dyscalculia (math)

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Information provided by The International DYSLE IAAssociation

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Tel: 410-296-0232 Fax: 410-321-5069 E-mail: [email protected] Website: http://www.interdys.org

This packet of information on

Dyscalculia (Math)

is comprised of the following articles from Perspectives (P), our quarterly

journal for members of IDA:

Mathematical Overview: An Overview for EducatorsDavid C. Geary

(P: Vol. 26, No. 3, Summer 2000, p. 6-9)

Mathematical Learning Profiles and Differentiated Teaching StrategiesMaria R. Marolda, Patricia S. Davidson

(P: Vol. 26, No. 3, Summer 2000, p. 10-15)

Translating Lessons from Research into Mathematics ClassroomDouglas H. Clements

(P: Vol. 26, No. 3, Summer 2000, p. 31-33)

You can always find copies of our Fact Sheets available for download on our website:

www.interdys.org -> "IDA Quick Links" (blue tab, right-hand side) -> Click on "Dyslexia Fact Sheets"


2/14

.JL

A MATHEMATICAL

ISORDERS:

IEE-I

lm

\M

AN

OVERVIEW

OR

EDUCATORS

17

BYDavid ' Geary

ff

uring

the

past

20 years,

we

have

l-/witnessed

enormous

advances n

our understandingof the genetic,

neurological,

and cognitive factors

hat

contribute

o reading

disorders, s

well

as advances

n the ability to

diagnose

and remediate

his form of

learung

disorder

(e.g.,

Torgesen

et a1.,7999).

We now

understand hat

most forms

of

readinsdisorder

esult rom a

heritable

risk and

have a phonolog icai

core; or

instance.manv of

these children

have

difficulties

associating

letters and

words with the associated

sounds,

which makes

learning to

decode

unfamiliar

words difficult

(Light,

DeFries,& Olson, 1998). At the same

time,

there have been

a handful of

researchers studying

children's

difficulties

with early mathematics,

difficulties

that emerge despite

low-

average

or better

intelligence and

adequate

nstruction

(Ceary,

Hamson,

& Hoard,

n press;

ordan

&

Montani,

1997). This essay

overviews

this

research,

ncluding discussion

of the

nrcvalence of chi ldren

wi th

mathematical

disorders and

their

diagnoses,

he approach

researchers

use to

study these children,

and some

malor

nnolnSs.

How common

s

a

Mathematical

Disorder

and

how is

it

diagnosed?

Although

there are

no definitive

answers,

studies conducted

in the

United States,Europe,

and Israel

al l

converqe on the

same conciusion:

About 6%. of school-age

children

and

adolescents have

some form

of

mathematical

disorder and

about one

half of these individuals also have

difficulty

in learning how to

read

(Gross-Tsur,

Manor, & Shalev,

199Q.

These studies

also suggest

that

mathematical

isorders reas

common

as

reading disorders

and that

a

common

deFicitmay contribute

to the

co-occurrence

of a mathematical

disorder nd

a reading isorder

n some

children

Ceary,1993).

Like

readingdisorders,

here s no

universally agreed

upon set of criteria

6

Perspectives, ummer

2000

Mathematical

Disorders:

Subtwe

I

SemantiiMemory

Cogiive &

Perfomana utures:

A. Low frequencyof

arithmetic ac[ retrieval

B. When

factsare

retrieved, here

s a high

error rate

C. Errors

are requendy

assoclatest tne

numbers n

the problem

D. Soluuon

imes or

correct etrieval

are

unsystematlc

N eu

o

sy holo

ica Featu

es

A. Appearso be

associated

#ith left hemispheric

dys unction,

n paru.cular,

postenor

egrons t tne

l p F t h p mic n 6 e ra

B. Possible

ubcortical

involvement,uch

as he

basalganglia

Cenetic eatures:

Preliminary tudies

and

the relatioirwith certain

formsof readrng

isorder

sugqest

hat hisdeficit

md'

be heritable

Relat

onshy tokadin

g D s rdets

Appears o

occurwith

n Hn n e t i r F^ * . n f

i -p a r l i n o d ic n rd c r

-_ - "^ ^ b * ^ ' " ^ - " '

for the diagnosis

of mathematical

disorders. n

our recent

work, we have

found a lower than expected based n

IO)

perfonnance

n math achievement

testsacross

t least wo

grade evels o

be

a usefuland

practical ndicator of

a

mathematical

isorder

Geary

et al.,

n

press).

This and other

studies ndicate

that children

with a

mathematical

disorder

rea

heterogeneousroup

and

show one or

more subrypes f

disorder

(Geary,1.993).

within each domain.

As an example,

the assessment f

computational

kills

in dyscalculia poor performanceafter

brain

injury) has often

been basedon

summary scores

or accuracyat solving

simple

(e.g.

9+Q and

complex

(e.g.

244+1,29) rithmetic

problems

(Geary,

1993).

These scores

ctually eflect

an

arcay of

component skills,

including

fact retrievaland

procedural

s

well as

conceptual

competencies,

making

inferences

about

the source of

poor

Mathematical

Disorders:

Subtvoe

2

Proc6dural

Cqnttive

Perfomanceeatures:

A. Relatively requentuse

of devel6pmentally

urunature

proceoures

B. Frequent nors

n the

executlon

t

procedures

C. Potential

developmental

delayn

the Lrndeisundmg

of le concepts

nderlyrg-

proceoural se

D. Difficulues equencing

t h p m r r l h n l p c t p n c r n

" ' "r" '

complex

proceoures

Ne o sy hoo

i

cal

Fe tu e

Unclear,

lthough

some

data uggest

nassociabon

with

left

hemispheric

dysfuncuon, nd n some

cises

prefronal

{ifurrtion

Mathematical

Disorders:

Subtrroe 3

Visuo'sfiatial

Cogrtwe

Perfotnanceutures:

A. Difficulties n spaually

rePresenunSumencar

rntormatlon uchas -fre

misalignment

f numerals

in mufticolumnarithmetic

problerns

r roatirg urnbels

B.

Misinterpretation

f

spatially'represented

numencal

ntormatlon,

suchas

place alueenors

C. Mav result n difficulties

in aieas ut

rely on spatial

abilities, uch

a3georhetry

N

e

u o sy hol ical Featu

e

Appears

o be associated

rlzrtt-r ght hemispheric

dystunchon,

m

partl.cular,

postenor

eqrons t

the

ipht herrusphere,ftnp$

the

panetal

ortex t the

left hemispheremay

be

implicated

as well.

-

Cenetic

eatures:

Unclear

RelationshiyRuding

Disordes:

Does

not appearo

be

related

Relationh

y n Readn D sorders

Unclear

How do researcherspproach

the study

of

Mathemat ical

Disorders?

The complexiry

of

the field of

mathematics

makes

the study of

mathematical

disorderschallenging.

In

theorv. mathematical

disorders

can

result'from

deficits

in the ability

to

represent

or process

nformation in

one

or all mathematical

domainsor

in one

or a set of

individual competencies

performance

on these

arithmetic

problems imprecise. Moreover, in

geometry

and

algebra,

not enough

is

known

about the

normal development

of

the associated

competencies

o

provide

a systematic

ramework

or the

study

of mathematical

disorders.

Iortunately,

enough

is now

known

about

normal development

n the

areas

of number

concepts,

counting

skills,

and early

arithmetic

skil ls

to

p r o v i d e

h e f r a m e w o r k

n e e d e d

continued n

Vage

7


3/14

Mathematical

Disorders:

n

Overview

or

Educators

continued

from Vage

6

to

systematicallystudy mathematical

disorders

see

Geary,1994, or a review).

The

sectionsbelow orovide an overview

of what we now know about

children's

developing

umber concepts, ounting

skills, and early arithmetic skills aiong

with a discussion of

any associated

learnins

disorder. The finai section

providJs a discussion

of

visuospatial

issues somet imes

associatedwi th

childrenwith

mathematical isorders.

Number

Concepts

Psychologistshave been

studying

children'sconceptual

understandingof

number, for instance that

"3"

is

an

abstract epresentation

f a collectionof

any three things, or

many decades. t is

now clear children's

understandinsof

small

auantities

and

number is evident

to some degree in

infancy. Their

understandingof larger

numbers and

related

skil ls, such as place value

concepts

e.g.,

he

"4"

in the numeral

"42"

represents our

groups of

10),

emerges slowly during

the

preschool

and early elementary

school

years

and

some times only with

instruction

(Iuson,

19BB;Ceary,

1994).

