Assessing preschool number sense: skills demonstrated by children prior to school entry

This article was downloaded by: [Ams/Girona*barri Lib]On: 14 November 2014, At: 04:57Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Educational Psychology: AnInternational Journal of ExperimentalEducational PsychologyPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/cedp20

Assessing preschool number sense:skills demonstrated by children prior toschool entrySally Clare Howell a & Coral Rae Kemp aa Macquarie University Special Education Centre , MacquarieUniversity , Sydney, New South Wales, AustraliaPublished online: 20 Jul 2010.

To cite this article: Sally Clare Howell & Coral Rae Kemp (2010) Assessing preschool number sense:skills demonstrated by children prior to school entry, Educational Psychology: An InternationalJournal of Experimental Educational Psychology, 30:4, 411-429, DOI: 10.1080/01443411003695410

To link to this article: http://dx.doi.org/10.1080/01443411003695410

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &

http://www.tandfonline.com/loi/cedp20

http://www.tandfonline.com/action/showCitFormats?doi=10.1080/01443411003695410

http://dx.doi.org/10.1080/01443411003695410

Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14

http://www.tandfonline.com/page/terms-and-conditions

http://www.tandfonline.com/page/terms-and-conditions

Educational PsychologyVol. 30, No. 4, July 2010, 411–429

ISSN 0144-3410 print/ISSN 1469-5820 online© 2010 Taylor & FrancisDOI: 10.1080/01443411003695410http://www.informaworld.com

Assessing preschool number sense: skills demonstrated by children prior to school entry

Sally Clare Howell* and Coral Rae Kemp

Macquarie University Special Education Centre, Macquarie University, Sydney, New South Wales, AustraliaTaylor and FrancisCEDP_A_470063.sgm(Received 17 July 2009; final version received 12 February 2010)10.1080/01443411003695410Educational Psychology0144-3410 (print)/1469-5820 (online)Original Article2010Taylor & Francis0000000002010Ms. [email protected]

Components of early number sense, as identified in two Delphi studies and in thenumber sense literature related to mathematics difficulties, were assessed for 176children in preschools and childcare centres across one local government area inSydney, Australia, using tasks or modifications of tasks reported in the numbersense literature. In addition, the children’s receptive vocabulary was measuredusing The Peabody Picture Vocabulary Test (third edition) and math reasoningwas measured using Woodcock-Johnson III Tests of Achievement. Although thechildren demonstrated a broad range of skills, there were no significant differencesbetween children attending childcare and preschools for any of the measures.However, boys performed significantly better than girls in quantitative conceptsand girls performed better than boys in subitising. In discussing the data, acomparison is made of the skills demonstrated by children and skills that werehighlighted in the two Delphi studies and in the early number sense literature asbeing essential components of number sense prior to school entry. Implications forkindergarten mathematics curricula and approaches to the teaching of earlynumber skills are discussed.

Keywords: preschool; assessment; number sense

Since the publication of the Curriculum and Evaluation Standards for School Mathe-matics (National Council of Teachers of Mathematics Commission on Standards forSchool Mathematics, 1989), ‘number sense’ has become a focus of much of the debateabout how mathematics can best be taught, and there has been considerable shift in theaccepted pedagogy for teaching mathematics. In the current mathematics teachingliterature, there is a strong emphasis on the need to capitalise on number sense in theearly years of schooling, and teachers are encouraged to provide children with expe-riences that will develop number sense and thus improve mathematics outcomes.Further, in many education systems, the development of number sense is viewed as ameans of improving mathematics achievement and, hence, the results of studentsparticipating in national and international testing regimes. Much of the mathematicsteaching literature reflects number sense, as it is demonstrated by competent studentsof mathematics, and describes learning environments in which children are providedwith opportunities to construct their own mathematical procedures and understandingsand share these with their peers (Cobb, Wood, & Yackel, 1991; Steffe, 1991; vonGlaserfeld, 1995). This constructivist paradigm now underpins both undergraduate

*Corresponding author. Email: [email protected]

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14

412 S.C. Howell and C.R. Kemp

teacher training and professional learning for teachers in the field and is reflectedthroughout much of the mainstream mathematics teaching literature.

