Joural of Educatioal Psychology 199, Vol. 9, No. 3,53-544 opyright 199 by the America Psychological Associatio, Ic. 22-3/9/$3. hildre's Kowledge of Tee Quatities as Tes ad Oes: omparisos of hiese, British, ad America Kidergarters oie Suk-Ha Ho hiese Uiversity of Hog Kog Kare. Fuso Northwester Uiversity Three studies were coducted to examie the effects of idividual differeces ad laguage differeces o childre's uderstadig of tee quatities ( s«< 19) as couted cardial tes ad oes (embedded-te cardial uderstadig). At age 4, most hiese childre, usig amed-te umber words (e.g., 12 is said as "te two"), did ot show such uderstadig o a task i which y quatities were added to 1 quatities. At age 5, half the childre of average or above itelligece who had high rote-coutig sequeces {M = 9) did show such uderstadig; those with lower rote-coutig sequeces did ot. Eglish-speakig 5-year-old childre i Eglad ad i the Uited States, whose tee words obfuscate the tes ad oes, showed o evidece of uderstadig tee quatities as cardial tes ad oes. Asia studets geerally attai higher mathematics achievemet tha Wester studets (e.g., Geary, Bow- Thomas, Liu, & Siegler, 199; Geary, Fa, & Bow-Thomas, 1992; Huse, 197; Lapoite, Mead, & Philips, 199; Steveso et al. t 199; Steveso, Lee, & Stigler, 19; Steveso & Stigler, 1992). This is true for some tasks eve before school ca have a great impact o mathematics learig. Asia studets perform better tha Wester studets i abstract coutig to 1 (Bryat & Lies, 1992; Miller, Smith, Zhu, & Zhag, 1995; Miller & Stigler, 197), i represetig the place values of umbers with testructured blocks early i first grade (Miura, Kim, hag, & Okamoto, 19; Miura et al., 1994; Miura, Okamoto, Kim, Steere, & Fayol, 1993), ad i metal additio i kidergarte (Geary, Bow-Thomas, Fa, & Siegler, 1993; Geary et al., 199), All of these early superiorities seem related to the regularity of umber words i East Asia laguages. hiese, Japaese, ad Korea (ad some other laguages) are regular for umbers betwee 1 ad 1 i the same way that Eglish is regular for the hudreds ad the thousads: 5,9 is said as "five thousad ie hudred" i Eglish ad East Asia laguages, but 59 also is said as "five te ie" i the latter. I cotrast, Eglish uses a decade structure (e.g., twety, thirty, forty, fifty), which obfuscates the meaigs of these umbers as two-te, three-te, four-te, ad five-te. I the Eglish umber words, learig the tees ad usig them i additio ad subtractio are particularly difficult because of the irregularities (eleve, twelve) ad the rever- oie Suk-Ha Ho, Departmet of Psychology, hiese Uiversity of Hog Kog, Hog Kog; Kare. Fuso, Departmet of Psychology, Northwester Uiversity. We would like to thak the participatio ad kid cooperatio of the studets, teachers, parets, ad head teacher of the reative Kidergarte i Hog Kog ad those of the participatig kidergartes ad schools i Eglad ad the Uited States. orrespodece cocerig this article should be addressed to oie Suk-Ha Ho, Departmet of Psychology, hiese Uiversity of Hog Kog, Shati, New Territories, Hog Kog. sals (say four first i fourtee but write it secod i 14; Geary et al., 199; Miller et al., 1995; Miller & Zhu, 1991). I cotrast, Fuso ad Kwo (1991) foud that most Korea childre by the middle of first grade used additio ad subtractio methods that ivolved thikig of tee umbers as a te ad some oes, ad Geary et al. (199) foud similar results for hiese childre. Geary et al. (199) foud also that childre i the Uited States rarely used such methods. A crucial missig lik: i our uderstadig of how East Asia childre come to use te-structured additio ad subtractio methods ad therefore how we might help Eglish-speakig childre lear these effective methods is whe ad how East Asia childre first uderstad that a coutig word "te x" is composed of a quatity of te plus the quatity x, what we term embedded-te cardial uderstadig. A aalogy i Eglish is that a substatial umber of urba U.S. first graders iitially do ot kow that forty plus six is forty-six (Fuso & Smith, 199). They cout o six more from forty to fid the sum of forty-six. After they have see this relatioship i some or may examples, the structure of the umber words (ad of the writte umerals) as "first-added word the secod-added word" helps them to geeralize this uderstadig, so that they o loger have to cout o each time (i.e., they have embedded-decade cardial uderstadig). For embedded-te or embeddeddecade cardial uderstadig, a child is required to shift from (a) the sequece meaig of the last couted word as just a rote sequece word ad the coutig meaig of that word as referrig to the last object couted to (b) the cardial meaigs of "te six" (or "forty six") as referrig to all of the objects ad of the embedded words "te" ad "six" (or "forty" ad "six") as referrig to groups of objects that axe embedded withi (that costitute) the total group of objects. Embedded-te cardial uderstadig is itself embedded i the more geeral theoretical issue of the Whorf hypothesis (Whorf, 195) that laguage affects thikig. We thik of this issue as part of the more geeral issue of facilitatios for directig thikig ad attetio ad for expressig relatioships. The presece of potetial facilitatios i the eviro- 53