The few studies

conducted with

children with

mathematical disorders

suggesthat basicnumber

competencies.

at least or

smail quantities, re ntact n

most of

these children

(Ceary,

1993;

Cross-Tsur

t al..1996'.

CountingSkills

During

the preschool

years.

children's

counting knowledge

can be

represented

v Celrnan

and Caliistel's

(1978)

five

implicit counting

principles.

These principles

include one-one

correspondence

one

and only oneword

tag, such as

"one,t' two,"

is assignedo

each counted object);

he stable order

principle

(the

order of

the

word

tags

must be inva riant acrosscounted sets):

the cardinality principie

(the

value of the

final word tag represents

he quantiry of

items n the counted

set); he abstraction

principie

(objects

of any kind

can be

collectedogether

and counted); nd, he

order-irrelevance

rinciple

(items

within

a

given set can

be tagged in any

sequence).

Children also

make

seeminglyapparent,

ut not necessarily

correct, inductions

about the basic

characteristics

f counting

by observing

standard counting

behavior

(Briars

&

S ieg le r , 984 ;F uson ,

9BB) . T hese

inductions

nclude

"

adjacency'

counting

must proceed onsecutively

nd n order

from one to the

next) and

"start

at an

end"

(counting

must proceed

rom left

to right).

Studiesof children

with concurent

mathematicai disorders

and readinq

disorders

or mathematical disorderi

alone indicate

that these children

understand

most of the

essential

features of counting, such

as stable

order. but consistentlv

err on tasks that

assess

adjacency"

and order-irrelevance

(Ceary

Bow-Thomas, 8t. Yao, 1992;

Geary et al. , in press).

In fact, these

children, at least in first

and second

grade, perform more poorly on

these

tasks han

do children with much lower

IO

scores, suggestinga very specific

deficit in

their counting knowledge. It

appears

hat thesechildren, egardless

f

I

Difficulties

n using

counting

procedures

an

thuscontribute o

later

arithmetic-fact etrieual

problems.

their

readingachievement, elieve

hat

counting

is constrained such that

counting procedures

can only be

executed

n the standard way

(i.e.,

objects

can oniy be counted

sequentially),which,

in turn, suggests

that

they do not fully understand

counrlng

concepB.

Other studiessuggest hat children

with

mathematicaldisordersalso have

difficulties keeping

information in

working memory while monitoring the

counting process or performing

other

mental

manipulations

(Hitch

&

McAuley,

1991),which, in tum, results

in more errors while

counting. Thus,

young children with mathematical

disorders show deficits

in countins

knowledge

and counting accuracy.

Arithmetic

Skills

When first leaming

to solve simple

arithmetic

roblems

e.g.,

+5).

children

gtprcally

rely

on their knowledge of

counting and use

counting procedures o

ftnd the answer

(Geary

et al., 1,992;

Siegler &

Shrager, I9B4). Their

procedures sometimes rely on finger

counting and sometimes

only on

verbal counting. Common counting

procedures

nclude he foilowing: sum

or

counting-all), here

chiidren count each

addend

starting from 1; max

(or

counting

on) where

children state he value

of

the

smaller addend and

then count the

larger addend; and, min

(counting

on),

where children

state the larger addend

and then count

the value of he smaller

addend, such

as stating 5 and counting

on 6.7. B

o solve3+5.

The derrelnnmeql

of eff icient

counting procedures for

simple

problems

(e.g.,

3+5) reFlects

gradual

shift from frequent use of the

sum and

max procedures

o frequent use of the

min procedure.

The repeateduse

of

counting procedures

also appears to

result

in the developmentof memory

representations

f basic acts

(Siegler

&

Shrager,1984),

hat is, with repeated

counting he

generated nswer

(..g.,

B)

eventually

becomes associated in

memoly'

with

the problem

(e.g.,

+5).

Difficulties in using counting procedures

can thus contribute to later arithmetic-

fact retrieval

problems.

Studies conducted in the United

Stites, Europe, and Israel

have

consistently found

that children with

mathematical isorders ave difficulties

solving simple

and complex arithmetic

problems

(e.g.,

Barrouillet,

Fayol, &

Lathuiibre,

1997;

Jordan

& Montani,

1997). These

differences nvolve both

procedural

and memory-baseddeftcits,

each of which is considered

n the

respective

ections elow.

Procedural

Deficits

Much

of the research n children

with mathematical disorders has

focused on their use of

counting

strategies

o solve simple arithmetic

problems

and indic ates that these

children commit more

errors than do

their normal peers

(Ceary

1993;

Jordan

& Montani, 1997).

They oftenmiscount

or iose

rack of the counting process.As

a

group, young children with


also rely on

finger counting

and use the sum

procedure

more frequently than

do

continued

n pag,e

Perspectives,

ummer

2000 7


4/14

Mathematical

isorders:

n

Overview

or

Educators

continued

from

Vage

7

normal

children.

Their use of

finger

counting

appears

to be a

working

memory

aid, in that

it helps these

children

to keep

track of the

counting

process.

Their prolonged

use of

sum

countingappears

o be related,

n part,

to their belief that "adjacency"is an

essential

eatureof counting

Geary

et

a1.,1992).

owever,

many, but not

all,

of these children

show more

efficient

orocedures

bv the middle

of the

bl.-..rtury

school

years

(Grades

4-6)

(Geary

1,993;

ordan&

Monuni,

1997).

Thus, for

these children,

their error-

prone

use

of immature

procedures

represents

a developmental

delay

rather than

a long-term

cognitive

deFicit.

There have

only been

a few

studies

of the ability

of children

with

mathematical

disorders

to

pursue

formal arithmetic

alsorithms

associated

with more iomplex

oroblems.

such as 126+537.

Th e

iesearch

that has been

conducted

suggestssome

specif ic diff icult ies.

Althoush some

studieshave attributed

calcuhtlon difficulties

to visuospatial

diff icult ies described

below,

other

studies suggest

hat these

calculation

difficulties

are likely not

due to the

spatial demands

of these

arithmetic

formats,

as most children

with

mathematical disordersdo not have

poor

spatial bil it ies

e.g.,

Gearyet al..

in

press).

Rather,

he errorsappeared

o

result

trom difficulties

in monitoring

the sequence

f steps

of the algorithm

and from

poor skill in detecting

and

then self-correcting

errors.

Thus,

proceduraldifficulties

associated

ith


are evident

when

these children

count to solve

simple

arithmetic problems

(e.g.,

3+5)

and use

algorithms to solve

more

complexproblems

e.g.,

126

+ 537).

RetrievalDeficits

Many

children with

mathematical

disorders

do not show

the shift from

direct

countingprocedures

o memory-

based

roductionof solutions

o simple

arithmetic

problems that is commoniy

found

in normal children.

It appears

that there

are tvvo different

foims of

retrieval deficit,

each reflecting

a

disruption to

different cognitive

and

neural systems

(Barrouillet

et aI.,

1997;

Geary,1993).

Cognitive

studies suggest

hat the

8 Perspectives,

ummer

2000

retrieval

deficits

are due,

in part, to

difficulties

in

accessingacts

rom long-

term memory.

In fact,

it appears hat

the memory

representations

for

arithmetic

facts

are supported,

n part,

by the same

phonological

and semantic

memory systems that support word

decodingand

reading

comprehension.

If this

is indeed

the case, hen

the

disrupted

phonological

processeshat

contribute

to reading

disordersmight

also be t he

sourceof the

fact retrieval

difficultiesof

childrenwirh

mathematical

disorders.

t mieht be

the sourceof

the

co-occurrence

of

mathematical

disorders and

reading disorders

n

many children

(Geary,

1993;

Light et

aI.. I99B\.

Recent

studies

suggest

a second

form of retrieval

deficit, specifically,

disruptions

n the retrievalprocess ue

to difficulties

in

inhibiting the retrieval

of irrelevant

associations.

his form

of

retrieval deficit

was discovered

by

Barrouillet

et al.

