This article was downloaded by:[Hoard, Mary K.] On: 2 July 2008 Access Details: [subscription number 793095007] Publisher: Psychology Press Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Developmental Neuropsychology Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t775653638 Mathematical Cognition in Intellectually Precocious First Graders Mary K. Hoard a; David C. Geary a; Jennifer Byrd-Craven a; Lara Nugent a a Department of Psychological Sciences, University of Missouri,

Online Publication Date: 01 May 2008 To cite this Article: Hoard, Mary K., Geary, David C., Byrd-Craven, Jennifer and Nugent, Lara (2008) 'Mathematical Cognition in Intellectually Precocious First Graders', Developmental Neuropsychology, 33:3, 251 — 276 To link to this article: DOI: 10.1080/87565640801982338 URL: http://dx.doi.org/10.1080/87565640801982338

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 DOI: 10.1080/87565640801982338 ISSN: 8756-5641print/1532-6942online Taylor &FrancisGroup,LLC 2008 Copyright © HOARD,MATHEMATICAL GEARY, COGNITIONINPRECOCIOUSCHILDREN BYRD-CRAVEN, NUGENT DEVELOPMENTAL NEUROPSYCHOLOGY, itdidvdas(uisi ebw eb lseRce,20;Lubinski, Mazzocco, 2006; Bleske-Rechek, Murphy, & mathematically press; Webb, of Benbow, contributions in (Lubinski, scientific individuals Delvin, long-term gifted the & 2007), Early, Mazzocco & Jordan, 2003; Hanich, 2004; Kaplan, DeSoto, & & Byrd-Craven, Hoard, Hanich, disabil Geary, learning (MLD; a with mathematics children in of deficits ity the of par understanding our been have in gains There ticular 2003). Royer, core 2006; the Geary, 2005; of Dowker, understanding children 2005; developing (Campbell, our of in competencies decade, mathematical school-taught past and the quantitative during made, been have Strides nvriyo isui scooyBidn,20S t t,Clmi,M 51-50 E-mail: 65211-2500. MO Columbia, St., 7th S. 200 Building, [email protected] Psychology Missouri, of University ftetrecmoet fwrigmmr otiue odfeetapcsof aspects different to contributed memory working of components skilled performanceonthemathematicstasks. in three each differences the that group revealed of analyses when differences addition Individual eliminated and controlled. or were line memory number reduced working the significantly on were children tasks precocious the strategy of processing. advantages of speed the to for and of differences problems group Many addition no were solve there to estimates; strategies line working of number of mix make mature components more all a for used peers Preco- and same-age memory, problems. over addition advantage complex an and had children simple es- cious solve line to number used at strategies skill assessed and that timation tasks and measures, processing of speed battery, a ( cal ( precocious intellectually Forty-six orsodnesol eadesdt ayK or,Dprmn fPyhlgclSciences, Psychological of Department Hoard, K. Mary to addressed be should Correspondence Jennifer Byrd-Craven isnow atthePsychologyDepartment ofOklahomaStateUniversity. Mathematical CognitioninIntellectually M Mary K.Hoard,David C.Geary, JenniferByrd-Craven, g 5mnh)cide eeamnsee tnadzdwrigmemory working standardized a administered were children months) 75 = age rccosFirstGraders Precocious Department ofPsychological Sciences University ofMissouri and LaraNugent M g 4mnh)ad20itletal typi- intellectually 250 and months) 74 = age 33 3,251–276 (3), - - Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 hlrnso ifrnilueo okn eoysseswe ovn mathe solving when systems memory working of use differential show children (see learning mathematical to Bull etal.,2008/thisissue). contributes and con intelligence mechanism fluid core general a the to is tributing is which memory, sophisticated, candidate working more of likely component of executive A use central strategies. the enabled problem-solving turn advanced in developmentally ad that an mechanisms had but these peers, for typical vantage intellectually to their mechanisms as cognitive problems same mathematics the the engaged advance solve developmental children precocious a if for found Evidence be (but issue). would mathematics 2008/this ad Wiebe, in & this advantages Espy, whether expected Bull, so, their see if to and, directly children contribute precocious the will whether for vantage known found not be is will it advantage but same work adolescents, visuospatial gifted enhanced mathematically for in evidence memory Hitch, ing found & (1991) Benbow (Baddeley and pad Dark sketch 1974). visuospatial cen difference the and systems; loop memory these working phonological executive, core if three tral the and of use children differential to mathematics typical related solved were intellectually children than precocious for- differently intellectually the if problems for Evidence provided learning. be mathematics would of mer study conceptu- a with this practice put into we alization and the advance, developmental on versus light difference shed velopmental not does itself of and in this IQ, more basicmechanismscontributing tothislearning. often average intellectual is of above children 5% an these top in to the common attributed is in that children learning represents accelerated 126 the of Although these ability. IQ mean into Our entrance IQ. children on precocious for based intellectually identified practice we programs, common enrichment) and and gifted the (gifted school-based Following in common programs. are been that has enrichment children research of little our samples but in using individuals, advances conducted achieving been have and typically there individuals and words, intelligent low-achieving other unusually in In in children. development 50 mathematical in of 1 knowledge to representing 20 individuals. common, in 10,000 more 1 in much about 1 is child about precocious only intellectually in includes “typical” found group The levels latter the IQ (2006); estimated mathemati al. with the et individuals from Lubinski ways by important studied in individuals differ gifted they cally group, a as (Gott but, adulthood 1997), in fredson, professionals educated among over-represented are be will and 1984), tests (IQ) intelligence standard on percentile 95th the common ingiftedandenrichmentprogramsthroughouttheUnitedStates. above typ are or who children at mathemati are ically These the children. on precocious focusing intellectually studies of are cognition cal absent Curiously (Galla 2005). mathematics Kaufman, in & differences sex gher of and 2001), Benbow, & Morelock, Webb, 252 ots hte nelculypeoiu hlrnaditletal typical intellectually and children precocious intellectually whether test To de- a of terms in is learning accelerated of study the conceptualize to way One Walberg, 2000; (Lubinski, settings academic in well do children Precocious HOARD, GEARY, BYRD-CRAVEN, NUGENT ------Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 on Working Memory and Working Memory mathematical and components these between sec relation learning. the potential in the and mechanisms discuss component ond underlying its re we and section, memory first working the view In memory. working is mechanisms cognitive these Among that cog- mechanisms and 1983) 2002) Hunt, underlie Engle, 1999; & Conway, Kane & 2000; Laughlin, Tuholski, al., (Engle, et nitive (Duncan brain the identifying to cades etdsic nweg ae htaecmoet of components are repre- These that of numerical. bases are, and knowledge spatial, There distinct verbal, memory. novelsent as working such learn abilities on and specific dependent solve, more strongly course, problem more reason, is flexibly and to information, intel- ability proce- Fluid memory. the and long-term represents on heuristics, dependent ligence is facts, and of lifetime, a base over acquired knowledge dures the represents intelligence ( lized intelligence crystallized factors: two Horn 1963; of (Cattell, that Horn proposed and 1966) mental Cattell Cattell, other intelligence, & all general on term performance his strong thus with and associated tests, be on to performance tends strong test that mental discovery one (1904) Spearman’s after decades six Nearly & Nugent, present anoverview ofthemathematical tasksandrelatedhypotheses. intelligence, general Byrd-Craven, of overview an Hoard, present or we (Geary, section, first pad the ex In central 2007). sketch Numtee, the both visuospatial to related and be to execu ecutive appears central latter the the to on related performance is number whereas task a tive, former and the on assessment Performance strategy task. estimation addition line an administered we problems, matics r soitdwt niiuldfeecsi okn eoy(.. Carpenter, (e.g., memory working in differences individual with associated are cen the between relation and executive A systems. tral visuospatial activated and information (phonological) of echoic manipulation the in the attentional epi control the the that of thus mechanisms composed inhibitory is and executive and systems central The two considered. first not the the is buffer of on sodic all conducted Nearly been buffer. has the Hitch, episodic work systems; experimental and & representational sketchpad, three Baddeley visuospatial and executive 2000, loop, central phonological (1986, a of Baddeley composed by is 1974), described as memory, Working g g n o nelgnei eae owrigmmr.I h eodscin we section, second the In memory. working to related is intelligence how and , uigpolmslig(arl,19) h ou a hfe nrcn de- recent in shifted has focus The 1993). (Carroll, solving problem during F g GENERAL INTELLIGENCEANDWORKING MEMORY n hsspotteaiiyt er nsho Gay 05 2007). 2005, (Geary, school in learn to ability the support thus and F g a enfrl salse:Promneo esrsof measures on Performance established: firmly been has F MATHEMATICAL COGNITIONINPRECOCIOUSCHILDREN g g smr cuaeydsrbda en composed being as described accurately more is g ) n li nelgne( intelligence fluid and C), g u r fe dependent often are but C g ) Crystal- F). 