KNOWLEDGE OF TEEN QUANTITIES 537 merit is complex; they do ot ecessarily mea that everyoe i that eviromet will be stimulated by the facilitatios. The regular te-words i East Asia laguages provide a facilitatio for te-structured additio ad subtractio methods. These methods are more difficult for Eglish-speakig childre because they have a extra fial traslatio step from "te ad five" to "fiftee." This souds trivial to adults, but may Eglish-speakig childre experiece cosiderable difficulty i costructig embedded-te uderstadig for tee words ad umerals. Kamii (195) reported that substatial umbers eve of U.S. fifth graders still do ot possess embedded-te cardial uderstadig (they choose 1 rather tha 1 objects as the meaig of the 1 i 1). Although the umber-word facilitatios for embedded-te uderstadig are available to East Asia childre ad ot to Eglish-speakig childre, two other facilitatios are available to both groups of childre. Te as a groupig is suggested both by our 1 figers ad by the structure of both types of coutig words as havig te as the first coutig stoppig poit. We pursued whe ad how embedded-te cardial uderstadig is differetially facilitated by these cultural (umber = word) ad material object (figer) supports for East Asia ad Eglish-speakig childre. We were iterested i whether embedded-te cardial uderstadig is built gradually for some umbers oly or comes all at oce. We also examied two attributes that seemed likely to affect childre's learig of embedded-te cardial uderstadig: IQ as a geeral measure of cogitive maturity (geeral learig potetial) ad the legth of the coutig sequece as a measure of specific learig i the umber area ad perhaps of the ease with which umber words are produced, which would facilitate reflectio o the cotet of the words. We costructed the hidde-object additio task, to assess embedded-te cardial uderstadig: First, x items were put oe by oe ito a box while they were couted out loud, the v items were put i as they were couted (begiig from 1), ad the the child was asked how may items were i the box (the objects were o loger visible). For each x + y problem, x was 4 or 1, ad v was 2, 5, 7, or 9. Embedded-te cardial uderstadig could be used o the 1 + y items but ot o the 4 + y items. If the child had full uderstadig, he or she would respod rapidly ad correctly without ay coutig. hildre's overt coutig methods were recorded to examie differeces i coutig methods (e.g., coutig o rather tha coutig all). I our first study, the performace of hiese childre with differet IQs ad differet mea legths of coutig sequece was followed logitudially from age 4 to age 5. I the secod study, the result that legth of coutig sequece affected embedded-te cardial uderstadig was followed up. Performace o the hidde-object additio task was compared for hiese-speakig ad Eglish-speakig childre with low ad with high coutig sequeces. I the third study, the result that Eglish-speakig childre i Eglad did ot have embedded-te cardial uderstadig was followed up by examiig the performace of Eglishspeakig childre i the Uited States. Participats Study 1 Method The 3 participats i Study 1 were selected from a group of childre participatig i aother logitudial study. They all lived i Hog Kog ad spoke oly hiese. Most of them came from middle-class families. The childre were first assessed at the age of 4 years (M = 4 years 4 moths, SD = 4.1 moths), whe they had just leared how to cout to 1 i their preschool but before they had leared how to do sums (age 4). All of the childre were assessed agai 1 year later, whe they were learig how to do very simple additio i their kidergarte, with sums of ot more tha 4 (age 5). Apart from the first warm-up trial (2 + 1), all other trials i the hidde-object additio task were far beyod the childre's curriculum at either age. Materials ad Procedures Before the study, the childre had bee assessed o the Staford-Biet Itelligece Scale: Form L-M (3rd editio, Terma & Merrill, 19) at the age of 3 years (mea IQ = 13, SD = 1, based o America orms). hildre were tested idividually o coutig sequece ad hidde-object additio at both age 4 ad age 5. outig sequece. The procedure for the coutig sequece task was take from Miller ad Stigler's (197) study. The childre were asked to cout aloud as high as they could ad were prompted at the begiig with "1, 2, 3" if ecessary. If they stopped coutig, they were prompted to cotiue by the questio "What comes after x [the last umber couted]?" If they still gave o respose, they were further prompted