=1B=2C=3D=4E=5,andsoforth



F + E =

A B C D E F A B C D E

Whatisthesum?



F + E = K

A B C D E F G H I J K



Now memorize the facts!!

G+ D

H+ FE+ H

D+ C

F+ F



H – C =

Trysubtractingby“takingaway.”



TryskipcountingbyB’stoT: B,D,...,T.

WhatisD×E?


Compared to Reading

Justasrecitingthealphabetdoesn’tteachreading,countingdoesn’tteacharithmetic.


•Rotecountingto100inkindergarten.

•Calendarsmisusedtoteachcounting.

•Countingonforaddition.(JackandJill)

•Countingbackforsubtraction.

•Numberlinesareabstractcounting.

•Skipcountingformultiplicationfacts.

•Doesnotworkwellformoneyorfractionsorreadinggraphs.

Counting-Based Arithmetic


Memorizing Math

Percentage RecallImmediately After1day After4weeks

Rote 32 23 8Concept 69 69 58

Mathneedstobetaughtso95%isunderstoodandonly5%memorized.

–Richard Skemp


Accordingtoastudywithcollegestudents,ittookthem:•93minutestolearn200nonsensesyllables.•24minutestolearn200wordsoftext.•10minutestolearn200wordsofpoetry.

Memorizing Math


•Oftenusedtoteachrote.

•Likedonlybythosewhodon’tneedthem.

•Givethefalseimpressionthatmathdoesnotrequirethinking.

•Oftenproducestress–childrenunderstressstoplearning.

•Notconcrete–useabstractsymbols.

•Causestress(maybecomephysicallyill).

•Resultinshort-termlearning.

•Mayleadtomathanxiety.

Flash Cards and Timed Tests


Visualizing

ApilotstudyoftheeffectsofRightStartinstructiononearlynumeracyskillsofchildrenwithspecificlanguageimpairment

RiikkaMononen,PirjoAunio,TuireKoponen

Abstract:....ThechildrenwithSLI[specificlanguageimpairment]begankindergartenwithsignificantlyweakerearlynumeracyskillscomparedtotheirpeers.Immediatelyaftertheinstructionphase,therewasnosignificantdifferencebetweenthegroupsincountingskills....

Published May, 2014


•Visualisrelatedtoseeing.

•Visualizeistoformamentalimage.

Visualizing


•Reading•Sports•Arts•Geography•Engineering•Construction•Biology

•Architecture•Astronomy•Archeology•Chemistry•Physics•Surgery•History

VisualizingVisualizingisalsoneededinotherfields:


Visualizing

Trytovisualize8identicalappleswithoutgrouping.


Visualizing

Nowtrytovisualize8apples:5redand3green.


Grouping in FivesEarly Roman numerals

123458

IIIIIIIIIIVVIII


Grouping in FivesMusical staff


Grouping in Fives Clocks and nickels


Grouping in FivesTally marks


Grouping in FivesSubitizing

•Instantrecognitionofquantityiscalledsubitizing.

•Groupinginfivesextendssubitizingbeyondfive.


Subitizing

•Five-month-oldinfantscansubitizeto1–3.

•Three-year-oldscansubitizeto1–5.

•Four-year-oldscansubitize1–10bygroupingwithfive.


Research on SubitizingKaren Wynn’s research


Research on SubitizingKaren Wynn’s research

Youcouldsaysubitizingismuchmore“natural”thancounting.


Research on SubitizingIn Japanese schools

•Childrenarediscouragedfromusingcountingforadding.

•Theyconsistentlygroupinfives.


Quantities 1–10Using fingers


Quantities 1–10Subitizing five


Quantities 1–10Subitizing five

5hasamiddle;4doesnot.


Quantities 1–10Tally sticks

Fiveasagroup.


Quantities 1–10Tally sticks


Quantities 1–10Number chart for remembering numerals

6!1 7!2 8!3 9!4 10!5!


Quantities 1–10Entering quantities

3


Quantities 1–10Entering quantities

7


Quantities 1–10Stairs


Quantities 1–10Adding



4 + 3 =



4 + 3 = 7

Japanesechildrenlearntodothismentally.


Games

•Gamesprovideinterestingrepetitionneededforautomaticresponsesinasocialsetting.