(1997)

and

was

recently confirmed

(Geary

et al., in

press). In the

latter study,

first and

second

srade

children

with concurrent

mathernatical

disorders

and reading

disorders,

mathematics

disorders

alone,or reading

disorders

lone were

compared o

their normal

peers. On

one of the

tasks, the

children

were

instructed

not to use

counting

orocedures

but onlv

use retrieval

iechniques

o find

solutions or

simple

addition

problems. Children

in all

iearning disorder

groups

committed

more retrievalerrors

han their normal

peersdid,

evenafter

controlling or

IO.

I he most common

error

was a

counting

string associate

f one

of the

addends.

lor instance,

common

retrieval errors

for the

problem 6+2

wereT and

3, the numbers

ollowing 6

and

2, respectively,

in the counting

seouence.

The

pattem

acrossstudies

suggests hat inefficient inhibition of

irrelevant associations

ontributes

to

the retrieval

difFiculties f children

with

mathematical

disorders. The

solution

process s

efficient

when irrelevant

associations

are

inhibited and

prevented

from entering

working

memory. InsufFicient

nhibition

results

in activation

of irrelevant

nformation,

which functionally

lowers

working

memory

capaciry.

In this

view.

children with

mathematical

disorders

may make retrieval

errorsbecause

hey

cannot

inhibit irrelevant

associations

from entering

working memory.

Once

in

working memory

theseassociations

either

suppressor compete

with

the

correct

association

i.e. ,

the correct

answer)

or expression.

Disruptions in the abii iry to

retrieve

basic facts

from long-term

memorv.

whether the cause

is

accessins

difficulties

or the lack

of

inhibit ion

of

jrrelevant

associations,

might,

in fact, be considered

a defining

feature

of mathematical

disorders

(Geary

1993).Moreover,

haracteristics

of these

etrievaldeficits

e.g.,

olution

times) suggest

hat for

many children

these

do not reflect

a simp le develop-

mental delay but

rather a

more per-

sistent

cognitivedisorder.

Visuospatialkills

In a variety

of neuropsychological

studies. soecif ic

diff icult ies

with

visuosoatial

killshave

beenassociated

with dyscalculia,

ith specific eference

to spatial

acalculia.

The particular

features

associated ith spatial

acalculia

include

the misaiisnment

of numerals

in multi-column

irithmetic

problems,

numeral

omissions,

numeral'

otation,

misreading

rithmetical

peration

siSns

and difficulties

with

place

value and

decimals

see

Geary

1993).Russell

nd

Ginsburg

(1984)

found

that fourth-

sfade

children

with mathematical

iisorders committed

more

errors han

thei r lO-matched

normal peers

on

complex

arithmetic

problems

(e.g.,

34x28).

These errors

involved the

misalignment

of

numbers

while writing

down

partial

answers

or errors

while

carrying

or borrowing

from

on e

column

o the next.

The children

with

mathematical

disorders

appeared

to

understand

he base-10 ysiem

as

well

as he normal

children did, and

thus the

errors

couldnot be attributed

o

a poor

conceptual

understanding

of the

structure

of the

problems

(see

also

Rourke

& Iinlayson,

I97B).

Other

studies uggest

hat spatial

deficits

will

also ntluence

the ability

to solve

other

types

of mathematics

problems,

such

as word

problems

and certain

ypes of

geometry

problems

Geary.

199Q.

In

elementary

school, however,

this

subwoe

of mathematical

isorderdoes

not

apoear o

be as common

as the

other subtvpes.

continued

n

yage

Q


5/14

MathematicalDisorders:An

Overview

or Educators

continued

from

yage 8

Conclusion

As a groupl children

with

mathematicaldisorders how a normal

understandingof number concepts,

conceptsunderlying arithmenc algorithms,

and most countingprinciples. At the

same ime. manv of thesechildrenhave

difficulty keeping information in

working memory while monitoring the

counting process and seem to

understand

ounting

onJy

as

a

rote,

mechanical rocess

i.e..

counting can

only

proceedwith

objects

counted n a

fixed order). When solving simpie

arithmetic problems,

young

children

with mathematical disorders use

developmentally mmature procedures

and commit many more errors in the

execution

of these procedures. Since

many of these children eventually

develop efficient counting procedures,

their difficulties n this area eDresenE

developmental elay. A defining eature

of mathematical isorders hat doesnot

appear o improvewith ageor schooling

is difficulty retrieving

basic

arithmetic

facts from

long-term

memory. This

memory deficit appears o result rom a

more generaldiff iculry n representing

information in or retrievine nformation

from

phonetic

and semanticmemory,as

weli as rom difficulties in inhibitinp the

retrieval of irrelevant

associations.

Finally,many childrenwith mathematical

disorders avediff icult iesn organizing

the sequenceof steps needed to

successfully ursue formal algorithms.

Future studies will, no doubt, clarify

thesepatterns

nd ay the foundation or

remedialstrategies.

References

Barrouilleg P., fayol, M., &

LathuLibre, .

(199n.

Selecting

between ompetiton multiplicanon

tasla: An explanation f the errors

produced y adolescentsith leaming

disabfities. ntemational

oumal

of

Behavioral

eveloVment,'l 253-275.

Brian,

D., &

Siegler,

.

S.

1984).

feanrral

analysisof preschoolers'ounting

knowledge.

DeveloVmenalsychology,

20,

fi7-618.

Iuson,

K.

C.

(1988).

Children'sluntingnd

anceps f

number

ew York Springer-

Verlag.

Geary,D. C.

(1993).

Mathematical

disabilities:ogutive, europsyctrologrcat

and genetic

omponents. sychologial

Bulletin,4

,

345-362.

Geary,D. C.

(1994).

Children's

athematial

develoVment:esearch

nd

Vractical

apyliations.

Washington,

DC:

American

sychologicalssociation.

Geary, . C.

(1996).

Sexual electionnd

sex differences n mathematical

abiliues.

Behaviual

nd

BrainSciences,

49,229-284.

Geary, . C.,Bow-Thomas,.C.,&Yao,Y

(1992). Countingknowledge nd

^ r . ; r f : - - ^ - ^ : L ; . . .

a d d i t i o n : AN t l l l l l r u 5 , 1 t l t l v (

^ ^ - - ^ - : ^ ^ -

^ r

n o r m a l a n d

v l l l P 4 r r D v r r

u r

mathematically disabled children.

Joumal

f Lweimenal ChildPsychology,

54,372-391.

Geary, . C..Hamson, . O.,& Hoard,

M.

K.

(in

press).Numericaland

arithmetical ogninonA longitudinal

studyof

process

ndconcept eficits

n

children ith leamrrg isabiliy. oumal

ofExVeimenalkh Psychology.

Gelman, .,& Gallistel, . R. (1978).

The child's

undersanding

f

number.

Cambridge,MA: HarvardUniversity

Press.

Cross-Tsur,, Manor,O., & Shalev, . S.

(1996).

Developmental

yscalculia:

Prevalence nd demographic

features.DeveloVmennlalicine nd

Chih

Neurology,8,25-33.

Hit h, G.

J.,

& McAuley, .

1991).

Workng

memory in childrenwith specific

arithmetical leaming disabfities.

British

ournal

f Psychology,2,375-

386.

Jordaq

N. C., & Montani,T. O.

$99n.

Cognitivearithmeticand probiem

solving: A comparison f children

with

specific ndgeneralmathematics

difficulties.

ournal

of Learning

Disabilities,

0,A4-634.

t-tght,l.C., Delries,

.

C., & Olsor5

R. K.

(998).

Multivanate ehavioralenetic

analysis f

achievementndcogmtive

measuresn reading-disablednd

control ivin paks.HumanBiology,

0,

215-237.

Rourke, . P, & Finlaysoq . A.

].

(1978).

Neuropsychologicalignificance f

variationsn pattemsof academic

performance:

erbal

nd

visual-spatial

abiliues.

ournal

of

Abnormal

Child

Ps ch lo y,b, 121-133.

Russell, .

L., & GinsburgH. P

(1984).

Cognitive analysis

of children's

mathematical ifficulties.Cognition

and nstruction,

,217

244.

Siegler,

.

S.,& Shrager,

.0984).

Suategy

choice n additionand subtraction:

How do children nowwhat o do?

n

C. Sophian

@d.),

Origins

of cognitive

. sbills (pp. 229-293).Hillsdale,NJ:

Erlbaum.

Torgeser5

.

K.,

Wagner, I(.,

Rose,

E.,

Lindamood,,Conway,

.,& Carvan,

C.

