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Sensitivity to Growth Over Time of the Preschool Numeracy Indicators With a Sample of Preschoolers in

Head Start

Robin L. HojnoskiLehigh University

Benjamin SilberglittTechnology and Information Educational Services, St. Paul, Minnesota

Randy G. FloydThe University of Memphis

Abstract. There has been increased attention to the development of measures forassessing mathematical skill and knowledge in young children. Most of theevidence supporting these measures is consistent with Stage 1 research in thedevelopment of progress monitoring measures (Fuchs, 2004) and consists of investigation of technical features of performance at one point in time. Thepurpose of the current study was to move into Stage 2 research and examinesensitivity to growth over time of the Preschool Numeracy Indicators (PNIs;Floyd, Hojnoski, & Key, 2006) in a sample of Head Start preschoolers through alongitudinal design. Results indicated the PNI Oral Counting Fluency, One-to-One Correspondence Counting Fluency, Number Naming Fluency, and QuantityComparison Fluency task scores are sensitive to growth over time and providepreliminary support for the promise of such measures in assessing early mathe-matical skill development. Consideration is given to implications for assessingearly mathematical skill development in the context of general outcome measure-ment.

There has been increased attention to thedevelopment of measures for assessing math-

ematical skill in young children (e.g., Chard etal., 2005; Clarke, Baker, Smolkowski, &

A faculty research grant from The University of Memphis to the rst author provided nancial support forstudy. The opinions expressed in this article do not necessarily reect those of the University of Memphis.We are grateful to Jessica Hall, who coordinated data collection and data entry efforts and participated inthe assessment; to Ashley Smith, Missy Flinn, James Ford, Lauren McDurmon, and the other students whoassisted with data collection; to the Head Start administrators and teachers who welcomed us into theircenter; and to the children who participated in data collection.

Correspondence regarding this article should be addressed to Robin Hojnoski, Department of Educationand Human Services, Lehigh University, 111 Research Drive, Bethlehem, PA 18015-4794; E-mail:roh206@lehigh edu

School Psychology Review,2009, Volume 38, No. 3, pp. 402418


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Chard, 2008; Floyd, Hojnoski, & Key, 2006;Methe, Hintze, & Floyd, 2008; Reid, Morgan,DiPerna, & Lei, 2006; VanDerHeyden, Brous-sard, & Cooley, 2006). Evidence is building towarrant such attention (e.g., National Mathe-matics Advisory Panel, 2008; National Centerfor Education Statistics, 2007) as mathemati-cal skill appears critical to later school suc-cess. A recent meta-analysis of six longitudi-nal data sets suggests that mathematical skillat kindergarten entry is a strong predictor of later academic achievement, stronger thanreading skills, attentional skills, or social be-havior at kindergarten entry (Duncan et al.,

2007). This suggests early mathematical com-petency is a primary target in improving aca-demic achievement, and thus an importantarea for early identication and intervention.

Effective assessment practices play acritical role in promoting early identicationand intervention in the development of math-ematical competency. Assessment tools spe-cically designed for young children and tar-geting elements thought to provide a founda-tion for later success can provide data aboutacquisition of key skills as well as growth overtime to inform effective instruction and inter-vention. In response to the limitations of moretraditional early childhood assessment prac-tices, curriculum-based assessment ap-proaches are increasingly being applied to theassessment of growth and development in veryyoung children with signicant potential forimproving outcomes (Bagnato, 2005; McCo-nnell, 2000; VanDerHeyden, 2005; VanDer-

Heyden & Snyder, 2006).

General Outcome Measurement

Curriculum-based assessment can beconceptualized as the umbrella term for anumber of assessment approaches character-ized by key features of authenticity, instruc-tional and intervention utility, reliability andvalidity, sensitivity to growth over time, anddecision-making utility (Hintze, 2008). Withinthe larger domain of curriculum-based assess-ment, assessment approaches can be grouped

(GOM). Although both specic subskill mas-tery measurement and GOM approaches re-ect the key features of curriculum-based as-sessment, there is a distinct difference in thedevelopment and content of each type of as-sessment. Whereas subskill mastery assess-ments typically and comprehensively sample adomain or hierarchy of skills that reect aninstructional sequence, in GOM a limitednumber of key skills are selected for measure-ment from the universe of possible skills andused as an indicator of global performance(Fuchs & Deno, 1991; Greenwood, Walker,Carter, & Higgins, 2006). In curriculum-based

measurement (CBM; Deno, 1985, 1986;Shinn, 1989), perhaps the most widely knownexample of GOM, the focus is on broad, long-term objectives as opposed to the short-termobjectives characteristic of a subskill masteryapproach (Hintze, 2008). Moreover, the em-phasis in CBM is on uency with the targetskill or behavior as measured across the as-sessment period, as opposed to mastery of specic skills in a hierarchy. CBM tasks, ortasks developed in the GOM framework, aredesigned as indicators of growth toward along-term objective; thus, they are sensitive togrowth over time and to the effects of instruc-tion and intervention (Fuchs & Deno, 1991;Hintze, 2008; McConnell, Priest, Davis, &McEvoy, 2002). These features make GOMwell suited for progress monitoring.

Development and Evaluation of ProgressMonitoring Tools

To demonstrate the utility and viabilityof GOM for progress monitoring, three re-search stages must be completed (Fuchs,2004). The rst stage consists of investigatingthe technical features of performance at onepoint in time. This stage may include examin-ing interscorer, alternate-form, internal consis-tency, and testretest reliability as well as con-current and predictive validity. Establishingthe technical features of scores from one pointin time is critical in ensuring that meaning canbe attributed to the scores. Stage 2 research

Preschool Numeracy Indicators


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the measures to growth over time and therelation between growth on the measures andimprovement in the target domain overall. Anessential feature of GOM is the degree towhich it is sensitive to performance changesthat result from skill acquisition. Stage 3 re-search examines the instructional utility of themeasures to determine whether the measurescan be used to inform instruction and inter-vention. Both Stage 2 and Stage 3 research areparticularly critical in considering the use of progress monitoring measures in tiered modelsof assessment and intervention (e.g., responseto intervention).

