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Learning and Instruction 29 (2014) 78e94

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Learning and Instruction

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Strategic counting: A novel assessment of place-value understanding

Winnie Wai Lan Chan a, *, Terry K. Au b, Joey Tang b

a Department of Psychological Studies, The Hong Kong Institute of Education, Hong Kong SAR, Chinab Department of Psychology, University of Hong Kong, Hong Kong SAR, China

a r t i c l e i n f o

Article history:Received 18 January 2013Received in revised form26 August 2013Accepted 2 September 2013

Keywords:Mathematical learningPlace-value understandingCountingAssessment

* Corresponding author.E-mail address: winniechan@ied.edu.hk (W.W.L. C

0959-4752/$ e see front matter � 2013 Elsevier Ltd.http://dx.doi.org/10.1016/j.learninstruc.2013.09.001

a b s t r a c t

Children’s counting strategies reflect how much they understand the place-value structure of numbers.In Study 1, a novel task, namely the strategic counting task, elicited strategies from kindergarteners andfirst graders that showed a trend of increasing place-value knowledge e from perceiving number as anundivided entity to seeing it as a collection of independent groups of powers of ten. In Study 2, first-graders’ strategic counting task scores at the end of fall semester were better predictors of year-endmathematical achievement than the traditional place-value tasks. In Study 3, a five-item subset ofstrategic counting was the best among 15 various cognitive predictors of end of second-grade mathe-matical achievement. Growth curve modeling revealed that low-mathematics achievers at the end ofsecond grade had been lagging behind their peers in strategic counting since early first grade. Impli-cations for early support for children with difficulties in place-value knowledge are discussed.

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1. Introduction

1.1. Place-value concept

Each digit in a multi-digit numeral carries a value that dependson its position e i.e., a place value. In the base-ten number system,eachposition represents a powerof ten. In an integer, the right-mostdigit is in the 100e or onese place; the digit to its left is in the 101eor tens e place; etc. For example, in the number 26, the digit 6 is inthe ones place, so it carries a value of 6 (6� 1), whereas the digit 2 isin the tens place, so it carries a value of 20 (2 � 10). The place-valueconcept seems important to mathematical learning and especiallymastering arithmetic (Chan & Ho, 2010; Wearne & Hiebert, 1994).

Understanding the base-ten structure of numbers is crucial tomathematical problem-solving (Collet, 2003; Dehaene & Cohen,1997; Fuson, Wearne, et al., 1997) and correlates with early math-ematical achievement (Miura & Okamoto, 1989). Subpar place-value understanding predicts mathematical difficulties (Chan &Ho, 2010; Hanich, Jordan, Kaplan, & Dick, 2001; Jordan & Hanich,2000). In early grades, it predicts subsequent poor performancein addition and subtraction (Ho & Cheng, 1997) as well as in morecomplex arithmetic problems (Hiebert & Wearne, 1996). Place-value concept training improves arithmetic performance (Fuson,1990; Fuson & Briars, 1990; Ho & Cheng, 1997; Jones, Thornton, &

han).

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Putt, 1994). Hence, a good grasp of the place-value concept seemscrucial in learning arithmetic.

Anyone who has difficulty with the place-value concept is apt tomake errors in the comprehension and production of numbers, e.g.,in counting as well as in reading and writing multi-digit numerals(McCloskey, 1992). Such errors tend to be syntactic, often leading toorder-of-magnitude mistakes (e.g., “one thousand two hundredfive” might be written as 10002005; Ginsburg, 1977). Proceduresinvolving the base-ten structure of numbers, such as carrying overin addition and borrowing in subtraction, also become error prone(Fuson, 1990).

In much of the world, children receive formal instruction on theplace-value concept in early elementary school. In Hong Kong, forinstance, students learn about two-digit numbers in first grade,three- and four-digit numbers in second grade, and five-digitnumbers in third grade. By the end of the third grade, studentsare expected to know how the place-value concept applies to allmulti-digit numbers. Teachers usually teach this concept usingbase-ten manipulatives (e.g., base-ten-blocks) to illustrate thedifferent places in a number concretely. They often assess place-value understanding by asking students to write down the num-ber represented by the base-ten manipulatives or to identify theplace value of a digit in a number.

1.2. Developmental stages of place-value understanding

By observing children’s strategies for solving addition andsubtraction problems, Fuson, Smith, and Lo Cicero (1997), Fuson,

W.W.L. Chan et al. / Learning and Instruction 29 (2014) 78e94 79

Wearne, et al. (1997) proposed the UDSSI model, which describesthe developmental sequence of conceptual structures of two-digitnumbers and offers insights into how the place-value concept de-velops. The model was named after five key conceptions: unitarymulti-digit, decade and ones, sequence-tens and ones, separate-tens and ones, and integrated sequence-separate-tens. As newconceptions evolve, old conceptions may co-exist and be used incertain situations.

The unitary multi-digit conception extends children’s concep-tion of single-digit to multi-digit numbers: like a single-digitnumber, a multi-digit number represents an undivided quantity,and its individual digits are not meaningfully separable. Forinstance, the digits 2 and 6 in “26” in this conception are notassociated with 20 and 6, or two groups of tens and six ones. Theabsence of a sense of partitioning differentiates this conceptionfrom the other four. This conception manifests itself in counting byones without grouping (1, 2, 3, ., 26).

Developing a sense of partitioning depends on the structure ofthe number-naming system. Languages with separate decadewords (e.g., twenty, thirty, forty in English) facilitate separating aquantity into decade and ones, resulting in the decade and onesconception. For instance, 26 is considered as twenty and six. Withmore experiences in counting by tens (e.g., ten, twenty, thirty,forty), children can develop the sequence-tens and ones concep-tion, with the decade further separated into groups of tens. Forinstance, children count the quantity of 26 as 10, 20, 21, 22, ., 26(Resnick, 1983).

In languages with a transparent base-ten number structure (e.g.,“26” is rendered literally as “two-ten six” in Chinese), the groups oftens are clearly separate (i.e., two-tens in 26), fostering theseparate-tens and ones conception (i.e., counting the groups of tensseparately with a single-digit sequence). For instance, counting“26” as 1, 2 tens and 1, 2, 3,., 6 ones. This conception, however, canbe difficult to develop in number-naming systems where thegroups of tens are not explicitly named (e.g., twenty-six in English).Finally, children may integrate these two latter conceptions (i.e.,integrated sequence-separate-tens-and-ones conception) andreadily shift back and forth between the two conceptions. Forexample, they can quickly tell that there are five tens in fifty, orseven tens make up seventy, without counting by tens.

The conceptions described in the UDSSI model e developingfrom whole-number quantity to base-ten partitioning e maysignify growing place-value knowledge. This model also relateschildren’s implicit knowledge about numbers with their countingbehaviors, highlighting potential connections between basic nu-merical concepts and mathematical operations. These importanttheoretical implications remain untested empirically.

In the UDSSI model, each conceptual structure theoreticallymaps onto a counting strategy, whereas such mappings may not beone-to-one in reality. For example, the count-by-tens strategy maybe used by children with the sequence-tens and ones conception(i.e., the theoretically associated conception) as well as those withthe unitary multi-digit conception using rote learning. Even so, astrategy may be favored more by children with the theoreticallyassociated conception than those without. Hence, we expected thatthe count-by-one strategy would be used by children with theunitary multi-digit conception more; the count-by-tens strategywould be used by those with the sequence-tens and ones concep-tionmore; the count-by-separate-tens-and-ones strategy would beused by those with the separate-tens and ones conception more.

Despite its usefulness, the UDSSI model needs elaboration. First,built upon informal observations of how children solved additionand subtraction problems, the UDSSI conceptual structures anddevelopmental sequence remain untested empirically. Second, themodel is silent on multi-digit numbers beyond two digits. Third, it

was situated in the English number-naming system. Does it workfor more regular systems in languages such as Chinese? Forexample, the numbers 11 and 12 are rendered respectively as “tenone” and “ten two” in Chinese e transparently mapped onto thebase-ten structure e but obscurely as “eleven” and “twelve” inEnglish. Previous studies have shown that a more regular number-naming system facilitates base-ten understanding (Fuson & Kwon,1991; Miller & Stigler, 1987; Miura, Kim, Chang, & Okamoto, 1988;Miura, Okamoto, Kim, Steere, & Fayol, 1993; Miura et al., 1994). Itremains to be seen whether the conceptions and developmentalsequence described in the UDSSI model are language specific oruniversal.

1.3. Assessments of place-value concept

For assessment and research purposes, several place-valueconcept tasks have been developed. As it turns out, though, suc-ceeding at these tasks is often possible without really under-standing the concept.

A typical test e namely the number representation task e askschildren to represent numbers with ten-blocks and unit-blocks(Miura et al., 1993; Naito & Miura, 2001; Saxton & Cakir, 2006;see Appendix A for other tests). Three kinds of constructions havebeen identified using this task (Miura et al., 1993; Naito & Miura,2001; Ross, 1986): (1) one-to-one collection, where only the unit-blocks are used (e.g., 28 unit-blocks for the number 28), (2) ca-nonical base-ten representation, where the numbers of ten-blocksand unit-blocks correspond to the base-ten structure (e.g., 2 ten-blocks and 8 unit-blocks for 28), (3) non-canonical representa-tion, where the numbers of ten-blocks and unit-blocks do notcorrespond to the base-ten structure (e.g., 1 ten-block and 18 unit-blocks for 28).

One problemwith this task is that many mathematics textbooksuse illustrations of the canonical base-ten representation with ten-blocks and unit-blocks to teach the place-value concept. Childrencan learn by rote to represent the left digit of a two-digit numberwith ten-blocks and the right digit with unit-blocks. Moreover, aone-to-one collection does not necessarily imply a lack of the base-ten concept. Indeed, childrenwho construct a one-to-one collectioncould construct a canonical base-ten representation whenreminded that a ten-block is equal to ten unit-blocks (Miura &Okamoto, 1989). Therefore, this task can under- as well as over-estimate children’s place-value understanding.

Another typical school assessment e intended to tap children’sconcept of base-ten partitioning e presents children with a pictureof objects and asks them how many groups of ten can be formed.However, dividing a set of objects into groups of ten does notnecessarily mean appreciating the base-ten structure of a multi-digit number (e.g., x units in the tens place stand for x groups often objects). Moreover, the task is limited by its assessing only asingle aspect of the place-value concept e the concept of base-tenpartitioningewithout touching on other important aspects such astrading ten for one.

