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7/25/2019 Dyscalculia (Math)
1/14
Information provided by The International DYSLE IAAssociation
promoting literacy through research, education and advocacyThe International Dyslexia Association 40 York Road Fourth Floor Baltimore MD 21204
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"
7/25/2019 Dyscalculia (Math)
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
7/25/2019 Dyscalculia (Math)
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
mathematical disorders
also rely on
finger counting
and use the sum
procedure
more frequently than
do
continued
n pag,e
Perspectives,
ummer
2000 7
7/25/2019 Dyscalculia (Math)
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
mathematical disorders
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
7/25/2019 Dyscalculia (Math)
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
7/25/2019 Dyscalculia (Math)
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
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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
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ummer
2000 11
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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
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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
7/25/2019 Dyscalculia (Math)
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
7/25/2019 Dyscalculia (Math)
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
7/25/2019 Dyscalculia (Math)
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
7/25/2019 Dyscalculia (Math)
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