Measuring Early Mathematical Development: Stage 1

Several research teams have begun toexamine the use of GOM within a framework for formative assessment in early mathemati-cal development. Creating a GOM for mathe-matics is challenging given the multiplestrands that comprise mathematics (e.g., num-bers and operations, algebra, geometry, mea-

surement, data analysis). In early mathemat-ics, the range of assessment targets is compli-cated further by the learning trajectoriesthrough which children progress and the vari-ability in young childrens development of mathematics (Clements, 2004). For example,in counting, roughly between the ages of 2and 5, young children progress from nonver-bally representing a collection of objects tocounting aloud in sequence from a number

other than 1 (Clements, 2004). Further, be-cause young children have not developed thecomputational skills that are typically the fo-cus of progress monitoring measures in ele-mentary school, measures developed foryoung children have focused primarily oncomponents of number sense .

There are several denitions of number sense , including uidity and exibility withnumbers, the sense of what numbers mean(Gersten & Chard, 1999, p. 19). The NationalCouncil for Teachers of Mathematics stan-dards for prekindergarten to Grade 2 suggest

tionships, patterns, operations, and placevalue (National Council for Teachers of Mathematics, 2000, p. 78). Independent of theprecise denition, there is general agreementthat number sense includes skills such ascounting, making numerical comparisons, ver-bal and nonverbal calculations, estimations,and facility with number patterns (Berch,2005); researchers investigating the use of GOM in early mathematical developmenthave typically included tasks that reect asubset of these skills.

Recently, 12 studies have been pub-lished detailing evidence for GOM tasks for

early mathematics with samples of kindergar-ten and preschool students. Seven studiespresent a variety of technical adequacy evi-dence consistent with Stage 1 research thatsupport the use and interpretation of measuresof early numeracy, whereas 4 studies alsopresent evidence of Stage 2 research. Onestudy presents preliminary evidence of sensi-tivity to intervention with preschoolers(VanDerHeyden et al., 2006). Two studies byClarke and colleagues (Chard et al., 2005;Clarke & Shinn, 2004) examined the inter-rater, alternate-form, and testretest reliabilityas well as predictive validity of four earlynumeracy CBM (EN-CBM) measures for kin-dergarten and rst-grade students. In the rststudy of the EN-CBM tasks (Clarke & Shinn,2004), although reliability coefcients varied,all were strong for each type of reliabilityinvestigated. In addition, median concurrentvalidity correlations ranged from .60 to .75,

and predictive validity coefcients were mod-erate to strong. Results from Chard et al.(2005) replicated the ndings of Clarke andShinn (2004) with regard to predictive andconcurrent validity and subsequent studies(Lembke & Foegen, 2009; Lemke, Foegen,Whittaker, & Hampton, 2008; Martinez et al.,2009) have supported the strong technicalproperties of the EN-CBM.

In another study with kindergarten stu-dents, Methe et al. (2008) investigated thetestretest reliability, internal consistency,concurrent validity, and decision accuracy of

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dinal Position Fluency and Number Recogni-tion Fluency, demonstrated reliability, valid-ity, and cutoff score data appropriate foraccurate classication decisions. Counting-onFluency and Match Quantity Fluency, al-though relatively accurate in grouping deci-sions, had lower reliability and validitycoefcients.

In a series of studies with preschool andkindergarten students, VanDerHeyden andcolleagues examined alternate-form reliabil-ity, criterion-related validity, convergent anddiscriminant validity, decision-making accu-racy, and predictive validity of a variety of

early numeracy measures (VanDerHeyden etal., 2006; VanDerHeyden, Witt, Naquin, &Noell, 2001; VanDerHeyden et al., 2004). Ingeneral, results from the studies indicatedstrengths in alternate-form reliability and cor-relations with teacher ratings. Mixed resultswere found for other evidence of validity. Forexample, for the preschool tasks, correlationswith the criterion measures ranged from low tomoderate, and for the kindergarten tasks, cor-relations with criterion measures ranged fromlow to moderate and were not domain specic.Discriminant functional analyses indicatedkindergarten measures were predictive of re-tention, problem validation procedures, andreferral to a team.

Reid et al. (2006) developed a set of tasks targeting early literacy and early nu-meracy for use with a Head Start sample, andthe internal consistency, item difculty, anditem discrimination of the probe scores were

investigated. According to the authors, for themathematics tasks, adequate item and scaleproperties were demonstrated and item-dis-crimination indices were positive and high.Adequate alternate-form reliability was alsodemonstrated and concurrent validity with thecriterion measure was moderate to strong.

Finally, Floyd et al. (2006) developedfour tasks as indicators of mathematical skillfor preschoolers. Technical features of themeasures were examined, including testretestreliability, and four types of validity evi-dencecontent, response processes, internal

preschool settings. Evidence of the technicalproperties discussed under Measures indicatedsufcient support to continue to explore addi-tional features of the measures consistent withStage 2, such as growth over time.

At least two patterns are evident acrossthis growing body of research. First, in termsof content, although the domain of mathemat-ics consists of various strands as reected inthe National Council for Teachers of Mathe-matics standards (e.g., numbers and opera-tions, geometry, algebra, measurement, anddata analysis and probability), tasks primarilyfocus on numbers and operations, with few

exceptions. This results in a great deal of sim-ilarity among tasks across research teams. Forexample, each of the research teams usedsome variation of a number identication task and a counting task. Second, in terms of sup-porting evidence for the measures, consistentwith the larger body of research on mathemat-ics CBM tasks (Foegen, Jiban, & Deno, 2007),much of the research on early mathematicsGOM tasks has focused on Stage 1 research,with some variability in the technical featuresinvestigated. In general, this collective body of research suggests that it is possible to developtechnically adequate measures of early mathdevelopment. A logical and necessary nextstep is to examine features of developed mea-sures that are consistent with Stage 2 research.Initial efforts have been made in this direction.