253 g F ------Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 ut hl,19;Egee l,19) h tegho hsrlto agsfrom ranges relation this of strength The 1999). al., et ( Engle moderate 1990; Shell, & Just, h neligcgiiessesi teto-rvnwrigmemory. working that attention-driven is argued define systems have reasoning cognitive 1999) abstract underlying and the Stanovich, solving 1988; problem strategic Horn, of 1990; measures many al., patterns, these et of basis (Carpenter the scientists On 2002). Minkoff, & Therriault, Bunting, Cowan, esn Jne,19)a atrsedo ontv rcsigi eae ohigher to related is processing of cognitive measures of on speed scores faster as 1998) (Jensen, cessing ombte ncmlxpolmsligtssta eur okn eoyre memory working require that tasks problem-solving complex per on and 1978) intellectually better Bobbitt, children, form & (Keating typical faster intellectually information process to children comparison precocious in that visuospatial surprising the not engage to appear line, number a of use esti Baddeley, sketch pad(Zorzi,Priftis,&Umilta,2002). in in & involved as processes quantity, (Logie other of counting whereas mation 1994), in Wynn, & as Gilhooly, Logie, articulation, 1987; number involve and quantitative Logie that in engaged problems. processes is loop word phonological problems algebraic the that solve multi-step demonstrated have to to colleagues ability addition the simple underlying has from component executive ranging crucial central a the as 2003), & Royer, implicated (Geary & been research Tronsky related 1978; and this Hitch, of 1992; much Widaman, In mathemat 2001). with Kirk, adults & Hitch, (Ashcraft and & anxiety 2001), McLean ics Sachse-Lee, 1991; John McAuley, & Bull, & Swanson Hitch 1997; 1993; 2004; Swanson, Johnston, al., 1999; et & Geary (Bull 1999; Roy, MLD & mathematical with ston, and children memory including achieve mathematics scores, working low ment with between individuals with relation conducted been the has competencies on research of Much andMathematics Working Memory mathemati- our of on contributions differences cal tasks. group independent and and individual overlapping to measures the corresponding assess the pro- to of able speed were issue, and we systems this memory cessing, resolve working cannot three We assessing 1999). simultaneously by al., executivebut et central (Engle the with & processing associated (Fry information focus processing speeds attentional neural the of whether speed or in 1996), memory differences Hale, working fundamental in more differences by individual driven whether are on center 2000). issues (Deary, and The memory, scores working debated. IQ processing, of highest consis speed the are among have relations who processes The Individuals these times. executing in multiple fast process tently same the executing of speed to lated 254 h te ehns hthsbe rpsda underlying as proposed been has that mechanism other The ie h eain mn okn eoy pe fpoesn,adI,i is it IQ, and processing, of speed memory, working among relations the Given HOARD, GEARY, BYRD-CRAVEN, NUGENT g h aiblt esr rvdsa seseto h ossec in consistency the of assessment an provides measure variability The . r .;Akra,Bir ol,20)t eyhg ( high very to 2005) Boyle, & Beier, Ackerman, 0.5; ˜ s g aiblt nsedo rcsigars rasi lore also is trials across processing of speed in Variability . g g ssedo pro of speed is F r r vigorously are F .;Conway, 0.8; > s g ,adoeof one and F, ------Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 hlrnslce ae npromneo esrsof measures on performance on of based Comparisons 2006) selected 1983). Swanson, children Zonderman, 1996; Kirk, Stanley, al., Benbow, et (>150; exceptional Robinson Robinson, to (˜115; par- average these 1993; of above Stumpf, Benbow, IQ from estimated & range & the ticipants 2006); Ablard, (Dark Swanson, Mills, 1996; se Busse, 1991; per & Berninger, Brown, IQ Abbott, & not Geary of and basis 1991; tests the 1990, achievement on selected mathematical been on pre have intellectually scores participants of adolescents, competencies quantitative and the children on cocious research the of all nearly In will memory working visuospatial enhanced if 1990, is Benbow, be foundforintellectuallyprecociouschildren. known & not (Dark is a adolescents What are 1994). gifted there 1991, noted, mathematically preco As highly understood. intellectually of well of working studies not advantage few is of tasks likely contributions mathematical the the on to children time, processing cious same of the speed At and 2003). memory Carr, & (Steiner sources omn ersnain htrsl na qa itnebtenscesv num involves successive contrast, between distance in equal line, an bers regardless ofpositionontheline(Siegler &Booth,2004). number in 72 mathematical result between that formal distance representations perceived the forming the of than Learning larger 73. is per 3 and the and that 2 such between magnitudes larger distance ( for ceived logarithm compressed are re natural representations system the the inherent to is, that The conform 2003). that Opfer, representations & in Siegler number sults 1992; mathematical Gelman, school-taught the & on (Gallistel based estimat line is for system second inherent the an and num on a magnitude based on is ing position representation esti correct of the line form in first number 32) The (e.g., make line. numeral ber to given a representation place of to is, forms that two mations, on rely to appear Children Number LineEstimation to performance their relate and choices, perfor- strategy num- the coreworking memorysubsystems. the addition of and of areas the estimation assessment in line tasks an cognitive ber with on children knowledge precocious intellectually of of lack mance cognitive this using redress competencies We quantitative measures. of of assessment performance fine-grained the not a examined have and have provided measures, date psychometric-type to on children studies and precocious the Benbow mathematically of and Dark most of Brown, exception and to the literature, Geary with Moreover, the knowledge. in our available of not best are the tasks cognition mathematical on peers typical MATHEMATICAL COGNITIONINPRECOCIOUSCHILDREN MATHEMATICAL COGNITION g n hi intellectually their and F ln ftenumber, the of ) 255 ------Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 256 rn n otesligo oecmlx(.. 7+6 rbes n assess and problems, 6) dur + engaged 17 system memory (e.g., working the complex in ing problemsolving. more differences group of are 7) solving chil there + younger whether the 4 to to analysis (e.g., this used problems and extend We children addition dren, peers. simple precocious typical intellectually solve intellectually their to did The counting than graders. less fourth and pat retrieval of this more just sample found a (1991) Brown in and tern Geary 1986). ad Kotovsky, developmentally are & but (Siegler peers vanced typical intellectually precocious their intellectually to domains, similar many are In children retrieval. and dominated decomposition becomes mix of the use or eventually by and sum frequently more the procedure of children min schooling, the use With use 1984). frequent Shrager, & and Siegler strategies, 1996; (Siegler, counting procedures max of use by dominated initially (e.g. decomposition re and direct 10 +2=12). memory support long-term from representations fact these the addends of formed, the trieval and Once counting 1984). of Shrager, means & by generated be- (Siegler proce- not) memory the long-term or counting in (correct stating of link answer associative involves the use an tween of which The formation addend. in used, result larger to is the appears dures counting procedure then max and the addend smaller Sometimes 1. from ing 92.Temnpoeueivle ttn h agrvle dedadthen and count- as addend such solv addend, to smaller valued 5 the of 4, larger value 3, the ing the to equal stating times of involves number Parkman, a procedure counting & com- Groen most min 1982; two (Fuson, The sum the and 1972). counting, min When termed are counting. procedures re- verbal used monly and and a counting involve strategies strategies finger counting counting of The mix of problems. these combination solve children. to a processes precocious trieval-based use intellectually is children of little choices achieving but strategy Typically 1984), addition Shrager, the & about Siegler known 1990; achieving (Geary, typically for children documented low-achieving well and addi been solve have to problems use subtraction children strategies and of tion mix the in differences group and Individual Addition Strategy Choices es line number advantage making at same skill the influence will timates. have advantage is such children any it whether precocious but so if 1991), intellectually and Benbow, young & if (Dark known memory not working visuospatial have in adolescents advantage precocious Mathematically an 2002). Bihan, al., Le et Piazza, Zorzi (Pinel, 2004; estimates Dehaene, line & number making at skill and abilities spatial h i fsrtge hlrnuet ov eso ipeadto rbesis problems addition simple of sets solve to use children strategies of mix The visuo between link a suggest studies neuropsychological and imaging Brain HOARD, GEARY, BYRD-CRAVEN, NUGENT e3+2.Thesum rcdr novscutn ohadnsstart- addends both counting involves procedure ,5+7=5+5= ------Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 ino esrso okn eoyadsedo rcsigealdu to us enabled processing of speed differences and individual in underlying memory mechanisms potential working the assess of simultaneously measures inclu of the Moreover, competencies. sion chil mathematical core precocious two assess intellectually that compares tasks mental that of few measure a the using of identified dren one is study current The mndi h urn td.Te4 2 os hlrnwt Qsoe >120 scores IQ with children boys) (27 46 group; The precocious study. intellectually the current composed the ex- in- was as in children classified recruited children amined these of 211 typical) (the chil- intellectually subset 311 or A precocious for population. tellectually received recruited were MLD the assent with of child 37% children and dren, consent in Parental