(1999).

revenbngeadngailuren

young childrenwith phonological

processing isabilities:

Group and

individual

esponseso instruction.

Joumal

of MuationalPsychology,

4,

s79-593.

David Geary

is a Professorof

Psychology nd Director of the training

Frogram

n Develoymentalsychology

t the

Universitv fMissouriat Columbia.

He has

more han 8J

publications

cross wide

range

of

toVics,ncluding earningdisorders

in elementary athematics.

is two books

have been

aublished

bv the American

Psvcholoeical Association, Children's

MathematicalDevelo?ment: esearch

nd

PracticalAy?lications

1994)

and Male,

Female: The Evolutionof Human Sex

Differences

1998).

Perspectives,

ummer

2000 9


6/14

Z\ MATHEMATICAL

EARNING

ROFILES

ND

a)l

\M\

DIFFERENTIATED

EACHING

TRATEGIES

lf-t

\V

ByMariaR.

Marolda ndPatricia .Davidson

Are

there

particular

mathe-

matical proTiles hat charac-

terize how students

learn

mathematics?

How does he understanding

of

mathematical

earninq

pro-

files translate

into

Setter

instrutionalopportunities

in

mathematics?-

primary consideration in

the

teaching of mathematics

is the

recognition that students bring

to the

mathematicsclassroom

a wide ranqe

of abil it iesand

learningapproaches.

Extensive

instructional and clinical

investigations

uring he past20years,

as

well as a detailed researchstudy

(Davidson,

1983),have

revealed hat

students'

eaming profiles are

marked

by different

constellations f relative

strengths

nd relativeweaknesses

ith

which students

face the worid of

mathematics. ndeed, t

is

this

study of

dffirences, rather than a definition

of

explicit deficits, hat provides

a more

useful approach to understanding

students'effectivenessr inefficiencies

in learrungmathematics.

Moreover,an

understandine of

differences s also

informative in fashioning instructional

approaches

hat are compatible

with

the various earningprofiles

hat exist

in the mathematics lassroom.

Child

World

System

Learning

profiles in mathematics

can best be understoodby considering

a

Child/1X/orld

ystem

(Bernstein

&

Waber.

1990) that characterizes

he

reciprocal elationshipof the

developing

child and the mathematical world in

which the child must function.

The

construction of a Child/Wold system

focusses n differences mong

earners

as well as differencesn

the demands

of what is to be eamed. It then expiores

instances f

natch andnisnatch. n the

consideration of differences among

students, he cntical

question becomes,

"When

does a learning

difference

rendera

student earningdisabled?"

A

10 Perspectives,

ummer

2000

leaming

"disability"

in mathematics

may be thought of as heoccurenceof

multiple

"mismatches"

and the

inability to overcome hose

mismatches.

A tantalizing

issue then becomes

whether specific

pproaches r sffategies

could be used so that

the mismatches

are minimized and the disabiliry

is

resolved r disappears.

It is imporiant

to recognize

hat

the diagnostic

process n education s

quite

different

from the diagnostic

process n medicine. Whereas

the

medical diagnostician

is lookrng to

uncover what

is wrong and what the

patient can't do, the educator must

strive to uncover

the student's

I

The

construction

f a

ChildlWorld

ystem

focuses n differences

among eamers

as well as

differencesn

the demands

of what s to be earned.

strengths nd

what the studentcan do.

The goal of the educator

s to find

those strengths

hat can be used

to

address

he weaknesses nd difficulties

inherent n st udents' earningprofiles.

In focussins

on the Child in

the

a

Child/\X/orld

ystem of mathematics,

a

multidimensional

view must be

taken

and a variew of

parameters

onsidered.

Specificaliy,

-the

following

factorsshould

be explored

n order to understand

a

student'sMathematicalLeaming

Protile:

o

the

presence f speciftc evelop-

mental eatures

hat are

prerequisiteo

specific

mathematics opics;

.

the preferredmodels

with which

mathematical opics

are nterpreted;

r

the

preferred approacheswith

which mathematical

opics are pursued;

.

memory skills that

affect

students'

abil ity to participate

in

mathematical activities;

o

language skills that affect

students'ability to participate n the

mathematical rena.

Developmental

Features

of

Mathematical

Learning

Profiles

A definit ion of a student's

mathematical

eaming profile

should

incorporate

an appreciation of

the

developmental

maturiw of

studentsat

varioui ages.There a..

-".ry develop-

mental

milestones n terms of

mathe-

matical

readiness for dealing

with

numerical, spatiai and logical topics.

For numerical concepts,

he develop-

mental

milestones consist of

an

appreciation

f number, he concept

of

number

(enumeration/cardinality),

conservation

of number, one to

one

correspondence

and the principles

of

class nclusion.

For spatialconcepts, he

construct

of space. conservation

of

length and conservation f

volume must

be considered.

Ior logical

thought,

deveiopmental

milestones include

the

concepts

underlying classif ication,

seriation, associativity,

eversibility

and

inference. Most chiidren between the

ages of four and

eight have acquired

thesemilestones.

Recent

ciinical

investigation and

teaching

practicehave

suggested hat

the concept of

place value might also

be

developmentaliy

mediated

Marolda

& Davidson,

1994).That

is, an appre-

ciation

of placevaluedepends

more on

the

state/ase of the child

than on

specific .ulhi.tg experiences.

f the

child is not cognitively

ready to deal

with place value, hen the concept

of

place value cannot be

formally or

meaningfullydeveloped,despite eaching

efforts.

The formal concept

of

place

value eems o be established

or most

children beNveen he ases

of six and

eight. The appreciation

of

formal

place value concepts

s of particular

importance

since they

are necessary

prerequisites or the

understanding

f

larger quantities

and the pursuit

of

multi-digit

computation

(onilnued

npage

|


7/14

Mathematical

LearningProfiles

and DifferentiatedTeaching

Strategies

continued

from

Vage

40

PreferredModels

and

Preferred

Approaches

of

Mathematical

Learning

Profiles

Mathematical situations

can be

interpretedwith concrete, ictorial,or

symbolic models.

For a particular

student, a specific nterpretation

might

be more comfortable

and meaningful.

Among concrete

models, further

distinctions can

be made. Within the

concrete

mode, students may prefer

set

(discrete)

models,suchas counters,

while others

appreciate perceptually

driven

(measurement)

odels,such as

Cuisenaire ods.

The ways in which

students

process or approach

mathematical

situations ollow

nvo distinct Datterns

(Marolda & Davidson, 1994). Some

studentsprocess

situations n a linear

fashion, building forward to an

exact

finai solution. Sometimes,

these

students are

so focuse d on the

individuai

eiements that the overall

thrust

or goal s obscured.This

styleof

processings often

characterized s a

sequential, tep-by-step

pproach. For

other students,a careful

building up

approach hoids

litt le inherent

meaning. Such

students

prefer

to

establish a

general overview of a

situation

first and then refine

that

overview successively ntil an exact

solution emerges.

Such studentsmay

be prone

to imprecision and

tend to

lack appreciation

f all relevantdetails.

l h r c c t \ / l P o t n r n c p s s i n o i q n F t c n

described s globai

or gestalt.

Incorporating

these inherent

preferences

n terms of models and

processing as

ed to the definitron of

two distinct

learning profi les in

mathematics, Mathematics

Learning;

Style andMathematics

earningStyle I

as reviewed in

Table 1

(Marolda

&

Davidson. 1994). Moreover,

it is

possible to describe mathematical

concepts and procedures

that are

inherently

compatible with each

I a r ' - i ^ - ^ ' ^ F i l .

To be full

and successful

participants

n

mathematics,students

must learn

to mobilize both

Mathematics

Learning

Style I and

Mathematics Leaming

Sryle II. The

student with

special eaming

needs,

however, is

often limited to one

learning style alone

and is unable to

Mathematics Leaming Style I

Preferred

odels

for

Number:

.

SetModels

Preferred yyroaches:

o

I i n p a r c fp n h r r c te n

.

Often relieson verbal

mediation

ToVics yVroached ith

Ease:

o

Counting orward & counting-on

o

Concepts f addition

& multiplication

r

Pursuit

of calculation rocedures

o

Fraction

oncepts terpreted n verbal erms

o

Ceometric hapes:

mphasis n rnming

Toyicsf Pantcukr hallenge:

.

Broader oncepts nd overarching nnciples

o

Estimation rrategies

.