In the early mathematics literature, the term ‘number sense’ is often used todescribe both the intuitive understanding of number that is prerequisite for success inschool-based mathematics (Gersten & Chard, 1999; Gersten, Clarke, & Jordan, 2007;Griffin, 2003; Griffin, Case, & Siegler, 1994; Van Luit & Schopman, 2000) and theinformal understanding of number displayed by children prior to formal instruction inmathematics (Fuchs, Fuchs, & Karns, 2001; Gersten & Chard, 1999; Gersten, Jordan,& Flojo, 2005; Ginsburg, 1997; Griffin et al., 1994; Van Luit & Schopman, 2000).While number sense remains an ill-defined construct (Berch, 2005; Gersten et al.,2005; Howell & Kemp, 2004; Laski & Siegler, 2007), and some authors use the terms‘numerosity’, ‘number competence’, ‘numerical proficiency’ or ‘mathematical profi-ciency’ rather than ‘number sense’ (Butterworth, 2005; Jordan, Kaplan, Ramineni, &Locuniak, 2009; Landerl, Bevan, & Butterworth, 2004; Ramani & Siegler, 2008), it isnonetheless considered a prerequisite for mathematical success.

The assessment of number sense as a predictor of early mathematics difficultieshas become a focus in special education research (Bryant, Bryant, Gersten, Scammacca,& Chavez, 2008; Fuchs et al., 2007; Gersten et al., 2005). Recognition that not allchildren begin school with the same level of number sense (Aunio, Hautamaki, & VanLuit, 2005; Fuchs et al., 2001; Gersten & Chard, 1999; Gersten et al., 2005; Ginsburg,1997; Griffin, 2003; Jordan et al., 2009; Laski & Siegler, 2007; Malofeeva, Day, Saco,Young, & Cianco, 2004; Van Luit & Schopman, 2000), coupled with recognition ofthe benefits of early intervention in addressing the educational needs of students ‘atrisk’, has resulted in research into early number sense as a measurable phenomenon(Bryant, 2005; Cowan, Dowker, Christakis, & Bailey, 1996; Gersten & Chard, 1999)and as a predictor of future mathematics difficulty (MD) (Bryant et al., 2008;Butterworth, 2005; Fuchs et al., 2007; Gersten et al., 2007).

In November 2005, the Special Series of the Journal of Learning Disabilities wasdevoted to early screening and intervention in mathematics. In this volume, recurringthemes included the relatively limited research on early screening for MD and the lackof agreement surrounding the concept of number sense. In this Special Series, Gersten,Jordan, and Flojo (2005) reported that ‘no two researchers have defined number sensein precisely the same fashion’ (p. 296), and Berch (2005) reported that the situation isfurther complicated by the fact that ‘cognitive scientists and math educators definenumber sense in very different ways’ (p. 334). In the same volume, Chiappe (2005)highlighted the need for research to establish the specific variable or variables thatplay a causal role in MD, and Mazzocco (2005) pointed out that while there are someassessment items that, when used in combination, predict later MD, it is not fullydetermined what skills these items measure.

A number of authors in the Special Series listed components of number senseor numerical ability that have been shown to be predictive of MD. Mazzoccoincluded reading numerals below 10, number constancy, using concrete materials toadd one-digit numbers and number magnitude. Dowker (2005) included countingsets, cardinal word principle, order irrelevance, repeated addition by one, repeatedsubtraction by one, number conservation and establishing equivalent sets. Chiappe(2005) included magnitude, counting and differences in quantities. Gersten et al.(2005) included counting and quantity discrimination with the former includingstrategies used for simple computation and the latter including use of the mentalnumber line.

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14

Educational Psychology 413

The Number Knowledge Test (Okamato & Case, 1996), which was one of the firstassessments designed to measure the number sense of young children, included itemsto assess counting, number magnitude and addition. Malofeeva et al. (2004) devel-oped a preschool number sense assessment that included items to assess counting,number identification, number–object correspondence, ordinality, comparison andaddition–subtraction. Clarke and Shinn (2004) and Chard et al. (2005) included itemsto assess number identification, quantity discrimination and missing number in theirscreening tool for Year 1 and kindergarten students. Gersten et al. (2007) reported onthese various assessments of early number sense and concluded that some show prom-ise in identifying students at risk of MD.