competencies 2007). mathematical al, et of (Geary, prospective development longitudinal a the in guardians of participate and/or parents to study their invited to home were sent schools letter a elementary via 12 from children kindergarten All Participants (devel same memory the working to difference) related (developmental are different systems. differences or group advantage) such opmental any whether and tasks, ru nldd20(1 os hlrnwt Qsoe ewe 1ad10 inclu 120, and 81 between scores IQ sive; with children boys) (110 250 included group ae h enae f7 ( 74 mixed of of ages or Asian mean were The white children mixed were race. remaining sample of of most typical or the intellectually and respectively, black the black, of were and 15% children and 69% remaining comparison, of In race. most the and respectively, Asian, p sn g-ae om Rvn or,&Rvn 93,adteewr converted were these and 1993), Raven, to scoresstandardizedwitha mean of100( & Court, (Raven, norms age-based measure using excellent an be to considered is of that test power non-timed a Matrices, sive Standardized Tests precocious andintellectuallytypicalgroups,respectively, didnotdiffer ( 0;8%ad1%o h nelculypeoiu apewr ht and white were sample precocious intellectually the of 11% and 80% .05; < g g Intelligence. o hlrn(esn 98.Apretl akn a bandfrec child, each for obtained was ranking percentile A 1998). (Jensen, children for F ,terptnilcnrbtost n ru ifrne ntemathematical the on differences group any to contributions potential their F, M 0 ( 107 = SD ) h tncbekondfee cosgroups, across differed breakdown ethnic The 8). = l hlrnwr diitrdteRvnsClue Progres Coloured Raven’s the administered were children All MATHEMATICAL COGNITIONINPRECOCIOUSCHILDREN SD CURRENT STUDY )ad7 ( 75 and 4) = g oteritletal yia er nexperi on peers typical intellectually their to F METHOD SD SD M = 15). )mnh o h intellectually the for months 4) = 2 ( 126 = SD ) h comparison The 5). = χ 2 6 14.7, = (6) p >.10). 257 ------Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 hl’ aki odtriei h eodls si h aeo ifrn re than order different or same the in is list second the the and if the first list. again, determine presented to child. then is the are order, to task different child’s presented a is in with level, possibly beginning successive but words, each words, same at of the The word by series one spoken adding a as and task, words order Matching two same List the Word in the tasks; words In span repeat experimenter. standard to Recall. are is List Recall task List child’s Nonword Nonword the and specifically, and Recall, Recall, List List Word Word Recall, Matching, Digit List Word Recall, Digit 258 a oeta n ouin swl sapcueo nietclmz ihapath a with maze identical an of picture a that as maze well a as with solution, presented one is the child than by the more presented task, as has order Memory same Mazes the blocks), in the of tapping the In the experimenter. series from duplicate to (or seen is ar be block task child’s “random” only a the a can and taps as that experimenter child side The the one to perspective. on experimenter’s appears numbers what have in blocks blocks The consist raised rangement. stimuli nine the with but board task, a span of another is Recall Block Sketchpad. Visuospatial altenme fcutddt tteedo h eiso ad.Bcwr Digit Backward cards. re- of to then series and the cards), of of end series (or the card at a dots on Recall isastandardformatbackward digitspan. dots counted 7 of the or requires number 6, Recall the 5, 4, Counting call of sentence. set each a in count word or to true last child is the sentences) of recall series then (or and sentence a false, if determine to child Listening the Recall. requires Digit Recall Backward and Recall, Counting Recall, Listening subtests, pnsoeadatil orc cr.Fo hs,sadr crsaedetermined are scores standard a these, generates From subtest score. correct Each for eachsubtestandthethreeworking memorysystems. trials Recall. a Digit and Backward score Matching, span Listening and List Memory, Recall, Word Mazes Recall, Counting Recall, Block Recall, Recall, Digit List was hardest: Nonword and Recall, to memory de List working Word easiest was of area subtests from component of arranged one order any generally over-tax The to increased subtest. not is the as remembered terminates so be signed items to three words) At Failing (e.g., level. next items one. the of to by subtest child number per the the moves span level level, maximum a span at the items each and four Passing level, 9. span to 6 each from memory at ranges of items working All six core sketchpad. have three visuospatial subtests the and the loop, assess phonological that executive, subtests central nine systems: of Gather & consists Pickering 2001) (WMTB-C; Children cole, for Battery Test Memory Working The Working Memory Visuospatial sketch pad. sketch Visuospatial loop. Phonological executive. Central HOARD, GEARY, BYRD-CRAVEN, NUGENT h eta xctv sasse sn he dual-task three using assessed is executive central The h hnlgcllo sasse sn orsubtests, four using assessed is loop phonological The lc ealadMzsMmr sesthe assess Memory Mazes and Recall Block - - - Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 tml,12o h rbessme o1 rls.Tecmlxsiuiwr 6+ 16 were stimuli complex The less. or 10 to 7, 3+18,915,174,6 19, and14+8. summed problems the of 1/2 stimuli, ino ln ubrln sn ecl hr a otm osritfrti task. this for constraint time no loca was appropriate There its pencil. mark a using to line asked number was blank child a the (2004). on task, tion Booth this and in Siegler number by target described each or procedures For logarithmic random the a following the in child, of presented each fitting trials. were for 24 stimuli a der yielding Experimental for 96, estimates. and allow child’s 90, the to 84, Following to over-sampled 21, 81, model 79, 17, pt). were 72, 12, 30 64, (72 8, below 61, 6, 57, font numbers 52, 4, The 48, large 3, 43, were 39, a target 33, numbers a 29, in target 25, with 1), 23, it 100 experiment of above (2004, endpoint Booth cm an and and Siegler 5 0 of approximately point printed start a number had line 1/2” number 8 Each standard a tation. of center the across printed lines, otly oisl.Temti speetdi ag ots htte5 tm take items letter the 50 for the reversals as that well so as 1/2” font errors 8 large sized a standard a in hori of or presented most (vertically is up adjacent matrix never The is itself. letter the to same in zontally) the times possible 11 that as to constraint 9 quickly the occurs (5 as with number) them matrix incidences (or name letter 50 Each to mistakes. of asked many is matrix each making and without for a numbers), tag with (or name letters presented the same then state these or is can letters five and child of recognizes The items child practice stimulus. the presented if is determine child processing to the assess task, numbers this (Denckla to In memory 1976). long-term used Rudel, from items were & learned tasks of retrieval with (RAN) associated Naming speed Automatized Rapid The Speed ofProcessing At shown. was she path and the removed booklet is response picture her The in outside. each level, themazesgetlarger byonewall. duplicate the to to is task center child’s the the from solution one of drawn es h rbescnitdo h nees2truh9 ihtecntan that constraint the with (e.g. 9, prob integers through addition the two 2 single-digit at integers same 14 time, the the were a of stimuli at consisted simple one problems The The tall), card. lems. 8’’ cm by 2 5’’ (about a font of large center a in presented horizontally were Mathematics Tasks and infrequent were reversals and (as Errors correct letters). number for and thus onlyRTs wereusedintheanalyses. RT reversals an of generates number task as the well numbers), or (letters stimulus of Addition strategy assessment. strategy Addition estimation. Numerical MATHEMATICAL COGNITIONINPRECOCIOUSCHILDREN ,2+2) tml o hsts eetet-or2 mnumber cm 25 twenty-four were task this for Stimuli × 1 ae.R smaue i tpac,and stopwatch, a via measured is RT paper. 11” eenvrue ntesm rbe.Across problem. same the in used never were sb&dandp&q,are ipeadcmlxadto problems addition complex and Simple × 1 ae nalnsaeorien- landscape a in paper 11” eodd o ahtype each for recorded; × 0 of 10) 259 - - - - Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 260 et.Ec eso vrgd3 o4 i.TeWT- a de otestudy the to added was WMTB-C The min. assess 40 between were to moved on 30 child room children averaged the testing of session if a van Each majority in testing ments. mobile occasionally The a and in grade. site, or mathematical first school campus university the their of the at and fall location kindergarten, quiet the a in of in session end tested single the a at in administered tasks was test IQ The Procedures chil of measure useful 1987). Siegler, a (e.g., provides choices method strategy the trial-by-trial this got dren’s they indicate how of studies but description child’s Previous answering, the of answer. on speed dependent and was behaviors coded re overt attempt strategy on child’s final initial based the an the made trial, then was every strategy for ambiguous, the words, was code other to In trial strategy. the the as experimenter If recorded strategy. the was then counting sponse overt, a was as counting it If the made. classified between was disagreement experimenter indicating the notation and a child then saw counting), (e.g., mouthing observations experimenter’s child re- the child’s the from the differed If it”) obtained. answer, knew was the “just answer (e.g., spoken the then sponse how had and on child 10 child get the the to After queried 8 experimenter (decomposition). the 12.” the just to “I get one added say to and might other 2s child the two the into added 8, 4 + broke “I 4 or for (retrieval) example, it” For knew problem. description the child’s solved the they from how came differentia- of retrieval The direct and trials). answer decomposition previous between the the tion on was this strategy spoke if retrieval-based decomposition movements, child