by repeatig the last three umbers couted i a risig itoatio. Hidde-object additio. Materials for this task icluded 2 oe-ich square blocks, a box with a lid, ad a stopwatch. There were ie x + y trials i the task. I each trial, the experimeter asked the child to cout with her while she was puttig x umber of blocks (i.e., the first added) ito the box. The procedure was repeated while a secod set of blocks (i.e., the secod added, y) was put ito the box. This was doe so that the child could ot cout o from x while the y blocks were put i. The experimeter the closed the lid ad asked, "First I put x blocks ito the box, ad the I put y more blocks i it. How may blocks altogether are i the box ow?" Feedback was give by askig the child to cout the total umber of blocks i the box (the blocks were dumped out of the box oto the table, so that they could be couted). The child's resposes ad observed solutio strategies were recorded. Overt coutig icluded coutig that could be heard ad lip movemets of the coutig words. Reactio times were recorded by startig the stopwatch at the ed of the questio ad stoppig it whe the child bega speakig. This method was accurate eough for our purposes of differetiatig coutig from more immediate resposes because coutig solutios took several secods. To elimiate the possible distractio of mior details, we rouded off reactio times to the earest secod i the tables. Small variatios i iterviewer iitiatig ad stoppig the stopwatch across trials or across iterviewers i the three studies thus are ot a issue. Of the ie trials, four were 4 + y trials, ad four were 1 + y trials. The ie trials were as follows: 2 + 1,4 + 2,4 + 5, 4 + 1, 4 + 9, 1 + 2, 1 + 5,1 + 7, ad 1 + 9. The first warm-up trial was always 2+1. From the results of pilot studies, we foud that 4-year-olds did ot perform very well o this task ad that it was
53 HO AND FUSON difficult to maitai their iterest ad attetio across all ie trials. Therefore, the 4-year-olds had oly five trials: the 2 + 1 trial plus either the extreme y trials or the middle y trials (i.e., either 4 + 2, 4 + 9, 1 + 2, ad 1 + 9 or 4 + 5, 4 + 7, 1 + 5, ad 1 + 7). At age 5, the childre received all ie trials. At both age 4 ad age 5, half of the participats had 4 + y trials first, ad half of them had 1 + y trials first. Groups formed. O the basis of the distributio of coutig sequece kowledge ad IQ, the childre were divided ito three groups accordig to the followig criteria. The low-s-av-iq childre were those with correct coutig sequeces of less tha 5 at age 5 (mea S at age 4 = 14.7, SD =.; mea S at age 5 = 39.3; SD =.4; mea IQ = 93.2, SD =.3).Thetwohigh-S groups were those who could cout up to 5 or more. hildre i the high-s-av-iq group had IQs lower tha 15 (mea S at age 4 = 35.9, SD = 24.7; mea S at age 5 = 91.4, SD = 13.9; mea IQ = 93., SD = 7.7), whereas those i the high-s-high-iqs group had IQs higher tha (mea S at age 4 = 39.4, SD = 14.9; mea S at age 5 = 7.5, SD = 19.5; mea IQ = 124.3, SD =.5). The high-s-av-iq group had sigificatly higher correct coutig sequeces tha the low-s-av-iq group at age 4,?(23) = 2.7, p <.5, ad age 5, r(23) = 1.93, p <.1, but the two groups did ot differ sigificatly o IQ. The high-s-high-iq group had sigificatly higher IQs tha did the high-s-av-iq group, f(23) = 9.39, p <.1, but these two groups did ot differ sigificatly o correct coutig sequeces at age 4 or age 5. Group sample sizes are give i the Tables 1 ad 2. riteria for embedded-te cardiality uderstadig. hildre who gave rapid (2 s or less) ad accurate resposes to all of the 1 + y trials without overt coutig behavior ad did ot do so o all 4 + y trials were defied to be uderstaders of embedded-te cardiality. To esure that uderstadig rather tha a rote patter from the umber words (i.e., just sayig "1/') was beig used, childre were so classified oly if they also did ot give ay resposes of the form "4y." Performace at Age 4 Results ad Discussio Two childre uderstood embedded-te cardiality, 1 i each of the high-coutig-sequece groups. Oly oe of the 4 + y tasks doe by both of these childre was doe correctly. Three other childre i the two high-coutigsequece groups did both of the 1 + y tasks correctly, but the childre seemed to be coutig covertly or overtly, because five of these six trials had log reactio times, ragig from 4 to 27 s. All but oe of the 4 + y tasks were doe icorrectly by these childre. May of the childre at age 4 foud the 2 + 1 trial solvable without the actual blocks but foud the other trials to be difficult. I Table 1, we preset the percetages of correct resposes across trials i the hidde-object additio task for each group ad the correspodig reactio times at age 4. From a Group X Problem aalysis of variace (ANOVA) o the rates of correct resposes, we foud that the mai effects of group, F(2, 33) = 4.37, p <.5, ad problem, F(l, 33) = 7.15, p <.5, were sigificat, but the iteractio effect was ot. From the post hoc comparisos by the Tukey a test, we foud that the two high-s groups each performed sigificatly better tha the low-s-av-iq group ad that the two high-s groups did ot differ sigificatly i performace. Thus, high correct coutig sequeces are more related to iitial success i additio tha is high IQ. Noe of the childre showed ay overt coutig behavior i solvig 4 + v problems correctly. However, 2 of the childre couted all ad aother 2 childre couted o overtly, to solve the 1 + y problems correctly. Performace at Age 5 All except I child solved the practice problem (2 + 1) correctly without overt coutig at age 5. This suggests that the childre might have relied o memorized additio facts to do the 2 + 1 trial. At age 5, 3 childre, all i the high-s-high-iq group, did all four 4 + y trials correctly ad all four 1 + y trials correctly. No child got all 4 + y trials correct without gettig all 1 + y trials correct, whereas 13 childre got all 1 + y trials correct but ot all 4 + y trials correct. Thus, childre leared to add 1 + y correctly before 4 + y t McNemar's x 2 (l, N = 3) =., p <.1. TTiese 13 childre were i the two high-s groups. There were,7, ad 7 embedded-te cardial uderstaders i the low-s-av-iq, high-s-av-iq, ad the high-shigh-iq groups, respectively. The uderstaders had sigificatly higher scores tha the ouderstaders o IQ (M = vs. 9), r(34) = 2.2, p <.5; correct coutig sequece at age 4 (M = 43 vs. 22), f(34) = 3.45, p <.1; ad correct coutig sequece at age 5 (M = 92 vs. 3), r(34) = 3.49, p =.1. From these results, we suggest that cogitive maturity as well as a certai level of coutig skills is required for embedded-te cardial uderstadig. However, whe we did these aalyses oly o childre i the two high-s groups, the uderstaders ad ouderstad- Table 1 Study 1: Age 4 Mea Percetage of orrect Resposes ad Mea Reactio Time for Each Group Problems 4 + y 1 + y Low S-av IQ 5 15 High S-av IQ 14 3 32 4 14 High S-high IQ 1 2 41 5 Note. S = coutig sequece; av = average; = mea percetage of correct resposes across trials; = mea reactio time, rouded off to the earest secod, for correct resposes; M mea percetage of correct resposes across groups. M 2
KNOWLEDGE OF TEEN QUANTITIES 539 ers did ot differ sigificatly o IQ or correct coutig sequeces. Thus, a miimum level of correct coutig sequece seems to be ecessary but ot sufficiet for cardial uderstadig. We coducted a discrimiat aalysis, to ascertai whether IQ or legth of correct coutig sequece was more importat i developig embedded-te cardial uderstadig. Whe we etered IQ ad correct coutig sequeces at age 4 ad age 5 as the idepedet variables, the discrimiat fuctio was sigificat i predictig the membership of embedded-te cardial uderstadig or ouderstadig, F(3, 32) = 5.7, p <.1. The discrimiat fuctio coefficiets for IQ ad correct coutig sequeces at age 4 ad age 5 were.2,.4, ad.5 respectively. I other words, correct coutig sequece at age 5 was the best discrimiator of embedded-te cardial uderstadig ad ouderstadig, ad IQ was the worst discrimiator. The resultig patter was cosistet with the percetage of uderstaders i each group; that is, % i the low-s-av-iq group, 5% i the high-s-av-iq group, ad 4% i the high-shigh-iq group. We also coducted a two-way ANOVA o the rates of correct resposes. Meas for each group are give i Table 2. From the ANOVA results, we foud that the mai effects of group, F(2, 33) - 1.53, p <.1, ad problem, f(l, 33) = 2.35, p <.1, were sigificat, but the iteractio effect was ot. As at age 4, the childre performed sigificatly better o 1 + v problems tha o 4 + v oes. O the basis of the post hoc comparisos by the Tukey a test, we foud that childre i each of the two high-s groups performed sigificatly better tha those i the low-s-av-iq group ad childre i the high-s-high-iq group performed sigificatly better tha those i the high-s-av-iq group. Summary Because all childre could cout through the tees but may, especially at age 4, did ot use embedded-te cardiality, the cardial meaig of coutig words followed rather tha preceded learig the sequece itself. This is similar to the coclusios of Briars ad Siegler (194) cocerig the relatioship betwee childre's coutig ad their uderstadig of coutig priciples: Uderstadig of some of the mai priciples followed rather tha preceded accurate coutig. Furthermore, both correct high coutig sequeces ad cogitive maturity are related to the accuracy of addig ad of usig embedded-te cardiality, but high correct coutig sequeces seem to be more strogly related to embedded-te cardiality use tha is cogitive maturity. Participats Study 2 Method There were four groups of 5-year-olds i Study 2: two hiese groups ad two Eglish groups. hildre i the hiese groups were selected from the sample i Study 1. Those i the Eglish groups were selected from the