•Gamesprovideanapplicationfornewinformation.

•Gamesprovideinstantfeedback.

GamesMath

BooksReading=


Objective:Tolearnthefactsthattotal10:

1+9 2+8 3+7 4+6 5+5

GamesGo to the Dump

ItisplayedsimilartoGoFish.


•English-speakingchildrenoftenthinkof14as14ones,not10and4ones.

•Thepatternthatisneededtomakesenseoftensandonesishidden!

Place ValueThe problem


Place ValueIts importance

•Placevalueisthefoundationofmodernarithmetic.

•Itiscriticalforunderstandingalgorithms.

•Itmustbetaught,notleftfordiscovery.

•Childrenneedthebigpicture,nottinysnapshots.


11=ten112=ten213=ten314=ten4....19=ten9

20=2-ten21=2-ten122=2-ten223=2-ten3........99=9-ten9

Transparent Number Naming


137=1hundred3-ten7or

137=1hundredand3-ten7



0

10

20

30

40

50

60

70

80

90

100

4 5 6 Ages (years)

Song, M., & Ginsburg, H. (1988). p. 326. The effect of the Korean number system on young children's counting: A natural experiment in numerical bilingualism. International Journal of Psychology, 23, 319-332. !

Korean formal [math way] Korean informal [not explicit] Chinese U.S.

Aver

age

Hig

hest

Num

ber C

ount

ed


Korean transparent Korean non-transparent Chinese U.S.


Math Way of Number Naming•Only11wordsareneededtocountto100themathway,28inEnglish.(AllIndo-Europeanlanguagesarenon-standardinnumbernaming.)

•Asianchildrenlearnmathematicsusingtransparentnumbernaming.

•Mathematicsisthescienceofpatterns.Numbernamesneedtobeanexample.

•Childrenwhoarehearing-impairedcandistinguishbetween14and40,and13and30.

•Learningtwolanguageshelpsbraindevelopment.


Compared to Reading

Justaswefirstteachthesoundoftheletters,wemustfirstteachthenameofthequantity,themathway.


Math Way of Number NamingRegular names

The“ty”meanstens.

4-ten = forty



The“ty”meanstens.

6-ten = sixty



“Thir”alsousedin1/3,13and30.

3-ten = thirty



Twousedtobepronounced“twoo.”

2-ten = twenty



Awordgame

fireplace place-fire

newspaper paper-news

box-mail mailbox



Suffix-teenmeansten.

ten 4 teen 4 fourteen



a one left a left-one eleven



Twosaidas“twoo.”

twelvetwo left


Composing Numbers

3-ten

3 0


Composing Numbers

3-ten 7

3 0

7


Composing Numbers

3 0 7

Notethecongruenceinhowwesaythenumber,representthenumber,andwritethenumber.

3-ten 7


Composing Numbers

1-ten 8

1 0

8


Composing Numbers

1-ten 8

1 0 8


Composing Numbers

10-ten

1 0 0


Place-Value Cards

3 0 0

3 0

3 0 0 0 3th-ou-sand

3hun-dred

3-ten


Place-Value Cards

4 0 0 0 4 0 0 0 4 0 0 0 2 0 0

2 0 0 2 0 0

5 0

5 0

5 0

7

7

7


Place-Value Cards

3 0 0 0 3 0 0 0 3 0 0 0 8

8 8


Static(Recording)•Valueofadigitisdeterminedbyposition.•Nopositionmayhavemorethannine.•Asyouprogresstotheleft,valueateachpositionistentimesgreaterthanpreviousposition.

•Representedbytheplace-valuecards.

Place ValueTwo aspects

Dynamic(Trading)•10ones=1ten;10tens=1hundred;10hundreds=1thousand,….

•Representedwithabacusandothermaterials.


Limitedsuccess,especiallyforstrugglingchildren,whenlearningis:

•Basedoncounting:whetherdots,fingers,numberlines,orcountingwords.

•Basedonrotememory:whetherflashcards,timedtests,orcomputergames.

•Basedonskipcounting:whetherfingersorsongs.

Achildisconsideredtoknowafactiftheycangiveitin2–3seconds.

Learning the Facts


Fact StrategiesComplete the Ten

9 + 5 =

Take1fromthe5andgiveittothe9.


