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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 Educators David C. Geary

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

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

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

Translating Lessons from Research into Mathematics Classroom Douglas 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:

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A MATHEMATICAL DISORDERS:IEE-Ilm

\M AN OVERVIEW FOR EDUCATORS17 BY David C' Geary

ff uring the past 20 years, we havel-/witnessed enormous advances inour understanding of the genetic,neurological, and cognitive factors thatcontribute to reading disorders, as wellas advances in the ability to diagnoseand remediate this form of learungdisorder (e.g., Torgesen et a1., 7999).We now understand that most forms ofreadins disorder result from a heritablerisk and have a phonologicai core; forinstance. manv of these children havedifficulties associating letters andwords with the associated sounds,which makes learning to decodeunfamiliar words difficult (Light,DeFries, & Olson, 1998). At the sametime, there have been a handful ofresearchers studying children'sdifficulties with early mathematics,difficulties that emerge despite low-average or better intelligence andadequate instruction (Ceary, Hamson,& Hoard, in press; Jordan & Montani,1997). This essay overviews thisresearch, including discussion of thenrcvalence of ch i ldren wi thmathematical disorders and theirdiagnoses, the approach researchersuse to study these children, and somemalor nnolnSs.

How common is aMathematical Disorder andhow is it diagnosed?

Although there are no definitiveanswers, studies conducted in theUnited States, Europe, and Israel allconverqe on the same conciusion:About 6%. of school-age children andadolescents have some form ofmathematical disorder and about onehalf of these individuals also havedifficulty in learning how to read(Gross-Tsur, Manor, & Shalev, 199Q.These studies also suggest thatmathematical disorders are as commonas reading disorders and that acommon deFicit may contribute to theco-occurrence of a mathematicaldisorder and a reading disorder in somechildren (Ceary,1993).

Like reading disorders, there is nouniversally agreed upon set of criteria

6 Perspectives, Summer 2000

Mathematical Disorders:Subtwe I

SemantiiMemory

Cogiive & Perfomana Futures:A. Low frequency of

arithmetic fac[ retrievalB. When facts are

retrieved, there is a higherror rate

C. Errors are frequendyassoclates ot tnenumbers in the problem

D. Soluuon times forcorrect retrieval areunsystematlc

N e u ro y sy c h olo gi ca I F eatu res :A. Appears to be associated

#ith left hemisphericdys fu nction, in paru.cular,postenor regrons ot tnel p F t h p m i c n 6 e r a

B. Possible subcorticalinvolvement, such as thebasal ganglia

Cenetic Features:Preliminary studies andthe relatioir with certainforms of readrng disordersugqest that this deficitmd'! be heritable

Relat ionsh iy to kadin g D is ordets :Appears to occur withn H n n e t i r F ^ * . n f

i - p a r l i n o d i c n r d c r' - _ - " ^ ^ b * ^ ' " ^ - " '

for the diagnosis of mathematicaldisorders. In our recent work, we havefound a lower than expected (based onIO) perfonnance on math achievementtests across at least two grade levels tobe a useful and practical indicator of amathematical disorder (Geary et al., inpress). This and other studies indicatethat children with a mathematicaldisorder are a heterogeneous group andshow one or more subrypes of disorder(Geary,1.993).

within each domain. As an example,the assessment of computational skillsin dyscalculia (poor performance afterbrain injury) has often been based onsummary scores for accuracy at solvingsimple (e.g. 9+Q and complex (e.g.244+1,29) arithmetic problems (Geary,1993). These scores actually reflect anarcay of component skills, includingfact retrieval and procedural as well asconceptual competencies, makinginferences about the source of poor

Mathematical Disorders:Subtvoe 2Proc6dural

Cqnttive & Perfomance Features:A. Relatively frequent use

of devel6pmentallyurunature proceoures

B. Frequent enors in theexecutlon ot procedures

C. Potential developmentaldelay n the Lrndeisundmgof fle concepts underlyrg-proceoural use

D. Difficulues sequencingt h p m r r l h n l p c t p n c r n" ' "r" " 'complex proceoures

N e u ro y sy cho I o gi cal F e atu r e s :Unclear, although somedata suggest an associabonwith left hemisphericdysfuncuon, and n somecises a prefronal {ifurrtion

Mathematical Disorders:Subtrroe 3

Visuo'sfiatial

Cogrtwe & Perfotnance Futures:A. Difficulties in spaually

rePresenunS numencarrntormatlon such as t-fremisalignment of numeralsin mufticolumn arithmeticproblerns or roatirg rurnbels

B. Misinterpretation ofspatially'representednumencal lntormatlon,such as place value enors

C. Mav result in difficultiesin aieas *ut rely on spatialabilities, such a3 georhetry

N e u ro y sy chol o gi cal F eatu r e s :Appears to be associatedrlzrtt-r nght hemisphericdystunchon, m partl.cular,postenor reqrons ot theipht herrusphere, uftnp$the panetal cortex ot theleft hemisphere may beimplicated as well.

-

Cenetic Features:Unclear

Relationshiy n Ruding Disordes:Does not appear to berelated

Relation sh iy n Read in g D i sorders :Unclear

How do researchers approachthe study of MathematicalDisorders?

The complexiry of the field ofmathematics makes the study ofmathematical disorders challenging. Intheorv. mathematical disorders canresult'from deficits in the ability torepresent or process information in oneor all mathematical domains or in oneor a set of individual competencies

performance on these arithmeticproblems imprecise. Moreover, ingeometry and algebra, not enough isknown about the normal developmentof the associated competencies toprovide a systematic framework for thestudy of mathematical disorders.Iortunately, enough is now knownabout normal development in the areasof number concepts, counting skills,and early arithmetic skil ls top r o v i d e t h e f r a m e w o r k n e e d e d

continued on Vage 7

Mathematical Disorders: An Overview for Educatorscontinued from Vage 6

to systematically study mathematicaldisorders (see Geary, 1994, for a review).The sections below orovide an overviewof what we now know about children'sdeveloping number concepts, countingskills, and early arithmetic skills aiongwith a discussion of any associatedlearnins disorder. The finai sectionprovidJs a discussion of visuospatialissues somet imes associated wi thchildren with mathematical disorders.

Number ConceptsPsychologists have been studying

children's conceptual understanding ofnumber, for instance that "3" is anabstract representation of a collection ofany three things, for many decades. It isnow clear children's understandins ofsmall auantities and number is evidentto some degree in infancy. Theirunderstanding of larger numbers andrelated skil ls, such as place valueconcepts (e.g., the "4" in the numeral"42" represents four groups of 10),emerges slowly during the preschooland early elementary school years andsome times only with instruction(Iuson, 19BB; Ceary, 1994).

The few studies conducted withchildren with mathematical disorderssuggest that basic number competencies.at least for smail quantities, are intact inmost of these children (Ceary, 1993;Cross-Tsur et al.. 1996'.

Counting SkillsDur ing the preschool years.

children's counting knowledge can berepresented bv Celrnan and Caliistel's(1978) five implicit counting principles.These principles include one-onecorrespondence (one and only one wordtag, such as "one,t' "two," is assigned toeach counted object); the stable orderprinciple (the order of the word tagsmust be invariant across counted sets):the cardinality principie (the value of thefinal word tag represents the quantiry ofitems in the counted set); the abstractionprincipie (objects of any kind can becollected together and counted); and, theorder-irrelevance principle (items withina given set can be tagged in anysequence). Children also makeseemingly apparent, but not necessarilycorrect, inductions about the basiccharacteristics of counting by observingstandard counting behavior (Briars &

S ieg le r , 1984 ; Fuson , i 9BB) . Theseinductions include " adjacency' (countingmust proceed consecutively and in orderfrom one to the next) and "start at anend" (counting must proceed from leftto right).