Measuring Early Mathematical Development: Stage 2

Four studies examine the EN-CBMtasks (Clarke & Shinn, 2004), or a modiedversion of these tasks, and provide evidencefor GOM tasks in early mathematics consis-tent with Stage 2 research. Clarke et al. (2008)examined the technical features of slope, in-cluding the predictive validity of slope overtime and its unique contribution to predictingperformance when examined with other mea-sures. EN-CBM (Clarke & Shinn, 2004) andcriterion measures were administered at thebeginning and end of the year to a sample of



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additional points during the year to a sub-sample of kindergarteners to compare the pre-dictive power of slope to the static score fromthe criterion measures. Results indicated thatonly one measure, Quantity Discrimination, ta linear growth curve, and similarly, growthon only Quantity Discrimination accounted forvariance above and beyond static criterionmeasures. In addition, Lembke and colleaguesinvestigated three of the Clarke and Shinn(2004) tasks as well as modied versions of the tasks. Lembke et al. (2008) reported thatQuantity Discrimination, Missing Number,and Number Identication demonstrated sen-

sitivity to growth when administered to 77kindergarten and 30 rst-grade students at ap-proximately 4-week intervals across sevenrounds of data collection. Both linear andpolynomialmodels revealed thatchildrensper-formance improved across time on the threemeasures. Results indicated signicant lineargrowth for only Number Identication.Growth in Quantity Discrimination and Miss-ing Number was signicant but nonlinear.Lembke and Foegen (2009) examined modi-ed versions of the tasks in a sample of 72kindergarten and rst-grade students. Differ-ences between fall and spring scores werecalculated to determine mean growth acrossthe time periods, and were tested using pairedsample t tests. Although both kindergarten andrst-grade students improved over the courseof time, mean differences were statisticallysignicant for kindergarten students only. In astudy with only kindergarten students using

three of the EN-CBM tasks (Clarke & Shinn,2004), students improved on all three tasksfrom fall to spring. Further, repeated-measuresanalysis of variance indicated a signicant andlarge effect for time (Martinez et al., 2009).

The results of these Stage 2 investiga-tions indicate some of the early numeracytasks show promise, although results are notconsistent across all tasks for all samples, andthe degree to which GOM tasks can be usedfor progress monitoring remains unclear. It isimportant to note that all of the Stage 2 inves-tigations were conducted with kindergarten

measures. Although these studies provide es-timates of growth over time to support the useof the EN-CBM, research is needed to under-stand whether sensitivity to growth is a char-acteristic of the early mathematics GOM tasksthat have been developed for preschoolers.Investigating sensitivity to growth over time inearly mathematics GOM tasks for preschool-ers is particularly important given that differ-ences in mathematical competencies appear bythe age of 3 (Case, Grifn, & Kelly, 1999) andthat mathematical competency at kindergartenentry is a strong predictor of later achievement(Duncan et al., 2007).

Purpose of the Study

The purpose of the present study was toadvance Stage 2 research on early mathemat-ics GOM tasks specically with preschoolersby examining the degree to which the taskscreated by Floyd et al. (2006) demonstratesensitivity to growth over time via a longitu-dinal study that purposefully included a rangeof ages. Measuring the same students over

time provides more denitive evidence regard-ing sensitivity to growth and represents thenext logical step in Stage 2 research on thedevelopment of a GOM for early numeracy. Inaddition, the present study focused on the abil-ity of the tasks to detect changes in perfor-mance over time in a sample of preschoolersfrom low-income ethnic minority families at-tending a Head Start program. Efforts to createmeasurement tools that can be used success-

fully with populations in which performancedisparities are evident (National Center forEducation Statistics, 2007; National Mathe-matics Advisory Panel, 2008) are critical toimproving mathematical outcomes for youngchildren. The general goal of the study was toexamine how the tasks developed by Floyd etal. (2006) functioned as early mathematicsGOM tasks. These measures were selected asthe focus for two reasons. First, the PNIs dem-onstrated sufcient technical properties to sup-port continued examination consistent withStage 2 research. Second, several of the tasks



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PNI tasks are similar in content to EN-CBM ina form modied to be appropriate for pre-schoolers. Demonstrating growth over time of the PNIs is an essential rst step in establish-ing a continuous measurement system thatlinks with EN-CBM. The specic researchquestions of the current study were as follows:(a) What are the basic descriptive features of each task (e.g., mean, standard deviation,skewness, kurtosis, and percent of zeroscores)? (b) What is the degree to which eachtask reects growth over a measurement pe-riod of 8 months, or approximately one schoolyear? (c) What is the degree to which each

task is sensitive to potential interactions withchild age and classroom placement?

Method

Participants and Settings

Participants were 139 children enrolledin an urban Head Start program located in themid-South. From this sample, 66 participantswere boys and 73 participants were girls. Atthe beginning of the study, the children rangedin age from 37 months to 60 months( M 51.0, SD 6.1 month). All of thechildren were African American, and Englishwas their primary language. None of the par-ticipants had been diagnosed formally with adisability condition.

The Head Start program was located in aprivate agency that also operated a day carecenter and a state-funded preschool program,although only children attending the Head

Start program participated in the study. TheHead Start program consisted of seven class-rooms, each serving approximately 20 chil-dren and each staffed with one teacher and oneteacher assistant. Children were assigned toone of seven classrooms by the family servicesmanager; no information was provided aboutthe specic procedures used to assign chil-dren. Because assignment to classrooms wasnot specied, classrooms included unequalnumbers of boys and girls as well as unequalnumbers of 3-, 4-, and 5-year-olds.