Fact StrategiesComplete the Ten

9 + 5 = 14

Take1fromthe5andgiveittothe9.


Fact StrategiesTwo Fives

8 + 6 =


Fact StrategiesTwo Fives

8 + 6 = 10 + 4 = 14

Thetwofivesmake10.


Fact StrategiesMissing Addend

9 + = 15

Startwith9;goupto15.


Fact StrategiesMissing Addend

9 + 6 = 15

Startwith9;goupto15.


Fact StrategiesSubtracting Part from Ten

15 – 9 =

Subtract5from5and4from10.


Fact StrategiesSubtracting Part from Ten

15 – 9 = 6

Subtract5from5and4from10.


Fact StrategiesSubtracting All from Ten

15 – 9 =

Subtract9from10.


Fact StrategiesSubtracting All from Ten

Subtract9from10.

15 – 9 = 6


MoneyPenny


MoneyNickel


MoneyDime


MoneyQuarter


MoneyQuarter

Fourquarters.


Part-Whole Circles

part part

whole


Part-Whole Circles

Whatisthewhole?

2 5


Part-Whole Circles

Whatistheotherpart?

7

10


Part-Whole CirclesMissing addend problem

Leereceived3goldfishasagift.NowLeehas5.HowmanygoldfishdidLeehavetostartwith?




3

Is3thewholeorapart?




3

5

Is5thewholeorapart?




3

5

Whatisthemissingpart?




3 2

5

Whatisthemissingpart?




Writetheequation.

3 2

5

2 + 3 = 53 + 2 = 55 – 3 = 2


Part-Whole Circles

•Researchshowspart-wholecircleshelpyoungchildrensolveproblems.Writingequationsdonot.

•Donotteach“key”words.Thechildneedstofocusonthesituation,notlookforspecificwords.


Multiplication Strategies Basic facts

6 × 4 = 24 (6taken4times.)



9 × 3 =



9 × 3 =30 – 3 = 27



7 × 7 =



7 × 7 = 25 + 10 + 10 + 4 = 49


Multiplication Strategies Commutative property

5 × 6 = 6 × 5


Multiplication Strategies Multiplication table


1000 100 10 1

TradingSimple adding

8+ 6


1000 100 10 1


8+ 614


1000 100 10 1


Toomanyones;trade10onesfor1ten.

8+ 614


1000 100 10 1


Sameanswerbeforeandaftertrading.

8+ 614


1000 100 10 1

TradingAdding 4-digit numbers

3658+ 2738

Enternumbersfromlefttoright.


1000 100 10 1


3658+ 2738

Addstartingattheright.Writeresultsaftereachstep.


1000 100 10 1


3658+ 2738

Trade10onesfor1ten.


1000 100 10 1


3658+ 2738

6

Write6.


1000 100 10 1


3658+ 2738

6

Write1fortheextra10.

1


1000 100 10 1


3658+ 2738

6

Addthetens.

1


1000 100 10 1


3658+ 2738

96

Writethetens.

1


1000 100 10 1


3658+ 2738

96

Addthehundreds.

1


1000 100 10 1


3658+ 2738

96

Trade10hundredsfor1thousand.

1


1000 100 10 1


3658+ 2738

396

Writethehundreds.

1


1000 100 10 1


3658+ 2738

396

Write1fortheextrathousand.

11


1000 100 10 1


3658+ 2738

396

Addthethousands.

11


1000 100 10 1


3658+ 2738

6396

Writethethousands.

11


1000 100 10 1


3658+ 2738

6396

11


Short Division

•Meanswedon’twritethestuffunderneath.

•Shouldalwaysbeusedforsingle-digitdivisors.

•Iseasiertounderstandthanlongdivision.

•Needstobetaughtbeforelongdivision.

•Ismuchmoreusefulinreallife.

•Ismuchquickertoperformfortests.


Short Division

3) 4 7 1


Short Division

3) 4 7 1

Thelittlelineshelpkeeptrackofplacevalue.


Short Division

3) 4 7 1 1

400÷3=?[100]Writethe1ontheline.Whatistheremainder?[100]Howmanytensisthat?[10]Howmanytotaltensdowehave?[10+7=17]


Short Division

3) 4 7 1 1

Showthe17tensbywritinga1beforethe7.