In most of other place-value tasks, children are given numbersand asked to do something with them (see Appendix A). But onetask takes a different tack, which typically appears in mathematicsexercises and textbooks. Children are shown sets of base-ten-blocks and asked to determine the numbers they represent. Thesets are always arranged in canonical groupings corresponding tothe base-ten structure (e.g., five ten-blocks and two unit-blocks forthe number 52) Note that children may simply learn by rote tocount the numbers of ten-blocks and unit-blocks and then stringthe two numbers together; here again, they may appear to under-stand the place-value concept more than they actually do (Ross,1989).

W.W.L. Chan et al. / Learning and Instruction 29 (2014) 78e9480

But with a crucial modification of this old test, we developed anew test e called strategic counting e that solves this problem.Childrenare still given the same task, but sometimes setsof blocks arearranged in non-canonical groupings. Thismeans that in order to getthe right answer, childrenneed to rearrange these setsephysically ormentallye based on their understanding of the place-value concept.We varied patterns of non-canonical groupings to discouragememorized strategies: some required rearranging blocks of the sameunit; others required trading ten for one; still others required both.Wealsoasked children for rationales behind their strategies to ensurethat they could make sense of what they did instead of just me-chanically implementing a sequence of “meaningless” steps.

Given a ten-block and 25 unit-blocks, for example, a child un-derstanding the place-value conceptmay first count by grouping theunit-blocks in tens e forming two groups of tens and five ones (i.e.,principle of grouping-in-tens). The two groups of tens are thentraded for two-ten-blocks (i.e., principle of trading ten for one) eresulting in three-ten-blocks andfive unit-blocks in total. In linewiththe Arabic base-ten structure, the child can map the blocks onto thenumerical notation to arrive at a total of 35. The processe frombase-ten partitioning the blocks to representing them with the conven-tional base-ten notation e is essentially an application of the place-value principles. Strategic counting was, hence, designed to assesschildren’s place-value knowledge through such process. Its effec-tivenesswas evaluated against two traditional place-value tasks: theplace-value representation task e which was very similar to thenumber representation task e and the grouping-in-ten task.

Several counting strategies are predicted by the UDSSImodel. Byexamining the strategies children used in our novel task, we eval-uated here aspects of the model empirically to see how well itcharacterizes the development of place-value understanding. Note,however, that we focused only on the unitary multi-digit,sequence-tens and ones, and separate-tens and ones conceptionsbecause: (1) in a pilot study of Chinese-speaking children, theircounting strategies could not distinguish between the integratedsequence-separate-tens conception and separate-tens and onesconception; (2) the decade and ones conception seems peculiar tolanguages with a separate list of decade words (e.g., ten, twenty,thirty,.; Fuson, Smith, et al., 1997; Fuson, Wearne, et al., 1997) andwas not apparent in our pilot study. Our studies focused on youngnative speakers of Chinese because decade words are very sys-tematic in Chinese (e.g., 20 is literally two-ten, 30 is three-ten, 40 isfour-ten), to contrast with the much-studied and far less trans-parent number-naming system in English.

1.4. Predicting mathematical achievement

Children’s place-value understanding has been found to predictlater mathematical achievement (Chan & Ho, 2010; Dehaene &Cohen, 1997; Hanich et al., 2001; Miura & Okamoto, 1989). Stra-tegic countingedesigned tooffer a better assessment of place-valueunderstanding than the traditional tests such as place-value rep-resentation and grouping-in-ten e should outperform the tradi-tional tests in predicting latermathematic achievement. Day-to-dayclassroom tasks such as counting, reading andwriting numbers, andsimple arithmetic, are also potential, handy indicators of children’smathematical learning. We therefore pitted strategic countingagainst performance on these widely used classroom tasks in pre-dicting later mathematical achievement e to assess the usefulnessof strategic counting as a classroom assessment tool.

1.5. Overview of the present research

The main goals of the present research were: (1) to provide thefirst systematic empirical test of the well-known UDSSI model

with a focus on the Chinese number-naming system; (2) todevelop an effective tool for assessing children’s place-value un-derstanding based on their counting performance; (3) to track thedevelopmental trajectories of place-value understanding usinggrowth curve modeling; and (4) to see how well the novel taskpredicts later arithmetic achievement among various cognitivemeasures.

2. Study 1

2.1. Overview

This study set out to address these questions: (1) Do childrencount in ways predicted by the UDSSI model? (2) Is the develop-mental sequence of the conceptual number structures languagespecific or universal? (3) Can the UDSSI model be extended beyondtwo-digit numbers? (4) Are children’s counting strategies useful forassessing their place-value understanding?

Based on the informal observations documented in priorstudies (Fuson, Smith, et al., 1997; Fuson, Wearne, et al., 1997), wehypothesized that children would count in ways as described bythe UDSSI model (Hypothesis 1). If the UDSSI model e developedin the context of English e was language universal, Chinese-speaking children’s counting strategies should appear in a similarsequence as that described in the model, despite Chinese having amore regular number-naming system than English (Hypothesis2a); if the model was language specific, our Chinese-speakingparticipants would reveal a different developmental sequence(Hypothesis 2b). In the strategic counting task, childrenwere askedto count two- and three-digit quantities. We hypothesized thatchildren followed similar principles for large and small multi-digitnumbers, so the developmental sequence of two-digit countingstrategies in the UDSSI model should also apply to three-digitquantities (Hypothesis 3). We also asked the children to com-plete two traditional place-value tests and several commonly usednumerical tasks. If the strategic counting task is more useful thanthe two traditional place-value tests in revealing children’s place-value understanding e a fundamental concept underlying a varietyof numerical abilities e the strategic counting task should beassociated more strongly with other numerical performance aswell (Hypothesis 4).

2.2. Method

2.2.1. ParticipantsWe recruited 72 kindergarteners (mean age ¼ 5 years 11

months, 41 boys and 31 girls) and 60 Grade 1 children (meanage ¼ 6 years 8 months, 37 boys and 23 girls), randomly selectedfrom a kindergarten and a primary school in Hong Kong. All thechildren were native speakers of Chinese and participated withwritten parental consent. The Grade 1 children also participated inStudy 3 as a follow-up.

2.2.2. Materials and procedureAt the end of the fall semester (December), children were asked

to complete the strategic counting task and a battery of tasks onsimple counting, number representation, place-value understand-ing, and arithmetic. They were assessed during lesson time atschool by trained researchers. Unless otherwise specified, one pointwas given for each correct item.

2.2.2.1. Strategic counting. This individually administered task wasdesigned to assess children’s conceptions of two- and three-digitnumbers. In the familiarization phase, a child was first shownthree types of magnets: small square (1.5 cm � 1.5 cm), bar

W.W.L. Chan et al. / Learning and Instruction 29 (2014) 78e94 81

(1.5 cm � 15 cm), and big square (15 cm � 15 cm). The child wasasked to count 10 small squares, line them up and observe that 10 ofthem were equivalent to one bar. A similar procedure was used toillustrate the equivalence between a big square and ten bars. In thetesting phase, the child was then shown 10 test items. Each was aline drawing showing some small squares, bars (each with 10 smallsquares outlined), and/or big squares (each with 100 small squaresoutlined). The line drawings were proportionally smaller than thephysical counterparts (see Appendix B). The child was asked tocount and write down the total number of small squares for eachtest item. The first five items were two-digit quantities, and theothers were three-digit quantities. For each item, the child wasencouraged to think aloud while counting. If the child did not thinkaloud or the counting procedure was unclear, the child would beasked at the end of the trial to explain the steps. The experimenterrecorded the child’s response verbatim and noted down his/herovert counting behavior for each item.

The two-digit items matched the learning objective of firstgraders, and the three-digit items tested how well the childrencould extend their place-value understanding to larger numbers.While the kindergarteners had no formal instruction on countingmulti-digit numbers, our pilot studies showed that they could dothat e perhaps from informal learning experience.

Among the two-digit items, Items 1 and 2 used a canonicalrepresentation of the base-ten numeration structure, whereasItems 3e5 used a non-canonical representation. According to theUDSSI model, efficient counting in Item 3 required rearranging thebars and small squares mentally. Item 4 required grouping thesmall squares in tens to form a bar. Item 5 required both rear-rangement and grouping-in-tens.

Among the three-digit items, Items 6 and 7 corresponded to acanonical structure. Item 7 included only big and small squares andno bars to see if children understood the concept of zero in the tensplace. Items 8e10 were shown in non-canonical structures, whichrequired mentally grouping small squares (i.e., Item 8), bars (i.e.,Item 9), or both (i.e., Item 10) in tens to carry over to a next place.(Note that rearranging procedure was not required in the three-digit items because pilot data suggested that it would have madethe task too complex for kindergarteners and first graders.) Duringthe task, whenever children tried to count the small squaresdepicted on a bar or a big square one-by-one, they were remindedof the equivalent quantities immediately (i.e., one bar ¼ ten smallsquares; one big square ¼ 100 small squares) and encouraged tocount in a quicker way. If they insisted on counting the smallsquares of a bar one-by-one, they were allowed to do so. However,if they insisted on counting the small squares of a big square one-by-one in the three-digit items, they were asked to move on to thenext item. This was because in the pilot study, childrenwho tried tocount the small squares of a big square one-by-one always endedup with an incorrect answer or failed to complete counting. To savetime and minimize frustration, children were allowed to skip thoseitems, which were marked as incorrect.

2.2.2.2. Simple counting. Children were assessed in two tasks:counting objects and counting in sequence.

2.2.2.2.1. Counting objects. There were four items, each showinga picture of dogs scattered randomly. Children were asked to countand write down the total number of dogs in each item. The quan-tities involved were 5, 7, 10, and 14 respectively.

2.2.2.2.2. Counting in sequence. There were 11 trials. In eachone, childrenwere told a start number and asked to count aloud upto a certain number. Each sequence involved a transition to a nextdecade (e.g., 29e30, 39e40). The first trial started with 1 and thelast trial ended with 120. A number generated in a correct sequencewould be given one point.

2.2.2.3. Number representation. Children were assessed for theirability to represent numbers in both written and verbal forms.

2.2.2.3.1. Reading Arabic numbers aloud. Children were showneight Arabic numbers (14, 16, 23, 47, 106, 128, 208, 259) and askedto read them out one-by-one.

2.2.2.3.2. Writing Arabic numbers. Children were shown pic-tures of pencil holders, each containing a different number ofpencils (13, 18, 26, 34, 108, 195, 207, 246). They were told thenumber of pencils in each holder and asked to write it down inArabic numerals.