as predominant (or counting-related retrieval the child’s as classified obvious If initially was without fingers. trial the as and then classified hesitation, initially without them, was quickly, counting of without number trial answer a an the out stated held then then child and addends the the she If represent how recorded. to to was fingers as re response child child’s counting, the the verbal probed and experimenter or counted, the counting trials, finger counting experi all as the On strategy spectively. trials, the these For classified movements. initially mouth menter or finger regular other. as or such counting, max, sum, using min, problem as the classified completed further but strategy, were trials one Counting was using strategy. trial started another mixed child A the 1984). which Shrager, in & one de (Siegler retrieval, strategy counting, other/mixed verbal or fingers, composition, fingers, 1 on counting into specifically, Based classified strategies; loud. was 6 out trial the answer of was observations, the too experimenter’s strategy speak the making whatever to and answer instructed use without child’s was the could possible and answer, child as the the get quickly that to as easiest emphasized pencil) was It problem and each mistakes. paper solve many of to asked use was the child The (without problems. complex the by immediately uigpolmslig h xeietrwthdfrpyia niain of indications physical for watched experimenter the solving, problem During followed presented were problems simple the problems, practice two Following HOARD, GEARY, BYRD-CRAVEN, NUGENT ------Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 ntetrewrigmmr aibe.Hge eta xctv crswr asso were scores executive central Higher variables. RT memory RAN working mean three the the regressed on simultaneously we memory, working and processing loop, cen ical the on performance better with executive, associated tral was processing of speed faster nam that number and ( letter significantly across were (averaged RTs ing) Mean numbers. than letters reading hssadrie ( We 2001). standardized than Gathercole, larger & was thus (Pickering children children 211 86 of of sample sub-sample our standardization study, the this in assessed range age the For andSpeedofProcessing Working Memory the are these because choice strategy addition and line line), number most complex ofthesetasks. (number the study, this Estimation on In focus Numerical Assessment. we Strategy order: Addition following Sets, Number the Knowledge, tasks. Counting in mathematical four given included (2007) mem were al. working et They the Geary of 2007). al., any et on (Geary performance tests ory with correlated not is administration ( of groups typical intellectually and precocious intellectually the urdwe h hl a o nsho.Tema gsa h ieo the of time the at ages mean The school. in oc not ( and 81 was min were 60 assessment child about WMTB-C required the assessment when The grade. in curred first administered was during typical battery van intellectually the testing and children, the precocious of majority intellectually the the For 44 in respectively. assess groups, children to able the were of we study, 167 this and for and kindergarten following summer the in .6 rfrtetp ygopinteraction, group significant, by type the for or 1.76, sdteefrsbeun nlss ie NV ihgopa h be- group, the for as effect group vari- significant with within-subjects a ANOVA the revealed as mixed system able memory A working and analyses. and sample variable subsequent our tween-subjects for for scales Executive these Central used and Pad, Sketch Visuospatial Loop, hlrnhdhge tnadsoe o l okn eoysses( systems memory working all as shown inTable 1 for scores standard higher had children tion, system, memory working for effects non-significant ie NV eeldn infcn fet ( effects significant no sec revealed ANOVA 43 and mixed means 41 respective were naming the number and for groups, intellectually-typical and lectually-precocious o etrnmn,tema T ee3 n 1sc epciey o h intel the for respectively, sec, 41 and 38 were RTs mean the naming, letter For F 2 1)<1 olwu NVscnimdteitletal precocious intellectually the confirmed ANOVAs Follow-up 1. < 418) (2, r 20 .9 esrs odtrieterltosbtensedof speed between relations the determine To measures. 0.39, = (210) F 1 7)=13.09, = 279) (1, r 20 .9 iusailsec pad, sketch visuospatial 0.49, = (210) MATHEMATICAL COGNITIONINPRECOCIOUSCHILDREN M . 100, = p 00)creae ihwrigmmr crs such scores, memory working with correlated .0001) < s SD p SD 01 h hlrnwr infcnl atrat faster significantly were children The .001. > 5 opnn crsfrtePhonological the for scores component 15) = RESULTS )ad8 ( 82 and 4) = gop y2(ye etr,numbers) letters, (type: 2 by (group) 2 A . F 1 7)<1 u h fetfrtp was type for effect the but 1, < 279) (1, F 1 0)=49.66, = 209) (1, SD p 1)frgroup, for .10) > s F )mnh,rsetvl,for respectively, months, 6) = r 2 1)<1 n h interac the and 1, < 418) (2, 20 .0 n phonolog and 0.30, = (210) p 1) g ttime at age .10); > p 00,but .0001, < p F .0001), < s 1 7)= 279) (1, 261 ------Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 262 rccosgop h iermdl( model linear the group, significant, precocious not was fit in difference the Numerical Estimation Numerical in was it significant, perfor not task was RT mathematics RAN for in the expected directionandthuswas includedintheseanalyses. difference differences group the group Although influenced mance. or memory processing working in differences of al group This which speed equation. to extent regression the the determine to to added us lowed was variable group in this a differences in then differences individual performance, group predicted significant variables were these there and of performance several mathematics-task or mem- one working and If RT RAN scores. on ory regressed were correct) % (e.g., tasks ( analy- mathematics standardized mechanism the were For RT. variables RAN all by indexed ses, was latter and the perfor- systems processing; memory this working of three speed to the were related explored and potentially we mechanisms individual mechanisms The then mance. cognitive and the performance in task differences in group differences group explored first We Mathematics Tasks to unrelated were variables pad sketch visuospatial RAN RT ( and loop phonological the on processing, of speed faster with ciated logarithmic one( oes ssoni iue1 o h nelculytpclgop ohteloga the both group, typical intellectually linear the and For ( logarithmic 1. rithmic the Figure to in fitted shown were as group models, each for estimates median (2003), visuospatial sketchpad (VSSP),andcentralexecutive (CE). T9 49 49 15 11 97 110 15 14 110 97 14 14 112 97 IP IT Group differences. Group Note oe T=itletal yia;I nelculypeoiu;Poooia op(PL), loop Phonological precocious; intellectually = IP typical; intellectually = IT Note. . HOARD, GEARY, BYRD-CRAVEN, NUGENT R 2 p 9)adlna oes( models linear and .95) = s >.10)whenperformanceonthecentralexecutive was controlled. DMS SD M SD M SD M R 2 EVS PL VSSP CE = .91), sn h ehiusdsrbdb ige n Opfer and Siegler by described techniques the Using t (23) =–2.29, Working Memory Scores Working Memory R AL 1 TABLE Working MemorySystem M R 2 ß 9)fte h ein etrta i the did than better medians the fitted .97) = 2 t 100, = 2)=0.92, = (23) 9)poie odftt h medians; the to fit good a provided .92) = .36, = p < .05. t SD 26 4.42, = (206) 5 n aaeesfo the from parameters and 15) = p 5.Frteintellectually the For .50. > p 00,btscores but .0001, < - - - Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 IUE1 FIGURE Logarithmic andlinearfits tomedian numberlineestimates. 263 Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 P4 52 4 8 7 12 3 4 5 7 24 19 35 48 40 33 IP IT % maynot=100duetorounding. 264 siaewscoe o4,tetilwscasfe slg hnteepce value expected + the than less When by the log. differed if as models and log classified linear, and was as linear trial the classified for the was trial 43, If the to 43. 43, closer is to was representation than estimate log 23 a to using closer is was is child representation estimate the linear the if a estimate using expected for is the child example, whereas the an 23, if As estimate group. expected typical the intellectually 23, numeral the the of medians the to fitted model olwu NVscnimda datg o h nelculyprecocious intellectually the linear, for both for advantage group an confirmed ANOVAs Follow-up strategy, 14.47, as = 258) ( log) mixed effects (linear, A significant strategy revealed children. variable and precocious within-subjects variable the intellectually between-subjects For the the as trials. for group 5 log as with ANOVA of 7, and of deviation average linear an a repre- by to value linear for linear compared a actual fit used the from group of deviated typical estimate degree intellectually their sentation, the absolute in children the the col- show when rightmost example, four 2 The Table trials. log of of frequency strat- umns for across difference varied largest difference the group with the egy, of magnitude the because emerged action ( effects group significant revealed ANOVAs ( strategy significant revealed and egy, variable variable within-subjects between-subjects the the as log) as or group (linear with ANOVA mixed A 2. ble lal etrfte ytelna rlgmdl(.. h eito rmtelgand log the from deviation the (i.e., model log or linear fits was linear <+ the by fitted better clearly sdteata antd o h ra,adfrtelgrpeetto eused we representation log the for we and representation, trial, linear the the for For 16 magnitude representation. log actual or the linear used a if estimate using expected an were mark—and the they placed each they for line estimate number child’s the each on trial—where between difference absolute the calculated we mate, Ln Note h ecnae ftil lsiida ier o,o miuu r hw nTa in shown are ambiguous or log, linear, as classified trials of percentages The oetmt ra-ytilvraini h ersnainue omk h esti the make to used representation the in variation trial-by-trial estimate To F ( x 2 5)=3.4 n ru ystrategy, by group and 30.44, = 558) (2, T=itletal yia;I nelculypeoiu.Ab=abgostil strategy trial; ambiguous = Amb precocious. intellectually = IP typical; intellectually = IT . ,where 7, – ) HOARD, GEARY, BYRD-CRAVEN, NUGENT ierLgAbLna Log Linear Amb Log Linear DMSD M SD M M M M 5) thenthetrialwas classified asambiguous. x taey%Fit Difference Strategy % h au ob siae.Ti qaini ae ntelog the on based is equation This estimated. be to value the = F F Number LineStrategy andFitAccuracy 1 5)=5.19, = 258) (1, 1 5)=3.2 n ru ystrategy, by group and 34.92, = 258) (1, AL 2 TABLE p 0,adlog, and .05, < p F 0)frec taey h inter- The strategy. each for .05) < s 2 5)=84,efcs Follow-up effects. 