Studies of children with concurentmathematicai disorders and readinqdisorders or mathematical disorderialone indicate that these childrenunderstand most of the essentialfeatures of counting, such as stableorder. but consistentlv err on tasks thatassess "adjacency" and order-irrelevance(Ceary Bow-Thomas, 8t. Yao, 1992;Geary et al., in press). In fact, thesechildren, at least in first and secondgrade, perform more poorly on thesetasks than do children with much lowerIO scores, suggesting a very specificdeficit in their counting knowledge. Itappears that these children, regardless of

IDifficulties in using

counting procedures canthus contribute to laterarithmetic-fact retrieual

problems.

their reading achievement, believe thatcounting is constrained such thatcounting procedures can only beexecuted in the standard way (i.e.,objects can oniy be countedsequentially), which, in turn, suggeststhat they do not fully understandcounrlng concepB.

Other studies suggest that childrenwith mathematical disorders also havedifficulties keeping information inworking memory while monitoring thecounting process or performing othermental manipulations (Hitch &McAuley, 1991), which, in tum, resultsin more errors while counting. Thus,young children with mathematicaldisorders show deficits in countinsknowledge and counting accuracy.

Arithmetic SkillsWhen first leaming to solve simple

arithmetic problems (e.g., 3+5). childrengtprcally rely on their knowledge of

counting and use counting procedures toftnd the answer (Geary et al., 1,992;Siegler & Shrager, I9B4). Theirprocedures sometimes rely on fingercounting and sometimes only onverbal counting. Common countingprocedures include the foilowing: sum (orcounting-all), where chiidren count eachaddend starting from 1; max (or countingon) . where children state the value of thesmaller addend and then count thelarger addend; and, min (counting on),where children state the larger addendand then count the value of ihe smalleraddend, such as stating 5 and countingon 6.7. B to solve 3+5.

The derrelnnmeql of efficientcounting procedures for simpleproblems (e.g., 3+5) reFlects a gradualshift from frequent use of the sum andmax procedures to frequent use of themin procedure. The repeated use ofcounting procedures also appears toresult in the development of memoryrepresentations of basic facts (Siegler &Shrager, 1984), that is, with repeatedcounting the generated answer (..g., B)eventually becomes associated inmemoly' with the problem (e.g., 3+5).Difficulties in using counting procedurescan thus contribute to later arithmetic-fact retrieval problems.

Studies conducted in the UnitedStites, Europe, and Israel haveconsistently found that children withmathematical disorders have difficultiessolving simple and complex arithmeticproblems (e.g., Barrouillet, Fayol, &Lathuiibre, 1997; Jordan & Montani,1997). These differences involve bothprocedural and memory-based deftcits,each of which is considered in therespective sections below.

Procedural DeficitsMuch of the research on children

with mathematical disorders hasfocused on their use of countingstrategies to solve simple arithmeticproblems and indicates that thesechildren commit more errors than dotheir normal peers (Ceary 1993; Jordan& Montani, 1997). They oftenmiscountor iose track of the counting process. Asa group, young children withmathematical disorders also rely onfinger counting and use the sumprocedure more frequently than do

continued on pag,e 8

Perspectives, Summer 2000 7

Mathematical Disorders: An Overview for Educatorscontinued from Vage 7

normal children. Their use of fingercounting appears to be a workingmemory aid, in that it helps thesechildren to keep track of the countingprocess. Their prolonged use of sumcounting appears to be related, in part,to their belief that "adjacency" is anessential feature of counting (Geary eta1.,1992). However, many, but not all,of these children show more efficientorocedures bv the middle of thebl.-..rtury school years (Grades 4-6)(Geary 1,993; Jordan& Monuni, 1997).Thus, for these children, their error-prone use of immature proceduresrepresents a developmental delayrather than a long-term cognitivedeFicit.

There have only been a fewstudies of the ability of children withmathematical disorders to pursueformal arithmetic alsorithmsassociated with more iomplexoroblems. such as 126+537. Theiesearch that has been conductedsuggests some specific diff icult ies.Althoush some studies have attributedcalcuhtlon difficulties to visuospatialdiff icult ies described below, otherstudies suggest that these calculationdifficulties are likely not due to thespatial demands of these arithmeticformats, as most children withmathematical disorders do not havepoor spatial abil it ies (e.g., Geary et al..in press). Rather, the errors appeared toresult trom difficulties in monitoringthe sequence of steps of the algorithmand from poor skill in detecting andthen self-correcting errors. Thus,procedural difficulties associated withmathematical disorders are evidentwhen these children count to solvesimple arithmetic problems (e.g., 3+5)and use algorithms to solve morecomplex problems (e.g., 126 + 537).

Retrieval DeficitsMany children with mathematical

disorders do not show the shift fromdirect counting procedures to memory-based production of solutions to simplearithmetic problems that is commoniyfound in normal children. It appearsthat there are tvvo different foims ofretrieval deficit, each reflecting adisruption to different cognitive andneural systems (Barrouillet et aI., 1997;Geary,1993).

Cognitive studies suggest that the

8 Perspectives, Summer 2000

retrieval deficits are due, in part, todifficulties in accessing facts from long-term memory. In fact, it appears thatthe memory representations forarithmetic facts are supported, in part,by the same phonological and semanticmemory systems that support worddecoding and reading comprehension.If this is indeed the case, then thedisrupted phonological processes thatcontribute to reading disorders mightalso be the source of the fact retrievaldifficulties of children wirh mathematicaldisorders. It mieht be the source of theco-occurrence of mathematicaldisorders and reading disorders inmany children (Geary, 1993; Light etaI.. I99B\.

Recent studies suggest a secondform of retrieval deficit, specifically,disruptions in the retrieval process dueto difficulties in inhibiting the retrievalof irrelevant associations. This form ofretrieval deficit was discovered byBarrouillet et al. (1997) and wasrecently confirmed (Geary et al., inpress). In the latter study, first andsecond srade children with concurrentmathernatical disorders and readingdisorders, mathematics disordersalone, or reading disorders alone werecompared to their normal peers. Onone of the tasks, the children wereinstructed not to use countingorocedures but onlv use retrievaliechniques to find solutions for simpleaddition problems. Children in alliearning disorder groups committedmore retrieval errors than their normalpeers did, even after controlling for IO.I he most common error was acounting string associate of one of theaddends. lor instance, commonretrieval errors for the problem 6+2wereT and 3, the numbers following 6and 2, respectively, in the countingseouence. The pattem across studiessuggests that inefficient inhibition ofirrelevant associations contributes tothe retrieval difFiculties of children withmathematical disorders. The solutionprocess is efficient when irrelevantassociations are inhibited andprevented from entering workingmemory. InsufFicient inhibition resultsin activation of irrelevant information,which functionally lowers workingmemory capaciry. In this view.children with mathematical disordersmay make retrieval errors because they

cannot inhibit irrelevant associationsfrom entering working memory. Oncein working memory these associationseither suppress or compete with thecorrect association (i.e., the correctanswer) for expression.

Disruptions in the abii iry toretrieve basic facts from long-termmemorv. whether the cause isaccessins difficulties or the lack ofinhibit ion of jrrelevant associations,might, in fact, be considered a definingfeature of mathematical disorders(Geary 1993). Moreover, characteristicsof these retrieval deficits (e.g., solutiontimes) suggest that for many childrenthese do not reflect a simple develop-mental delay but rather a more per-sistent cognitive disorder.