According to the education manager, the

Dodge, Colker, & Heroman, 2002), which hascurriculum objectives aligned with the Na-tional Council for Teachers of Mathematicsstandards in number concepts, patterns andrelationships, geometry and spatial sense,measurement and data collection, organiza-tion, and representation. The objectives arebroad and many are the same across the con-tent of the standards. For example, one objec-tive indicates the child will observe objectsand events with curiosity whereas another in-dicates the child will approach problems ex-ibly. In addition, all of the classrooms fol-lowed Head Start performance standards in

designing classroom activities and instruction.Standards were reviewed regularly at staff meetings to ensure teachers were implement-ing instruction consistent with the standardsacross specic domains. Teachers followedsimilar general classroom schedules and infor-mal observations suggested teachers usedmany of the same types of learning activities(i.e., circle time, learning centers); however,there was some variability in implementation.For example, some teachers made differentlearning centers available at different times,and different materials were evident in each of the classrooms. In terms of assessment, class-room teachers assessed child progress at threepoints in time during the school year using theLearning Accomplishment ProleThirdEdition (Sanford, Zelman, Hardin, & Peisner-Feinberg, 2003).

Measures

Four PNIs (Floyd et al., 2006) were ad-ministered to all children to assess early math-ematical development. 1 All PNIs but OralCounting Fluency include demonstration andsample items designed specically to mini-mize scores of 0 that are not reective of trueskill levels.

One-to-One Correspondence CountingFluency targets the ability to count objectsuently and requires children to point to andcount circles approximately 1-inch in diameterprinted on a page. Circles are presented in four



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score results from multiplying 30 times thenumber associated with the last circle countedonly once in sequence and dividing the prod-uct by the childs time of completion. Time of completion for all children who do not cor-rectly count all circles is 30 s. Prior researchrevealed testretest reliability coefcients of .62 across a 2- to 4-week interval and .96 acrossa 5- to 7-week interval. Corrected correlationsbetween One-to-One Correspondence CountingFluency and scores from the Bracken Basic Con-cept ScaleRevised (Bracken, 1998) and theWoodcock-Johnson III Applied Problems test(Woodcock, McGrew, & Mather, 2001) ranged

from .29 to .38. Corrected correlation with theTest of Early Mathematics AbilityThird Edi-tion (Ginsburg & Baroody, 2003) total scoreswas .64 (Floyd et al., 2006). A 1-month testretest reliability coefcient from the current sam-ple (from the rst assessment to the second) was.77.

Oral Counting Fluency targets the abil-ity to produce numbers uently in sequence,beginning with the number 1. Children areasked to state numbers in sequence until theyreach the highest number they can producein 1 min. The Fluency score represents thenumber of numbers stated correctly in se-quence from 1 in 1 min. Prior research re-vealed testretest reliability coefcients of .90across a 2- to 4-week interval and .82 across a5- to 7-week interval. Corrected correlationsbetween Oral Counting Fluency and scoresfrom the Bracken Basic Concept ScaleRe-vised and the Woodcock-Johnson III Applied

Problems test ranged from .31 to 45. Its cor-rected correlation with Test of Early Mathe-matics AbilityThird Edition total scores was.55 (Floyd et al., 2006). The 1-month testretest reliability coefcient from the currentsample was .71.

Number Naming Fluency targets theability to name numerals uently. It requireschildren to say the names of the numerals020 presented one at a time. Examinerspresent these pages in rapid succession for 1min, and children have 3 s to respond to eachpage. The 21 numerals included in the task are

tions. The Fluency score represents the numberof numerals named in 1 min. Prior researchrevealed testretest reliability coefcients of .91across a 2- to 4-week interval and .88 across a 5-to 7-week interval. Corrected correlations be-tween Number Naming Fluency and scores fromthe Bracken Basic Concept ScaleRevised andthe Woodcock-Johnson III Applied Problemstest ranged from .29 to .40. Its corrected corre-lation with the Test of Early Mathematics Abil-ityThird Edition total scores was .70 (Floyd etal., 2006). The alternate-form reliability coef-cient at a 1-month interval from the current sam-ple was .72.

Quantity Comparison Fluency targetsthe ability to make judgments uently aboutdifferences in the quantity of object groups. Itrequires children to identify which of twoboxes of circles printed on a page containsmore circles. Children respond by touchingthe box with more circles. Each side contains16 circles, and each quantity of circles isrepresented in a standard fashion across pages.Examiners present, in rapid succession, upto 30 pages with sets of circles on each sidefor 1 min. Children have 3 s to respond to eachpage. The Fluency score results from multi-plying 60 times the number of correct re-sponses and dividing the product by thechilds time of completion (1 min or less).Prior research revealed testretest reliabilitycoefcients of .89 across a 2- to 4-week inter-val and .94 across a 5- to 7-week interval.Corrected correlations between QuantityComparison Fluency and scores from the

Bracken Basic Concept ScaleRevised andthe Woodcock-Johnson III Applied Problemstest ranged from .38 to .58. Its corrected cor-relation with the Test of Early MathematicsAbilityThird Edition total scores was .58(Floyd et al., 2006). The alternate-form reli-ability coefcient at a 1-month interval fromthe current sample was .65.

Procedure

Training and reliability. Three under-graduate students in psychology and seven



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measures collected the data. Initial trainingconsisted of two 1-hr group sessions, one in-dividual session, and a practice administration.Monthly group meetings were held to discussany difculties, questions, or concerns thatarose during administration and to minimizeassessor drift. In addition, the rst author per-formed at least one assessment integrity check with each examiner during the data collectionprocess using a procedural checklist devel-oped by the rst author. All administrationsobserved were completed with 100% integrity.

Data collection. The PNIs were ad-ministered as part of a larger assessment thatincluded the Individual Growth and Develop-ment Indicators (McConnell et al., 2002) and aletter-naming task. Only results pertaining tothe PNIs are reported here. All tasks wereindividually administered by the trained exam-iners and completed in the childs classroom ata child-sized table apart from other classroomactivities. Assessment batteries were counter-balanced across children to minimize ordereffects in performance. The entire administra-

tion was completed in approximately 10 min.Children received a sticker when the assess-ments were completed.