2.2.2.3.3. Writing Chinese numbers. This was analogous to thetask of writing Arabic numbers, except that children were asked towrite down the number in Chinese words (14, 16, 25, 43, 104, 179,201, 235).

2.2.2.3.4. Transcoding Chinese numbers to Arabic. This task alsoused the pictorial representations of pencil holderswith pencils, butwith the number of pencils written in Chinese. Childrenwere askedto write the number in Arabic (12, 19, 27, 38, 102, 156, 203, 248).

2.2.2.3.5. Transcoding Arabic numbers to Chinese. Children weregiven the number of pencils in Arabic numbers (15, 17, 28, 49, 103,267) and asked to transcode it into written Chinese.

2.2.2.4. Place-value understanding. Two traditional place-valuetasks, namely grouping-in-ten and place-value representation,were administered.

2.2.2.4.1. Grouping-in-ten. After being told that a school buscould seat 10 students, children were shown four pictures ofvarious-sized groups of students: 12,16, 24, 36. For each they had todecide (1) howmany school buses could be filled and (2) howmanystudents, if any, would be left behind. They had to answer both (1)and (2) correctly in order to obtain one point.

2.2.2.4.2. Place-value representation. In each question, a teennumberwas shown in Arabic. Childrenwere first asked to representits quantity using the small squares and bar magnets describedearlier. One point was given for each canonical base-ten repre-sentation (e.g., “14” represented by a bar and four small squares).The experimenter would help any child who failed this part toconstruct a correct representation before moving on. The teennumbers used were 17, 11, 13, 14.

The digit in either the tens or ones place was then replaced by alarger digit (i.e., 17e18; 11e21; 13e15; 14e24). Childrenwere askedto represent the new number by making changes to the originalbase-ten construction. One point was given for a correct change inthe canonical representation (e.g., “14” being changed to “24”required adding a bar to the original representation).

2.2.2.5. Arithmetic. Childrenwere assessed for self-paced symbolicand non-symbolic addition skills.

2.2.2.5.1. Symbolic addition. Children were shown four addi-tions (13þ 2; 24þ 3; 15þ 6; 26þ 7) in vertical format and anotherfour (15 þ 3; 22 þ 4; 14 þ 7; 25 þ 8) in horizontal format. Childrenwere asked to compute the additions with no time limit. They wereallowed to calculate in their own ways and write out any roughwork. One point was given for each correct answer.

2.2.2.5.2. Non-symbolic addition. Children were shown the pic-tures of students and school buses used earlier in the grouping-in-ten task and asked to add up the number of students in each pictureand write down the total. There were four items, all involving two-digit quantities (15, 34, 20, 33).

2.3. Results and discussion

The number of children who completed the tasks and theirmean scores are shown in Table 1. (Incompletes were due to schoolabsences or time clashes with other activities.) Grade 1 children

W.W.L. Chan et al. / Learning and Instruction 29 (2014) 78e9482

scored significantly higher than kindergarteners in all tasks (allps < .01), except grouping-in-ten, F(1,126) ¼ .007, p > .05. The highaccuracy in counting objects suggests the children’s mastery ofbasic counting principles e prerequisites of strategic counting.

Simple correlations among the tasks are presented in Table 2.Significant correlations were found between strategic counting andthe traditional place-value tasks (r ¼ 0.36 for grouping-in-ten;r ¼ 0.52 for place-value representation), suggesting convergingvalidity of this novel task. In particular, the place-value represen-tation task had higher correlations with the canonical items ofstrategic counting (r ¼ 0.59) than with the non-canonical items(r ¼ 0.44). This was consistent with what we had noted earlier:unlike the place-value representation task and the canonical itemse both of which could be completed by rote-learnt procedures e

the non-canonical items relied more on genuine understanding ofthe place-value concept.

2.3.1. Children’s counting strategiesThe strategies children used in the strategic counting task were

coded by two raters based on the experimenter’s real-time recordsof the children’s counting behavior and the verbatim records of theirexplanations. If there was a conflict between the two records for achild, the real-time record of the child’s counting behaviorwould beused as the primary basis for classification of strategies, because theaccuracy and clarity of some children’s verbal report might belimited by their developing verbal ability. The raters reviewed50%ofthe records and agreed on the classifications of strategies listed inTable 3; they then independently coded the strategies item by itemfor all children. Even items with incorrect answers were coded aslong as the strategies used were clear and sensible. Otherwise, theywould be classified as errors. The kappa for inter-rater reliability forthe overall classification of children’s strategies was .92, and theraters resolved any discrepancy through discussion.

When presented with a two-digit quantity such as the 54 smallsquares in Table 3, children used five types of strategies e

Table 1Means scores on the tasks in the three studies.

Task Study 1

Kindergarteners First Gr

Mean Score (SD) N Mean S

Raven’s Standard Progressive Matrices e e e

Memory SpanDigit span forward e e e

Digit span backward e e e

Chinese word readingStrategic counting 3.56 (1.81) 72 6.23 (2Simple CountingCounting objects 3.62 (0.73) 68 3.98 (0Counting in sequence 53.76 (12.61) 72 59.97 (6

Number RepresentationReading Arabic numbers aloud 5.25 (1.51) 71 6.49 (1Reading Chinese numbers aloud e e e

Writing Arabic numbers 5.32 (1.71) 69 6.37 (1Writing Chinese numbers 4.88 (2.64) 67 6.30 (1Transcoding Chinese numbers to Arabic 4.06 (2.50) 66 5.32 (2Transcoding Arabic numbers to Chinese 2.60 (2.16) 65 4.33 (1

Place-Value UnderstandingGrouping-in-ten 1.51 (1.37) 68 1.53 (1Place-value representation 5.97 (1.96) 71 7.47 (0

ArithmeticSymbolic addition 4.64 (2.90) 70 6.93 (1Non-symbolic addition 2.42 (1.22) 67 3.05 (1

LAMK e e e

a LAMK was administered at the end of first grade.b Unless otherwise specified, the scores under this column were measured at the endc Brief strategic counting task was administered. Mean scores measured at Phases 2ed LAMK was administered at the end of second grade (i.e., Phase 4).

consistent with the ones described by the UDSSI model (Hypothesis1): (1) count all: counting by ones e e.g., 1, 2, 3, 4, ., 54 (unitarymulti-digit conception); (2) count on: counting by tens for someparts, then falling back to one-by-one counting e e.g., 10, 20, 21, 22,23, 24,., 54 (in transition to sequence-tens and ones conception); (3)sequence: counting by tens and ones in a continuous sequence e

e.g., 10, 20, 30, 40, 50, 51, 52, 53, 54 (sequence-tens and onesconception); (4) separate: counting by tens and ones in separatesequences e e.g., 1, 2, 3, 4, 5 tens; 1, 2, 3, 4 ones (separate-tens andones conception); (5) addition: counting by adding groups of smallsquares one-by-one from left to right or from right to left withoutrearranging the small squares into groups of tens e e.g., 10, 20, 21,31, 41, 43, 44, 54. Some children failed to count or counted erro-neously and randomly e e.g., 10, 20, 30, 40, 50, 60, 70.

When presented with a three-digit quantity such as the 124small squares in Table 3, children used four different strategies: (1)count on: counting by hundreds and tens for some parts, thenfalling back to one-by-one countinge e.g., 100,101,102,103,104,.,124 (in transition to sequence-tens and ones conception); (2)sequence: counting by hundreds, tens, and ones in a continuoussequence e e.g., 100, 110, 120, 121, 122, 123, 124 (sequence-tens andones conception); (3) sequence-separate: counting by hundreds andtens in a continuous sequence, but adding the left over ones in aseparate sequencee e.g., 100,110,120; 1, 2, 3, 4 ones (in transition toseparate-tens and ones conception); (4) separate: counting by hun-dreds, tens and ones in separate sequences e e.g., 1 hundred; 1, 2tens; 1, 2, 3, 4 ones (separate-tens and ones conception). Somechildren attempted to count by ones, or failed to count, or countederroneously and randomly e e.g., 100, 200, 300, 400.

2.3.2. Developmental pattern of counting strategiesTo identify sub-groups of childrenwho adopted similar patterns

of counting strategies, a two-step cluster analysis was performedon the frequencies in which various counting strategies were used.Three clusters of childrenwere identified (Cluster 1:N¼ 36; Cluster

Study 2 Study 3

aders First Graders First Gradersb

core (SD) N Mean Score (SD) N Mean Score (SD) N

e 109.30 (13.71) 582 107.33 (14.27) 186

e e e 4.50 (1.60) 182e e e 3.60 (1.40) 182

14.50 (13.99) 190.54) 60 6.11 (2.64) 581 1.86 (1.74)c 190

.13) 60 e e e e

.70) 60 60.04 (6.65) 581 58.66 (7.65) 190

.62) 59 6.65 (1.46) 582 6.16 (1.63) 189e 7.13 (1.49) 582 e e

.74) 60 e e 5.96 (1.76) 190

.99) 60 e e 5.62 (2.30) 190

.58) 60 e e 4.96 (2.67) 190

.48) 60 e e 3.63 (1.86) 189

.20) 60 2.27 (1.54) 582 1.40 (1.29) 189

.93) 60 7.38 (1.24) 582 7.15 (1.34) 190

.48) 60 6.93 (1.80) 582 6.46 (1.97) 190

.35) 60 2.67 (1.53) 581 2.48 (1.60) 189e 438.87 (192.33)a 557 555.84 (240.67)d 181

of the fall semester of first grade (i.e., Phase 1).4, were 2.92 (1.83), 4.15 (1.31), and 4.45 (1.16) respectively.

Table 2Correlations among the tasks in Study 1.

Task 1 2 3 4 5 6 7 8 9 10 11 12

1. Strategic counting 12. Count objects 0.18* 13. Count sequence 0.44** 0.33** 14. Reading Arabic numbers aloud 0.51** 0.15 0.35** 15. Writing Arabic numbers 0.52** 0.22* 0.29** 0.49** 16. Writing Chinese numbers 0.39** 0.31** 0.40** 0.27** 0.42** 17. Transcoding Chinese numbers to Arabic 0.45** 0.19* 0.31** 0.42** 0.58** 0.42** 18. Transcoding Arabic numbers to Chinese 0.50** 0.25** 0.37** 0.45** 0.50** 0.62** 0.58** 19. Grouping-in-ten 0.36** 0.10 0.29** 0.34** 0.27** 0.19* 0.33** 0.27** 110. Place-value representation 0.52** 0.16 0.45** 0.35** 0.24** 0.34** 0.26** 0.40** 0.27** 111. Symbolic addition 0.48** 0.17 0.47** 0.35** 0.27** 0.25** 0.31** 0.37** 0.21* 0.44** 112. Non-symbolic addition 0.45** 0.16 0.32** 0.22* 0.21* 0.21* 0.36** 0.28** 0.38** 0.29** 0.35** 1

*p < 0.05, **p < 0.01.