Appreciationof appropriateness

f solution

generarco

.

Selection

f aritimetic operationn word

problems

dif iculty swrtthing

tretween perauonsn a setot word problerns

.

Concept f a fraction

o

More

sophisucatedeometnc opics

.

Requirementor flexibleor

alteirranve ooroaches

Mathematics

Learning Style II

Preferred odels

or

Number:

.

Perceptual

Measurement)

odels

Preferred yyroaches:

.

Deductive,

global

.

Often elies n successive

ooroximations

Toyics yyroached ith

Ease:

o

Counting backward

.

Concepts f subtraction

& division

.

Estimatlon

.

Fracbon onceptsrterpreted

n a

vanery

of

\,1SUalooets

.

Ceometric hapes: mphasis n spatiai

relationshios

ird manibulauors

ToyrcsfPattrularChallenge:

.

Appreciation f all salient etails f multi-

stip proceduresr word

problems

.

Pursuit f multi-step alculation rocedures

.

Relevance f exactsolutiorx; prefers o guess

.

Follow throush to exact

solutions n

word

problems, eipite

conectchoiceof operation

. Formal racuon perations,espite omfon

wlth underlymg racEon

oncept

o

Requiremento descnbe pproachn

exacung erbal erms

o

Insistence

n a single, pecific pproach

Table

mobilize skil ls and strategies

associated ith the altemative

eamrng

stvle. For success.

eachers must

trinslate

activities into the student's

operating ryle,building

a scaffold hat

integrates he areas

of strengthsand

weaknesses

o that thev comolement

oneanother nd ead o he

acouisit ion

of mathematical conceoti an d

proceduresn a meaningfulway.

The

foliowing charts

flables

2

&

3: as shown on pages13 and

14)offer

more explicit featuresof

each of the

Mathemitical LearningStyles

and can

be helpful in recognizing

them and

teaching o

them.

Memory

Skillsas a

Feature

of Mathematical earning

Profiles

Often students are characterized

as having difficulry in mathematics

because hey

"can't

remember."

The

attribution of mathematical

difficulties

to a global memory

deficit is

somewhat simplistic. Cognitive

psychologists

Hoimes,

19BB) uggest

that

memory issues, n general. are

very comprex.

In evaluating

a child's recall of

materials, the clinician

should

recogtize

he

various

components

of the process oosely

called

4

memory:

registration of the

stimulus,encoding,organization,

storage and retrieval....Learning

disabled children, however,

are

constantly described in the

psychologicaland educational

literature

as having memory

deficits of various types, usually

.visualor auditorv (shorc erm or

otherwise). In almost all cases,

the impairment

involves either

the initial encodins or the

effective etrievalof iriformation.

(p.1Be)

In mathematics, it is particularly

important to consider

the distrrctron

between encodingand

retrieval aspects

of memory. Is the student having

difficulty remembering the f act or

procedure

becauset was neverproperly

understood and therefore not encoded

for storage n memory? Or is the student

having difficulty remembering he fact

or procedure because t

cannot

be

aCcessedrom the student's epertoireof

leamedskills?

Four specitic memory skills are

important in mathematics

o

retrievalof solut ions o one dieit

facts;

.

the recall f the sequence

f

multi-stepprocedures;

continued n page

| 2

Perspec-tives,

ummer

2000 11


8/14

Mathematical

Learning

Profilesand

Differentiated

TeachingStrategies

continued

from Vage

4 4

.

visual memory

of perceptual

geometric

stimuli;

.

recall

of mathematical

data

presented uditorially.

In terms of retrievaldifficulties n

the

productionof solutions

o one digit

facts, mathematically

it may be

more

important

to consider

f the solution

s

produced

efficiently

rather than

automatically.

The distinction

that is

important

is whether

the retrieval

s

auiomatic

or efficient.

Difficuities

with the

retrieval of

one digit facts

may be supported

by alternative

strategies

hat are compatible

with

a

student's

inherent

learning style

and

result n more

efficient

production

of

solutions.

In the example, B

+

6, a

student with Mathematics Learning

Style I would be most

efficient tuming

to counting on strategies:

9,10,1I,12,

13...14 r strategies

hat build 10s:

B +

(2+4)

=

1.0+ 4

=

l4l A student

with

Mathematics

Leamrng Sryle

II would

be most efficient turrung

to related

facts,

.g.doubles, +B=16,

o 8+6=14

. . .2

Less Or 6+6=72,

so 8+6=14. . .2

More

In dealing

with multi-step

orocedures.

the recall

of the

organizationof the

specific teps

elies

on

an understanding

f the conceptual

foundations

driving he

procedure.By

offering

alternative

approaches hat

appeal o a specific

earningstyle,

he

orocedure is better

understood

and

more

easily pursued. In

dealingwith

the

multiplication problem

23 x 14,

a

student

with Mathematics

Learning

Style I

would tum to a

successrve

addition

approach or the

formal

algorithm.

Iurther

supports to

remembering he

steps of procedures

include encouraging

erbai mediation

techniques,

developing

verbai and

visual flow charts hat

can be used

as

referents,

and developing

mnemonics

to cue each step.

In contrast,

the

student

with Mathematics

Leamins

Sryle

I would turn to the definition

oT

multiolication as

an area and

would

then-combine the

area of the four

subregions

to determine

the final

solution.

Further supports

would be

estimation

echniquesmade

teratively

o once

an initial estimate

s made,

he

use of

a calculator for an

exact

solution.

With firm

understanding

12 Perspectives, ummer

2000

established.

he

procedure

s more

effectively ncoded.

That understanding,

however, may

emerge

from different

a0Droacnes.

- -

In dealing

with geometric designs,

students need to use visual memory

skills. With

visualmemorv

difficulties,

students may

find the buiiding

and

copylrrg of geometric

designs

hallengng.

To support

visuai memory

difficulties

students might

be encouraged

o

interpret

geometric

designs n

verbal

terms. Difficulties

in visual

memory

can also manifest

themselves

n non-

geometric situations,

such as difficulties

orienting

written digits,

difficulties

aligning

numerals n

written procedures,

and difficulty

organizing a page

of

problems. Copying

problems rom

the

text and the board or interpretingdata

presented on a computer

screen may

also

be difftcult. In

response, opying

requirements

should

be minimized,

while graphic

organizersmay

be offered

to support

he copying

hat is required.

Students

with

auditorv memory

difficulties

are challenged

when required

to remember

ail relevantdata

presented

in

instruction,

remember the

overall

outcome sought,

remember

directions,

or remember

all the relevant

infor-

mation

in word

problem

situations

presented erbally. These

students

may

be supported by offering directions n

visual formats

as well

as by offering

written directions and

/ or allowing

students

o write down

the directions

and then referring

to the

written text

asneeded.

Interestingly tudents

with

apparentauditory

memory

issuesare

often confused

with students

whose

primary difficulties

are in

language

where memory

difficulties

are

secondary

to specific

language

pro-

cessing

ssues.

Languagessues

Language

skills,

both oral and

written. are

imoortant in

mathematics

in terms of:

.

word retrievalskills;

o

verbal ormulation

esuirements;

r

comprehension

equirements.

They

become an

issue when

students are

required to

retrieve the

names

of coins,'geometric

shapes

or

other mathematical

terms,

when they

are asked

o explain heir

solutionsor

approaches,

hen they

must deal

with

lengthy

verbal presentations

ypical of

classroom

nstruction,and

when they

are

acedwith

word problems.

These

language emands

have become

more

prominent in mathematicsaseducation

curricula

and textbooks

have encour-

aged teachers

to ask students

fo r

eiplanations

or

iustifications

of their

approaches.

Moreover,

eachers ave

been encouraqed

o

ask students

to

take

t.rpot Jbility

for their

own

learning by

reading printed

materials

or texts.

These newer

emphases ose

particular

challenges or

students

with

language

ifficulties.

In orde r to address

word retrieval

difficulties

in mathematics,

students

might focusprimarily

on the

valuesof

the coins rather than their specific

names,

might be encouraged

o

draw

geometric

shapes

rather than

name

them

and might

be offered ecognition

formats

when dealing

with mathe-

matical definitions.

Retrieval

issues

are

further supported

by minimizing

confrontational,

astanswer

ituations.

Students

with verbal

formulation

issuesoften have

difficulty describing

their approaches

r in

portfolio work

where

approachesmust

be written

down.