Data collection was part of a larger ef-fort initiated by the Head Start program tomonitor the childrens development in the keypreacademic areas of early literacy and earlynumeracy. All children who attended the HeadStart center participated in the data collection,and caregivers were notied of the process by

the Head Start program. No parents objectedto their childrens participation and thus nochildren were excluded from the study unlessthey left the program while data were beingcollected.

Data were collected monthly on all chil-dren who were in attendance on data collec-tion sessions from October to May. The as-sessments were scheduled at equal intervalsacross the entire data collection period. Amonthly schedule of assessments was selectedto determine a precise rate of growth for shorttime intervals. In a typical benchmark assess-

However, although benchmark assessmentstrategies are quite common in the kindergar-ten through 12th-grade setting, there is nocorresponding standard for GOM assessmentin a preschool setting. Monthly assessmentsprovided an appropriate balance between thefrequency of assessment that might typicallyhappen in a preschool setting and collecting asufcient number of data points to limit thestandard error of the slope estimate ( SE b).

Data Analysis Plan

One goal of this study was to examinehow the PNIs functioned as a progress moni-toring tool according to basic descriptive fea-tures of the tasks as well as growth over time.First, descriptive statistics were examined forthe potential early mathematics GOM tasksused in the study. Previous research on chil-dren in the preschool-age range has found asignicant number of zero scores for the Indi-vidual Growth and Development Indicators,indicating a oor effect for these measures(Missall et al., 2007). It was important, then,

to determine whether a oor effect might alsobe present for the PNIs with this sample.

Second, to explore the growth of eachtask, sensitivity to growth and potential inter-actions with child age and classroom place-ment were examined using a linear mixedmodel (LMM). It is important to use an anal-ysis tool that allows for explicit modeling of individual differences in growth (Raudenbush& Bryk, 2002). LMM, of which hierarchical

linear modeling is a well-known example, areespecially suited to the type of problem ad-dressed by this study (Fitzmaurice, Laird, &Ware, 2004). LMMs are useful for analyzinglongitudinal data, where the assumption of independence between measurements cannotbe met. Measurements are nested within chil-dren, and in this case, children are nestedwithin their age and classroom placement. Therandom effects structure allows individualchildren to vary on all parameters being esti-mated while still providing group-level esti-mates that are more appropriate than those



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In this analysis, the Level 1 equationsmodel individual growth over time. For eachindividual i at time j, the Level 1 linear equa-tion is as follows:

yij 0 i 1 it ij e ij (1 )

In Equation 1, linear growth is modeledfor each student, with eij representing error.The Level 1 quadratic equation is as follows:

yij 0i 1it ij 2it ij2 eij (2 )

The Level 2 equations then model groupdifferences in key beta values. For the pur-poses of explanation, Level 2 equations willbe presented that model group differences in

1 i, the linear slope estimate for individual i inEquation 1. Level 2 equations could also beexamined for all beta values in any of theLevel 1 equations. Equations 3 and 4 representLevel 2 equations for differences in classroomgroup mean estimates (similar equations wereused for differences in age). In these equa-

tions, c is a dummy-coded variable reectingwhether the child participated in that class-room, and b is the error term in each equation.

0i 0 2c1i 3c2i ncni b0i

(3)

1i 1 10c1i 11c2i ncni b1i

(4)

LMMs also allow for missing data, suchas those from attrition. Missing data are acommon occurrence when conducting longi-tudinal studies. The percentage of missingdata in this study ranged from 0.48% to 0.98%for each measure. Data were assumed to bemissing at random. Similar approaches to han-dling missing data have been used in otherstudies investigating early mathematics perfor-mance over time (e.g., Jordan, Kaplan, Olah,& Locuniak, 2006). LMMs also assume inde-

from the population. Although this sample isfrom a single region of the country, we believethis assumption has been met for the sake of these analyses and recognize the limitations of our sample. A nal assumption of LMMs isthat the response variable be normally distrib-uted. A more thorough discussion of the dis-tributional properties of each measure is ad-dressed later in this article. In those caseswhere the measure varied signicantly from anormal distribution, analyses were still con-ducted, keeping in mind the limitations of thedata set.

Results

Results are presented rst in terms of thedistributional properties of each measure.Next, the measures are examined for sensitiv-ity to growth. Last, group-level differences forclassroom and age are described.

Descriptive Statistics

The distributional properties of eachmeasure were rst explored. High standarddeviations relative to the mean, large positiveskewness values, large positive kurtosis val-ues, and a high number of zero scores all raiseconcerns about a measures ability to discrim-inate well across children. In this particularsample, some of these concerns are presentwith the PNIs. All data, regardless of wave of data collection, were compiled and both per-centage of zero scores and descriptive statis-tics were explored. Students who did not suc-

cessfully complete the sample items (i.e., erroror no response), and students who successfullycompleted the sample items but did not cor-rectly complete a single item, both received azero score in this study. These data are pre-sented in Table 1.

Next, students were divided as towhether they had reached 4 years of age at thetime of measurement, and the percentage of zero scores across these two groups were com-pared. This analysis was to better understandwhether the percentage of zero scores de-creased when measuring older versus younger



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(37.8%), with large differences between olderand younger groups of children (30.9 vs. 60.2,respectively). The percentage of zero scoresfor all other measures was below 10% for theoverall group and below 20% for the youngergroup.

Descriptive statistics of the measureswere rst examined with zero scores included,as shown in Table 1. These values revealed

high standard deviations relative to the meanfor most measures, although for QuantityComparison Fluency, standard deviationswere around half of the mean. Quantity Com-parison Fluency also demonstrated the skew-ness values closest to zero, although skewnesswas moderately positive (between 1.5 and 2.0)for all other PNI tasks. In addition, both Quan-tity Comparison Fluency and Number NamingFluency demonstrated highly leptokurtoticdistributions.