Visuospatial SkillsIn a variety of neuropsychological

studies. soecific diff icult ies withvisuosoatial skills have been associatedwith dyscalculia, with specific referenceto spatial acalculia. The particularfeatures associated with spatial acalculiainclude the misaiisnment of numeralsin multi-column irithmetic problems,numeral omissions, numeral' rotation,misreading arithmetical operation siSnsand difficulties with place value anddecimals (see Geary 1993). Russell andGinsburg (1984) found that fourth-sfade children with mathematicaliisorders committed more errors thanthei r lO-matched normal peers oncomplex arithmetic problems (e.g.,34x28). These errors involved themisalignment of numbers while writingdown partial answers or errors whilecarrying or borrowing from onecolumn to the next. The children withmathematical disorders appeared tounderstand the base-10 sysiem as wellas the normal children did, and thus theerrors could not be attributed to a poorconceptual understanding of thestructure of the problems (see alsoRourke & Iinlayson, I97B). Otherstudies suggest that spatial deficits willalso intluence the ability to solve othertypes of mathematics problems, suchas word problems and certain types ofgeometry problems (Geary. 199Q. Inelementary school, however, thissubwoe of mathematical disorder doesnot apoear to be as common as theother subtvpes.

continued on yage Q

Mathematical Disorders: An Overview for Educatorscontinued from yage 8

ConclusionAs a groupl children with

mathematical disorders show a normalunderstanding of number concepts,concepts underlying arithmenc algorithms,and most counting principles. At thesame time. manv of these children havedifficulty keeping information inworking memory while monitoring thecounting process and seem tounderstand counting onJy as a rote,mechanical process (i.e.. counting canonly proceed with objects counted in afixed order). When solving simpiearithmetic problems, young childrenwith mathematical disorders usedevelopmentally immature proceduresand commit many more errors in theexecution of these procedures. Sincemany of these children eventuallydevelop efficient counting procedures,their difficulties in this area reDresenE adevelopmental delay. A defining featureof mathematical disorders that does notappear to improve with age or schoolingis difficulty retrieving basic arithmeticfacts from long-term memory. Thismemory deficit appears to result from amore general diff iculry in representinginformation in or retrievine informationfrom phonetic and semantic memory, asweli as from difficulties in inhibitinp theretrieval of irrelevant associations.Finally, many children with mathematicaldisorders have diff iculties in organizingthe sequence of s teps needed tosuccessfully pursue formal algorithms.Future studies will, no doubt, clarifythese patterns and lay the foundation forremedial strategies.

ReferencesBarrouilleg P., fayol, M., &

LathuLibre, E. (199n. Selectingbetween competiton n multiplicanontasla: An explanation of the errorsproduced by adolescents with leamingdisabfities. Intemational Joumal ofBehavioral DeveloVment, 2'l , 253-275.

Brian, D., & Siegler, R. S. (1984). A feanrralanalysis of preschoolers' countingknowledge. DeveloVmenal Psychology,20, ffi7-618.

Iuson, K. C. (1988). Children's clunting andanceps of number New York Springer-Verlag.

Geary, D. C. (1993). Mathematicaldisabilities: Cogutive, neuropsyctrologrcatand genetic components. PsychologialBulletin, 4 4 4, 345-362.

Geary, D. C. (1994). Children's mathematialdeveloVment: Research and Vracticalapyliations. Washington, DC:American Psychological Association.

Geary, D. C. (1996). Sexual selection andsex differences in mathematicalabiliues. Behaviual and Brain Sciences,49,229-284.

Geary, D. C.,Bow-Thomas, C. C., &Yao,Y(1992). Counting knowledge and^ r . ; r f : - - ^ - ^ : L ; . . . a d d i t i o n : AJ N t l l l l l r u 5 , 1 t l t l v (^ ^ - - ^ - : ^ ^ - ^ r n o r m a l a n dl v l l l P 4 r r D v r r u r

mathematically disabled children.Joumal of Lweimenal Child Psychology,54,372-391.

Geary, D. C.. Hamson, C. O., & Hoard,M. K. (in press). Numerical andarithmetical cogninon A longitudinalstudy of process and concept deficits inchildren with leamrrg disabili ty. J oumalof ExVeimenal Ckh Psychology.

Gelman, R., & Gallistel, C. R. (1978).The child's undersanding of number.Cambridge, MA: Harvard UniversityPress.

Cross-Tsur, V, Manor, O., & Shalev, R. S.(1996). Developmental dyscalculia:Prevalence and demographicfeatures. DeveloVmennl Malicine andChih N eurology, 38, 25-33.

Hit h, G. J., & McAuley, E. (1991). Workngmemory in children with specific

arithmetical leaming disabfities.British Journal of Psychology, 82,375-386.

Jordaq N. C., & Montani, T. O. $99n.Cognitive arithmetic and probiemsolving: A comparison of childrenwith specific and general mathematicsdifficulties. Journal of LearningDisabilities, 30, A4-634.

t-tght,l. C., Delries, I. C., & Olsor5 R. K.(998). Multivanate behavioral geneticanalysis of achievement and cogmtivemeasures in reading-disabled andcontrol tivin paks. Human Biology, 70,215-237.

Rourke, B. P, & Finlaysoq M. A. ]. (1978).Neuropsychological significance ofvariations in pattems of academicperformance: Verbal and visual-spatialabiliues. Journal of Abnormal ChildP s y ch ol o gy, b, 121-133.

Russell, R. L., & Ginsburg H. P (1984).Cognitive analysis of children'smathematical difficulties. Cognitionand Instruction, 4, 217 -244.

Siegler, R. S., & Shrager, J.0984). Suategychoice in addition and subtraction:How do children knowwhat to do? InC. Sophian @d.), Origins of cognitive

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

Torgeser5 J. K., Wagner, R I(., Rose, E.,Lindamood, P, Conway, T., & Carvan,C. (1999). Prevenbng readng failure inyoung children with phonologicalprocessing disabilities: Group andindividual responses to instruction.Joumal of Muational Psychology, 94,s79-593.

David Geary is a Professor ofPsychology and Director of the training

Frogram in Develoymental Psychology at theUniversitv of Missouri at Columbia. He hasmore than 8J publications Across a widerange of toVics, including learning disordersin elementary mathematics. His two bookshave been aublished bv the AmericanPsvcholoeical Association, Children'sMathematical Develo?ment: Research andPractical Ay?lications (1994) and Male,Female: The Evolution of Human SexDifferences (1998).

Perspectives, Summer 2000 9

Z\ MATHEMATICAL LEARNING PROFILES ANDa)l\M\ DIFFERENTIATED TEACHING STRATEGIES

lf-t

\V By Maria R. Marolda and Patricia S. Davidson

Are there particular mathe-matical proTiles that charac-terize how students learnmathematics?

How does the understandingof mathematical learninq pro-files translate into Setterinstrutional opportunities inmathematics?-

-

primary consideration in theteaching of mathematics is the

recognition that students bring to themathematics classroom a wide ranqeof abil it ies and learning approaches.Extensive instructional and clinicalinvestigations during the past 20years,as well as a detailed research study(Davidson, 1983), have revealed thatstudents' leaming profiles are markedby different constellations of relativestrengths and relative weaknesses withwhich students face the worid ofmathematics. Indeed, it is this study ofdffirences, rather than a definition ofexplicit deficits, that provides a moreuseful approach to understandingstudents' effectiveness or inefficienciesin learrung mathematics. Moreover, anunderstandine of differences is alsoinformative in fashioning instructionalapproaches that are compatible withthe various learning profiles that existin the mathematics classroom.