To

rupport these

students,

alternate

forms

of communication

should be encouraged, including

demonstrations

with physical

modeis

and

use of

pictures or diagrams

o

describe

olution

processes.

Students

with

comprehension

difficulties

often

have difficulties

with

directions

and with

reading texts

or

word

problems. They otten

can't get

started

with classwork,

mistakenly

suggesting

they have

attentional

difficulties.

These

students benefit

from

careful monitoring

of

new

presentations

and having

word

problems

ead o them.

Suchstudents

iun be supported by teachers

presenting content

in

meaningful

"chunks"

that

are then carefully

linked

together.

In t erms of

word problems,

the

situations

can be

presented

verbally

rather han

requiring eading.

Students hould

hen be encouraged

o

draw

pictorial interpretations

to

reDresent

the situation

and data

involved.

continued

n yage

43


9/14

Mathematical

LearningProfiles

and DifferentiatedTeaching

Strategies

continued

from

page 4

2

Mathematics Learning Style I

Coenitive & Behavioral

"

Correlates

Mathematical Behaviors Teachins Implications

& Strat6gies

.

Highly eliant

n

verbal

kills

.

Approachesituations sing ecipes;

t

; ^ 1 1 . , L - ^ , , - L r , ^ ^ t .

6'^J L'uuuBl LaJNr

.

Interprets geometric designs verbally

.

Emphasize

he meaning of each

concept or

procedure

n v?rbal terms.

.

Build on subvocalization strategies

tn Ai rerr nrnrpd' ' .ac

.

Tends o

focuson individualdetails r

single spectsf a situation

.

Seeshe

"trees,"

but overlooks

the

"forest"

.

Approachesmathematics n a mechanical,

rddtine based ashion

.

Overwhelmed in situations in which there

are,multiple

considerations, uchas n

multl-step tasl(S

.

Can generate orrect.solu tions, ut may not

recognrze when solutrons

are

lnappropnate

.

Diff icu lt ies."checking'

ork;

must re-do

entlre

problem

.

Diff icu lr ies.choosingn approachn

word problems

Difficulties.

appreciating arger geometric

constructsbecause f an emphasls

on

componenr

pans

.

Highlight concept /overall

goal.

.

Break down

complex tasks nto salient

units and make li hkase berween

units

expuclr.

.

Bui ld simpleestrmarion

rrategies;

encourageNvo hnal steps o eachcalculabon

problem:

Does

this arsiver the

quesuon?"

and

"Does

the

solurjon seem ighr?"

.

Encouragestudents

to rewrite or state

problerns

n their own words.

.

Develop metacognitiv e trategi eso

analyze

word

problem

Srtuatrons.

.

Encourage

arts

o

wholes

approach n

build ing eometr ic igures.and xpl ic i t

descnp[ons ot the overalldeslgn hat emerges.

.

Prefers

OW to

WHY

.

Prefers

umericalapproachover

manipulative

moddls

.

{\e.eds

rill and pracrice.to

establish

rocedure

Derore

onstoennS ppllcauons

r nroaoer

conceprualmeanlng

o

Link manipulativemodel on a step-by-

step basis o the numerical

procedur6.

.

Once

procedure

s secure, elatemath

topics ' to e l evant eal i fe s ituat ions.

.

Relieson a definedsequence f steps

ro pursue

a

goal

.

Relianton teacher or THE approach

.

Lack of versatility

.

Prefers

xplicit elineationf each tepof a

procedure

nd

linkage

f steps ne o another

.

Vulnerable hen

here remultiple pproaches

f n : c i n o l e t n n r r" - " "_ b ' - - " r ' -

.

Overwhelmed by multiple models or

multiple

approaihes

.

Prefers inear approaches or ari thmetic opics

.

Offer flow chart approaches.

.

Help students reat6 andbooks

i th

proiedures

escribedn thei r own

words.

.

Choose one manipulativemodel or

approach o.developa wide. angeof

toprcs; vord wrtchlnSmodels

or

approaches oo

qurckly.

.

Don't emphasizespecial ases; ather

develop h over-r id ing ule hat applies

to a l l cases; .g. or t f ie

addit ion nd

subtractionof lractions

with

unlike

denominators,develop

a single

process

using he

product

of the den6minatorsn

all cases, ven f i t is not the Ieast

common denominator.

.

Cive explanations efore or after

procedure,

ut not while st udent

ls

pursulng rocedure.

.

Us'e ount.-Gon techniques or addition

factsand

missinS

addend echniques

lor subtractlon acts.

.

lnterpret multiphcatronas successive

adchtlons.

.

Challenged y perceptual

demands

.

Difficulties

with

more sophisticated

erceptual

models, uchas Cuisenaiie ods

.

Ceometr ic.act iv i t ies.maye challenging.

esDecral lv

n hreedrmensrons.

.

Difficulties nterpretinganalogclocks

.

Di f f icu l t ies

ist inguishingoins, specia l ly

rucKel no ouaner

.

Difficulties brganizing written formats

.

Emphasize et

(discrete)

models

or

counthg,

such

as money or counting hips.

.

I ranslate

erceptual

ues n

terms of

verbal'descriptions.

.

Prefers

uizzes

or unit

tesrs o more

comprenenstve nal

exams

.

Mav be able to comolete the most

difficult

exampie n a set of dxam ples elyi ng on the

same concepVskjll, ut has diffiiulry switching

to a new topic or new approach

.

Spiral ll opics

o

keep

hemcurrent,

continued n yage

411

Perspectives,

ummer

2000

I AAte

I J


10/14

MathematicalLearningProfilesand Differentiated

Teaching

Strategies

continued

from

yage 43

Mathematics Learning Style Il

Coenitive & Behavioral

Correlates

Mathematical Behaviors

Teaching Implications

ag

trategres

.

Prefersperceptualstimuli and often

relnterprets abstract sltuahons

visual$ or

pictorially

.

Benefits rom manipulatives

o

Lovesgeometric opics

.

Offer a variew of models: ntroduce

perceptr-ral

models, sud-ras BaseTenBlocks r

Cuisinaie Rols, o suppon calculations.

.

Emphasize eometry'dsa

viul

part

ot

fhe

curnculum.

.

Likes o deal

wrth

big deas; oesn

want

to be botherefwith details

.

Prefersconcepts to algorithms

.

Tolerates mbiguity

and imprecision

.

Offers impulsirie

guesses

s'solutions

.

Usesestimationstrategies pontaneously

. Skimsword problems irst but must be

encouraged o re-read for salient details

.

Perceives ver all shapeof geometnc

configurations t the expenseof an appreciatior

of th e individual comoonents

.

Relate manipulative models to

procedures

efore

practicing

algorithms

.

Keward

approachas

well

as precrse

solunons.

.

Develop an appreciation f how much

preclslona sltuatlon

warrants.

.

Reward/encourage

estimation strategies

as hrst step.

.

Encouragediagramsas a technique to

organtzedata n

problem

solvrng

situations.

. Aloy calculators o support problem

solvmg.

o

Encouragemultiple refinements

when

building

geomet'ric

esigns n order to

incorpola-"tell the indiv-idual

arts.

.

Prefers

WHY

to HOW

o

Reouires

a definition of overview before

dealingwith exacting

procedures

.

Requir?s

manipulatiriemodeling before

developinea cbnceptor algorithm

.

Likes t6 se"t p proSlems, b"ut esists oilowing

throush to a conclusron

.

Offer

opportunitieso work in

cooperatlve

roups.

.

Prefers

onsequential

pproaches,

.

involving

pattems

and intenelationships

.

Prefers uccessive pproximationsapproach

to formal alsorrthms

.

Addition an"dmultiplication acts nvolvlng 9s

more readily

generited

because f underlying,

patterns

hat are recogruzed ut not

verbauzed

. Not t roubled bv mixe-doracticeworksheets

.

Comfortable with horizontal formats for

long calculations

.

Can offer a

variery of alternative nswers

or approaches o I single

problem

.

Can appreciate perationneeded

n

a

word

problcimbut has difficulry

following through

to an exact solutron

.

Likes ogicalproblem solvrng

n the form

or

general

easonlnS

roolems

.

Allow

altemativecalculationprocedures.

.

Help

students

to create heir own

handbooksof rypical

problems.

o

Ceneratearithm6tic dcs through

relationships to known facts;

e.e.doubles or + facts.