W.W.L. Chan et al. / Learning and Instruction 29 (2014) 78e94 83

2: N ¼ 40; Cluster 3: N ¼ 56). The majority of kindergarteners wasin Clusters 1 (41.7%) and 2 (36.1%), whereas the majority of firstgraders was in Cluster 3 (66.7%). Only a small minority of firstgraders was in Cluster 1 (10%). Therefore the developmental trendappeared to go from Cluster 1, through Cluster 2, to Cluster 3.

Fig. 1aed shows the mean frequencies of strategy use for two-and three-digit items by grade and by cluster. One-way ANOVAswere performed to compare the frequencies for each kind ofstrategy among the three clusters of children. The three clustersdiffered significantly for all five strategies in counting two-digitquantities (Counting from one: F(2,129) ¼ 34.5, p < .001,hp2 ¼ .35; Count on: F(2,129) ¼ 42.8, p < .001, hp2 ¼ .40; Sequence:

F(2,129) ¼ 71.7, p < .001, hp2 ¼ .53; Separate: F(2,129) ¼ 130.7,p < .001, hp2 ¼ .67; Addition: F(2,129) ¼ 6.4, p ¼ .002, hp2 ¼ .09;Error: F(2,129) ¼ 32.1, p < .001, hp2 ¼ .33). Post hoc contrasts sug-gested that, compared to children in other clusters, children inCluster 1 showed significantly more erroneous counting and usedcount all strategy more e compatible with unitary multi-digitconception; those in Cluster 2 used count on and sequence stra-tegies more e compatible with developing sequence-tens and onesconception; and those in Cluster 3 used addition and separatestrategies more e compatible with separate-tens and onesconception. (Note that addition strategy was prominent in childrenwith separate-tens and ones conception probably because thesechildren had a firmer grasp of the place-value structure thatfacilitated carrying over in addition.) In general, the developmentaltrend of conceptions of quantity appeared to go from unitarymulti-digit through sequence-tens and ones to separate-tens andones, offering empirical support for the UDSSI model (supportingHypothesis 2a instead of Hypothesis 2b). Likewise, the threeclusters also differed significantly in their counting strategies forthree-digit quantities (Count on: F(2,129) ¼ 15.1, p < .001, hp2 ¼ .19;Sequence: F(2,129) ¼ 8.7, p < .001, hp

2 ¼ .12; Separate:

Table 3Counting strategies in the strategic counting task.

Counting Strategy

Example: How many small squares are there?

Two-Digit Quantities Three-

Count all 1, 2, 3, 4, ., 54 e

Counton 10, 20, 21, 22, 23, 24, ., 54 100, 1Sequence 10, 20, 30, 40, 50, 51, 52, 53, 54 100,11Sequence-separate e 100,11Separate 1, 2, 3, 4, 5 tens 1, 2, 3, 4 ones 1 hundAddition 10, 20, 21, 31, 41, 43, 44, 54 e

Error 10, 20, 30, 40, 50, 60, 70 100, 2

F(2,129) ¼ 55.2, p < .001, hp2 ¼ .46; Count from one or error:F(2,129) ¼ 35.7, p < .001, hp

2 ¼ .36), except for the sequence-separate strategy (F(2,129) ¼ 2.4, p ¼ .09, hp2 ¼ .09). Post hoc con-trasts suggested that, compared to children in other clusters,children in Cluster 1 showed more erroneous counting and moreattempts to use count all strategy (compatible with unitary multi-digit conception); those in Cluster 2 adopted count on andsequence strategies more frequently (compatible with developingsequence-tens and ones conception); those in Cluster 3 countedwith a separate strategy more often (compatible with separate-tens and ones conception). The developmental shift revealedhere by children’s counting strategies for three-digit quantities e

from unitary multi-digit through sequence-tens and ones toseparate-tens and ones conception e was consistent with that fortwo-digit quantities (Hypothesis 3).

2.3.3. From counting strategies to numerical performanceStrategic counting correlated significantly with all numerical

tasks. Compared with the place-value tasks, strategic countingshowed higher correlations with all the numerical tasks, exceptcounting in sequence (r ¼ 0.44 for strategic counting and r ¼ 0.45for place-value representation). Strategic counting seemed to beassociatedmore stronglywith numerical performance than the twotraditional tasks were (Hypothesis 4). The three clusters of childrenwere further compared on their numerical performance (Fig. 2),with Grade (Kindergarten vs. Grade 1) as a covariate. Children inCluster 3 performed significantly better than their peers in Cluster 1in strategic counting, counting in sequence, writing Arabicnumbers, transcoding Arabic numbers to Chinese and vice versa,grouping-in-ten, place-value representation, symbolic and non-symbolic additions (all ps < .05). In particular, children in Cluster3 were the best performers among the three clusters in strategiccounting, transcoding Arabic numbers to Chinese and place-value

Conception of quantityDigit Quantities

Unitary multi-digit01, 102, 103, 104, ., 124 In transition to sequence-tens and ones0,120,121, 122, 123, 124 Sequence-tens and ones0,120 1, 2, 3, 4 ones In transition to separate-tens and onesred1, 2 tens 1, 2, 3, 4 ones Separate-tens and ones

e

00, 300, 400 e

Fig. 1. a. Mean frequencies of strategy use for two-digit items (shown by grade level). b. Mean frequencies of strategy use for three-digit items (shown by grade level). c. Meanfrequencies of strategy use for two-digit items (shown by cluster). d. Mean frequencies of strategy use for three-digit items (shown by cluster).

W.W.L. Chan et al. / Learning and Instruction 29 (2014) 78e9484

Fig. 2. Estimated marginal mean scores on different mathematical tasks of the three clusters of children.

W.W.L. Chan et al. / Learning and Instruction 29 (2014) 78e94 85

representation (all ps < .05). Children in Cluster 1 were the worstperformers in strategic counting, counting in sequence and sym-bolic addition (all ps < .05). Hence, using more advanced countingstrategies (revealed in the upper clusters) was associated withbetter performance in the strategic counting task. Moreover, higherlevels of place-value knowledge e as revealed by the strategiccounting task e were associated with better performance incounting in sequence, number representation, and arithmetic, aswould be expected from the hypothesized connection betweenplace-value understanding and various numerical skills.

2.4. Conclusions of Study 1

This study was the first attempt to evaluate the well-knownUDSSI model on the developmental sequence of conceptual struc-tures of number, each of which was mapped onto a certain way ofcounting. Here, children were observed to count with strategies inline with these hypothesized conceptual structures, namely unitarymulti-digit, sequence-tens and ones, and separate-tens and onesconceptions (Hypothesis 1). As children developed, they shiftedtheir counting strategies from unitary multi-digit to separate-tensand ones, as predicted by the UDSSI model. This was so eventhough children in this study used a more regular number-namingsystem (i.e., Chinese) than the children with whom the model wasoriginally developed (i.e., English). Hence, the number conceptionsas well as their developmental sequence appeared to be universal(Hypothesis 2a) rather than language specific (Hypothesis 2b).

Although the UDSSI model was originally developed for two-digit quantities, it extended well to larger quantities. The devel-opmental pattern that went from unitary, through sequence, toseparate conception was consistently found for both two- andthree-digit quantities, suggesting that children went throughsimilar developmental stages to make sense of two-digit numbersand beyond (Hypothesis 3).

In general, the present findings supported the UDSSI model, butmuch remains to be done. For one thing, how the conceptualstructures of number subserve counting strategies could be furtherunderstood by analyzing the number knowledge represented byeach strategy.

In this study, a novel task e strategic counting ewas created forassessing children’s place-value knowledge. It revealed how wellchildren could apply and integrate various place-value componentsin an everyday activity: counting. Its validity in assessing

place-value knowledge was supported by its association withtraditional place-value tasks, namely grouping-in-ten and place-value representation. Importantly, strategic counting was associ-ated more strongly with other numerical performance, andhence seemed to be a better tool for assessing the place-valueknowledge e the fundamental element for a variety of numericalabilities e than the two traditional tests (Hypothesis 4).

In sum, the present findings suggest that children’s countingstrategies reflect their underlying place-value knowledge. Childrenusing more advanced counting strategies also performed better invarious numerical tasks. However, as a classroom assessment forplace-value understanding, the novel strategic counting task mightbe too time-consuming for teachers to use on a whole class. Recallthat in this task, children who counted with more advanced stra-tegies also counted more accurately. Study 2 therefore explored aquicker way to assess children’s place-value understanding by us-ing their accuracy scores in the strategic counting task.

3. Study 2

3.1. Overview

This study addressed two research questions: (1) Canwe use theaccuracy scores of strategic counting to assess children’s place-value understanding? (2) Can the task be streamlined to includefewer items? Given that children who counted with a moreadvanced strategy showed higher accuracy on the strategiccounting task in Study 1, we hypothesized that children’s accuracyscores of correct responses would work as well as the painstakingclassification of counting strategies as an indicator of place-valueknowledge. Specifically, we hypothesized that first-graders’ accu-racy score on the task at the end of the fall semester could predicttheir mathematical achievement at the end of the spring semester(Hypothesis 1). To make the task even shorter, we developed a briefversion through item analysis, which was predicted to be a validshortcut of the full version (Hypothesis 2).

3.2. Method

3.2.1. ParticipantsFive hundred and eighty two Grade 1 children (mean age ¼ 6

years 7 months, 336 boys and 246 girls) participated in the studywith written parental consent. The children, all native speakers of

Table 4Correlations among the tasks in Study 2.