Child / World SystemLearning profiles in mathematics

can best be understood by consideringa Child/1X/orld system (Bernstein &Waber. 1990) that characterizes thereciprocal relationship of the developingchild and the mathematical world inwhich the child must function. Theconstruction of a Child/Wold systemfocusses on differences among iearnersas well as differences in the demandsof what is to be leamed. It then expioresinstances of natch and nisnatch. in theconsideration of differences amongstudents, the cntical question becomes,"When does a learning differencerender a student learning disabled?" A

10 Perspectives, Summer 2000

leaming "disability" in mathematicsmay be thought of as the occurence ofmultiple "mismatches" and theinability to overcome those mismatches.A tantalizing issue then becomeswhether specific approaches or sffategiescould be used so that the mismatchesare minimized and the disabiliry isresolved or disappears.

It is imporiant to recognize thatthe diagnostic process in education isquite different from the diagnosticprocess in medicine. Whereas themedical diagnostician is lookrng touncover what is wrong and what thepatient can't do, the educator muststrive to uncover the student's

IThe construction of aChildlWorld system

focuses on differencesamong leamers as well asdifferences in the demandsof what is to be learned.

strengths and what the student can do.The goal of the educator is to findthose strengths that can be used toaddress the weaknesses and difficultiesinherent in students' learning profiles.

In focussins on the Child in thea

Child/\X/orld system of mathematics, amultidimensional view must be takenand a variew of parameters considered.Specificaliy,

-the following factors should

be explored in order to understand astudent's Mathematical Leaming Protile:

o the presence of speciftc develop-mental features that are prerequisite tospecific mathematics topics;

. the preferred models with whichmathematical topics are interpreted;

r the preferred approaches withwhich mathematical topics are pursued;

. memory skills that affectstudents' abil ity to participate inmathematical activities;

o language skills that affectstudents' ability to participate in themathematical arena.

Developmental Features ofMathematical LearningProfiles

A definit ion of a student'smathematical leaming profile shouldincorporate an appreciation of thedevelopmental maturiw of students atvarioui ages. There a.. -".ry develop-mental milestones in terms of mathe-matical readiness for dealing withnumerical, spatiai and logical topics.For numerical concepts, the develop-mental milestones consist of anappreciation of number, the concept ofnumber (enumeration/cardinality),conservation of number, one to onecorrespondence and the principles ofclass inclusion. For spatial concepts, theconstruct of space. conservation oflength and conservation of volume mustbe considered. Ior logical thought,deveiopmental milestones include theconcepts underlying classification,seriation, associativity, reversibility andinference. Most chiidren between theages of four and eight have acquiredthese milestones.

Recent ciinical investigation andteaching practice have suggested thatthe concept of place value might alsobe developmentaliy mediated (Marolda& Davidson, 1994). That is, an appre-ciation of place value depends more onthe state/ase of the child than onspecific t.ulhi.tg experiences. If thechild is not cognitively ready to dealwith place value, then the concept ofplace value cannot be formally ormeaningfully developed, despite teachingefforts. The formal concept of placevalue seems to be established for mostchildren beNveen the ases of six andeight. The appreciation of formalplace value concepts is of particularimportance since they are necessaryprerequisites for the understanding oflarger quantities and the pursuit ofmulti-digit computation

(onilnued on page | |

Mathematical Learning Profiles and Differentiated Teaching Strategiescontinued from Vage 40

Preferred Models andPreferred Approaches ofMathematical LearningProfiles

Mathematical situations can beinterpreted with concrete, pictorial, orsymbolic models. For a particularstudent, a specific interpretation mightbe more comfortable and meaningful.Among concrete models, furtherdistinctions can be made. Within theconcrete mode, students may preferset (discrete) models, such as counters,while others appreciate perceptuallydriven (measurement) models, such asCuisenaire rods.

The ways in which studentsprocess or approach mathematicalsituations follow nvo distinct Datterns(Marolda & Davidson, 1994). Somestudents process situations in a linearfashion, building forward to an exactfinai solution. Sometimes, thesestudents are so focused on theindividuai eiements that the overallthrust or goal is obscured. This style ofprocessing is often characterized as asequential, step-by-step approach. Forother students, a careful building upapproach hoids l itt le inherentmeaning. Such students prefer toestablish a general overview of asituation first and then refine thatoverview successively until an exactsolution emerges. Such students maybe prone to imprecision and tend tolack appreciation of all relevant details.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 as globai or gestalt.Incorporating these inherent

preferences in terms of models andprocessing has led to the definitron oftwo distinct learning profi les inmathematics, Mathematics Learning;Style I and Mathematics Learning Style IIas reviewed in Table 1 (Marolda &Davidson. 1994). Moreover, it ispossible to describe mathematicalconcepts and procedures that areinherently compatible with eachI a r ' - i ^ - ^ ' ^ F i l .

To be full and successfulparticipants in mathematics, studentsmust learn to mobilize bothMathematics Learning Style I andMathematics Leaming Sryle II. Thestudent with special leaming needs,however, is often limited to onelearning style alone and is unable to

Mathematics Leaming Style I

Preferred Models for Number:. Set Models

Preferred Ayyroaches:o I i n p a r c f p n h r r c t e n

. Often relies on verbal mediation

ToVics AyVroached with Ease:o Counting forward & counting-ono Concepts of addition & multiplicationr Pursuit of calculation procedureso Fraction concepts hterpreted in verbal termso Ceometric Shapes: Emphasis on rnming

Toyics of Pantcukr Challenge:. Broader concepts and overarching pnncipleso Estimation srrategies. Appreciation of appropriateness of solution

generarco. Selection of aritimetic operation in word

problems : dif fi culty swrtthingtretween operauons ln a set ot word problerns

. Concept of a fractiono More sophisucated geometnc topics. Requirement for flexible or

alteirranve aooroaches

Mathematics Learning Style II

Preferred Models for Number:. Perceptual (Measurement) Models

Preferred Ayyroaches:. Deductive, global. Often relies on successive aooroximations

Toyics Ayyroached with Ease:o Counting backward. Concepts of subtraction & division. Estimatlon. Fracbon concepts rrterpreted in a vanery of

\,1SUal mooets. Ceometric Shapes: Emphasis on spatiai

relationshios aird manibulauors

Toyrcs of Pattrular Challenge:. Appreciation of all salient details of multi-

stip procedures or word problems. Pursuit of multi-step calculation procedures. Relevance of exact solutiorx; prefers to guess. Follow throush to exact solutions in word

problems, deipite conect choice of operation. Formal fracuon operations, despite comfon

wlth underlymg rracEon concepto Requirement to descnbe approach in

exacung verbal termso Insistence on a single, specific approach

Tablemobilize skil ls and strategiesassociated with the altemative leamrngstvle. For success. teachers musttrinslate activities into the student'soperating sryle, building a scaffold thatintegrates the areas of strengths andweaknesses so that thev comolementone another and lead to ihe acouisit ionof mathematical conceoti andprocedures in a meaningful way.

The foliowing charts flables 2 &3: as shown on pages 13 and 14) offermore explicit features of each of theMathemitical Learning Styles and canbe helpful in recognizing them andteaching to them.