.

Einphasize area model

for multiplication.

.

Start

with real-life situation and tease

out more

formal

arithmetic

opics.

.

Use simulations, relating simildr concepts/

approaches

n a varieryofdifferent situau6rs.

.

Mbdel complex

problems

wi th s imi la r

problems

n simirler orms.

.

Cive Nvo grades on word problem

activiries; ne

for

correctapproach:

one for exact final solution.

.

lnclude

general

easoning xamples

n

logical

p"roblem

olving ctivities.

.

Challengedby demands

or deuils or

the reou-=iremintf orecise olutions

.

Difficultieswith

precise

alculations

. Difficultiesofferiire ationale or

correct olutions

-

.

Encourase tudentso describehe

approach r conceptual.nderpinnig

even t hev cannotmobllrze

n

exacung

roceoure.

Prefersperformance

based or

portfolio

t 'pe assessmento typlcal tests

Prifers comorehensivd exams to

. l

qulzzes

and unlt tests

More comfortable ecognizing orrect

soiuttons han

generattng

hem

.

May be overwhelmed when faced with

multiple examples

.

Corsider a

variery

of assessment

echniques

.

Allow oral Dresentations.

.

Do not alw'avs reouire exact

solution but

sometimes

g'rade

homework and tests

only for corlect

approach.

.

Indude somemuluirlechoice tems on tess.

14 Perspectives,

ummer

2000

Table

continued n page

'l

5


11/14

Mathematical

LearningProfiles

and Differentiated

Teaching

Strategies

continued

from

page44

Conclusion:

The

Child/World System allows

teachers

o achievean understanding

of the dynamic interplay

that affects a

student's

earning n mathematics.

It

leads to the delineation of specific

Mathematical Learning

Profiles.

Extensive

clirucal investigations

and

classroom instruction,

along with

rigorous research

efforts, have

corroborated

he presence

f speciFic

Mathematical

Leaming

Profiles. hose

learning

profiles

nvolve differencesn

deveiopment swell

as

preferences

or

modelsand preferences

or approaches.

Complicating the

consideration of

iearrung

profiles in mathematics

are

more

general memory and

language

issues that intrude

on

efforts in

mathematical ctivities.

The understanding

f Mathematical

Learning Profiles

helps teachers

offer

specificapproaches

nd strategies

hat

make use of students'

areasof relative

strengths, that

minimize

areas of

vulnerabiliry

and that support

areasof

specif ic

deficit,

ensuring the

comfortable

participation

and growth

of all students

in the mathematical

arena.

The importance

o teachersof

understanding

Mathematical

Learning

Profiles is that

they lead to the

development

of more effective

learning

strategieswhich, in

turn,

allow

more students to

experience

successn the domain

of ma*rematics.

References

Bernstein,

.H.

& WabeS

D

(1990).

Developmental

europsychological

assessment:The systematic

approach.

n A.A. Boulton,

G.B.

Baker &

M. Hiscoc

(Eds.),

N euromethods

pp.31I-7

).

Clifton,

NJ:HumanPress.

Davidson,P.S.

1983).

Mathematics

learningiewed

from

neurobiological

model

for

ntellectual

funaioning

NIE

Crant No.

NIE-G-79-0089).

Washington,

DC: National

Institute f Education.

Holmes,

.M.

(1988).

Testing.

n R.

Rudel,

.M.

Holmes&

J.R.

Pardes

(Eds.),

ssessmentf Developmental

Disorders:

A neuroysychological

aVproach

pp.1,1,6-201).

ew

York:

Basic ooks.

Marolda

M.R.& Davidson .S.

1994).

Assessingmathematical

bilities

and learning

approaches. n

Windowsfo47ortuility.

eston,

A:

National

Councilof Teachers

f

Mathematics.

Maria

Marolda and Paticia

Daddson

have collaborated

n teachine

materials,

curriculumdevelopment

anl

staff develoVnentn nathematics

ince

4

968

and on assessmelttesearch

nd

diagnosticesting ince 977. Marolda s a

Research ssociate t Harvard

Medkal

School

nd

seniormathematicspecialist

t

the Learning Disabilities

Program at

Children'sHosVital n Boston.

Davidson s

ViceProvost

for

Academic

uyyortServices

and yrofessor

of mathematics t the

Universityof Massachusetts,

oston,and

has

beena chiefexaminern

mathematics

and chair of the Examining

Board of the

International

Baccalaureate

diVloma

Frograffi.

BothMarolda and Davidson re

authorsof articles,mathematics

urricula

and yrograms as well

as alternative

assessmentnstruments.

Visit he

IDAWebsite

ItlXtlt.it

terdys,ofg

Permission

To

Copy From

PERSPECTIYES

Perspectives,

ummer2000 15


12/14

TRANSLATING

ESSONS ROM

RESEARCHNTO

.JL

MATHEMATICSCIASSROOMS

ZII

\Un

Mathematics nd Special

Needs

Students

17

ByDouglas

.

Clements

Too often students with leaming

I disabiiities eceive imited mathematici

instruction.This is due n

part

to special

education

teachers feeling uncom-

fortable

teachins mathemitics. This

leads to an overimphasis on

training

skills. There are three reasons or thii

focus

on skills. First, there is a major

misconception

hat skill learning s ihe

bedrock of mathematics.uooriwhich

all further mathematicsmubt

be built.

Second,

kills are

easier

o measure nd

teach.

Third, teachers ften believe hat

students'

perceived memory deficits

imply t he need for constant epetition

ano orlll.

Lessonsrom Research

Decadesof research

ndicate that

students anand should solve

problems

before

they have

mastered

procedures

or algorithms traditionally

used o solve

these

problems

(National

Council

of

Teachers f Mathematics,

000). f they

are given opportunities o do so, their

conceptual nderstanding nd abiliry

to

transfer knowledge is increased

e.g.,

Carpenter,Franke,

acobs,

ennema,&

Empson,1997).

-T ndeed . som e o f t he m os t

consistendy successfulof the reform

curriculahive been

orosrams

hat

o

build directlv Jn students

strategles;

.

provide

opportunities

or both

tnventron

ancl

practlce;

o

have children analyzemultiple

strategles;

.

ask o-rexolanations.

Research valuatioi-rsf these

programs

show that these curricula

'facilitate

conceptual

qrowth

without sacrificinq

skil ls

'and

ilso helo students ieari

concepts

(ideas)

arid skil ls

while

problem solving Hiebert,1999).'

What is re"markable

s that similar

principles

apply to students

with

learning disabiliiies. Many children

classified s eamins disabledcan earn

effectively with

qrialiry

conceptually-

oriented nstruction

Paimar

& Cawley,

1997).As the Pinciyles

and Standards

or

SchoolMathematics TTustrates

National

Council of Teachersof Mathematics,

2000), a balanced and

comprehensive

instruction,using he child's-abilitieso

shore up weak-iesses,

rovides

better

long-term esults.

or exlmple, st udents

beneficial in meeting the needs

of all students. Studdnts who use

manipulatives

in their mathematics

classesusually outperform those

who

do not

(Drisioll,

19ffi; Greabell,

1978;

Raphael

& Wahlstrom, 1989; Sowell,

1989: Suvdam. 1986). Maninulatives

can be

pirticularly

heipful to students

with

learnine disabilities.

Somewhat

sumrising,manipulatives

do not necessarilv'have"io e

'phvsical

objects. Computer. manipulatives can

provrde

representatronshat are

ust

as

bersonally

meaningful to students.

Paradoxicilly, comp,iter representations

may even be more manageable,

f leiible, and extensible than" their

physical

counterparts

(Clements

&

Miuilt.n. 1996,.'students who use

phvsical

and software manipulatives

bemonstrate a

greater

mathematical

sophistication

"than

do control

gr6up

students

who

use

physical

frranipulativesalone

(Olson,

IgBB).

Good

manipulativesare those that are

meaningful- to the learner,

provide

control

"and

flefbility to the

'learner,

have character istics' hat mirror, or

are

consistent with coenitive and

mathematicalstructures, id assist he

leaJnern making,connectionsetween

variouspiecesand types of knowledge.

lor example, computer sorrware can

dynamically

connett

pictured

obiects,

slch

as bise ten bloiks, to symbolic

representations. omputer

manipulatives

cah

play those oles.They he lp children

generalize and abstracl expe riences

ivith physicalmanipulatives.