Task 1 2 3 4 5 6 7 8 9 10

1. Raven’s Standard Progressive Matrices 12. Strategic counting 0.22** 13. Count sequence 0.13** 0.41** 14. Reading Arabic numbers aloud 0.16** 0.57** 0.46** 15. Reading Chinese numbers aloud 0.11* 0.34** 0.39** 0.41** 16. Grouping-in-ten (Robot Version) 0.15** 0.47** 0.30** 0.35** 0.30** 17. Place-value representation 0.14** 0.38** 0.43** 0.36** 0.32** 0.33** 18. Symbolic addition 0.15** 0.38** 0.38** 0.31** 0.34** 0.29** 0.39** 19. Non-symbolic addition (Robot Version) 0.12** 0.44** 0.32** 0.34** 0.26** 0.47** 0.34** 0.32** 110. LAMK 0.25** 0.50** 0.34** 0.40** 0.36** 0.32** 0.30** 0.28** 0.30** 1

*p < 0.05, **p < 0.01.

Table 5Hierarchical regression predicting mathematical achievement at the end of Grade 1(N ¼ 528).

Step Predictor B b Total R2 R2 change

1 General predictors 0.07 0.07**Age 1.51 0.37Nonverbal intelligence 3.81 0.27**

2 Numerical measures 0.29 0.22**Counting in sequence 2.14 0.07Reading Arabic numbers aloud 26.65 0.20**Reading Chinese numbers aloud 22.43 0.17**

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Chinese, were randomly recruited from 21 primary schools in HongKong.

3.2.2. Materials and procedureChildren were assessed at school twice, once at the end of the

fall semester in December and again at the end of the spring se-mester in June. In the first assessment, children completed anonverbal intelligence test, the strategic counting task, and a bat-tery of tasks on simple counting, number representation, place-value understanding, and arithmetic. In the second assessment,they were asked to complete a mathematics achievement test.

3.2.2.1. Nonverbal intelligence test. Children’s nonverbal intelli-gence was used here as a control variable in predicting mathe-matical outcomes due to its association with mathematicalperformance (Kyttälä & Lehto, 2008). It was assessed by the shortform of Raven’s Standard Progressive Matrices with a local normChildrenwere asked to complete the first three sets of 12 questionsin the original form. On each question, children were asked tochoose a pattern to fill in the missing part of a matrix.

3.2.2.2. Strategic counting. The same materials and procedureswere used as in Study 1.

3.2.2.3. Simple counting. The counting-in-sequence task in Study 1was used.

3.2.2.4. Number representation. As in Study 1, children were askedto read Arabic numbers aloud. In addition, they were asked to readeight Chinese numbers aloud (15,17, 24, 36, 109, 184, 205, 263). Onepoint was given for each number correctly read.

3.2.2.5. Place-value understanding. The grouping-in-ten and place-value representation tasks in Study 1 were used here with onemodification. In the grouping-in-ten task, children were asked tobuild robots1 (rather than to fill up school buses). A robot was madeup of 10 blocks, and children were asked how many robots theycould build with a certain number of blocks shown on a piece ofpaper, and how many blocks, if any, would be left over.

3.2.2.6. Arithmetic. The symbolic and non-symbolic addition tasksin Study 1 were used here with one modification. Namely, the non-symbolic one used the theme of building robots rather than fillingschool buses.

1 In Study 1, grouping-in-ten was presented as a task to group students forschool-bus allocation (10 students for a bus). The task appeared too difficult for firstgraders (mean score ¼ 1.53; max score ¼ 4). To make the task more child-friendly, anew version using a theme that children can more readily make sense of, namelybuilding robots, was created.

3.2.2.7. Mathematics achievement test. The Learning AchievementMeasurement Kit 2.0 for mathematics (LAMK) is a standardized testforHongKong children inGrades1e6,with localnorms for eachgradelevel (Hong Kong Education Bureau, 2008). The test for Grade 1 stu-dents consists of 37 items, covering simplearithmetic,wordproblems,shape and space, andmoney transactions. Childrenwere given45minto complete the test. Raw scores were converted into ability levelsaccording to the local norms calibrated by the Rasch Model.

3.3. Results and Discussion

Some children did not complete all of the tasks due to schoolabsences or time clashes with other activities. The mean scores forthe children who did complete the tasks are shown in Table 1.Simple correlations among the tasks are presented in Table 4. All ofthe correlations were significant and positive. Importantly, amongall the predictors, strategic counting showed the strongest relationwith LAMK (r ¼ 0.50).

3.3.1. Predicting mathematical achievementA three-step hierarchical regression revealed that, with respect

to mathematical achievement at the end of Grade 1 (i.e., LAMKresult), strategic counting contributed an additional unique vari-ance of 5% over and above the control variables (i.e., age andnonverbal intelligence) (Step 1) and various numerical measures(Step 2; Table 5) (Hypothesis 1).

Strategic counting taps important components of theplace-valueknowledge perhaps not revealed by the grouping-in-ten and place-value representation tasks widely used in educational assessmentand research. Grouping-in-ten probably only taps a single compo-nent of the place-value concept, namely, base-ten partitioning.Place-value representation may primarily evaluate children’sconstructions of base-ten manipulatives rather than more

Grouping-in-ten (robot version) 12.78 0.10*Place-value representation 5.95 0.04Symbolic addition 7.02 0.06Non-symbolic addition(robot version)

9.85 0.08

3 Strategic counting 22.65 0.31** 0.34 0.05**

*p < 0.05, **p < 0.001.

Table 6Proportion correct and corrected item-total correlations of the items in strategiccounting.

Proportion Correct Corrected Item-Total Correlation

Two-Digit QuantitiesItem 1 0.98 0.13Item 2 0.90 0.37Item 3 0.78 0.54Item 4 0.86 0.32Item 5 0.78 0.38Three-Digit QuantitiesItem 6 0.43 0.72Item 7 0.51 0.63Item 8 0.31 0.73Item 9 0.29 0.74Item 10 0.26 0.70

All corrected item-total correlations were significant (p < .05).

W.W.L. Chan et al. / Learning and Instruction 29 (2014) 78e94 87

comprehensive place-value understanding. By contrast, strategiccounting requires children to integrate multiple components of theplace-value concept, such as grouping-in-tens, carrying over,generating a base-ten count sequence, and mapping betweenquantities and count names e thereby seems to offer a morecomprehensive evaluation of children’s place-value understanding.

Another plausible explanation for strategic counting out-performing the two traditional place-value tasks is that theformer e tapping knowledge of two- and three-digit numerals e isrelatively broader andmore difficult than the latter twoe accessingknowledge of two-digit numerals only. To explore this possibility,we compared the two-digit items on the strategic counting taskwith the two traditional place-value tasks. The two-digit strategiccounting task showed the strongest correlation with LAMK(r ¼ 0.35) when compared with the grouping-in-ten task (r ¼ 0.32)and the place-value representation task (r ¼ 0.30). A four-step hi-erarchical regression revealed that, with respect to mathematicalachievement, the two-digit items on the strategic counting task(Step 3) contributed an additional unique variance of 1.4% over andabove the control variables (Step 1) and the rest of the numericalmeasures including the two place-value tasks (Step 2). The three-digit items on the strategic counting task (Step 4) further accoun-ted for an additional unique variance of 3.9% on top of all thesepredictors. Hence, evenwhen the difficulty level was controlled for,the two-digit strategic counting task remained a stronger predictorof mathematical achievement than the grouping-in-ten and place-value representation tasks e suggesting that the nature of strategiccounting indeed rendered the task more effective in tapping chil-dren’s place-value understanding and thereby in predictingmathematics learning outcome. Including the more challenging,three-digit items in the task further increased its predictive power.

An important difference between the strategic counting taskand the other two widely used place-value tasks was that theformer allowed for the use of different strategies. Previous findings(e.g., Torbeyns, Verschaffel, & Ghesquière, 2004) suggested that theadaptiveness of children’s strategy choices was related to theirmathematical ability. Hence, perhaps the unique varianceexplained by the task had partly to do with children’s ability tochoose the most efficient strategye namely the separate strategyewhich requires a sound knowledge of various components of theplace-value system (e.g., carrying over) and brings about more ac-curate counting performance.

3.3.2. Streamlining the strategic counting taskItem analysis on the strategic counting task helped select the

most effective and reliable items to create a brief version. TheCronbach’s alpha reliability coefficient of the original task was .84.Each item was evaluated in terms of item difficulty and item

discrimination. Item difficulty was indicated by the proportion ofchildren who got the item correct. Items with higher proportioncorrect are easier. A testworksbestwhen it contains itemsof varyingdifficulty with an average proportion correct of 0.5 (Ghiselli,Campbell, & Zedek, 1981). Items with the extreme proportion-correct values are not useful for differentiating between in-dividuals because all test takers either pass (proportion correct¼ 1)or fail (proportion correct ¼ 0) the items. Item discrimination wasestimated by the corrected item-total correlation. The higher thecorrelation, the better the item was at determining the children’sabilities. Traditionally, a value of at least 0.3 is considered to beacceptable (Everitt, 2002). Table 6 shows the proportion correct andcorrected item-total correlations of the items.

First consider item difficulty. Items with proportion-correctvalues closer to either 1 or 0 were less useful. In the initial step,we dropped Items 1, 2, and 10. The first twowere probably too easy(proportion correct ¼ 0.98 and 0.90 respectively), and the last onewas probably too hard (proportion correct ¼ 0.26). Next consideritem discrimination. Except for the first, all items had a correcteditem-total correlation of at least 0.3, suggesting that they were allacceptable discriminators. But to streamline the test, we only keptthe five items with the highest corrected item-total correlations:Items 3, 6e9. These items had varying difficulty levels (rangingfrom0.29 to 0.78) with an average proportion correct of 0.46, whichwas very close to the optimal value 0.5. The Cronbach’s alphareliability coefficient of the five-item brief version was .85, com-parable to that of the full version (.84). The brief version correlatedstrongly with the full version (r ¼ 0.96), indicating that the formerwas a valid shortcut of the latter (Hypothesis 2).

3.4. Conclusions of Study 2

This study showed that, without painstaking analysis of chil-dren’s counting strategies, the scores of correct strategic countingresponses provided a valid shortcut to assessing children’s place-value understanding. Specifically, first-graders’ strategic countingscores at the end of the first term uniquely predicted their math-ematical achievement at the end of the second term, over andabove their age, nonverbal intelligence, and performance in variousnumerical domains (Hypothesis 1). More importantly, strategiccounting turned out to be a stronger predictor of mathematicaloutcomes than grouping-in-ten and place-value representation,two direct and widely used place-value tests.

A brief version of the strategic counting task was developed toprovide a quicker tool for assessing children’s place-value under-standing in class. The number of items was cut by half, to just fivethat could be completed in 10 min. Yet, the brief version showedhigh internal consistency and correlated strongly with the fullversion, indicating that the five-itemversionwas a reliable shortcutof the full version (Hypothesis 2). Study 3 further evaluated thepsychometrics of this brief version.

