Teaching Math to Children with Special...

Joan A. Cotter, Ph.D.JoanCotter@RightStartMath.com

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!!

H+ FE+ H

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

Youcouldsaysubitizingismuchmore“natural”thancounting.

Research on SubitizingIn Japanese schools

•Childrenarediscouragedfromusingcountingforadding.

•Theyconsistentlygroupinfives.

Quantities 1–10Using fingers

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

Quantities 1–10Stairs

Quantities 1–10Adding

4 + 3 =

4 + 3 = 7

Japanesechildrenlearntodothismentally.

•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

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.

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 7

Composing Numbers

Notethecongruenceinhowwesaythenumber,representthenumber,andwritethenumber.

3-ten 7

Composing Numbers

1-ten 8

Composing Numbers

1-ten 8

Composing Numbers

10-ten

Place-Value Cards

3 0 0 0 3th-ou-sand

3hun-dred

Place-Value Cards

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

2 0 0 2 0 0

Place-Value Cards

3 0 0 0 3 0 0 0 3 0 0 0 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.

9 + 5 =

9 + 5 = 14

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

Fourquarters.

Part-Whole Circles

part part

Part-Whole Circles

Whatisthewhole?

Part-Whole Circles

Whatistheotherpart?

Part-Whole CirclesMissing addend problem

Leereceived3goldfishasagift.NowLeehas5.HowmanygoldfishdidLeehavetostartwith?

Is3thewholeorapart?

Is5thewholeorapart?

Whatisthemissingpart?

Writetheequation.

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

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

1000 100 10 1

3658+ 2738

1000 100 10 1

3658+ 2738

1000 100 10 1

3658+ 2738

1000 100 10 1

3658+ 2738

Addstartingattheright.Writeresultsaftereachstep.

1000 100 10 1

3658+ 2738

Trade10onesfor1ten.

1000 100 10 1

3658+ 2738

Write6.

1000 100 10 1

3658+ 2738

Write1fortheextra10.

1000 100 10 1

3658+ 2738

Addthetens.

1000 100 10 1

3658+ 2738

Writethetens.

1000 100 10 1

3658+ 2738

Addthehundreds.

1000 100 10 1

3658+ 2738

Trade10hundredsfor1thousand.

1000 100 10 1

3658+ 2738

Writethehundreds.

1000 100 10 1

3658+ 2738

Write1fortheextrathousand.

1000 100 10 1

3658+ 2738

Addthethousands.

1000 100 10 1

3658+ 2738

Writethethousands.

1000 100 10 1

3658+ 2738

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.

Short Division

3) 4 7 1 1

Dividethetens:17tens÷3=?[5tens]

Short Division

3) 4 7 1 1 5

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

Short Division

3) 4 7 1 1 5

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

Short Division

3) 4 7 1 1 5

Dividetheones:21÷3=?[7]

Short Division

3) 4 7 1 1 5 7

Dividetheones:21÷3=?[7]Write7ones.

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 7 1 8

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

6 1 7 1

7 1 7 1 8 1

8 1 8 1

8 1 9 1

9 1 9 1

9 1 10 1

10 1 10 1

10 1 10

Fraction Chart1

1 7 1 8

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

6 1 7 1

7 1 7 1 8 1

8 1 8 1

8 1 9 1

9 1 9 1

9 1 10 1

10 1 10 1

10 1 10

Howmanyfourthsareinawhole?

Fraction Chart1

1 7 1 8

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

6 1 7 1

7 1 7 1 8 1

8 1 8 1

8 1 9 1

9 1 9 1

9 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, Ph.D.JoanCotter@RightStartMath.com

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...

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