Because previous research on other pre-school GOMs (i.e., Individual Growth and De-

ing analyses, descriptive statistics were againexamined with zero scores removed, alsoshown in Table 1. The key descriptive statis-tics of the PNIs remained remarkably stablewhen comparing before and after the removalof zero scores; the largest change in skewnessand kurtosis was in Number Naming Fluencyand the largest change in standard deviationwas in Quantity Comparison Fluency.

Sensitivity to Growth

The purposes of further analyses in thisstudy were to examine sensitivity to growthand whether group differences could be mod-eled across age and classroom placement. In apractical setting, these measures will be ad-ministered to all children, not just those chil-dren who already demonstrate scores greaterthan zero. Thus, characteristics such as sensi-tivity and group differences pertained to allstudents in the sample, regardless of whether

Table 1Descriptive Statistics of the Preschool Numeracy Indicators

Statistic OOCCF OCF NNF OCF

Percentage of zero scoresTotal 3.8 1.6 37.8 8.14 0.8 0.5 30.9 4.8

4 14.5 5.5 62.0 19.8Descriptive statisticsAll students

M 16.6 20.3 5.1 15.7 Mdn 12 15 2 15SD 16.3 15.1 7.1 8.4Skewness 1.5 1.7 2.0 0.1Kurtosis 1.3 5.1 4.2 0.5

Descriptive statisticsNo zero scores M 17.3 20.6 8.1 17.1 Mdn 13 15 6 16SD 16.3 15.0 7.5 7.3Skewness 1.5 1.7 1.6 0.2Kurtosis 1.2 5.2 2.6 0.5

Note. OOCCF One-to-One Counting Correspondence Fluency; OCF Oral Counting Fluency; NNF NumberNaming Fluency; QCF Quantity Comparison Fluency; 4 children 4 years old and older; 4 children underthe age of 4 at time of measurement.



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duct all further analyses on the data set withzero scores included.

The sensitivity to growth of all measureswas examined using the LMM. First, themodel explored whether the measures demon-strated a signicant linear growth component.Next, the model examined whether a signi-cant quadratic component was present. Param-eter estimates and signicance values aregiven for both the linear-only condition andthe quadratic condition in Table 2. Signi-

cance values of the linear component must beexplored in a linear-only condition, as signif-icance tests of lower order parameter estimatesare not appropriate in the presence of higherorder parameter estimates (Fitzmaurice et al.,2004).

All measures demonstrated a signicantlinear growth component. Linear parameterestimates were around 1 item correct per waveof data collection for One-to-One CountingCorrespondence Fluency, Oral Counting Flu-ency, and Quantity Comparison Fluency. Av-erage growth was around 0.5 items correct per

nent was found for Quantity Comparison Flu-ency whereas a nonsignicant but large posi-tive quadratic component was found for OralCounting Fluency. Finding a signicant lineargrowth component is strong, supporting evi-dence for the sensitivity of the measures togrowth over time. It indicates that the rate of change of childrens scores on these measuresover time was nonzero and positive, and thatthis difference from zero was greater thanwhat could simply be accounted for by thestandard error of the slope estimate ( SE b).

Classroom and Age Effects

An exploratory analysis was conductedto evaluate the effects for classroom and ageseparately using a LMM with a random effectscovariance structure.

Classroom effects were examined usinga series of omnibus tests across all classroomconditions, to examine whether any signicantdifferences across classrooms might be foundfor each measure. Although no interventionwas used and the curriculum used in each

classroom ostensibly was the same, informalobservations suggested instructional variationsoccurred across classrooms because of differ-ences in teaching styles. However, these in-structional variations were not controlled ordocumented, and no attempt was made to sup-port an expectation that a particular classroomwould outperform another; therefore, this ele-ment of the data analysis plan remains explor-atory. In the case of signicant omnibus tests,

individual classroom comparisons could beconducted, but were not. The purpose of thisanalysis was only to establish whether class-room differences in slope estimates might rea-sonably be examined using this model, so as toinform future research. Classroom assignmentwas assumed to be random.

After controlling for family-wise errorusing a Bonferroni correction ( p .05/4 .0125), signicant omnibus tests for classroomdifferences were found for Quantity Compar-ison Fluency ( p .001). Omnibus tests werenear signicance for Oral Counting Fluency

Table 2Parameter Estimates for Linear andQuadratic Growth Models for thePreschool Numeracy Indicators

Parameter OOCF OCF NNF OCF

Linear modelIntercept 12.4 16.5 3.5 12.2Linear 1.37 1.15 0.52 1.09Linear pvalue .001* .001* .001* .001*

Quadratic modelIntercept 13.0 17.3 3.3 11.3Linear 0.9 0.3 0.7 2.0

Quadratic 0.07 0.13 0.03 0.14Quadratic pvalue .47 .08 .15 .004*

Note. OOCCF One-to-One Counting CorrespondenceFluency; OCF Oral Counting Fluency; NNF NumberNaming Fluency; QCF Quantity Comparison Fluency.* Statistically signicant with Bonferroni correction.



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of the PNI tasks to differentiate growth differ-ences across classrooms and interventions.

To examine the effect of age, childrensinitial age was entered into the model, usingage at October 1 of the year the study began.After controlling for family-wise error againusing a Bonferroni correction ( p .05/4

.0125), a signicant effect for child age wasnot found for any measure, with the exceptionof Number Naming Fluency ( p .004). Thus,for all measures except Number Naming Flu-ency, slope is unlikely to be signicantly re-lated to the age of the student, for the popu-lation represented by this sample. For Number

Naming Fluency, older children demonstratedsignicantly steeper slopes. This indicates thatgrowth rates tended to be higher for olderversus younger participants and that these dif-ferences were greater than that which might beexplained by the standard error of the slopeestimate ( SE b). A signicant effect for age onslope would indicate that the measure was lesssensitive to growth over certain age ranges andmay be less appropriate for certain portions of the age range of participants in the study.