4. Study 3

4.1. Overview

This study addressed three research questions: (1) How useful isthe five-item strategic counting task for predicting children’smathematics learning outcome beyond their first year of elemen-tary school? (2) Do children who end up as low achievers in math-ematics follow a different path in their place-value development,compared with children who end up as on-track-mathematicsachievers? (3) How sensitive is the task for screening potential lowachievers in mathematics?

Table 7Hierarchical regression predicting mathematical achievement at the end of Grade 2(N ¼ 165).

Step Predictor B b Total R2 R2 change

1 General predictors 0.39 0.39***Age 6.74 0.13*Nonverbal intelligence 5.40 0.32***Digit span forward 12.99 0.09Digit span backward 29.83 0.17**Chinese word reading 5.00 0.29***

2 Numerical measures 0.59 0.20***Counting in sequence 3.77 0.11Reading Arabic numbers aloud 29.25 0.20**Writing Arabic numbers �2.75 �0.02Writing Chinese numbers 16.18 0.15*Transcoding Arabic numbersto Chinese

�3.58 �0.03

Transcoding Chinese numbersto Arabic

12.68 0.14

Grouping-in-ten 30.64 0.17**Place-value representation 3.27 0.02Symbolic addition �4.39 �0.03Non-symbolic addition 22.37 0.15*

3 Strategic counting(brief version)

29.02 0.21* 0.61 0.02*

*p < .05, **p < .01, ***p < .001.

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We first evaluated the predictive value of the five-item strategiccounting against those of general and numerical predictors. Spe-cifically, first-graders’ performance on the task at the end of the fallsemester was expected to be the strongest predictor of theirmathematical achievement at the end of second grade (i.e., 18months later) (Hypothesis 1).

We then used growth curve modeling to track children’s place-value understanding over four time points from the fall semester offirst grade to the spring semester of second grade. The childrenwere divided into three groups according to their mathematicalachievement at the end of second grade, and their developmentalpaths of place-value understanding were compared. Given theimportance of place-value understanding in mathematicallearning, we hypothesized that second graders who were lowachievers in mathematics had already lagged behind their coun-terparts in their early place-value understanding (Hypothesis 2).

We also performed discriminant analyses to test how well thestrategic counting task could identify childrenwhowould end up aslow achievers in mathematics at the end of second grade. Toestablish a cutoff score for the task, receiver operating character-istic curve analysis was performed.We hypothesized that the cutoffscore could reliably identify potential low achievers inmathematics(Hypothesis 3).

4.2. Method

4.2.1. ParticipantsOne hundred ninety three Grade 1 children from five primary

schools in Hong Kong (mean age ¼ 6 years 9 months, 116 boys and77 girls) participated in the studywith written parental consent. Allof them were native speakers of Chinese. They were randomlyrecruited from five classrooms in each school. Sixty of them hadparticipated in Study 1 as first graders.

4.2.2. Materials and procedureChildren were assessed at the end of four consecutive semes-

ters: fall (December) and spring (June) semesters of first grade andsecond grades. The assessments took place during lesson time atschool. In the first phase, children were given the Raven’s StandardProgressive Matrices, memory span tasks (i.e., digit span forwardand backward), a Chinese word reading test, the brief version ofstrategic counting, and a battery of numerical tasks on simplecounting, number representation, place-value understanding, andarithmetic. For grouping-in-ten and non-symbolic addition, theschool-bus version used in Study 1 was adopted in this study.2

In each subsequent phase, children were given the brief versionof strategic counting again. In the fourth phase, children were alsogiven a mathematical achievement test. Most tests were the sameas those described in Studies 1 and 2. Exceptions were as follows.

4.2.2.1. Raven’s Standard Progressive Matrices. The full form of Ra-ven’s Standard Progressive Matrices with local norms was admin-istered to assess children’s nonverbal intelligence. It consisted offive sets of 12 questions on matrix reasoning.

4.2.2.2. Memory span. Memory span for digits is related to arith-metic (e.g., Ashcraft, Donley, Halas, & Vakali, 1992; Noël, Désert,Aubrun, & Seron, 2001), and hence may predict mathematicalachievement (e.g., Bull, Espy, & Wiebe, 2008). Children’s forwardand backward digit spans were assessed with the standard

2 In a study not reported here, children performed comparably in the school-busand robot versions of grouping-in-ten and non-symbolic addition. Hence, theschool-bus version, which was the first version, was used in this study.

procedure for theWechsler Intelligence Scale for Childrene FourthEdition (Hong Kong) (WISC-IV (HK); Wechsler, 2010).

For digit span forward, children listened to a sequence of two-digit numbers and were asked to repeat them as heard. Therewere seven test items, starting with two numbers in the first item,three in the second, and so on. Each test item consisted of twodifferent trials. The task was ended if children failed on both trialsof any item. Backward digit span was similar to forward digit span,except that children listened to single-digit numbers and repeatedthem in a backward sequence. In both tasks, one point was given foreach trial when a child gave the correct numbers and sequence.

4.2.2.3. Chinese word reading. This test was developed for HongKong children in Grades 1e6 by the Hong Kong Education Depart-ment (now the Hong Kong Education Bureau). Children were askedto read some two-character Chinese words aloud, 100 in all. Theyreceived one point whenever they read both characters in a wordcorrectly. The test was ended if they failed on 10 consecutive words.

4.2.2.4. Mathematics achievement test. This was the same as forStudy 2 except we used the LAMK for Grade 2, which consisted of33 items.

4.3. Results and discussion

Some children did not complete all of the tasks due to school ab-sences or time clashes with other activities. The mean scores for thechildren who did complete the tasks are shown in Table 1. All of thesimple correlations among LAMK in Phase 4 and the tasks in Phase 1were positive and significant. Importantly, the brief version of stra-tegic counting showed the strongest correlationwith LAMK(r¼ 0.64).

The Cronbach’s alpha reliability coefficients of the brief versionof strategic counting were .83, .77, .77, and .81 in Phases 1 through 4respectively; internal consistencies were invariably high.

4.3.1. Predicting mathematical achievementA three-step hierarchical regression revealed that, with respect

to mathematical achievement at the end of Grade 2 (i.e., LAMKresult), strategic counting contributed an additional unique varianceof 2% over and above the control variables (Step 1) and various

Fig. 3. Average empirical growth trajectories on brief version of strategic counting byachievement group.

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numericalmeasures (Step2; Table 7) (Hypothesis 1).While the extravariance explained by strategic counting might appear small, it isnoteworthy that the task could still make significant contribution tolater mathematical achievement even after a cluster of general aswell as traditional numericalmeasures had been taken into account.Strategic counting, then, seems indeed useful in tapping uniquemathematical knowledge e probably the place-value knowledge e

apart from assessing some other conceptual and procedural skills(e.g., counting skills and executive control) that overlapped withother measures. As in Study 2, strategic counting was the strongestunique contributor to later mathematical achievement e confirm-ing its usefulness in predicting mathematical outcomes 18 monthslater. Hence, if teachers can only afford the time to carry out just onetask in class to predict children’s later mathematical performance,strategic counting appears to be the best option.

Other predictors also made significant contributions to latermathematical achievement. Chinese number reading (Study 2)and Arabic number reading (Study 2 and this study) remainedsignificant predictors even after other measures had been takeninto account. By contrast, Chinese word reading was a significantpredictor in the initial step, but no longer so when numericalmeasures were added to the model. Although these three taskswere all related to phonological processing skills, which areimportant in solving computation problems (Bull & Johnston,1997; Geary, 1993; Rourke & Conway, 1997), reading non-numerical symbols failed to contribute significantly to mathe-matical outcomes in the presence of other predictors. Thiscorroborated the prior finding that early word reading skills do notsignificantly predict calculation fluency in second grade (Locuniak& Jordan, 2008). Reading numerical symbols, be they Arabic orChinese, involves an understanding of digit relations and numberstructure, probably explaining its importance in predicting math-ematical outcomes.

Other significant predictors included backward digit span,grouping-in-ten, and non-symbolic addition. Consistent with pre-vious findings (Locuniak & Jordan, 2008), backward but not forwarddigit span predicted later mathematical achievement. Unlike for-ward digit span, which measured passive short-term recall fornumber sequences (Locuniak & Jordan, 2008), backward digit spantapped active sequential memory or working memory, whichshould be necessary for carrying out counting procedures in cal-culations (LeFevre, DeStefano, Coleman, & Shanahan, 2005). Anassociation between weak working memory and mathematicaldifficulty has also been documented among Chinese children (Chan& Ho, 2010). Grouping-in-ten and non-symbolic addition signifi-cantly predicted mathematical outcome in second grade (thisstudy) but not first grade (Study 2), perhaps because they wereassociated with multiplication and division concepts (e.g., threeschool buses, each containing 10 students, altogether had threetimes 10, i.e., 30 students; Mulligan, 2002), which were taught andtested in second grade.

4.3.2. Developmental trajectories of place-value understandingGrowth curve modeling was performed on the brief version of

strategic counting to examine intra-individual differences in rates ofgrowth. It was also used to examine whether childrenwho ended upwith differentmathematical achievement in second grade had variedgrowthpaths for earlyplace-valueunderstanding. For this,wedividedchildren into three achievement groups and compared their growthpaths. The achievement groups varied in their scores on LAMK at theend of second grade: high, at or above the 75th percentile; average,between the 25th and 75th percentiles; low, at or below the 25thpercentile. The 25th percentile is considered an appropriate cutoff foridentifying children with learning difficulties (Fletcher et al., 1994;Locuniak & Jordan, 2008; Swanson & Beebe-Frankenberger, 2004).

Note that the intra-class correlation (ICC) for the schools was0.01 and the design effect e a function of ICC indicating how largethe deviation is from the assumption of independence betweenobservations e was estimated to be 1.38. A design effect greaterthan 2 indicates that the clustering in the data (i.e., school) needs tobe taken into account in a multilevel analysis (Muthén & Satorra,1995). Since the design effect here was smaller, and it was notour focus to compare the developmental patterns between schoolsin this study, multilevel analysis was not performed. Note also thatthe size of a sample required to detect any effect in a growth modeldepends on the characteristics of the research design (Curran,Obeidat, & Losardo, 2010). Generally speaking, size approaching100 is often preferred (Curran et al., 2010). Growth models similarto ours (with covariates and missing data), however, have beenreliably fitted to samples as small as 22 (Huttenlocher, Haight, Bryk,Seltzer, & Lyons, 1991). The sample size here was much larger(N ¼ 193) and it successfully detected the characteristics of growthcurves for different achievement groups (explained below), sug-gesting that the sample size was not a problem in this model.