8.41, = 558) (2, rtecidsetmt a not was estimate child’s the or 5 p 02 o group, for .002) < s F 1 5)=15.39, = 258) (1, F 1 5)=10.51. = 258) (1, p 01,strat- .001), < s F p (1, < - - Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 taeyetmts u ihrcnrlexecutive, central higher but estimates, strategy p >.05. differ group the non-significance, to equation, reduced final was p strategy the linear In the strategy. of use log in the ence of use frequent more with Addition Strategy Choices ance explained bygroupwas reducedby73%. cuayo o rasrmie significant, in remained difference trials group predicted The log strategy. the on log comparing the accuracy using difference when estimate) smaller child’s the a and (i.e., value estimates accurate more with eec nacrc fteetmtsfrteesrtge.Hge eta executive, central Higher strategies. these dif mean for and estimates strategies the log of and linear accuracy the in of ference use percent of predictors as used were iusailsec pad, sketch visuospatial n hnlgclloop, phonological and a rlgrtmcsrtg.Teitrcineegdbcuetegopdifference group the because emerged lin in fit was larger forlogthanlineartrials. interaction a The used strategy. they the logarithmic whether than or of independent ear accurate task, more this on were children children normal intellectually precocious intellectually The trials. .0001, hra oe eta executive, central lower whereas 1.75, 177) =12.69, verbal and finger the for procedure min the of strat counting strategies arepresentedinFigure2. use The re and analyzed. percentages, counting, were error these verbal mix, only egy fingers, thus and counting strategies, the decomposition or using trieval, solved were problems complex rea oefeunl hnteitletal rccoschildren, precocious intellectually the 4.61, than frequently more trieval 4.41, = 251) peers, their frequently more position n;teitletal rccoscide omte ee erea errors, retrieval fewer committed children precocious intellectually the ing; e ee rosta h nelculytpclcide o onigfingers, counting for children typical intellectually the than errors fewer ted F finger when frequently more procedure min 1 0)=7.31, = 200) (1, .19, = 2,a a h ru ifrnei s ftelgstrategy, log the of use in difference group the was as .25, > oeo h aibe mre sasgiiatpeitro cuayo linear of accuracy of predictor significant a as emerged variables the of None Group differences. Group mechanisms. Cognitive hnsligcmlxpolm,teitletal yia hlrnue re used children typical intellectually the problems, complex solving When decom used children precocious intellectually problems, simple solving When p p 0,soe eeascae ihmr rqetueo h ierstrategy, linear the of use frequent more with associated were scores .08, = 0,btterma ro aeo 9 niae hywr agl guess largely were they indicated 79% of rate error mean their but .05, < t 25 1.82, = (205) p p 0,vra counting, verbal .05, < < .001,but notfordecomposition p 0 onig h nelculypeoiu hlrncommit children precocious intellectually The counting. .01 < MATHEMATICAL COGNITIONINPRECOCIOUSCHILDREN p 0,adhge iusailsec pad, sketch visuospatial higher and .07, = –.15, = o hlrni ohgop,ms ftesml and simple the of most groups, both in children For –.17, = A Tadtetrewrigmmr variables memory working three the and RT RAN t t 19 –1.75, = (199) 25 –2.04, = (205) –.18, = F F 1 7)=15.43, = 279) (1, 120 4.62, = (1,200) F F 1 9)=4.90, = 198) (1, t 1 5)=10.27, = 251) (1, 25 –1.79, = (205) p p –.23, = F 0,soe eeassociated were scores .08, = (1, 68)<1. 0,soe eeassociated were scores .05, < p 0,adretrieval, and .05, < p t 19 –2.40, = (199) 00,adue the used and .0001, < p p p 0,bttevari- the but .05, < F 0,adlower and .07, = 0,adverbal and .01, < F 1 0)=3.15, = 204) (1, 1 0)=1.27, = 204) (1, .15, = F 1 7)= 279) (1, t p 25 = (205) .02, < F F F 265 (1, (1, (1, ------Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 266 IUE2 FIGURE ple andcomplex additionproblems. taeycocs ro ecnae n s ftemnpoeuefrsligsim solving for procedure min the of use and percentage, error choices, Strategy - Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 ( position, 0,addecomposition, and .05, < p t eeascae ihmr rqetueo eblcounting, verbal of use frequent more with associated were t 7 10.86, = 77) 0.Drn igrcutn,teitletal rccoscide sdtemin the used children precocious often, intellectually more the procedure counting, finger During children, .05. typical < intellectually the did than frequently more tion t .0 n ee erea ros( errors retrieval fewer and –.20) crswr soitdwt ee igrcutn errors, counting finger fewer with associated were scores pe fpoesn lmntdtegopdfeecsfrfne onig verbal counting, finger for ( differences errors retrieval group and the counting, eliminated processing of speed ope diinpolm;hge crswr soitdwt esfeun use frequent less with associated retrieval, were of scores higher problems; addition complex count when procedure min the of use in difference ing verbally ( group the ( for significant trend remained a was counting There finger for procedure min the and position t tdwt esfeun s ftemnpoeuewe eblcounting, verbal when procedure min the of use frequent less with ated oitdwt oefeun roswe hlrnue igrcounting, finger used children when errors frequent more with sociated oefne onigadtemgiueo hssitwslre o h intellectu- the and for decomposition larger was less shift in this of resulted magnitude ally precociouschildrenthanfortheirintellectuallytypicalpeers. problems the and complex counting to finger more simple from shift the ing, 12.30, = 279) 4.08, tion, counting, finger strategies: counting, four verbal all for problems complex and variable between-subjects the ( as significant group confirmed and strategy variable com each within-subjects with ANOVAs (group) the for 2 as ANOVAs by plexity (complexity) measures 2 Specifically, repeated 2004). al., multiple et (Geary of means by examined for minusage,and 12 1.94, = (182) 5)=–2.85, = (56) 12 –2.32, = (142) 12 2.13, = (182) p 0,admr rqetueo h i rcdr hnfne counting, finger when procedure min the of use frequent more and .02, < 0) ihrsoe eeascae ihls rqetvra onig( counting verbal frequent less with associated were scores higher .08); < s Cognitive mechanisms. Cognitive h eta xctv mre sasgiiatpeitro taeycocsfor choices strategy of predictor significant a as emerged executive central The hfsi h taeymxcmaigtesml n ope rbeswere problems complex and simple the comparing mix strategy the in Shifts F F p 1 7)= 279) (1, 1 7)=6.5 ru ifrne mre o s fdecomposition, of use for emerged differences Group 60.35. = 279) (1, 0.Tesm a on o eblcounting; verbal for found was same The .05. < .43, = p p p p –.20, = p 0.Teitletal rccoscide gi sddecomposi used again children precocious intellectually The .01. < p = .08). p 0,vra counting, verbal .05, < 04 lwrsedo rcsig(.. ihrR aus a as was values) RT higher (i.e., processing of speed Slower .054. = 01 n h opeiyb ru neato o igrcount finger for interaction group by complexity the and .001, < 0.Tevsopta kthpdddnteeg sasignificant a as emerge not did pad sketch visuospatial The .01. < p F t 0.Teol fet o h eta xctv eetotrends two were executive central the for effects only The .05. < 4)=2.20, = (45) 1 7)=9.7 retrieval, 92.37, = 279) (1, 0,addecomposition and .06, < F MATHEMATICAL COGNITIONINPRECOCIOUSCHILDREN (1,115) =6.09, t F 25 –2.01, = (205) p 1 0)=4.20, = 205) (1, 00)cagsi taeisuaears h simple the across usage strategies in changes .0001) < s .18, = p p 1) u h ru ifrne nueo decom of use in differences group the but .15), > s o ipepolm,hge iusailscores visuospatial higher problems, simple For 0.Soe pe fpoesn a loassoci also was processing of speed Slower .05. < t 25 2.14, = (205) .2.Cnrligfrwrigmmr and memory working for Controlling –.22). = p p < .02,forerrorpercent. 0,adfwrrtivlerrors, retrieval fewer and .05, < p .45, = 0,adwt ee errors, fewer with and .05, < F F 1 7)=14.38, = 279) (1, t 12 4.58, = (142) 1 7)=7.3 n decomposi and 72.53, = 279) (1, p 0.Hge hnlgclloop phonological Higher .05. < F 1 1)=3.37, = 115) (1, –.21, = .18, = p 01 n decom and .001, < F F p 1 7)=90.79, = 279) (1, 1 7)=3.97, = 279) (1, .0.I short, In <.001. t t 25 2.09, = (205) 12 –2.42, = (182) F 1 0)= 205) (1, p .01). < s p –.25, = –.58, = .18, = .08, < .19, = F 267 (1, = p p ------Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 268 l tde yBno n olausaepoonl nelculypeoiu and individu precocious the intellectually that profoundly given are colleagues surprising and not Benbow are by as studied Benbow study als and our Dark for of findings different those The to systems. compared memory working three all for tage h i rcdr hnfne counting, finger when procedure min the igrcutn errors, counting finger ecns ahr u nelculypeoiu hlrnhdaruhy1 roughly a Benbow’s had and children precocious Dark intellectually ado in our precocious Rather, verbally implicated lescents. and precocious as mathematically of loop), contrasts phonological 1994) (1991, the (i.e., work verbal memory to ing relative memory working visuospatial for of evidence enhancement find however, differential not, did We memory. an working had visuospatial also in mathe did advantage sample had for precocious than (1991) intellectually children Benbow our memory and adolescents, precocious Dark working precocious by matically intellectually of found As component our peers. executive typical 1999), intellectually central their al., the et on Engle scores higher 1990; and memory al., working et between relation ter the of studies with keeping In andSpeedofProcessing