1


Short Division

3) 4 7 1 1

Dividethetens:17tens÷3=?[5tens]

1


Short Division

3) 4 7 1 1 5

Tofindtheremaindergoup:3×5=15.Howfaris15from17?[2]Howmanyonesdowehave?[20+1=21]

1


Short Division

3) 4 7 1 1 5

Tofindtheremaindergoup:3×5=15.Howfaris15from17?[2]Howmanyonesdowehave?[20+1=21]

1 2


Short Division

3) 4 7 1 1 5

Dividetheones:21÷3=?[7]

1 2


Short Division

3) 4 7 1 1 5 7

Dividetheones:21÷3=?[7]Write7ones.

1 2


Short Division

3) 4 7 1 1 5 7 1 2


Long Division•Childrenwithlearningdisabilitiesshouldnotbeexpectedtolearnlongdivision.Itcantakemonthstolearnit—anunwiseuseoftime.

•Itisamostlymemorizinganumberofstepswithlittleunderstanding.

•Itisnotaskillneededinadvancedmath.Dividingpolynomialsdoesnotentailguessingatrialdivisor.

•Rarelyperformedonthejoborineverydaylife.Whatisimportantisestimatingananswerandknowingwhattodowithanyremainders.


Fractions in the Comics


Meaning of a Fraction

•Oneormoreequalpartsofawhole.

•Oneormoreequalpartsofaset.

•Divisionoftwowholenumbers.

•Locationonanumberline.

•Ratiooftwonumbers.


Meaning of a FractionWhichmeaningsaremostmathematical?•Oneormoreequalpartsofawhole.






Meaning of a FractionWhichmeaningsareusedineverydaylife?•Oneormoreequalpartsofawhole.






Meaning of a FractionWhichmeaningisusedinelementarytexts?•Oneormoreequalpartsofawhole.






Fraction Chart1

1 2 1

2 1 3

1 4

1 5

1 6

1 7 1 8

1 9

1 10

1 3 1

3 1 4

1 5

1 6

1 7

1 8

1 9

1 4

1 5

1 6

1 7

1 8

1 4 1 5 1 6 1 7 1 8 1 9

1 5

1 6 1

6 1 7 1

7 1 7 1 8 1

8 1 8 1

8 1 9 1

9 1 9 1

9 1 9 1

9 1 10 1

10 1 10 1

10 1 10 1

10 1 10 1

10 1 10


Fraction Chart1

1 2 1

2 1 3

1 4

1 5

1 6

1 7 1 8

1 9

1 10

1 3 1

3 1 4

1 5

1 6

1 7

1 8

1 9

1 4

1 5

1 6

1 7

1 8

1 4 1 5 1 6 1 7 1 8 1 9

1 5

1 6 1

6 1 7 1

7 1 7 1 8 1

8 1 8 1

8 1 9 1

9 1 9 1

9 1 9 1

9 1 10 1

10 1 10 1

10 1 10 1

10 1 10 1

10 1 10

Howmanyfourthsareinawhole?


Fraction Chart1

1 2 1

2 1 3

1 4

1 5

1 6

1 7 1 8

1 9

1 10

1 3 1

3 1 4

1 5

1 6

1 7

1 8

1 9

1 4

1 5

1 6

1 7

1 8

1 4 1 5 1 6 1 7 1 8 1 9

1 5

1 6 1

6 1 7 1

7 1 7 1 8 1

8 1 8 1

8 1 9 1

9 1 9 1

9 1 9 1

9 1 10 1

10 1 10 1

10 1 10 1

10 1 10 1

10 1 10

Whichismore,one-thirdorone-fourth?


Compared to Reading

•Justasreadingismuchmorethandecoding,phonics,andwordattackskills,mathematicsismuchmorememorizingfactsandlearningalgorithms.

•Justasthegoaloflearningtoreadisreadingtolearnandenjoyment,thegoalofmathissolvingproblemsandexperiencingwonder.


Joan A. Cotter, [email protected]

Sioux Empire Christian Home EducatorsHomeschool Conference

Sioux Falls, SDSaturday, May 2, 20153:00 p.m.– 4:00 p.m.

Teaching Math to Childrenwith Special Needs

teaching math to children with special needs · • rote counting to 100 in kindergarten. •...

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