Three growth curve models were computed. First, a baselinemodel (Model 0) traced the general developmental path of place-value understanding, with parameters including the intercept,slope, and quadratic term. The intercept was interpreted as theaverage score for strategic counting at the end of fall semester infirst grade (i.e., Phase 1); the slope, as the average linear growthover the next 18 months; and the quadratic term, as the accelera-tion or deceleration in growth rates across the period. Thenachievement group effects were added (Model 1) to see if therewere any group differences. The average math group was used asthe reference group for comparison. Finally, the effects of time-invariant predictors were added (Model 2) to investigate theirimpact, if any, on the parameters. These included age at the end offall semester in Grade 1, gender, and nonverbal intelligence. Thepredictors were coded so that the intercept, slope, and accelerationvariables referred to males with average age and nonverbal intel-ligence in the average math group. The missing data were handledbased on full-information maximum likelihood in the Mplus pro-gram (Muthén & Muthén, 2007). The average empirical growthtrajectories by achievement group are shown in Fig. 3. The results ofthe three models are presented in Table 8. In Model 0, the averagescore at the end of fall semester in first grade was 1.83 and theaverage growth was 1.52 points over the two years. The rates ofgrowth slowed over time as evidenced by the negative accelerationparameter (�0.22).

Model 1 showed the effects of adding achievement groupmembership to the model. The children who ended their secondgrade with low (high) mathematical achievement had alreadyshown a low (high) level of place-value understanding early in

Table 8Parameter estimates for the place-value understanding growth curve models(N ¼ 193).

Estimate Model 0 Model 1 Model 2

Intercept 1.83*** 1.75*** 1.74***Slope 1.52*** 1.84*** 1.85***Var (intercept) 2.53*** 1.31*** 1.02***Var (slope) 0.24*** 0.18*** 0.16***Acceleration variable �0.22*** �0.28*** �0.28***Intercept on low-math group �1.34*** �0.95***Slope on low-math group �0.62* �0.68*Acceleration on low-math group 0.19* 0.19*Intercept on high-math group 1.53*** 1.21***Slope on high-math group �0.61* �0.56*Acceleration on high-math group 0.05 0.05Intercept on age at fall first grade 0.06***Slope on age at fall first grade �0.02Acceleration on age at fall first grade 0.00Intercept on female �0.61**Slope on female 0.45*Acceleration on female �0.09Intercept on nonverbal intelligence 0.03***Slope on nonverbal intelligence 0.00Acceleration on nonverbal intelligence 0.00

Note. Var ($) stands for the variance of the parameters in parentheses.*p < 0.05, **p < 0.01, ***p < 0.001.

Fig. 4. ROC curve for the screening of later low-math using strategic counting scoresmeasured at the end of fall and spring semesters in first grade.

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first grade. The high-math group showed smaller linear growthin place-value understanding over the two years than theaverage math group probably because their starting point washigher, limiting the room for improvement. However, the low-math group appeared to have difficulty acquiring the place-valueknowledge. They already lagged behind their peers early in firstgrade and made slower progress over the next 18 months (Hy-pothesis 2).

Model 2 showed the effects of adding predictors of age at theend of fall semester in Grade 1, gender, and nonverbal intelligence.Holding the predictors constant, comparisons among the achieve-ment groups yielded the same patterns as in Model 1. Comparedwith the average math group, the low-math group had lower andthe high-math group had higher scores at the end of fall semester infirst grade. Both the low- and high-math groups had significantlysmaller average growth than the average math group. While thelow-math group had a significantly lower rate of deceleration thanthe average math group, no significant differences were foundbetween the high and average math groups.

Both age and nonverbal intelligence were significant predictorsof the intercept. As expected, older children with higher nonverbalintelligence scored better on the brief version of strategic counting.However, the effects of age and nonverbal intelligence did notsignificantly affect the slope and acceleration. Females had signif-icantly lower average scores at the end of fall semester in first gradethan males. However, they had higher average growth than males.No significant differences were found between females and malesin the rate of deceleration.

As shown inModels 1 and2, the three achievement groups variedin their place-value understanding early in first grade. To see if suchdifferences persisted at the end of second grade, we compared theperformance of the three groups on the brief version of strategiccounting at the end of second grade using a one-way ANCOVAwithnonverbal intelligence as the covariate. The group difference wassignificant, F(2,175) ¼ 27.94, p < .0001, hp2 ¼ .24. Post hoc meancomparisons showed that while the average math group caught upwith the high-math group by the end of second grade (p > .05), thelow-math group still lagged behind the other two groups(ps< .0001). As shown in Fig. 3, by the end of second grade, the low-math group performed at a level comparable to the high-mathgroup’s 18 months before. That is, they lagged behind by 1.5 years.

4.3.3. The screening utility of the strategic counting taskDiscriminant analysis was conducted using the scores on

strategic counting at the end of the fall semester of first grade topredict membership in the low-math group at the end of secondgrade. High and average math groups were collapsed into onegroup as on-track-math children. Since one low-math and two on-track-math children did not complete the strategic counting taskin the first phase, the following analysis was based on the per-formance of 178 children, with 44 being low-math and 134 beingon-track math.

Results showed that strategic counting scores in early first gradecould differentiate low-math children from their on-track-mathpeers at the end of second grade, Wilks’ L ¼ 0.79, c2 (1,N ¼ 178) ¼ 40.65, p < .0001. With prior probabilities based ongroup sizes for classification, the discriminant function correctlyclassified 75% of the children.

To establish a strategic counting cutoff score to use in identifyingwhich children would be low-math at the end of second grade,receiver operating characteristic (ROC) curve analysis was per-formed. For each possible cutoff point, the proportion of low-mathchildren being correctly identified (i.e., sensitivity) and the pro-portion of on-track-math children being correctly identified (i.e.,specificity) were calculated. A ROC curve shows the tradeoff be-tween sensitivity and specificity: If the cutoff score is too high, itwill correctly identify most of the low-math children, but errone-ously identify too many on-track children as low-math. Theopposite will be the case if the cutoff score is too low. The optimalcutoff score finds the happiest possible medium e correctly iden-tifying as many low-math children as possible without mis-identifying too many on-track children.

Fig. 4 shows the ROC curve used for determining the cutoff scoreat the end of the fall semester in first grade. If the cutoff was set at 2,all low-math children were correctly identified (i.e.,sensitivity ¼ 100%). But less than half (38%) of the on-track-mathchildren were correctly identified (i.e., specificity ¼ 38%). Theoptimal cutoff scorewas determined to be 1.With this criterion, 95%of children who turned out to be low-math in second grade werecorrectly identified in early first grade (i.e., sensitivity ¼ 95%), and59% of children who ended up as on-track-math in second gradewere correctly predicted in early first grade (i.e., specificity ¼ 59%).The overall correct classification rate was 68%. With this reasonablyhigh sensitivity, the task identified most of the childrenwho endedup as low-math in second grade. At the same time though, it mis-identified as low-math a fairly large number of children (41%) who

W.W.L. Chan et al. / Learning and Instruction 29 (2014) 78e94 91

actually turned out to be on-track 18 months later. This is under-standable since most children at this stage are still trying to masterthe place-value knowledge, so making errors does not necessarilyimply a potential problem in mathematical achievement. Indeed,many children eventually figure out the principles of the base-tensystem and demonstrate on-track-mathematical achievement bythe second grade. These analyses were repeated using the scores atthe end of spring semester in first grade, to see if scores in late firstgrade were useful for follow-up screening.

The following analyses were based on the performance of 181children; 45 and 136, respectively, turned out to be low-mathand on-track-math at the end of second grade. Discriminantanalysis confirmed the usefulness of strategic counting scoresobtained at the end of spring semester in first grade inpredicting low-math in second grade, Wilks’ L ¼ 0.70, c2 (1,N ¼ 181) ¼ 63.12, p < .001. With prior probabilities based ongroup sizes for classification, the discriminant function correctlyclassified 80% of the children.

Fig. 4 shows the ROC curve used for determining the cutoff scoreat the spring semester in first grade. The optimal cutoff was iden-tified as the score of 2, yielding a reasonably good sensitivity of 87%and an acceptable specificity of 66% for the screening of later low-math status (Hypothesis 3). Hence, with this cutoff, 87% of childrenwho turned out to be low-math in second grade were correctlyidentified in late first grade, while only 34% of on-track secondgraders were mis-identified as low-math. The overall correct clas-sification rate was 71%. Thus, strategic counting scores in late firstgrade turned out to be useful for follow-up screening.

4.4. Conclusions of Study 3

As with the full version in Study 2, the brief version of stra-tegic counting showed the largest correlation with later mathe-matical outcomes, compared with other numerical measuresincluding two traditional place-value tests, grouping-in-ten andplace-value representation. Moreover, it was the strongest pre-dictor of mathematical achievement 18 months later (Hypothesis1). These findings confirmed that strategic counting is a moreinformative assessment tool than the commonly used place-value tests.

Children with different levels of mathematical outcomes at theend of second grade had followed different paths in developingplace-value understanding. Children who were the highest per-formers in mathematics in second grade had shown the best un-derstanding of the place-value knowledge in early first grade.Children with average performance in mathematics in secondgrade had started off with a lower level of place-value under-standing, but had caught upwith the highest performers by the endof second grade. Children with the lowest mathematical perfor-mance in second grade, however, had been struggling with theplace-value knowledge since early first grade and, unlike theaverage performers, had failed to catch up with their peers by theend of second grade (Hypothesis 2).

Strategic counting scores inearlyand latefirst gradewereuseful inpredicting low-mathematics achievement at the endof secondgrade.In particular, scores at the end of fall semester in first grade could beused to identify potential low-mathematics achievers (Hypothesis 3).Teacherswould dowell tomonitor these children’s learning progressregularly and provide appropriate learning support when needed. Bythe end of first grade, children could be given the strategic countingtask again for follow-up screening. A clearer prediction would bepossible by then because the learning gap between childrenwith andwithout difficulty inplace-value understandingwould probably havewidened. The identified children would be potential targets forremedial programs in place-value understanding.