Recommendations

or

Classroom

Practice

Research

provides

several ecom-

mendations for meeting the needs of

all students n mathematlcseducation.

4. KeeV exyectationsreasonable,

but not low.

Low exoectations are esoeciallv

oroblematic 6.."ur" students *tro ti"'.

in

poverry,

studentswho are not native

spiakers-of English, students

with

disabiiities, females, and many non-

white students have traditionally been

far more likely than their counterparts

in other

.

demog.raphicgroups

to be

the vrctrms ot low exDectatlons.

Exoectationsmust be raisdd

because

"rnathematics

can and must be learned

continued n

page

32

Perspectives,

ummer

2000 31

may benefit iess rom intensive drill and

prattice

and

more

from

help searching

For, finding., a.nd us]ng

fatterns

ii

learruns he baslc

numbercomblnatlons

and aiithmetic

strategies

(Baroody,

1996).

Many of the lessons we have

learned

-from

research for

general

educa t i on s t uden t s

app l y , - w i t h

modificationsof course,

'tb 'students

with soecial needs as

well.

A

particulaily

important one is

"less

is

more."

That

ii. in mathematics and

science,

we

have found

that

sustained

time on fewer bey coftce4ts

eads to

sreateroverall student achievement n

I

Decades f

research

indicate hat students an

andshould

olue

roblems

before

heyhouemastered

procedures

r algorithms

traditionallyused o solue

these

roblems.

the long run. Compared to other

countries"hat sienificindy outperform

us on tests, LJ1S. urri?ula do not

challenge students to learn important

topics in depth

(National

Cenier

for

Education Statistics, 199Q. We state

many more ideas n an ave rage esson,

but

develop

ewer of them, dompared

to other countries

(Stieler

&

Hibbert,

Y

Iyyyl. Inus. u.5. studentswould De

betteroff focusing

on in-depth study on

fewer importanf concepis. Such an

aooroach s critical with studentswith

ldarning disabil it ies. They

,

need to

concentrate

n mastenng he Key cleas,

and these ideas are iot arifhmetic

algorithms. Even

proficient

adults

use

relationships nd itrategies o

produce

basic fact's. Thev ten-d not to use

traditional paper-and-pencil

algorithms

when computlng.

Anoth'er research esson is that a

variety of instructional materials is


13/14

Translating essons

rom Research

nto Mathematics

lassrooms

continued

from

yage 34

by all students"

(NCTM,

2000).

Raisine

standards

ncludes increased

empha"sis n conductingexperiments,

authentic

problem

iolving, and

proiect-based

earninq

(McLIuehlin.

Nolet, Rhim, & Hende"rson, 999\.

2. Partently

hely students

develoV concepttal understanding

and

skills.

Students who

have difficulw in

mathematics may need

additibnal

resources o support

and consolidate

the

underlying- oncepts

and skil ls

beine learied."

They' benefit from

multiple experiences ith

models and

reiterition bf the

linkage of models

with

abstract,numerical

ranipuiations.

Expand time

for mathematics. n

general,

the tradit ional

curriculum

Zioes ot

allow adequate ime for

the

many. instructional ,and .learning

strategles

ecessaryor the mathernahcal

succeis of leamine

disabled students

Qemer,1997).

Students with

disabilities mav

also

need increased ime to

compleie

assignments.

Finally, they may also

benetlt trom more tlme or tewer

examples

on tests or from

the use of

oral rither

than written assessments.

3. Build on

children's

?.tr?ngths.

This statementoften is

litde hore

than a trite

pronouncement.

Bu t

teachers an reinvigorate

t when they

make a conscientioirseffort to build oh

what

children know how

to do,

relying

on children'sown

strengths

o

addreis heir deficits.

4.

Build on children's

infotmal

sttategies,

Even severely earrung

disabled

childrencan nvent quite

sofhisticated

counting

strategies

Baroody.

1996).

lntormal

strategres

rovlde

a startlng

place or develoipingboth

oncepts nil

proceoures.

5.

Develop skills in a

meaningfal

and yuryoseful

fashion.Practice s important,

but

practice

a t t h e

p r o b l e i n

s o l v i n g ' l e v e l

is prefer ied whenever p-ossib le.

Meamngfuf purp,oseful

racti.S

gives

us two tor th e Drlce ot one.

Meaninsless

drill mhv actuallv be

harmful"to

these children

(Bar6ody,

1999:

Swanson& Hoskyn,

1998).

6. Use manipulatives

wisely.

Manipulatives

can help leaming

disabled

siudents eam both

concept:

and skills

(Mastropieri,

Scruggs,

-&

32

Perspectives,

pummer

000

I

In

general,

he traditional

curriculumdoes otallow

adequateime or the

manu nstructional

nd

leamingstrategies

necessaru

or

the

mathematical uccess

f

learning

disabled tudents.

Shiah, 1991).

However, students

should not learn to use

manipulatives

rn a

rote manner

(Clements

&

McMillen, 1990. Make

sure students

explain what they

are doine and link

th6ir work wit6 manioulatives to

underlying conceptsand Formalskills.

7. .

U:,

technologt

dyly.

It rs rmDortant that all students

have

opportunities o use technology

in appropriateways

so that they have

access

o interesting and important

mathematical

ideis. Acceis to

technology must not

become yet

another

dimension of educational

inequity (NCTM, 2000). Computers

can

servemany

purposes

Clements

&

Nastasi, 1992; Mastropieri

et aI., 1991.;

Pagl iaro.

99B: Shaw, Durden,

&

h Y , h ^ ^ \

Baker,1998).Computerswith voice-

recognition

or voice-creation

oftware

can

6ffer teachers nd

peers

access

o

the mathematical

deasand arquments

developed

v studentswith disabilities

who would

otherwise be unable to

share their thinking.

Computers can

also

serve as a val"uable xtens ion

o

traditional manipulatives

hat might be

particularly

helbful to

special eeds

itudents

i . f .

Weir, 1987). '

Students should learn

countins

and arithmeticalstrategies

ut should

also earn to use calculators

or some

pfrposes

(Leqer,

1.997). or students

who

can demonstra te a clear

understanding of an operation,

the

calculatormieht be the

prrmary

means

of computation

(Parmir

& Cawley,

1,997\.

8. Make connections.

Integrate concepts

and skil ls.

Help children link

iymbols, verbal

d e s c r i p t i o n s .

a n d w or k w i t h

manipulatives.

Use every

possible

socialsituation

o

provide

me'anineful

situations for mathematical oroblem

solving

opporturuties

(Baroody,

1999;

Parmai & Caw\ey,1997).

,. Adiust instructional formats

to individual learning

styles or

sVecrfic earning

needs.

Formats

might include

modeling,

demonstration,

nd feedback;guidinlg

and teaching

strategies;mnemonji

s t ra t e . g i es o r l ea rn i ng

num ber

com b lna t l ons : nd Dee rm edra t l on

(Gers t en ,

1 , 985 ; Le rne r ,

1 , 997 ;

M as t rop i e r i

e t a l . , 199 I ) .

Use

prolects and

,games

to help

, the

teacher

gurcle

eamlng, rather

than

relying

iolelv on

"tellinq"

Garoodv,

. ^ ^ , v - 1

.

Iyyy). Ihe tradlt lonal

sequence f

direct teacher explanations,.

trategy

lnstructron, relevant practrce, an d

feedback

and reinforcement s often

effective.but the

potential

of students

to learn through

problem

solving

should not be'ignored..Too

often]

direct instruction approaches

queeze

out other

possibil idies.

Use

'direct

instruction

-onlv

when students

are

unable

o inverit their own srraregles.

In all cases, elp

them make stratdgies

explicit

(Kame'enui

& Camine,

199).

10.

EmVhasizestatistks, geome\,

and flteasurenteflt as well as

arithmertc

toyics

(Parmar

& Cawley,

4997).

A11 tudents eed

accesso varied

topics n mathematics.Topics

beyond

arithmetic are increasingly

important

in our dav-to-dav ives.

Oveiall,

solve

problems,

ncourage

reasoning,

,and

r-ise

modeling.Wi[h

Dattence

nclsupport.

nese Drocesses

ire aiso n the rdath of mosr hildr.n.

References

Baroody,A.

L.

0996).

An investigative

approach

to the mathemat

dyscalculia (math)

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