Working Memory and group and processing, of speed individual differences forperformanceonthesetasks. memory, working between to relation contribute potential to mecha- appear basic processing—that the in in of variation differences speed individual and and memory group nisms—working In review 2006). we Swanson, 1991; section, Brown, first & Geary the 1994; per 1991, IQ than 1990, Benbow, rather & scores (Dark ability se and also achievement mathematics study of iden- basis potential current been the the have on The that tified of adolescents areas. and studies children few these precocious on the in research of complements their advantages one of and their study line, underlying first number mechanisms a the on problems, intel estimate by addition used to strategies solve abilities of to mix the children of precocious studies few lectually the of one provides research The ( counting finger fre when and usage retrieval, min of and The groupdifference inuseofretrieval errorsremainedsignificant ( use errors in counting finger differences of group quency the eliminated processing of speed .55, rdco faysrtg aibe u ihrpoooia opsoe eeasso counting, were verbal scores frequent loop more phonological with higher ciated but variable, strategy any of predictor pe fpoesn a soitdwt oefeun igrcutn errors, counting finger frequent more with associated was processing of speed counting, t 17 5.99, = (147) HOARD, GEARY, BYRD-CRAVEN, NUGENT –.29, = g .I h eodscin efcso h ahmtc ak n the and tasks mathematics the on focus we section, second the In F. p t 01 n esfeun s ftemnpoeuewe finger when procedure min the of use frequent less and .001, < 17 –2.71, = (147) –.18, = t 17 –1.92, = (147) DISCUSSION p 0.Cnrligfrwrigmmr and memory working for Controlling .01. < .23, = .18, = p 07 n oefeun s of use frequent more and .057, = t 17 2.07, = (147) t 25 1.92, = (205) p p 0.Slower .05. < 07 fewer .057, = p g < .005). (Carpen F SD p .15). > s advan = ------Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 alwl bv h o %i nelgne hra u ru fitletal pre intellectually of group our whereas intelligence, in cocious childrenincludesthoseinthetop5%to10%ofcognitive ability. 1% top the above well fall fpoesn esrsadmaue of speed measures between and correlation measures found not often processing did the differences of group so, Even the but significance. memory, statistical long-term reach from let accessing names of number children speed and for precocious peers ter typical intellectually intellectually the their than that faster found slightly were and expected we 1998), Jensen, Mathematical Cognition colleagues and Engle of model memory working the representational (Engle, 2002;Engleetal.,1999;Kane&Engle,2002). with visuospatial keeping and (phonological) in representation echoic information systems, the of speed in the manipulation influence and associated may control executive inhibitory central data, and the attentional these the with from that determined suggests be results of cannot pattern effect perfor the and lon and no cause RT, were Although variables RAN significant. pad by sketch ger visuospatial measured and loop as phonological the processing, on of controlled, mance was speed variable between executive central relation equation the the regression on simultaneous performance the once working that for three indicated results the pro of the However, of each for speed systems. performance faster memory enhanced that with indicated associated correlations was order cessing Zero sample. our for firmed o ersnaint aeteeetmtsi ossetwt s fa neetbut inherent an of use with Tsivkin, consistent is & estimates the these on Stanescu, make reliance heavy to children’s representation Pinel, typical log intellectually Spelke, The 2002). al., (Dehaene, et Zorzi estimates 1999; in estimates, line magnitude making number for system cluding visuospatial and consis cortex areas is implicated parietal estimates have the line that of studies number neuropsychological the and make neuroimaging to with representations tent log and linear of use and log the both using when estimates their in linear representations. preco accurate intellectually more the strategy, were same children the oppo cious used the groups the the showed than when children Even linear pattern. typical site the intellectually of the use whereas in frequent representations, the individuals log more linear as and showed and log children group of a precocious mixture as tellectually a placements, of line use number for making groups in strategies both peers. in precocious evidence intellectually found their we and Although be children differences typical by intellectually mirrored 2004, our was Booth, pattern tween & grade-related (Siegler The grades 2003). Opfer, later & in Siegler representation rely and mathematical placements linear line the number make on to representation logarithmic the on rely ers ntebsso h eainbtensedo rcsigand processing of speed between relation the of basis the On Number line estimation. line Number h idn htidvda ifrne nvsopta okn eoysupport memory working visuospatial in differences individual that finding The MATHEMATICAL COGNITIONINPRECOCIOUSCHILDREN ige n olaushv on htfrtgrad- first that found have colleagues and Siegler g eg,Fy&Hl,19)wscon was 1996) Hale, & Fry (e.g., F g Hn,1978; (Hunt, F 269 ------Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 hss oprsn cosdfeetgae,o eieeto h akisl n as and itself em task curriculum the in of sociated improvement refinement insensitivity totheuseofdecompositionisnotknown. changes or to grades, due different across is comparisons studies phasis, across decompo difference of the frequency we low Whether intellectually retrieval, very sition. a the direct reported among Brown of strategy and Geary favored use children; a precocious frequent as their emerged decomposition in that reflected found in largely However, preco was mix. mathematically children of strategy advantage cious the mature that finding Brown’s more and Geary a to contrast used children in strate- the precocious solving decomposition—but problem tellectually and of retrieval, mix counting, same verbal stud- the counting, both used gies—finger In children graders. precocious fourth and for (1991) typical Brown ies, and Geary by reported findings use eosrtdta nelculypeoiu hlrnue eaieymr mem more relatively we used case, this children In precocious answer. correct intellectually the that produce to demonstrated likely involve most choices adaptive are changes, that problem strategies the using strategy of adaptive complexity process when of is, this that termed use choice; (1996) the Siegler from problems. in shift complex the to shifted to simple response successfully and in strategies simple more counting-based solve to to they processes peers memory-based problems, their did addition than complex mix strategy more mature more a used curately 270 lypeoiu hlrno hsts eerltdt hi datgsi h central the in advantages their intellectu- to the related of were executive advantages andvisuospatialsketch pad. task the this case, on any children precocious In and ally representation. construction linear pre- explicit the and of and use natural representation later the of log inhibition invoked for automatically important sumably are associ- executive competencies central that the suggest We with representations. ated representations log linear of of use frequent use central less frequent Higher and more it: with with associated consistent were nonetheless scores the are executive as direct but of a hypothesis, systems for this allow control not of do attentional sessment data the current The on 2007). part, 2005, (Geary, have in executive but central dependent, known, be not are to line predicted number been the of representation mathematical linear natural the of string the to relation in repre and itself digit digit each the “space” numbers. with the line of number correspondence the one-to-one on a sents map skills, to counting yet rote have acquired newly but representa with analogue conjunction intellectually in an the magnitude, using relative estimates words, of their tion ad other making children be In to the counting. appear of children through some typical estimates that observation initial our their with justed of consistent accuracy is with estimates associated log were the loop phonological that the finding for The differences 1992). individual Gelman, & (Gallistel system estimation magnitude inexact naycs,itletal rccoscide ntecretsuyntol ac only not study current the in children precocious intellectually case, any In h coln-ae ehnssta otiuet h eeomn n s of use and development the to contribute that mechanisms schooling-based The Addition strategy choices. strategy Addition HOARD, GEARY, BYRD-CRAVEN, NUGENT hssuyrpiae n xeddtestrategy the extended and replicated study This ------Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 oeacrt onigpoess kle s fcutn oslecmlxprob complex solve to counting of of lems, inturn,was dependenton speed ofprocessingandthephonologicalloop. use use Skilled the processes. to contributed counting of likely accurate inhibition thereby of more the and use process for retrieval in important inaccurate as differences largely emerged the individual executive to central The contributed prob- counting. also addition verbal complex loop more phonological solving fin- the were of children lems; frequency the and when counting errors finger of counting use ger in differences group and individual of of findings the with keeping in is the strategies Logie andcolleagues(Logie&Baddeley, 1987;Logieetal.,1994). counting in of differences use group and and loop individual phonological these between For relation addition. strategy consistent simple of to for the speed rates related problems, error Rather, in consistently study. differences more group current to were the and loop characteristics in phonological emerge et the not (Geary and did problems processing relation addition this simple but solve strate to 2007), in used al., differences rates, group error to