Discussion

The goal of the study was to examinehow the PNIs functioned as an early numeracyGOM in terms of sensitivity to growth overtime with a sample of children from predom-inantly low socioeconomic backgrounds. Interms of general descriptive statistics, thePNIs demonstrated relatively low percentages

of zero scores across three of the four tasks.The exception was the high percentage of zeroscores for Number Naming. The lower per-centage of zero scores for three of the PNItasks is encouraging in that a high percentageof zero scores coupled with large standarddeviations relative to the mean call into ques-tion the ability of the task to differentiatebetween children on the lower end of the dis-tribution, a necessary characteristic for an ad-equate screening measure. It is important tonote that although there is a rationale for se-lecting a sample that may be at risk for math-

of the tasks in the context of the utility of thePNIs for the larger preschool population. Ad-ditional research with diverse samples isneeded to determine if a low percentage of zero scores is consistent across samples and if the high percentage of zero scores for NumberNaming Fluency is an artifact of the sample ora characteristic of the task.

Despite the low percentage of zeroscores for Quantity Comparison Fluency, OralCounting Fluency, and One-to-One CountingCorrespondence Fluency, there remain someconcerns. The large standard deviations rela-tive to the mean, combined with moderatepositive skewness for all but the QuantityComparison Fluency, and the high leptokurto-sis for the Oral Counting Fluency and NumberNaming Fluency, are often indicative of aoor effect. In addition, large standard devia-tions raise the issue of measurement error andthe extent to which a true score can be iden-tied. Especially in light of the standard errorof measurement, in such distributions it be-comes difcult to make meaningful statementsabout differences across students whose scoresare at the bottom of the range of performance.

However, the skewness values weremoderate, not severe, and the demographics of the current sample raise the possibility thatperformance was more homogenous and po-tentially different from that of the general pop-ulation. Distinct patterns of performance havebeen demonstrated in research with samples of kindergarten children similar in demographics(Jordan et al., 2006; Jordan, Kaplan, Locu-niak, & Ramineni, 2007). In addition, childrendid demonstrate signicant growth on themeasures across months in the Head Start en-vironment. This lends further support to thenotion that these distributional properties maybe a function of the sample, as exposure to arich instructional environment led students togrow up and off of any oor in the distri-bution. Thus, further research should explore

the distributional properties of these tasks witha larger and more representative sample to



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Sensitivity to Growth Over Time

Results suggest each PNI task is sensi-tive to growth over time, with growth rates for

three of the tasks (i.e., Quantity ComparisonFluency, Oral Counting Fluency, and One-to-One Counting Correspondence Fluency) cal-culated at 1 item per month. Number NamingFluency demonstrated growth over timebutat a rate of 0.5 items per month. These growthrates are encouraging in that they are of suf-cient magnitude to be meaningful forprogress monitoring. That is, if the PNIs wereused on a benchmarking schedule, one couldexpect a growth rate of between 2 and 4 itemsper benchmark period, which is a growth ratelarge enough to be visually detected whengraphing data. Additional research with morediverse samples is needed to determinewhether growth rates differ. Steeper growthrates would facilitate the use of the tasks asprogress monitoring measures that can be usedmore frequently than the typical benchmark schedule. Further, research is also needed todetermine how growth over time is related to

important early mathematical outcomes, sim-ilar to the research conducted by Clarke et al.(2008).

Classroom and Age Effects

Demonstrated differences for QuantityComparison Fluency and Oral Counting Flu-ency across classrooms might be consideredan initial indication of sensitivity to interven-tion in that, assuming assignment of children

to classrooms is mostly random, differentgroup mean slope estimates indicate that chil-dren may be experiencing different instruc-tional environments. This nding suggests thatQuantity Comparison Fluency and OralCounting Fluency may be useful tools in eval-uating the effects of instruction and curricu-lum, although caution should be used in inter-preting this nding. Differences in instructionand curriculum were not systematically docu-mented in this study, only informally noted.Stage 3 research efforts are needed to experi-mentally explore this area and demonstrate the

assignment to classrooms and intervention im-plementation and integrity. Sensitivity to in-tervention is critical for formative assessmentmeasures to be used in tiered models of ser-vice delivery. Response to intervention is in-creasingly becoming part of the dialogue inearly education, and the need for tools that aresensitive to growth over time and the effects of intervention are critical (VanDerHeyden &Snyder, 2006).

Differences in slope across initial agewere generally not found, with the exceptionof Number Naming Fluency. Children demon-strated growth over time on these tasks, and

typically this growth was similar for botholder and younger children. This fact, com-bined with the relatively low number of zeroscores for these tasks, suggests Oral CountingFluency, One-to-One Correspondence Count-ing Fluency, and Quantity Comparison Flu-ency can reasonably be given to assess perfor-mance over time across the entire age rangerepresented by this sample. Further researchshould explore performance across the agerange with a broader sample, perhaps in across-cohort longitudinal design using lineargrowth modeling, to understand more fully if there are specic age ranges where a givenPNI demonstrates steeper growth and thusgreater sensitivity to changes in early nu-meracy skill.

Limitations

There are a number of limitations of the

current study, the existing evidence for thePNIs, and aspects of the tasks that need to beaddressed through future research. First, thesample for this study was highly specic interms of geographic location, educational set-ting, race, at risk, and socioeconomic status.The specic characteristics of the sample pre-vent estimation of typical performance; it ispossible that patterns of performance as wellas growth over time may differ in other sam-ples. To determine whether the measures holdpromise for a more general population, furtherresearch is needed to establish whether similar



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eral population. Further research is alsoneeded to better understand the potential dif-ferences between children who do not success-fully complete the sample items and childrenwho complete the sample items but do notcorrectly respond to any items in the task, bothof whom received a score of zero in this study.