Eglish sample i aother logitudial hiese-eglish study. I the origial logitudial studies, by 5 years 3 moths, the average correct coutig sequece of the hiese childre was {SD = 32), whereas it was oly 45 (SD = 33) for the Eglish childre. Earlier researchers also have reported such learig differeces (Miller & Stigler, 197, for U.S. childre, ad Bryat & Lies, 1992, for British childre), but IQ was ot cotrolled i these earlier studies. I our sample, the mea IQ levels were similar: 15 for the hiese childre ad 13 for the Eglish childre. Therefore, this cross-atioal differece is ot due to some sample differece i IQ. I Study 1, it was foud that high coutig skills are ecessary for hiese childre to uderstad the cardial meaig of umber words ad apply it i additio. Therefore, Eglish groups were formed of high correct coutig sequeces (higher tha 4) ad of low correct coutig sequeces (with correct coutig sequeces less tha 4). hiese groups were selected to match them as close as possible o the group meas of coutig sequeces ad IQ levels as well as o the overall distributio. O the basis of these criteria, a total of 2 childre i the high-coutig-sequeces groups (high-s groups) ad 1 hi the low-coutig-sequeces groups (low-s groups) were selected. The two high-s groups were matched quite well: The hiese group did ot differ sigificatly from the Eglish group o age (M = 4. vs. 7. moths, respectively), correct coutig sequece (Af = 5 vs. 92), or IQ (M = 3 vs. 4), all fs(1) < 1.9, Table 2 Study 1: Age 5 Mea Percetage of orrect Resposes, the Mea Reactio Time, ad the Mea Percetage of hildre Usig Overt outig Strategies for Each Group 4 + y 1+y Problem Embedded-te cardial uderstaders Low S-av IQ 14 34 4 O 2 12 High S-av IQ 43 71 5 3 14 7 5 O 3 High S-high IQ Note. S = coutig sequece; av = average; = mea percetage of correct resposes across trials; = mea reactio time, rouded off to the earest secod, for correct resposes; O = mea percetage of childre usig overt coutig strategies for correct resposes. 91 4 2 7 4 O 32 M 41
54 HO AND FUSON all ps >.7. However, childre i the Eglish low-s group were sigificatly older tha those i the hiese low-s group (M =. vs 1.5 moths), *(14) = 5.9,/? <.1, although the two low-s groups did ot differ o correct coutig sequece (M =33vs.3)oroIQ(M = 99vs.91),alltt(14)<1.7,alIps>.. The hiese ad Eglish high-s groups were sigificatly higher tha their correspodig low-s groups o correct coutig sequeces, all fs(1) > 4., allps <.1, ad o IQs, all ts(1) > 3., all/>s <.1, but they did ot differ o age, all is(1) < 1.7, all ps >.12. Materials ad Procedures As reported i Study 1, the hiese childre's levels of itelligece had bee assessed i aother logitudial study. Similarly, the Eglish childre had bee assessed o the Staford- Bie t Itelligece Scale: Form L-M (3rd editio) at the age of 4 (mea IQ = 17, SD = 12, based o America orms) i the origial logitudial study. The Eglish childre were tested idividually o coutig sequece ad hidde-object additio i exactly the same way as were the hiese childre at age 5 i Study 1, except that they were give the test i Eglish. hiese test istructios i Study 1 had bee first traslated ito Eglish ad the back-traslated, to esure that the istructios were equivalet i the two laguages. Results ad Discussio All of the hiese ad Eglish childre did the 2 + 1 trial correctly without overt coutig. Thus, by the age of 5, both hiese ad Eglish childre are able to add small umbers without great difficulty, o matter whether their coutig sequeces are high or low. We preset the results of the four groups i the hiddeobject additio task i Table 3. Because the childre performed differetly for sums with large ad small secod addeds, mea scores for 4 + (2 or 5), 4 + (7 or 9), 1 + (2 or 5), ad 1 + (7 or 9) were computed ad are show i Table 3 ad Figure 1. Accordig to the criteria of te-* cardiality uderstaders stated i Study 1, there were 7 uderstaders (7%) i the hiese high-s group but oe i the other three groups. Agai, o restatig errors (4y) were foud amog the 4 + y trials for the hiese uderstaders. High-outig Sequece Groups A 2 (laguage: hiese vs. Eglish) X 2 (first added: 4 vs. 1) X 2 (secod added: 2 or 5 vs. 7 or 9) ANOVA o the score of correct resposes for the high-s groups was computed. The laguage, first added, ad secod added mai effects ad the First Added X Secod Added iteractio effect were sigificat, all Fs(l, 1) > 5.2, all ps <.5. Although the two high-s groups were matched o age, IQ, ad coutig sequece, childre i the hiese high-s group did sigificatly better tha those i the Eglish high-s group (M - 74% vs. 4%). hildre i both groups did sigificatly better o 1+ y trials tha o 4 + y trials (M = 9% vs. 51%) ad o x + (2 or 5) trials tha o x + (7 or 9) trials {M = 7% vs. 44%). From the post hoc comparisos by the l\ikey a test, we foud that childre i the high-s groups did ot differ o 1 + y