Memory Skills as a Featureof Mathematical LearningProfiles

Often students are characterizedas having difficulry in mathematicsbecause they "can't remember." Theattribution of mathematical difficultiesto a global memory deficit issomewhat simplistic. Cognitivepsychologists (Hoimes, 19BB) suggestthat memory issues, in general. arevery comprex.

In evaluating a child's recall ofmaterials, the clinician shouldrecogtize the various componentsof the process ioosely called

4memory: registration of thestimulus, encoding, organization,storage and retrieval....Learningdisabled children, however, areconstantly described in thepsychological and educationalliterature as having memorydeficits of various types, usually

.visual or auditorv (shorc rerm orotherwise). In almost all cases,the impairment involves eitherthe initial encodins or theeffective retrieval of iriformation.(p.1Be)

In mathematics, it is particularlyimportant to consider the distrrctronbetween encoding and retrieval aspectsof memory. Is the student havingdifficulty remembering the fact orprocedure because it was never properlyunderstood and therefore not encodedfor storage in memory? Or is the studenthaving difficulty remembering the factor procedure because it cannot beaCcessed from the student's repertoire ofleamed skills?

Four specitic memory skills areimportant in mathematics :

o retrieval of solutions to one dieitfacts;

. the recall of the sequence ofmulti-step procedures;

continued on page | 2

Perspec-tives, Summer 2000 11

Mathematical Learning Profiles and Differentiated Teaching Strategiescontinued from Vage 4 4

. visual memory of perceptual /geometric stimuli;

. recall of mathematical datapresented auditorially.

In terms of retrieval difficulties inthe production of solutions to one digitfacts, mathematically it may be moreimportant to consider if the solution isproduced efficiently rather thanautomatically. The distinction that isimportant is whether the retrieval isauiomatic or efficient. Difficuitieswith the retrieval of one digit factsmay be supported by alternativestrategies that are compatible with astudent's inherent learning style andresult in more efficient production ofsolutions. In the example, B + 6, astudent with Mathematics LearningStyle I would be most efficient tumingto counting on strategies: 9,10,1I,12,13...14! or strategies that build 10s: B +(2+4) = 1.0 + 4 = l4l A student withMathematics Leamrng Sryle II wouldbe most efficient turrung to relatedfacts, e.g. doubles, B+B=16, so 8+6=14. . .2 Less! Or 6+6=72, so 8+6=14. . .2More!

In dealing with multi-steporocedures. the recall of theorganization of the specific steps relieson an understanding of the conceptualfoundations driving the procedure. Byoffering alternative approaches thatappeal to a specific learning style, theorocedure is better understood andmore easily pursued. In dealing withthe multiplication problem 23 x 14, astudent with Mathematics LearningStyle I would tum to a successrveaddition approach or the formalalgorithm. Iurther supports toremembering the steps of proceduresinclude encouraging verbai mediationtechniques, developing verbai andvisual flow charts that can be used asreferents, and developing mnemonicsto cue each step. In contrast, thestudent with Mathematics LeaminsSryle II would turn to the definition oTmultiolication as an area and wouldthen-combine the area of the foursubregions to determine the finalsolution. Further supports would beestimation techniques made iterativelyo! once an initial estimate is made, theuse of a calculator for an exactsolution. With firm understanding

12 Perspectives, Summer 2000

established. the procedure is moreeffectively encoded. That understanding,however, may emerge from differenta0Droacnes.- -

In dealing with geometric designs,students need to use visual memoryskills. With visual memorv difficulties,students may find the buiiding andcopylrrg of geometric designs challengng.To support visuai memory difficultiesstudents might be encouraged tointerpret geometric designs in verbalterms. Difficulties in visual memorycan also manifest themselves in non-geometric situations, such as difficultiesorienting written digits, difficultiesaligning numerals in written procedures,and difficulty organizing a page ofproblems. Copying problems from thetext and the board or interpreting datapresented on a computer screen mayalso be difftcult. In response, copyingrequirements should be minimized,while graphic organizers may be offeredto support the copying that is required.

Students with auditorv memorydifficulties are challenged when requiredto remember ail relevant data presentedin instruction, remember the overalloutcome sought, remember directions,or remember all the relevant infor-mation in word problem situationspresented verbally. These students maybe supported by offering directions invisual formats as well as by offeringwritten directions and / or allowingstudents to write down the directionsand then referring to the written textas needed. Interestingly students withapparent auditory memory issues areoften confused with students whoseprimary difficulties are in languagewhere memory difficulties aresecondary to specific language pro-cessing issues.

Language lssuesLanguage skills, both oral and

written. are imoortant in mathematicsin terms of:

. word retrieval skills;o verbal formulation resuirements;r comprehension requirements.They become an issue when

students are required to retrieve thenames of coins,'geometric shapes orother mathematical terms, when theyare asked to explain their solutions or

approaches, when they must deal withlengthy verbal presentations typical ofclassroom instruction, and when theyare faced with word problems. Theselanguage demands have become moreprominent in mathematics as educationcurricula and textbooks have encour-aged teachers to ask students foreiplanations or iustifications of theirapproaches. Moreover, teachers havebeen encouraqed to ask students totake t.rpot Jbility for their ownlearning by reading printed materialsor texts. These newer emphases poseparticular challenges for students withlanguage difficulties.

In order to address word retrievaldifficulties in mathematics, studentsmight focus primarily on the values ofthe coins rather than their specificnames, might be encouraged to drawgeometric shapes rather than namethem and might be offered recognitionformats when dealing with mathe-matical definitions. Retrieval issuesare further supported by minimizingconfrontational, fast answer situations.

Students with verbal formulationissues often have difficulty describingtheir approaches or in portfolio workwhere approaches must be writtendown. To rupport these students,alternate forms of communicationshould be encouraged, includingdemonstrations with physical modeisand use of pictures or diagrams todescribe solution processes.

Students with comprehensiondifficulties often have difficulties withdirections and with reading texts orword problems. They otten can't getstarted with classwork, mistakenlysuggesting they have attentionaldifficulties. These students benefitfrom careful monitoring of newpresentations and having wordproblems read to them. Such studentsiun be supported by teacherspresenting content in meaningful"chunks" that are then carefully linkedtogether. In terms of word problems,the situations can be presentedverbally rather than requiring reading.Students should then be encouraged todraw pictorial interpretations toreDresent the situation and datainvolved.

continued on yage 4 3

Mathematical Learning Profiles and Differentiated Teaching Strategiescontinued from page 4 2

Mathematics Learning Style I

Coenitive & Behavioral" Correlates

Mathematical Behaviors Teachins Implications& Strat6gies

. Highly reliant on verbal skills . Approaches situations using recipes;t ; ^ 1 1 . ^ , L - ^ , , - L r , ^ ̂ t . ̂6 '^J L' uuuBl LaJNr

. Interprets geometric designs verbally

. Emphasize the meaning of eachconcept or procedure in v?rbal terms.

. Build on subvocalization strategiestn A i re r r n rnrpd ' ' .ac

. Tends to focus on individual details orsingle aspects of a situation

. Sees the "trees," but overlooksthe "forest"

. Approaches mathematics in a mechanical,rddtine based tashion

. Overwhelmed in situations in which thereare,multiple considerations, such as inmultl-step tasl(S

. Can generate correct.solutions, but may notrecognrze when solutrons are lnappropnate

. Di f f icul t ies."checking' work; must re-doentlre problem

. Di f f icul r ies.choosing an approach inword problems

Difficulties. appreciating larger geometricconstructs because of an emphasls oncomponenr pans

. Highlight concept /overall goal.