Currently, the evidence to support thePNIs is limited in scope, although growing.The National Center on Response to Interven-tion (n.d.) outlines necessary criteria forscreening and progress monitoring tools. Tocertify the PNIs as a screening and progressmonitoring tool, the following needs to be

established: (a) classication data, (b) predic-tive validity, (c) benchmarks, (d) improvedteacher planning or student achievement as aresult of using the PNIs, (e) predictive valid-ity, (f) sensitivity to the effects of intervention,and (g) reliability of slope. This informationwill inform recommendations as to which PNItasks are most useful in an early warning sys-tem designed to improve outcomes for youngchildren.

Finally, the PNIs, like most of the earlynumeracy measures developed across researchteams, are limited in their focus. The PNIsrepresent tasks thought to relate to numbersense with no attention to other areas of math-ematical development (e.g., algebra, geome-try, measurement, data analysis, and probabil-ity). Understanding how number sense relatesto other areas is critical in validating numbersense tasks as GOM for mathematical compe-tency. To be considered a GOM, growth in

single-skill measurement tasks, like those fornumber sense, must correspond to globallearning in the broader domain of interest(Fuchs, 2004). In addition, future researchshould consider other areas of mathematicaldevelopment for potential GOM tasks.

Implications for Practice

The purpose of the present study was tofurther Stage 2 research on early mathematicsGOM tasks by examining the sensitivity togrowth over time of the PNI tasks in a sample

status and ethnicity. Although the results of the study have limited generalizability, thereare important implications for early educators,school psychologists, and other educationalpersonnel in developing and implementingsystems-level approaches to promote positiveacademic and social outcomes for children.First, although additional research is necessarywith more diverse samples, this study providespreliminary evidence that the PNIs may be auseful progress monitoring tool for preschool-age children. Results indicate all of the PNImeasures demonstrate sensitivity to growthover time with a sample of preschoolers at risk

for mathematics difculties, with Oral Count-ing Fluency, One-to-One Counting Corre-spondence Fluency, and Quantity Discrimina-tion Fluency demonstrating greater growthrates than Number Naming Fluency. Further,although conrmatory research is needed, thegrowth rates from this study suggest promisefor using Quantity Comparison Fluency andOral Counting Fluency to monitor progressmore frequently to evaluate response to inter-vention and inform data-based decision mak-ing about childrens educational program-ming.

Second, this study used a longitudinaldesign that purposefully included the full agerange of preschoolers (i.e., 3- to 5-year-olds),and results provide support for the use of OralCounting Fluency, One-to-One Correspon-dence Counting Fluency, and Quantity Com-parison Fluency with the full age range, elim-inating the need for specic tasks to be used

only with children of a specic age. Useacross the preschool-age range facilitatestracking childrens growth and development inearly mathematics at a critical time point,given research that suggests differences inchildrens growth trajectories in mathematicsemerge by 3 years of age and increase throughthe preschool years (Case et al., 1999; Na-tional Mathematics Advisory Panel, 2008).More research is needed to explore age differ-ences in Number Naming Fluency and its util-ity across the age range.

Finally, this study was conducted pur-



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search with a broader sample may providemore information about typical performanceand growth over time on the PNI tasks, dem-onstrating sensitivity to growth over time witha population at risk for mathematical difcul-ties based on demographics is critical. Thesechildren may not exhibit the same growth ratesas the broader population, and thus tasks thatdetect growth for children at risk for mathe-matical difculties are needed to inform in-struction and intervention to efforts. Differ-ences in performance between children fromlow-income households and their peers fromhigher income households provide evidence of

the need for attention to these disparities at theearliest point possible (Jordan et al., 2006;National Council for Education Statistics,2007; National Mathematics Advisory Panel,2008).

In general, a set of early mathematicsGOM tasks will provide educators and parentswith important information about a childsgrowth trajectory and a means to better deter-mine whether a child is developing the skillsneeded for mathematical competency. Thisearly warning system can allow for early, in-tensive intervention, to help children gain theskills necessary and a means for monitoringprogress toward valued outcomes in mathe-matical development. It remains unclearwhich of the PNI measures is the most usefuland demonstrates the strongest properties as aGOM for use in screening or progress moni-toring. At this point in the development of thePNIs, evidence has been demonstrated for al-

ternate-form reliability, testrest reliability,some forms of validity (Floyd et al., 2006),and sensitivity to growth over time with prom-ise for sensitivity to the effect of instruction.Collectively, this evidence supports continuedexploration of the PNIs in a data-based deci-sion-making system through careful practicalapplications in the context of research andmeasurement development.

Footnotes

1 Following collection of the data used inFloyd et al. (2006), PNI materials were revised as

administration after completion of sample itemswere added; (3) test records and directions for eachprobe were separated; (4) boxes were added toQuantity Comparison Fluency to eliminate the con-

cept of side of the page. Eight alternate forms of Number Naming Fluency and Quantity ComparisonFluency were prepared for repeated measurement.(No alternate forms are needed for Oral CountingFluency and One-to-One Correspondence CountingFluency.) Technical properties are reported for thetasks used in Floyd et al. (2006), except wherenoted.

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Robin Hojnoski is Assistant Professor in the School Psychology Program at LehighUniversity. She is interested in the application of school psychology principles andpractices to early education. Her research focuses on assessment and intervention to

improve educational and social outcomes for preschool children and their families.

Benjamin Silberglitt completed his PhD in Educational Psychology at the University of MinnesotaTwin Cities. His areas of interest include student assessment, school orga-nization and systems-level change, problem solving, response to intervention, and data-based decision making. He is Director of Software Applications for TIES, a technologycooperative of 39 Minnesota school districts.

Randy G. Floyd is Associate Professor of Psychology at The University of Memphis. Hereceived his doctoral degree in school psychology from Indiana State University. Hisresearch interests include the structure, measurement, and correlates of cognitive abilities;the technical properties of early numeracy measures; and the process of professional

publication.



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