trials with small ad large secod addeds. However, they did sigificatly better o 4 + y trials with small secod addeds tha with large secod addeds (75% vs. 27%; q =.2, p <.5). Eve though childre i both high-s groups did 1 + y trials better tha 4 + y trials, the solutio strategies of the two groups were differet. The childre i the Eglish high-s group did the majority of trials for both 1 + y ad 4 + y by coutig (see Table 3). The childre i the hiese high-s group also did the 4 + y trials with some coutig (couted o most of the 4 + 7 ad 4 + 9 trials), but they gave quick ad accurate resposes with o or little overt coutig to most 1 + y trials. This suggests that the regular hiese umber-word patter facilitates their calculatio ad memory of the additio facts for 1 + y sums, eve whe they do ot have full embedded-te cardial uder- Table 3 Study 2: Age 5 Mea Percetage of orrect Resposes, the Mea Reactio Time, ad the Mea Percetage of hildre Usig Overt outig Strategies for Each Group Problems 4+ (2 or 5) 4 + (7or9) 1+ (2 or 5) 1+ (7 or 9) Embedded-te cardial uderstaders % 5 4 5 5 hiese highs 3 1 7 7 oc 35 92 5 15 7 35 Eglish highs 17 47 1 oc 7 7 44 7 19 3 31 hiese lows 1 5 1 O 7 25 3 13 5 19 Eglish lows Note. S = coutig sequece; = mea percetage of correct resposes across trials; = mea reactio time, rouded off to the earest secod, for correct resposes; O = mea percetage of childre usig overt coutig strategies for correct resposes. 9 1 O 1 7
KNOWLEDGE OF TEEN QUANTITIES 541 hiese High S Group Eglish High S Group hiese Low S Group Eglish Low S Group 9, ru - X=4 X=1 % - - 5-4- 3-2- 1- - 1 Irl 1. 1. Y=2 Y~7 or 5 or 9 Figure 1. Percetages of correct resposes across trials for sums x + y o the hidde-object additio task for each group i Study 2. S = coutig sequece.
542 HO AND FUSON stadig (immediate rapid correct respodig o all 1 + y trials). Low-outig-Sequece Groups A 2 (laguage: hiese vs. Eglish) X 2 (first added: 4 vs. 1) X 2 (secod added: 2 or 5 vs. 7 or 9) ANOVA o the score of correct resposes for the low-s groups was computed. The first added ad secod added mai effects ad the Laguage X First Added iteractio effect were sigificat, all Fs(l, 14) > 5.9, all p$ <.5. Overall, the childre i the low-s groups did sigificatly better o 1 + y trials tha o 4 + y trials (M = 42% vs. 25%) ad o x + (2 or 5) trials tha o x + (7 or 9) trials (M = 5% vs. 17%). However, from the post hoc comparisos by the Tukey a test, we foud that childre i the hiese low-s group performed sigificatly better o 1 + y trials tha o 4 + y trials {q -.291, p <.5) but that those i the Eglish low-s group performed equally well o 1 + y ad 4 + y trials. hildre i the Eglish low-s group were, o average, 7 moths older tha those i the hiese low-s group. These Eglish childre made more correct resposes o 4 + y trials (predomiatly with much overt coutig) tha did the hiese childre. Thus, the older Eglish childre seemed to use coutig strategies with small secod addeds with more success tha did their hiese couterparts. However, the childre i the hiese group surpassed those i the Eglish group o the 1 + y trials. There was o embedded-te cardial uderstader i either of these two groups. Eve though the childre i the hiese low-s group did ot uderstad embedded-te cardiality cosistetly, they did fid 1 + y sums more easily tha 4 + y sums, ad they did ot cout overtly to fid most 1 + y sums. Thus, childre may uderstad this gradually rather tha all at oce, at least childre with shorter coutig-word sequeces. Participats Study 3 Method Twelve participats were selected to match the age ad socioecoomic backgroud of the Eglish childre. All were ative speakers of America Eglish. The childre raged i age from to 7 moths, with a average of moths (SD = 3.2). The rage of accurate rote-coutig sequece was 29 to 2, with a mea of 9. (SD = 5). Materials ad Procedures The childre were tested idividually o the hidde-object additio task ad the o the coutig sequece task, i exactly the same way as were the Eglish childre i Study 2. Results ad Discussio All of the childre did the 2 + 1 warm-up trial correctly without overt coutig. hildre receivig the 4 + y trials first icreased their use of overt verbal or figer coutig (see Table 4). hildre receivig the 1 + y trials first did all trials with o overt coutig ad rapidly gave correct aswers. I cotrast, of the childre receivig the 4 + y trials first, 4 overtly couted verbally or with figers o 1 or more trials. These 4 childre couted o of the 4 + y trials (4 of which were correct) ad o of the 1 + y trials (5 of which were correct). Rote-coutig skill (mea coutig sequece above or below 5) did ot affect performace, except that all childre with a coutig sequece above 5 did the 4 + 2 trial accurately whereas oly half the childre with a coutig sequece betwee 29 ad 49 did so. Thus, as with the hiese childre (although i a differet