. Break down complex tasks into salientunits and make lihkase berween unitsexpuclr.

. Bui ld s imple estrmar ion srrategies;encourage Nvo hnal steps to each calculabonproblem: "Does this arsiver the quesuon?"and "Does the solurjon seem righr?"

. Encourage students to rewrite or stateproblerns in their own words.

. Develop metacognitive strategies toanalyze word problem Srtuatrons.

. Encourage parts to wholes approach inbui ld ing geometr ic f igures.and expl ic i tdescnp[ons ot the overall deslgn that emerges.

. Prefers HOW to WHY . Prefers numerical approach overmanipulative moddls

. {\e.eds

drill and pracrice.to establish procedureDerore constoennS appllcauons or nroaoerconceprual meanlng

o Link manipulative model on a step-by-step basis io the numerical procedur6.

. Once procedure is secure, relate mathtopics ' to re levant real l i fe s i tuat ions.

. Relies on a defined sequence of stepsro pursue a goal

. Reliant on teacher for THE approach

. Lack of versatility

. Prefers explicit delineation of each step of aprocedure and linkage of steps one to another

. Vulnerable when there are multiple approachesf n : c i n o l e t n n r r' " - " " _ b ' - - " r ' -

. Overwhelmed by multiple models ormultiple approaihes

. Prefers linear approaches for arithmetic topics

. Offer flow chart approaches.

. Help students creat6 handbooks wi thproiedures descr ibed in their own words.

. Choose one manipulative model orapproach to.develop a wide. range oftoprcs; avord swrtchlnS models orapproaches too qurckly.

. Don't emphasize special cases; ratherdevelop ah over-r id ing ru le that appl iesto a l l cases; e.g. for t f ie addi t ion indsubtraction of lractions with unlikedenominators, develop a single processusing the product of the den6minators inal l cases, dven i f i t is not the Ieastcommon denominator.

. Cive explanations before or afterprocedure, but not while studentls pursulng procedure.

. Us'e count.-G on techniques for additionfacts and missinS addend techniqueslor subtractlon tacts.

. lnterpret multiphcatron as successiveadchtlons.

. Challenged by perceptual demands . Difficulties with more sophisticated perceptualmodels, such as Cuisenai ie rods

. Ceometr ic .act iv i t ies.may be chal lenging.esDecral lv rn three drmensrons.

. Difficulties interpreting analog clocks

. Di f f icul t ies d ist inguishing coins, especia l lyrucKel ano ouaner

. Difficulties brganizing written formats

. Emphasize set (discrete) models forcounthg, such as money or counting chips.

. I ranslate perceptual cues lnterms of verbal'descriptions.

. Prefers quizzes or unit tesrs to morecomprenenstve nnal exams

. Mav be able to comolete the most difficultexampie in a set of dxamples relying on thesame concepVskjll, but has diffiiulry switchingto a new topic or new approach

. Spiral all topics to keep them current,

continued on yage 411

Perspectives, Summer 2000

I AAte I

I J

Mathematical Learning Profiles and Differentiated Teaching Strategiescontinued from yage 4 3

Mathematics Learning Style Il

Coenitive & Behavioral- Correlates

Mathematical Behaviors Teaching Implicationsag )trategres

. Prefers perceptual stimuli and oftenrelnterprets abstract sltuahonsvisual$ or pictorially

. Benefits from manipulatives

o Loves geometric topics

. Offer a variew of models: introduceperceptr-ral models, sud-r as Base Ten Blocks orCuisinaie Rols, to suppon calculations.

. Emphasize geometry'ds a viulpart ot fhe curnculum.

. Likes to deal wrth big ideas; doesn twant to be botherefwith details

. Prefers concepts to algorithms

. Tolerates ambiguity and imprecision

. Offers impulsirie guesses as'solutions

. Uses estimation strategies spontaneously

. Skims word problems first but must beencouraged io re-read for salient details

. Perceives overall shape of geometncconfigurations at the expense of an appreciatiorof the individual comoonents

. Relate manipulative models toprocedures before practicing algorithms

. Keward approach as well as precrsesolunons.

. Develop an appreciation of how muchpreclslon a sltuatlon warrants.

. Reward/encourage estimation strategiesas hrst step.

. Encourage diagrams as a technique toorgantze data ln problem solvrngsituations.

. Aloy calculators to support problemsolvmg.

o Encourage multiple refinements whenbuilding geomet'ric designs in order toincorpola-"te all the indiv-idual parts.

. Prefers WHY to HOW o Reouires a definition of overview beforedealing with exacting procedures

. Requir?s manipulatirie modeling beforedevelopine a cbncept or algorithm

. Likes t6 se"t up proSlems, b"ut resists foilowingthroush to a conclusron

. Offer opportunities to work incooperatlve Sroups.

. Prefers nonsequential approaches, .involving pattems and intenelationships

. Prefers successive approximations approachto formal alsorrthms

. Addition an"d multiplication facts involvlng 9smore readily generited because of underlying,patterns that are recogruzed but not verbauzed

. Not troubled bv mixe-d oractice worksheets

. Comfortable with horizontal formats forlong calculations

. Can offer a variery of alternative answersor approaches to I single problem

. Can appreciate operation needed in a wordproblcim but has difficulry following throughto an exact solutron

. Likes logical problem solvrng in the formor general reasonlnS proolems

. Allow altemative calculation procedures.

. Help students to create their ownhandbooks of rypical problems.

o Cenerate arithm6tic fdcs throughrelationships to known facts;e.e. doubles for + facts.

. Einphasize area model for multiplication.

. Start with real-life situation and teaseout more formal arithmetic topics.

. Use simulations, relating simildr concepts/approaches in a variery ofdifferent situau6rs.

. Mbdel complex problems wi th s imi larproblems in simirler forms.

. Cive Nvo grades on word problemactiviries; one for correct approach:one for exact final solution.

. lnclude general reasoning examples inlogical p"roblem solving ictivities.

. Challenged by demands for deuils orthe reou-=iremint of orecise solutions

. Difficulties with precise calculations

. Difficulties offeriire rationale forcorrect solutions

-

. Encourase students to describe theapproach or conceptual. underpinni ngeven rt lhev cannot mobllrze anexacung proceoure.

Prefers performance based or portfoliot!'pe assessment to typlcal testsPrifers comorehensivd exams to

. lqulzzes and unlt testsMore comfortable recognizing correctsoiuttons than generattng them

. May be overwhelmed when faced withmultiple examples

. Corsider a variery of assessment techniques

. Allow oral Dresentations.

. Do not alw'avs reouire exact solution butsometimes g'rade homework and testsonly for corlect approach.

. Indude some muluirle choice items on tess.

14 Perspectives, Summer 2000

Table 3 continued on page 'l 5

Mathematical Learning Profiles and Differentiated Teaching Strategiescontinued from page 44

Conclusion:The Child/World System allows

teachers to achieve an understandingof the dynamic interplay that affects astudent's learning in mathematics. Itleads to the delineation of specificMathematical Learning Profiles.Extensive clirucal investigations andclassroom instruction, along withrigorous research efforts, havecorroborated the presence of speciFicMathematical Leaming Profiles. Thoselearning profiles involve differences indeveiopment as well as preferences formodels and preferences for approaches.Complicating the consideration ofiearrung profiles in mathematics aremore general memory and languageissues that intrude on efforts inmathematical activities.