associated contributor and important gies, an of be revealed to Contrasts have executive the children (MLD). central achieving with mathematics the typically and emerged in achieving, have disability low MLD, learning that with a children patterns with from children ways of strat some in study accuracy in and differed mix strategy execution the egy of sophistication the and memory, working their to comparison precocious in and their problems than complex errors the own performancewhenthey solved thesimpleproblems. more solve many to committed attempting when result peers a as of use and the shift is more adaptive than is problems complex counting solving processes of to when use memory-based answer counting correct because a adaptive, more yield is relatively to shift likely and The problems problems. simple complex solve solve to processes ory-based nsrtgcapoce otenme ieadadto ak r ossetwt a with consistent are tasks addition and line number the to differences approaches group strategic The prob in complexity. of problem mix in the changes in with shifts strategies strategies adaptive make sophisticated lem-solving easily more more and use intellectu problems also more addition solve children their to precocious than The frequently peers. more typical representa placements ally line linear number and make intel mathematical to that formal tion found the We use processes. children cognitive precocious related in lectually advantage the this and underlie Kotovsky, areas that & processes academic (Siegler cognitive specific peers the typical about more known their is aca than Less learn 1986). pace children faster precocious a intellectually at that material time demic some for known been has It pe fpoesn n h hnlgcllo eandcnitn predictors consistent remained loop phonological the and processing of Speed processing, of speed in differences group and individual among relations The MATHEMATICAL COGNITIONINPRECOCIOUSCHILDREN SUMMARY ANDCONCLUSION . h nelculytpclcide i o aethis make not did children typical intellectually The 271 ------Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 scat .H,&Kr,E .(01.Terltosisaogwrigmmr,mt nit,and anxiety, math memory, working among relationships The (2001). P. E. Kirk, & H., M. Ashcraft, nteaiiyt naeoesse raohrcnign nts demands. task be to on remains difference systems contingent developmental memory a another determined. working or or advance of developmental constellations system a different represents one of use engage contingent advantage to an Whether have ability to appeared the they working is, in that re each tasks, these for allocating mathematical differentially the advantage at across working better an sources be of to had appeared components also only three they system not the memory children of each precocious to The related memory. mathemat differentially the were on children tasks precocious ics intellectually mathematical chil the across of deficit of studies core advantages unlike a areas—the is and executive time, central same the MLD—where the develop with At a dren difference. with a keeping not in and systems, advance board memory the mental working across core an three showed all sample for our advantage Rather, sample. memory precocious working our visuospatial advanced for especially system an find not did we adolescents, typical intellectually the of ahead accu and years mix several strategy were children, but the didnotdiffer inkind. execution mix; strategy strategy the dif of of but accuracy racy problems, the and solve maturity to the strategies of in types chil fered same because the used is groups This both difference. in developmental dren a not and advance developmental 272 cemn .L,Bir .E,&Bye .O 20) okn eoyaditliec:Tesm or same The intelligence: and memory Working (2005). O. M. Boyle, & E., M. Beier, L., P. Ackerman, Shocklee, Jennifer Smith,andAshley Stickney. Amanda Lemp, O’Connor, Ford Mary Catherine Hale, Numtee, Chatavee Rebecca Maloney, Patrick Haggard, Desoto, Larissa Catherine Felton, as Andersen, Matthew research Kendra Ensenberger, and Lab: Sara staff Development thank Cognitive also the We encouraging of schools. in the sistants in instrumental Linda research been our administrators. has coordinating and coordinator, and parents, mathematics students, K–5 teachers, the Pub many Columbia Coutts, the the thank and We Sciences. Schools, Education lic for Institute co-funded of the HD045914 Institute and R37 National NICHD and by the (NICHD), Development from Human HD38283 and R01 Health grants Child by supported was study This performance. different constructs? nietefnig fDr n ebw(91 o ahmtclygifted mathematically for (1991) Benbow and Dark of findings the Unlike HOARD, GEARY, BYRD-CRAVEN, NUGENT Journal ofExperimentalPsychology: General, 130 Psychological Bulletin,131 ACKNOWLEDGMENTS REFERENCES , 30–60. , 224–237. ------Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 ean,S,Sek,E,Pnl . tnsu . svi,S 19) ore fmteaia think mathematical of Sources (1999). S. Tsivkin, & R., Stanescu, P., Pinel, E., Spelke, S., Dehaene, (2000). J. I. Deary, working-mem between relationship the mediates stimulus of Type (1994). P. C. Benbow, & J., V. Dark, ul .&Jhso,R .(97.Cide’ rtmtcldfiute:Cnrbtosfo processing from Contributions difficulties: arithmetical Children’s (1997). S. R. Johnston, & R. Bull, ex and memory, working memory, Short-term issue). (2008/this A. S. Wiebe, & A., K. Espy, intel in R., intelligence Bull, of Structure (1983). B. A. Zonderman, & K., M. Kirk, C., J. Stanley, P., C. Benbow, apne,P . ut .A,&Sel .(90.Wa n nelgnets esrs hoeia ac theoretical A measures: test intelligence one What (1990). P. Shell, & A., M. Just, A., P. Carpenter, Campbell, J.I.D.(ed.)(2005). ecl,M . ue,R 17) ai uoaie aig(A) ylxadfeetae from differentiated (2005). Dyslexia A. Dowker, (RAN): naming automatized Rapid (1976). R. Rudel, & B., M. Denckla, mathematical with memory working of enhancement Differential (1991). P. C. Benbow, & J., V. Dark, experiment. critical A intelligence: crystallized and fluid of Theory (1963). B. R. Cattell, (1993). B. J. Carroll, ak .J,&Bno,C .(90.Ehne rbe rnlto n hr-emmmr:Compo- memory: short-term and translation problem Enhanced (1990). P. C. latent Benbow, A & (2002). J., B. V. R. Dark, S. Minkoff, & J., D. Therriault, F., M. Bunting, N., Cowan, A., R. A. Conway, adly .D,&Hth .J 17) okn eoy nG .Bwr(ed.), Bower Baddeley, A.D.(1986). H. G. In memory. Working (1974). J. G. Hitch, & D., A. Baddeley, ul . ontn .S,&Ry .A 19) xlrn h oe ftevsa-pta kthpdand pad sketch visual-spatial the of roles the Exploring (1999). A. J. Roy, & S., R. Johnston, R., Bull, memory? working of component new A buffer: episodic The (2000). A. Baddeley, r,A . ae .(96.Poesn pe,wrigmmr,adfuditliec:Eiec o a for Evidence intelligence: fluid and memory, working speed, Processing (1996). S. Hale, & F., A. Fry, for basis neural A (2000). al. et A., Ahmed, H., Herzog, D., Bor, J., Kolodny, S., J. Rüdiger, J., Duncan, nl,R .(02.Wrigmmr aaiya xctv attention. executive as capacity memory Working (2002). W. R. Engle, nl,R . uosi .W,Luhi,J . owy .R .(99.Wrigmemory, Working (1999). A. R. A. Conway, & E., J. Laughlin, W., S. Tuholski, W., R. Engle, UK: OxfordUniversity Press. ory performanceandtypeofprecocity. pe,ie dniiain n hr-emmemory. short-term and identification, item speed, 7 age at achievement mathematical of predictors years. Longitudinal preschoolers: in functioning ecutive lectually precociouschildrenandtheirparents. other learningdisabilities. ing: Behavioral andbrain-imaging evidence. versus verbal precocity. bridge University Press. Test. Matrices Progressive Raven the in processing the of count nents ofmathematicaltalent. and speed, processing capacity, memory short-term general fluidintelligence. capacity, memory working of analysis variable ucational Psychology, 54 Press. 8 Vol. motivation: and learning eta xctv ncide’ rtmtclsil:Vesfo onto n eeomna neuro developmental and cognition from Views psychology. skills: arithmetical children’s in executive central 1–24. Sciences, 4, developmental cascade. general intelligence. education hr-emmmr,adgnrlfuditliec:Altn-aibeapproach. latent-variable A intelligence: mental Psychology: General, 128 fluid general and memory, short-term logical Science, 11 Developmental Neuropsychology, 33, . Hove, UK:PsychologyPress. 417–423. Developmental Neuropsychology, 15 niiuldfeecsi rtmtc mlctosfrpyhlg,nuocec and neuroscience psychology, for Implications arithmetic: in differences Individual okn ono ua nelgne rmpyhpyist h brain the to psychophysics From intelligence: human on down Looking , 19–23. ua ontv blte:Asre ffco nltcstudies. analytic factor of survey A abilities: cognitive Human Science, 289 MATHEMATICAL COGNITIONINPRECOCIOUSCHILDREN Working memory. Journal ofEducationalPsychology, 83 Psychological Science, 7 , 1–22. Intelligence, 30, Neuropsychologia, 14 Handbook ofmathematicalcognition Journal ofEducationalPsychology, 82 . dacsi eerhadtheory and research in Advances , 309–331. , 457–460. Intelligence, 19 Oxford: OxfordUniversity Press. 163–183. 205–228. Science, 284 Intelligence, 7 , 471–479. , 237–241. , 421–442. ora fEprmna hl scooy 65 Psychology, Child Experimental of Journal , 337–357. , 970–974. , 129–152. scooia eiw 97, Review, Psychological , 48–60. p.4–9.NwYr:Academic York: New 47–89). (pp. . New York: PsychologyPress. , 420–429. urn ietosi Psycho in Directions Current rnsi Cognitive in Trends ora fExperi of Journal h scooyof psychology The e ok Cam- York: New ora fEd- of Journal 404–431. Oxford, . 273 ------, Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 Jensen, A.R.(1998) Hunt, E.(1978).Mechanismsofverbal ability. Hunt, E.(1983).Onthenatureofintelligence. ae .J,&Ege .W 20) h oeo rfotlcre nwrigmmr aaiy execu capacity, working-memory in cortex prefrontal of role The (2002). W. R. longi Engle, A & children: J., young M. in Kane, mastery fact Arithmetic (2003). D. Kaplan, & B., L. Hanich, C., N. Jordan, (eds.), Cattell B. R. & Nesselroade R. J. In abilities. human about Thinking (1988). L. J. Horn, on .L,&Ctel .B 16) eieetadts fteter ffudadcytlie general crystallized and fluid of theory the of test and Refinement (1966). B. R. Cattell, & L., J. Horn, ic,G . cue,E 19) okn eoyi hlrnwt pcfcaiheia learning arithmetical specific with children in memory Working (1991). E. McAuley, & J., G. Hitch, ic,G .