fashio), kowledge of the coutig sequece cosiderably exceeds additio performace. hildre lear a very log sequece before they ca do eve simple additio of much smaller umbers whe objects are hidde. A Good Match Is Hard to Fid Oe may questio the matchig of hiese ad Eglish participats i this study. We uderstad that matchig participats o oe variable may produce groups of participats who are umatched o some other relevat variables (e.g., Meehl, 197; Stigler & Miller, 1993). We metioed i the Participats sectio that hiese childre of comparable IQ ted to lear coutig sequeces earlier tha Eglish childre. By matchig both IQ ad coutig sequeces of the two groups i this study, we might have selected Eglish childre with better educatio ad learig motivatio tha their hiese couterparts. Yet, the hiese childre i this study still showed greater embedded-te cardial uderstadig tha did the Eglish childre. Therefore, the atioal differeces i the developmet of embedded-te cardiality seemed to be robust. Table 4 Study 3: Mea Percetage of orrect Resposes, Mea Reactios Times, ad Mea Percetage of Overt outig for hildre i the Uited States by Order of First Added Problems 4 + (2 or 5) 4 + (7or9) 1 + (2 or 5) 1 + (7 or 9) 4 + y items first 7 9 7 17 9 25 1 15 O 3 1 3 1 1 c 5 25 9 + y items first 2 3 a 2 O c 59 4 4 13 Total 25 2 O 21 1 27 7 Note. = mea percetage of correct resposes across trials; = mea reactio time, rouded off to the earest secod for correct resposes; O = mea percetage of childre usig overt coutig strategies for correct resposes. "Oe child reported usig the wall clock for 1 + 2 ad had a reactio time of 14 s; this reactio time was omitted as a outlier of time ad method.
KNOWLEDGE OF TEEN QUANTITIES 543 There was o evidece of embedded-te cardial uderstadig or of kowledge of embedded-te memorized facts. No child met the criterio for embedded-te cardial uderstader; the oly child to get all of the 1 + y trials correct couted with figers for three of those four trials. Performace was affected primarily by the size of the secod umber added. hildre did well o the trials i which 2 objects were added, coutig icreased performace o the trials i which 5 objects were added, ad the trials with 7 or 9 objects added were difficult ad were rarely solved correctly. Whether the first added was 4 or 1 did ot affect the correctess of aswers very much. hildre did ot use the feedback of coutig at the ed of each trial, to suggest coutig usig substitute objects (i.e., coutig did ot icrease over trials), ad did ot use ay patter they might have uderstood i the te words (e.g., sevetee is te ad seve). Geeral Discussio I summary, hiese childre surpass their Eglish ad America couterparts ot oly i rote coutig ad place value umeratio, as foud i previous studies, but also i embedded-te cardial uderstadig ad i applyig this kowledge to solvig simple additio problems. Furthermore, the earlier learig of the rote-coutig sequece, as foud i earlier studies, was foud here ot to deped o IQ. Most hiese 4-year-olds have ot yet developed geeral embedded-te cardial uderstadig. May, but ot all, 5-year-olds with a average or above level of itelligece ad a well-automatized coutig sequece (ear 1) uderstad embedded-te cardiality ad apply this kowledge to solvig additio problems. Differet researchers have suggested that either cultural factors or laguage factors accout for the atioal differeces i mathematics achievemet (e.g., Geary et al., 1993; Steveso, Lee, he, et al., 199; Steveso, Lee, & Stigler, 19; Steveso & Stigler, 1992). These both might be importat, however. Yag ad obb (1995) foud that the iitial arithmetical learig activities i which hiese childre egaged at home ad i school supported the developmet of composite multiuit umerical coceptios (e.g., the emphasis of a decade as a coutig uit ad the use of the up-over-1, dow-over-1, ad subtract-from-1 methods i solvig simple additio ad subtractio problems). Furthermore, hiese mothers ad teachers seemed to believe that it is atural for hiese childre to develop the cocept that umbers are composed of tes ad oes early, whereas the America childre were iitially ecouraged to costruct uitary umber cocepts based o coutig by oes. Therefore, apart from the direct ifluece of the umber-word regularities o hiese childre's coceptual developmet of umbers, the ifluece also may be mediated by hiese adults' beliefs about childre's arithmetical developmet ad resultat culturally supported learig activities at home ad i school. School traiig was ulikely i the preset studies to be a cotributig factor for the hiese superiority i embeddedte cardial uderstadig ad i solvig 1 + v problems. The hiese childre i Study 1 ad Study 2 leared how to cout i school begiig at the age of 3. Rote coutig was taught i class by askig childre to follow teachers' recitals ad was practiced durig may school activities (e.g., whe