The understanding of MathematicalLearning Profiles helps teachers offerspecific approaches and strategies thatmake use of students' areas of relativestrengths, that minimize areas ofvulnerabiliry and that support areas ofspecific deficit, ensuring thecomfortable participation and growthof all students in the mathematicalarena. The importance to teachers ofunderstanding Mathematical LearningProfi les is that they lead to thedevelopment of more effectivelearning strategies which, in turn,

allow more students to experiencesuccess in the domain of ma*rematics.

ReferencesBernstein, ].H. & WabeS D (1990).

Developmental neuropsychologicalassessment: The systematicapproach. In A.A. Boulton, G.B.Baker & M. Hiscoc (Eds.),N euromethods (pp.31I-7 1). Clifton,NJ: Human Press.

Davidson, P.S. (1983). Mathematicslearning viewed from a neurobiologicalmodel for intellectual funaioning (NIECrant No. NIE-G-79-0089).Washington, DC: NationalInstitute of Education.

Holmes, ].M. (1988). Testing. In R.Rudel, J.M. Holmes & J.R. Pardes(Eds.), Assessment of DevelopmentalDisorders: A neuroysychologicalaVproach (pp.1,1,6-201). New York:Basic Books.

Marolda M.R. & Davidson P.S. (1994).Assessing mathematical abilitiesand learning approaches. InWindows of o47ortuility. Reston, VA:National Council of Teachers ofMathematics.

Maria Marolda and PaticiaDaddson have collaborated on teachinematerials, curriculum development , anlstaff develoVnent in nathematics since4 968 and on assessmeltt research anddiagnostic testing since 4 977. Marolda is aResearch Associate at Harvard MedkalSchool and senior mathematics specialist atthe Learning Disabilities Program atChildren's HosVital in Boston. Davidson isVice Provost for Academic Suyyort Servicesand yrofessor of mathematics at theUniversity of Massachusetts, Boston, andhas been a chief examiner in mathematicsand chair of the Examining Board of theInternational Baccalaureate diVloma

Frograffi. Both Marolda and Davidson areauthors of articles, mathematics curriculaand yrograms as well as alternativeassessment tnstruments.

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Perspectives, Summer 2000 15

TRANSLATING LESSONS FROM RESEARCH INTO.JL MATHEMATICS CIASSROOMS

ZII

\Un Mathematics and Special Needs Students

17 By Douglas H. Clements

Too often students with leamingI disabiiities receive limited mathematici

instruction. This is due in part to specialeducation teachers feeling uncom-fortable teachins mathemitics. Thisleads to an overimphasis on trainingskills. There are three reasons for thiifocus on skills. First, there is a majormisconception that skill learning is ihebedrock of mathematics. uooriwhichall further mathematics mubt be built.Second, skills are easier to measure andteach. Third, teachers often believe thatstudents' perceived memory deficitsimply the need for constant repetitionano orlll.

Lessons from ResearchDecades of research indicate that

students can and should solve problemsbefore they have mastered proceduresor algorithms traditionally used to solvethese problems (National Council ofTeachers of Mathematics, 2000). If theyare given opportunities to do so, theirconceptual understanding and abiliry totransfer knowledge is increased (e.g.,Carpenter, Franke, Jacobs, Fennema, &Empson,1997).-Tndeed .

some o f t he mos tconsistendy successful of the reformcurricula hive been orosrams that

o build directlv Jn studentsstrategles;

. provide opportunities for bothtnventron ancl practlce;

o have children analyze multiplestrategles;

. ask fo-r exolanations.Research evaluatioi-rs of these programsshow that these curricula

'facilitate

conceptual qrowth without sacrificinqskil ls

'and i lso helo students ieari

concepts (ideas) arid skil ls whileproblem solving (Hiebert, 1999).'

What is re"markable is that similarprinciples apply to students withlearning disabiliiies. Many childrenclassified as leamins disabled can learneffectively with qrialiry conceptually-oriented instruction (Paimar & Cawley,1997). As the Pinciyles and Standards forS chool Mathematics rTTustrates (NationalCouncil of Teachers of Mathematics,2000), a balanced and comprehensiveinstruction, using the child's-abilities toshore up weak-iesses, provides betterlong-term results. For exlmple, students

beneficial in meeting the needsof all students. Studdnts who usemanipulatives in their mathematicsclasses usually outperform those whodo not (Drisioll, 19ffi; Greabell, 1978;Raphael & Wahlstrom, 1989; Sowell,1989: Suvdam. 1986). Maninulativescan be pirticularly heipful to studentswith learnine disabilities.

Somewhat sumrising, manipulativesdo not necessarilv'have"io be

'phvsical

objects. Computer. manipulatives canprovrde representatrons that are lust asbersonally meaningful to students.Paradoxicilly, comp,iter representationsmay even be more manageable,fleiible, and extensible than" theirphysical counterparts (Clements &Miuilt.n. 1996,.'students who usephvsical and software manipulativesbemonstrate a greater mathematicalsophistication

"than do control

gr6up students who use physicalfrranipulatives alone (Olson, IgBB).Good manipulatives are those that aremeaningful- to the learner, providecontrol

"and flefbility to the

'learner,

have characteristics' that mirror, orare consistent with coenitive andmathematical structures, aid assist theleaJner in making,connections betweenvarious pieces and types of knowledge.lor example, computer sorrware candynamically connett pictured obiects,slch as bise ten bloiks, to symbolicrepresentations. Computer manipulativescah play those roles. They help childrengeneralize and abstracl experiencesivith physical manipulatives.

Recommendations forClassroom Practice

Research provides several recom-mendations for meeting the needs ofall students in mathematlcs education.

4. KeeV exyectations reasonable,but not low.

Low exoectations are esoeciallvoroblematic 6.."ur" students *tro ti"'.in poverry, students who are not nativespiakers- of English, students withdisabiiities, females, and many non-white students have traditionally beenfar more likely than their counterpartsin other . demog.raphic groups to bethe vrctrms ot low exDectatlons.Exoectations must be raisdd because"rnathematics can and must be learned

continued on page 32Perspectives, Summer 2000 31

may benefit iess from intensive drill andprattice and more from help searchingFor, finding., a.nd us]ng fatterns iilearruns the baslc number comblnatlonsand aiithmetic strategies (Baroody,1996).

Many of the lessons we havelearned

-from research for general

educa t i on s tuden ts app l y , - w i t hmodifications of course,

'tb 'students

with soecial needs as well. Aparticulaily important one is "less ismore." That ii. in mathematics andscience, we have found that sustainedtime on fewer bey coftce4ts leads tosreater overall student achievement in

IDecades of research

indicate that students canand should solue problemsbefore they houe masteredprocedures or algorithmstraditionally used to solue

these problems.

the long run. Compared to othercountries" that sienificindy outperformus on tests, LJ1S. curri?ula do notchallenge students to learn importanttopics in depth (National Cenier forEducation Statistics, 199Q. We statemany more ideas in an average lesson,but develop fewer of them, domparedto other countries (Stieler & Hibbert,. Y

Iyyyl. Inus. u.5. students would Debetter off focusing on in-depth study onfewer importanf concepis. Such anaooroach is critical with students withldarning disabil it ies. They , need toconcentrate on mastenng the Key rcleas,and these ideas are iot arifhmeticalgorithms. Even proficient adults userelationships and itrategies to producebasic fact's. Thev ten-d not to usetraditional paper-and-pencil algorithmswhen computlng.