(98.Terl fsottr okn eoyi etlarithmetic. mental in memory working short-term of role The (1978). J. G. Hitch, re,G . aka,J .(92.Acrnmti nlsso ipeaddition. simple of analysis chronometric A (1972). M. J. Parkman, & J., G. Groen, Gottfredson, L.(1997).Why er,D . or,M . ydCae,J,Ngn,L,&Nme,C 20) ontv mechanisms Cognitive (2007). C. Numtee, & L., Nugent, J., Byrd-Craven, K., M. Hoard, C., D. and Geary, componential of convergence the On cognition: Numerical (1992). F. K. Widaman, & C., D. Geary, er,D .(07.A vltoayprpcieo erigdsblt nmathematics. in disability learning on perspective evolutionary An (2007). C. D. Geary, er,D .(2005). C. D. Geary, er,D .(06.Dvlpeto ahmtcludrtnig nD ul&R .Selr(vol. Siegler S. R. & Kuhl D. In understanding. mathematical of Development (2006). C. D. Geary, mathematics. in deficit learning early an of analysis componential A (1990). C. D. Geary, er,D . or,M . ydCae,J,&Dst,M 20) taeycocsi ipeadcom- and simple in choices Strategy (2004). M. Desoto, & J., Byrd-Craven, K., M. Hoard, C., D. Geary, dif speed-of-processing and choice Strategy addition: Cognitive (1991). C. S. Brown, & C., D. Geary, alse,C . emn .(92,Peebladvra onto n computation and cognition verbal and Preverbal (1992), R. Gelman, & R., C. Gallistel, alge,A . afa,J .(d. (2005). (eds.) C. J. Kaufman, & M., A. Gallagher, uo,K .(92.A nlsso h onigo ouinpoeuei diin nT .Cretr J. Carpenter, P. T. In addition. in procedure solution counting-on the of analysis An (1982). C. K. Fuson, 274 etn,D . obt,B .(98.Idvda n eeomna ifrne ncognitive-process in differences developmental and Individual (1978). L. B. Bobbitt, & P., D. Keating, ieatnin n eea li nelgne nidvda-ifrne perspective. individual-differences An intelligence: fluid general and attention, tive psychology experimental multivariate of Handbook tudinal investigation. intelligence. num. difficulties. ogy, 10 view, 79 neligaheeetdfct ncide ihmteaia erigdisability. learning mathematical with ment children in deficits achievement underlying psychometric models. child psychology eds.), ington, DC:AmericanPsychologicalAssociation. Experimental ChildPsychology, 49 43–74. lxadto:Cnrbtoso okn eoyadcutn nweg o hlrnwt mathe- with children for knowledge counting and matical disability. memory working of Contributions addition: plex children. Psychology, disabled 27 mathematically and normal, intellectually-precocious, in ferences Neuropsychology, 32 psychological approach. Hillsdale, NJ:Erlbaum. .Msr .A obr (eds.), Romberg A. T. & Moser, M. Bulletin &Review, 9 ing componentsofmentalability. , onto,preto,and perception, Cognition, 78 , 302–323. HOARD, GEARY, BYRD-CRAVEN, NUGENT , 329–343. , 1343–1359. rts ora fPyhlg,82, British Journal ofPsychology, Journal ofEducationalPsychology, 57 , 398–406. (6th ed.).New York: JohnWiley &Sons. h rgno id vlto fban onto,adgnrlintelligence general and cognition, brain, of Evolution mind: of origin The Journal ofExperimentalChildPsychology, 88 The gfactor:scienceofmentalability , 637–671. , 471–519. Journal ofExperimentalChildPsychology, 85 Intelligence, 16 Cambridge UK:University Press. g matters: Thecomplexity ofeveryday life. language, Child Development, 49 , 363–383. diinadsbrcin ontv perspective cognitive A subtraction: and Addition , 47–80. o.2 vol. Science, 219 Psychological Review, 85 375–386. edrdfeecsi ahmtc:A integrative An mathematics: in differences Gender p.7780.W ao gn ed.), (gen. Damon W. 777–810). (pp. , 253–270. 2de. p 4–8) e ok Ple York: New 645–685). pp. ed., (2nd , 141–146. , 155–167. . Westport, CT: Praeger. , 121–151. , 103–119. , 109–130. Intelligence, 24 ontv Psychol Cognitive scooia Re- Psychological onto,44 Cognition, . hl Develop- Child Developmental Developmental Psychonomic adokof Handbook , 79–132. p.67–81). (pp. ora of Journal Wash . ------, Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 ikrn,S,&Gtecl,S (2001). S. Gathercole, & S., Pickering, sense number rational in Gaps “holes”: and Parts press). (in T. K. Devlin, & M., 10-year M. A M. 10,000: Mazzocco, in 1 Top (2001) P. C. Benbow, & J., M. Morelock, M., R. Webb, D., Lubinski, uisi .(00.Sinii n oilsgiiac fassigidvda ifrne:“Sinking differences: individual assessing of significance social and Scientific problem (2000). arithmetic D. in memory Lubinski, working on Counting (1994). V. Wynn, & J., K. counting. Gilhooly, H., in R. processing Logie, Cognitive (1987). D. A. Baddeley, & H., R. Logie, ae,J . or,J . ae,J (1993). J. Raven, & H., repre- J. cerebral Court, overlapping C., and J. Distributed Raven, (2004). S. Dehaene, & D., Bihan, Le D., Piazza, P., Pinel, characteristics Cognitive (2007). C. M. Early, & B., L. Hanich, M., M. M. Mazzocco, young M., talented M. Murphy, academically in differences Gender (1993). H. Stumpf, & E., K. mathe Ablard, specific J., with C. children in Mills, impairments memory Working (1999). H. G. Hitch, & F., J. McLean, human exceptional Tracking (2006). A. Bleske-Rechek, & M., R. Webb, P., C. Benbow, D., Lubinski, oisn .M,Abt,R . enne,V . us,J 19) h tutr faiiisin abilities of structure The Royer, (1996). J.M.(ed.)(2003). J. Busse, & W., V. Berninger, D., R. Abbott, M., N. Robinson, ige,R .(97.Sm eea ocuin bu hlrnssrtg hiepoeue.SeilIs Special procedures. choice strategy children’s about conclusions general Some (1987). S. R. Siegler, ige,R . ot,J .(04.Dvlpeto ueia siaini on children. young in estimation numerical of Development (2004). L. J. Booth, & S., R. Siegler, ige,R .(1996). S. R. Siegler, ige,R . oosy .(96.Tolvl fitletal-rccoses hl vrtetwain the ever Shall intellectually-precociousness: of levels Two (1986). K. Kotovsky, & S., R. Siegler, tnvc,K .(99.Tescoscoerc flann disabilities. learning of sociopsychometrics The (1999). E. K. Stanovich, measured. and determined objectively intelligence, know General children (1904). do C. How Spearman, subtraction: and addition in choice Strategy (1984). multiple J. Shrager, for & Evidence S., estimation: R. Siegler, numerical of development The (2003). E. J. Opfer, & S., R. Siegler, among childrenwithvs.withoutmathematicallearningdisabilities. intellectually-precocious. profoundly 718–729. the of follow-up shafts atafew criticalpoints.” solving. Psychology: Learning, Memory, andCognition, 13 sentations ofnumber, size,andluminanceduringcomparative judgments. London: PsychologicalCorporationLtd. used criterion cutoff the of function a as to define vary MLD. (MLD) disability learning mathematics with children of subskills. and age chology, across 85 Patterns reasoning: mathematical students’ matics learningdifficulties. capital over two decades. ahpeoiu on hlrn edrsmlrte n differences. and chology, 88 similarities Gender children: young math-precocious lary Scales University Press. u:TenoPaeintere fcgiiedvlpet oada integration. an Toward development: cognitive of Journal ofPsychology, theories 22 neo-Piagetian The sue: et nR .SenegadJ .Dvdo (eds.), Davidson E. J. and Sternberg J. R. In meet? Development, 75 ities, 32 Psychology, 15 what todo?InC.Sophian(ed.), representations ofnumericalquantity. 417–435). Cambridge,England:CambridgeUniversity Press. , 350–361. Memory &Cognition, 22 . London:H.K.Lewis &Co. , 340–346. , 341–352. , 201–293. Journal ofLearningDisabilities,40 , 428–444. mrigmns h rcs fcag ncide’ thinking. children’s in change of process The minds: Emerging MATHEMATICAL COGNITIONINPRECOCIOUSCHILDREN Mathematical cognition Psychological Science, 17 , 729–749. Journal ofExperimentalChildPsychology, 74, Annual Review ofPsychology, 51 Origins ofcognitive skills , 395–410. okn eoyTs atr o hlrn(MBC manual (WMTB-C) Children for Battery Test Memory Working Psychological Science, 14 aulfrRvnsPorsieMtie n Vocabu- and Matrices Progressive Raven’s for Manual . Greenwich,CT: InformationAgePublishing. ocpin fintellectually-precociousness of Conceptions , 310–326. , 194–199. , 458–478. (pp. 229–293).Hillsdale,NJ:Erlbaum. ora fApidPyhlg,86 Psychology, Applied of Journal , 405–444. , 234–243. Developmental Science ora fEuainlPsy Educational of Journal ora fLann Disabil Learning of Journal ora fEuainlPsy- Educational of Journal ora fExperimental of Journal ern 41 Neuron, 240–260. mrcnJunlof Journal American e ok Oxford York: New , 1–20. International . Child 275 (pp. - - - - , . Downloaded By: [Hoard, Mary K.] At: 21:11 2 July 2008 oz,M,Pits . mlá .(02 a ) elc irpstemna ubrline. number mental the disrupts Neglect 9). May (2002, C. Umiltá, & K., Priftis, M., Zorzi, rnk,L . oe,J .(03.Rltosisaogbsccmuainlatmtct,working automaticity, computational basic among Relationships (2003). M. J. Royer, & N., L. Tronsky, chil in memory working and solving problem Mathematical (2001). C. Sachse-Lee, & L., chil H. young Swanson, in precociousness mathematical underlie that processes Cognitive (2006). L. H. subgroups. Swanson, disability learning in memory Working (1993). L. H. Swanson, abr,H .(94.Ipoigtepoutvt fAeiasschools. America’s of productivity the Improving (1984). J. H. Walberg, tie,H . ar .(03.Cgiiedvlpeti nelculypeoiu hlrn Toward children: intellectually-precocious in development Cognitive (2003). M. Carr, & H., H. Steiner, 276 417 19–27. eoy n ope ahmtclpolmslig htw nwadwa ene oko.In know. to need we (ed.), what Royer and know M. we J. What solving: problem mathematical complex and memory, important. are processes phonological Experimental ChildPsychology, and 79 executive Both disabilities: learning with dren dren. view, 15 lishing. Child Psychology, 56 oepeieudrtnigo mrigdfeecsi intelligence. in differences emerging of understanding precise more a , 138. Journal ofExperimentalChildPsychology, 93 HOARD, GEARY, BYRD-CRAVEN, NUGENT , 215–246. ahmtclcognition Mathematical , 87–114. , 294–321. p.1715.Genih T nomto g Pub Age Information CT: Greenwich, 117–145). (pp. , 239–264. dctoa scooyRe Psychology Educational dctoa edrhp 41 Leadership, Educational ora fExperimental of Journal ora of Journal Nature, - - - - ,