goig to the toilet or whe liig up). The coutig sequece was legtheed gradually to 1 by the age of 5. However, these coutig activities did ot attach to ay objects adthus had o cardial meaigs, so embedded-te cardial meaigs were ot supported by reported school activities. By the age of 5, the hiese childre had leared how to do simple additio with sums less tha 5 i school. Thus, apart from the 2 4-1 warm-up trial, all other problems had ot bee taught i school. Arithmetical traiig at home was ot examied i the preset studies. Because the hiese participats i Study 1 ad Study 2 came from the same kidergarte, school factors were less varied tha home factors. Whether the hiese parets of embedded-te cardial uderstaders support the developmet of this uderstadig at home is of iterest for future research. How they capitalize o the regular-tes umber words could be a model for use i schools for those hiese-speakig childre who are ot yet embedded-te cardial uderstaders, as well as for methods that might exted to Eglish-speakig childre. Refereces Briars, D. J., & Siegler, R. S. (194). A featural aalysis of preschooler's coutig kowledge. Developmetal Psychology, 2, 7-1. Bryat, P. E., & Lies, S. (1992). Number-amig systems i Eglish ad hiese: Liguistic effects o umber uderstadig ad basic mathematical skill Upublished mauscript. Fuso, K., & Kwo, Y. (1991). hiese-based Europea irregular systems of umber words: The disadvatages for Eglish speakig childre. I K. Durki & B. Shire (Eds.), Laguage i mathematical educatio: Research ad practice (pp. 2-22). Buckigham, PA: Ope Uiversity Press. Fuso, K., & Smith, S. T. (199). [ardial kowledge i coutig to 1.] Upublished raw data. Geary, D., Bow-Thomas,., Fa, L., & Siegler, R. S. (1993). Eve before formal istructio, hiese childre outperform America childre i metal additio. ogitive Developmet,, 517-529. Geary, D., Bow-Thomas,., Liu, F., & Siegler, R. S. (199). Developmet of arithmetical competecies i hiese ad America childre: Ifluece of age, laguage, ad schoolig. hild Developmet, 7, 222-244. Geary, D., Fa, L., & Bow-Thomas,.. (1992). Loci of ability differeces comparig childre from hia ad the Uited States. Psychological Sciece, 3, 1-15. Huse, T. (197). Iteratioal study of achievemet i mathematics: A compariso of twelve coutries (Vol. 2). Stockholm: Almqvist & Wiksell. Kamii,. (195). Youg childre reivet arithmetic: Implicatios ofpiaget's theory. New York: Teachers ollege Press. Lapoite, A. E., Mead, N. 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544 HO AND FUSON the philosophy of sciece (Vol. 4, pp. 373-42). Mieapolis: Uiversity of Miesota Press. Miller, K. F., Smith,. M., Zhu, J., & Zhag, H. (1995). Preschool origis of cross-atioal differeces i mathematical competece: The role of umber-amig systems. Psychological Sciece,, 5-. Miller, K. F., & Stigler, J. W. (197). outig i hiese: ultural variatio i a basic cogitive skill. ogitive Developmet, 2, 279-35. Miller, K. F., & Zhu, J. (1991). The trouble with tees: Accessig the structure of umber ames. Joural of Memory ad Laguage, 3, 4-. Miura, I. T., Kim,., hag,. M., & Okamoto, Y. (19). Effects of laguage characteristics o childre's cogitive represetatio of umber: ross-atioal comparisos. hild Developmet, 59, 1445-145. Miura, I. T., Okamoto, Y., Kim,., hag,. M., Steere, M., & Fayol, M. (1994). omparisos of childre's cogitive represetatio of umber: hia, Frace, Japa, Korea, Swede ad the Uited States. Iteratioal Joural of Behavioral Developmet, 17, 41-4. Miura, I. T., Okamoto, Y., Kim,., Steere, M., & Fayol, M. (1993). First graders' cogitive represetatio of umber ad uderstadig of place value: ross-atioal comparisos Frace, Japa, Korea, Swede, ad the Uited States. Joural of Educatioal Psychology, 5, 24-3. Steveso, H. W. ( Lee, S. Y., he,, Lummis, M., Stigler, J., Fa, L., & Ge, F. (199). Mathematics achievemet of childre i hia ad the Uited States. hild Developmet, 1, 153-1. Steveso, H. W., Lee, S. Y, & Stigler, J. W. (19). Mathematics achievemet of hiese, Japaese, ad America childre. Sciece, 231, 93-99. Steveso, H. W., & Stigler, J. W. (1992). The learig gap: Why our schools are failig ad what we ca lear from Japaese ad hiese educatio. New York: Summit Books. Stigler, J. W., & Miller, K. F. (1993). A good match is hard to fid: ommet o Mayer, Tajika, ad Staley (1991). Joural of Educatioal Psychology, 5, 554-559. Terma, L. M., & Merrill, M. A. (19). Staford-Biet Itelligece Scale: Form L-M. Lodo: Houghto Miffli. Whorf, B. L. (195). Laguage, thought, ad reality: Selected writigs of Bejami Lee Whorf (J. B. arroll, Ed.). ambridge, MA: MIT Press. Yag, M. T.-L., & obb, P. (1995). A cross-cultural ivestigatio ito the developmet of place-value cocepts of childre i Taiwa ad the Uited States. Educatioal Studies i Mathematics, 2, 1-33. Received November 2, 199 Revisio received December, 1997 Accepted December 15,1997