Anoth'er research lesson is that avariety of instructional materials is

Translating Lessons from Research into Mathematics Classroomscontinued from yage 34

by all students" (NCTM, 2000).Raisine standards includes increasedempha"sis on conducting experiments,authentic problem iolving, andproiect-based [earninq (McLIuehlin.Nolet, Rhim, & Hende"rson, 1999\.

2. Partently hely studentsdeveloV concepttal understanding andskills.

Students who have difficulw inmathematics may need additibnalresources to support and consolidatethe underlying- concepts and skil lsbeine learied." They' benefit frommultiple experiences with models andreiterition bf the linkage of modelswith abstract, numerical iranipuiations.

Expand time for mathematics. Ingeneral, the traditional curriculumZioes not allow adequate time for themany. instructional , and .learningstrategles necessary for the mathernahcalsucceis of leamine disabled studentsQemer,1997).

Students with disabilities mavalso need increased time to compleieassignments. Finally, they may alsobenetlt trom more tlme or tewerexamples on tests or from the use oforal rither than written assessments.

3. Build on children's ?.tr?ngths.This statement often is litde hore

than a trite pronouncement. Butteachers can reinvigorate it when theymake a conscientioirs effort to build ohwhat children know how to do,relying on children's own strengths toaddreis their deficits.

4. Build on children's infotmalsttategies,

Even severely learrung disabledchildren can invent quite sofhisticatedcounting strategies (Baroody. 1996).lntormal strategres provlde a startlngplace for develoipingboth concepts anilproceoures.

5. Develop skills in ameaningfal and yuryoseful fashion.

Practice is important, but practicea t t h e p r o b l e i n s o l v i n g ' l e v e lis prefer ied whenever p-ossib le.Meamngfuf purp,oseful practi.S givesus two tor the Drlce ot one.Meaninsless drill mhv actuallv beharmful"to these children (Bar6ody,1999: Swanson & Hoskyn, 1998).

6. Use manipulatives wisely.Manipulatives can help leaming

disabled siudents leam both concept:and skills (Mastropieri, Scruggs,

-&

32 Perspectives, Spummer 2000

IIn general, the traditionalcurriculum does not allow

adequate time for themanu instructional and

leaming strategiesnecessaru for the

mathematical success oflearning disabled students.

Shiah, 1991). However, studentsshould not learn to use manipulativesrn a rote manner (Clements &McMillen, 1990. Make sure studentsexplain what they are doine and linkth6ir work wit6 manioulatives tounderlying concepts and Formal skills.

7. . U:, technologt dyly.It rs rmDortant that all students

have opportunities to use technologyin appropriate ways so that they haveaccess to interesting and importantmathematical ideis. Acceis totechnology must not become yet

another dimension of educationalinequity (NCTM, 2000). Computerscan serve many purposes (Clements &Nastasi, 1992; Mastropieri et aI., 1991.;Pagl iaro. l99B: Shaw, Durden, &h Y , h ^ ^ \

Baker, 1998). Computers with voice-recognition or voice-creation softwarecan 6ffer teachers and peers access tothe mathematical ideas and arqumentsdeveloped bv students with disabilitieswho would otherwise be unable toshare their thinking. Computers canalso serve as a val"uable extension totraditional manipulatives that might beparticularly helbful to special ieedsitudents (i.f. Weir, 1987).'

Students should learn countinsand arithmetical strategies but shouldalso learn to use calculators for somepfrposes (Leqer, 1.997). For studentswho can demonstrate a clearunderstanding of an operation, thecalculator mieht be the prrmary meansof computation (Parmir & Cawley,1,997\.

8. Make connections.Integrate concepts and skil ls.

Help children l ink iymbols, verbald e s c r i p t i o n s . a n d w o r k w i t h

manipulatives. Use every possiblesocial situation to provide me'aninefulsituations for mathematical oroblemsolving opporturuties . (Baroody, 1999;Parmai & Caw\ey,1997).

,. Adiust instructional formatsto individual learning styles orsVecrfic learning needs.

Formats might include modeling,demonstration, Ind feedback; guidinlgand teaching strategies; mnemonjis t ra te .g ies f o r l ea rn ing numbercomb lna t l ons : and Deer medra t l on(Gers ten , 1 ,985 ; Le rne r , 1 ,997 ;Mas t rop ie r i e t a l . , 199 I ) . Useprolects and ,games to help , theteacher gurcle leamlng, rather thanrelying iolelv on "tellinq" Garoodv,. ^ ^ , v - 1

.

Iyyy). Ihe tradltlonal sequence ofdirect teacher explanations,. strategylnstructron, relevant practrce, andfeedback and reinforcement is ofteneffective. but the potential of studentsto learn through problem solvingshould not be'ignored..Too often]direct instruction approaches squeezeout other possibil idies. Use

'direct

instruction -onlv

when students areunable to inverit their own srraregles.In all cases, help them make stratdgiesexplicit (Kame'enui & Camine, 199€).

10. EmVhasize statistks, geome\,and flteasurenteflt as well asarithmertc toyics (Parmar & Cawley,4997).

A11 students need access to variedtopics in mathematics. Topics beyondarithmetic are increasingly importantin our dav-to-dav lives.

Oveiall, solve problems, encouragereasoning, ,and r-ise modeling. Wi[hDattence ancl support. tnese Drocessesire aiso ln the rdath of mosr ihildr.n.

References

Baroody, A. L. 0996). An investigativeapproach to the mathematlcsinslruction of children classified aslearning disabled. In D. K. Reid &W. P. Flresko & H. L. Swanson(Eds.), Cog,nitive approaches Iolearning dishbilities (pp. 5a7-615).Austin. TX: Pro-ed.

Baroody, A.l. (1999). The developmentot. ,baslc. countlng,, numbrer, andarithmetic knowledge amongchildren classified as mentallthandicapped. In L. M. Glidden (Id.),Interuatiitial review of research in ieniilretardarion $lol. 22, fp. 5 I - I 03). NewYork: Academic Prds's.

continued on yage 33

Translating Lessons from Research into Mathematics Classroomscontinued from Vage 32

Carpenter, T. P., Franke, M. L., Jacobs,-V. R., Fennema, E., & Empson, S.

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Note:Time ro nrenare this material

was partially provided by tvvoNational Science Foundation (NSF)Research Crants, ESI-9230804,"Building Biocks-Foundations forMathematical Thinking, Pre-Kindergarten to Grade 2: Research-based Materials Development" andESI-981,42I8: "Pianning forProfessional Development in Pre-School Mathematics: Meeting theChallenge of Standards 2000." Anyopinions, findings, and conclusions orrecommendations expressed in thispublication are those of the author anddo not necessarily reflect the views ofthe NSF.

Douglas H. Clements, Professor ofMathematics, Early Childhood, andComputer Education at SUNY/Buffalo,has conducted research and publishedwidely in the areas of the learning andteaching of geometry comyuter aVplicationsand the effeas of sotial interactions onlearnina. He has co-directed seueral NSFyroiects, Vroducing Logo Ceometry,Investigations in Numbel Data, andSyace, and more than /0 referred researcharticles. Active in the NCTM, he is editorand author of the NCTM Addendamaterials and was an author d NCTM'sPrinciyles and Standards for SchoolMathematics (2000). He was chair of theEditorial Panel of NCTM's researchjournal, the Journal for Research inMathematics Education. In his currentNSF-funded project, Building Blocbs-Foundations for Mathematkal ThinbingPre-Kinderparten to Crade 2: Research-based Matirials Develoyment, he andJulieSarama are develoying mathematicssoftware and activities for young, children.

Perspectives, Summer 2000 33