5. General discussion

5.1. Children’s conception of number

Study 1 was the first systematic investigation to evaluate theUDSSI model with children speaking a language with a ratherregular number-naming system. In English e the linguistic contextfor developing the UDSSI model e number names do not show adirect correspondence with the place-value numeration system,but in Chinese they do. While theoretically the more regularnumber-naming system in Chinese could facilitate place-valueunderstanding (e.g., Fuson & Kwon, 1991; Miller & Stigler, 1987;Miura et al., 1988, 1993, 1994), Chinese-speaking children in thepresent studies developed through unitary multi-digit, sequence-tens and ones, and separate-tens and ones conceptions just likeEnglish-speaking children. These conceptions as well as theirdevelopmental sequence are probably universal rather thanlanguage-dependent. However, since the present studies did nottest English-speaking children, future investigations mightcompare different number-naming systems systematically.

The decade and ones conception seemsmost likely to develop inchildrenwho speak irregular languages (like English) with separatedecade words (Fuson, Wearne, et al., 1997). As expected, it was notevident in Chinese-speaking children in Study 1. Children with anintegrated sequence-separate conception are marked by theirability to integrate and shift between sequence-tens and ones andseparate-tens and ones conceptions flexibly and rapidly. The stra-tegic counting task did not identify children with this conception,so future studies should explore other tasks that might.

Study 1 went beyond the existing UDSSI model e formulatedwith two-digit numbers e to examine children’s conceptualstructures of three-digit numbers. Although the kindergartenersand first graders in this study had not received formal instructionon three-digit numbers, they applied their unitary, sequence, andseparate conceptions to them as they did of two-digit numbers.Children do not seem towait for formal instruction to construct anduse meaningful conceptions of multi-digit numbers.

These findings have implications for educational practices. First,children should be helped at an early stage to clarify the conceptualstructure of two-digit numbers since this lays the foundation forworkingwith largermulti-digit numbers. Second, informal learningcan play a crucial role in children’ facility with multi-digit numbers.Playing board- and card-games, for instance, may be a fun way forchildren to become familiar with the place-value knowledge.

5.2. Strategic counting as a novel place-value assessment tool

The strategic counting task offers a new and better way ofassessing children’s place-value understanding. On traditionalplace-value tasks, children may appear to know more than they doby relying on rote learning. For strategic counting, by contrast,children have to integrate and apply their knowledge of countsequence, grouping-in-tens, and place-value representation of non-symbolic quantities e essential elements of their place-valueknowledge. In other words, the strategic counting task appears tobe a comprehensive test of children’s underlying place-valueknowledge. Further evaluation of its validity against the place-value items on some widely used mathematical tests (e.g., theTest of Early Mathematics Ability-3) is needed. Furthermore, eval-uating strategic counting against a conceptual place-value taskwould help establish the concurrent validity of the strategiccounting as indeed ameasure of conceptual place-value knowledge.

A brief version of the strategic counting task was developed inStudy 2. With the cutoff scores identified in Study 3, this task canidentify first graders who would turn out to perform in the bottom

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quartile in mathematics at the end of second grade. In classroompractice, teachers can use the task at the end of first semester infirst grade as an initial screening for potential low-mathematicsachievers in need of more help in understanding the place-valueconcept. Teachers can use the task again at the end of first gradeas a follow-up screening to help them more confidently identify atarget group for a remedial program in place-value understanding.

5.3. Predicting mathematical achievement

Among the potential predictors examined, strategic countingwas the single strongest predictor of later mathematical out-comes. Compared with the two commonly used place-valuetasks, namely grouping-in-ten and place-value representation,strategic counting had the strongest correlation with, and wasthe strongest predictor of, mathematical achievement 6 monthsand 18 months later. These promising findings warrant furtherevaluation of the predictive value of strategic counting againstthe traditional place-value tasks (e.g., with comparisons con-trolling for the number of two- and three-digit items and thenumerals used for each item).

One may wonder whether the exceptionally strong correlationbetween strategic counting and mathematical performance wassimply due to the inherently complex nature of this novel task e

involving cognitive processes such as working memory, executivecontrol, and other basic numerical knowledge (e.g., counting prin-ciples). But note that all numerical tasks in the present studies relyon some common cognitive processes and basic numerical knowl-edge, in addition to the specific knowledge domain targeted forassessment. The grouping-in-ten task, for example, also depends onworkingmemory, executive control, and knowledge of the countingprinciples to keep track of the counting and grouping processes.What made strategic counting unique, then, was perhaps its rathercomprehensive scope in assessing place-value understanding.

Another potential predictor of later mathematical outcomes ischildren’s school performance in first-grade mathematics.Although its predictive value was not directly examined here, someof the numerical tasks included in Studies 2 and 3 (e.g., symbolicand non-symbolic addition, transcoding Arabic numbers to Chineseand vice versa) together can serve as a proxy of a typical schoolmathematics test for first graders. Importantly, strategic countingwas found to be a better predictor than such numerical tasks formathematical achievement in second grade. Moreover, in-housemathematics tests in individual schools are not standardizedtests. Unlike the strategic counting task, these tests do not have areliable and valid cutoff for differentiating children’s later mathe-matical achievement. Therefore strategic counting should be abetter tool for predicting later low-mathematics achievers.

Other predictors of mathematical abilities have also been iden-tified in the literature. For example, a subset of items from the Test ofEarlyMathematics Ability-2 (TEMA-2)emeasuring skills in readingnumerals, number constancy, magnitude comparison of one-digitnumbers, mental addition of one-digit numbers e predicts whichkindergarteners are at risk for mathematics learning disability insecond and third grade (bottom 10% on TEMA-2 and Woodcock-Johnson-Revised (WJ-R) Math Calculations subtest), with sensi-tivity and specificity being close to 80% (Mazzocco & Thompson,2005). Children’s number sense in kindergarten (number knowl-edge and number combinations) predicts calculation fluency diffi-culties in second grade (bottom 25% on 1-min addition andsubtraction), with sensitivity and specificity being 52% and 84%respectively (Locuniak & Jordan, 2008). Compared with strategiccounting, these measures are useful for making earlier prediction e

at kindergartene of later mathematical outcomes. Yet, the fact thatthese predictors are based on different outcome variables and

classification criteria makes it difficult to compare their predictivepower simply through the reported sensitivity and specificityvalues.Moreover, whether these predictors are universally sensitiveacross children learning numbers in different languages is not clear.Similar concerns apply also tofirst-grade tasks that have been foundto predict mathematical learning difficulties (e.g., the Number SetsTaske tapping the ability to subitize andmap Arabic numerals ontosmall sets of quantities; see Geary, Bailey, & Hoard, 2009). Furtherstudies can compare strategic countingwithother knownpredictorssystematically and rigorously to evaluate its validity in screening forlater mathematical difficulties.

The predictive value of strategic counting was better whenadministered at the end of the seconde rather than firste semesterin first grade. This task probably works the best within a certainwindow, and further study is needed to determine what that win-dow is. Since strategic counting is supposed to assess place-valueknowledge e which is critical for later mathematics learning e thetask is potentially useful for predicting mathematical outcome wellbeyond the second grade. This too needs to be followed up.

5.4. Developmental trajectories of place-value understanding

Children with low achievement in second grade had consis-tently lagged behind their peers in place-value understanding overthe preceding 18 months. By the end of second grade, they still hadnot yet adequately mastered the place-value knowledge of two-and three-digit numbers, falling behind the curriculum. In fact, theyhad only reached the level their high-achieving peers had alreadyreached 18 months before. An important message here is thatchildren’s diversity in mathematical achievement at the end ofsecond grade was associated with a diversity in place-value un-derstanding earlier on. Therefore, early identification of, andintervention for, children having difficulty with the place-valueknowledge may be one key to narrowing the learning gap andsupporting their long-term mathematical development.

6. Conclusions

The place-value knowledge is an important building block ofchildren’s mathematical development. A fast and effective tool, thestrategic counting task developed in these three studies, can helpteachers assess children’s place-value understanding. Strategiccounting has emerged as a stronger predictor of later mathematicalachievement than some commonly considered potential pre-dictors. Children’s developmental trajectories revealed that earlydifficulty in understanding the place-value system could persist toat least the end of second grade, impeding mathematical devel-opment. The present findings highlight the importance of earlyidentification of, and remedial support for, children with problemsin the place-value understanding.

Acknowledgments

We thank the staff at the Educational Psychology Services (NewTerritories) Section of Hong Kong Education Bureau for their lo-gistic support in this research. We are also grateful to Karen Ravnfor copyediting the draft.

Appendix A

A typical way to test children’s place-value concept in schoolassessment and educational research is to ask them to identify theplaces and values of the digits in a number. The position knowledgetask (Cawley, Parmar, Lucas-Fusco, Kilian, & Foley, 2007; Hanichet al., 2001; Naito & Miura, 2001; Ross, 1986), for instance, shows

W.W.L. Chan et al. / Learning and Instruction 29 (2014) 78e94 93

children a number and asks them to identify which digits are inwhich places. Children may be shown “26” and asked, “Whichnumeral is in the tens place?” A variation of the task requireschildren to show the values of the digits with base-ten-blocks(Miura & Okamoto, 1989) (e.g., children may be asked to use thebase-ten-blocks to represent the place value of digit 2 in thenumber “26”). The problem with this type of task is that childrenwithout a genuine understanding of the base-ten structure can stillget it right by rote learning the place-value rules.

Another commonly used test is the digit correspondence task(Hanich et al., 2001; Miura et al., 1993; Naito & Miura, 2001; Ross,1986). Children are shown a two-digit number represented by acertain construction of groups of objects and asked how the digitsof the number correspond to the groups. The construction may beeither canonical base-ten or non-canonical. Consider a canonicalbase-ten construction: the number 26 is represented by two groupsof 10 and a group of 6. Note that the prominent base-ten groupingsstrongly prime a correct mapping between the digits and thegroups (i.e., the digit 2 corresponds to the two groups and the digit6 corresponds to the six objects; Ross, 1989).

Consider instead a non-canonical representation: the number26 is represented by six groups of 4 and a group of 2. Such a rep-resentation can mislead children to focus on the face values of thedigits (i.e., the digit 2 corresponds to the two objects in a group andthe digit 6 corresponds to the six groups of objects).

Appendix B. Strategic counting task

1. How many small squares are there? _______________________

2. How many small squares are there? _______________________

3. How many small squares are there? _______________________

4. How many small squares are there? _______________________

5. How many small squares are there? _______________________

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