In their review of recent research in the area of early mathematics screening, Fuchset al. (2007) reported correlations between some early mathematics screening toolsand outcome measures of mathematics achievement and concluded that assessmentssuch as the Number Knowledge Test show promise in predicting specific areas offuture MD. A number of the screening tools cited by Fuchs et al. and appearing incurrent literature (Jordan, Kaplan, Locuniak, & Ramineni, 2007; Locuniak & Jordan,2008) use the term ‘number sense’ to describe the skills being measured. Skillsappearing in the number sense literature that also appear in the recently developedResearch-based Early Mathematics Assessment (REMA) (Clements, Sarama, & Liu,2008), and which feature in the Building Blocks preschool mathematics intervention(Clements & Sarama, 2008), include rote counting, comparison of quantity, 1:1 corre-spondence, cardinal value, simple addition and subtraction, subitising, ‘number after’and ‘number closer to’. Number sense skills that appear in the I Can Do Maths assess-ment (Doig, McCrae, & Rowe, 2003) include comparison of quantity, ordinal value,1:1 correspondence, cardinal value, simple addition and subtraction, and number after.Despite some commonality across assessments, exactly which components of thenumber sense/early mathematics assessments appearing in the literature are crucial formathematics success has not yet been fully determined. In this context, there is a needfor further research that examines whether or not particular components of numbersense are prerequisite for success in the early mathematics assessment of the class-room. Examination of the number sense and early mathematics assessment literatureclearly reveal some potential ‘pieces of number sense’ that may be prerequisites forsuccess with early mathematics in the number domain, but large-scale longitudinalresearch is needed to verify whether there is a causal relationship between anyproposed number sense skills and future mathematics success.

From the literature, it is apparent that most authors view number sense as somethingthat applies across all ages, and that different aspects of number sense are relevant todifferent areas of mathematics. The challenge for the research community is to identifywhich components of number sense are related to which pieces of mathematics at whichstages of learning. Only then will it be possible to determine which number sense taskswill be useful in instruction. In order to inform teaching practices in the early years ofschooling, there is a need to identify skills and tasks that reflect early number sense asa prerequisite for early mathematics competence. While the focus of the current researchis in the area of number, it is important to acknowledge that number sense is not theonly area of early mathematics assessment that is thought to be predictive of mathematicslearning. A sense of pattern and structure (Mulligan, Mitchelmore, & Prescott, 2005)is another component of mathematical development identified as a possible marker forfuture mathematical success. The research reported in this paper is confined to compo-nents of number sense that appear in the MD literature related to the number domain.

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


On the basis of a review of international literature in the areas of ‘number sense’and ‘early mathematics’, Howell and Kemp (2005) identified a number of skills andtasks that may reflect components of early number sense essential for early arithmeticcompetence. These skills and tasks were included in an Australasian Delphi study(Howell & Kemp, 2005) and, with slight modifications, in an international Delphistudy (Howell & Kemp, 2006). The purpose of these two studies was to establish aconsensus on just which skills are considered essential components of early numbersense and how these components could best be assessed with young children (Howell& Kemp, 2009). Both the proposed components and tasks in the questionnaires foreach Delphi study were limited to the number domain.

On the basis of the results of the Australasian Delphi study (Howell & Kemp,2005) and the early mathematics assessment literature, a number of componentnumber sense skills were identified for children prior to their commencement inkindergarten. With no clear consensus on essential components of number sense beingestablished in the Australasian Delphi study, a decision was made to include compo-nents described as predictive of MD in the early mathematics assessment literatureand components that are reflected in the early number sense literature. This meant thatwhile components, such as number magnitude, did not reach consensus in theAustralasian Delphi study, they were nonetheless included. This decision could bejustified on the basis that each item assessed would be analysed to establish anypotential link with mathematics performance. Thus, the components of number senseexamined in the research reported in this paper were limited to components of numbersense that have been proposed as having predictive value or which were deemed‘essential’ in the two Delphi studies. No attempt was made to include the full range ofskills described as number sense covered within the large body of early mathematicsassessment literature.

The purpose of the pilot study conducted from 2004 to 2006 was to trial identifiedmeasures of number sense with children at the end of their preschool year and againafter one and two years of formal schooling. Further, the aim was also to see whetherany of the components of number sense assessed prior to school entry were related tosubsequent measures of number sense or formal measures of mathematics achieve-ment. The participants in the pilot study were 30 children (11 boys and 19 girls), witha mean age of 60 months (range 54–71 months) who were assessed just prior to start-ing school in a range of privately run childcare centres in the Sydney metropolitanarea. Of those 30 children, 18 (five boys and 13 girls) took part in the secondassessment during the final month of their first year of formal schooling and 12 (fourmales and eight females) took part in a third assessment at the end of their second yearof formal schooling, with 11 of these children having participated in the secondassessment. Thus, 11 children were assessed three times over a period of three years,seven children were assessed twice over the first two years of the study, and one childwas assessed in the first and third years of the study but not in the second year.

Components of number sense were measured at the end of preschool and at the endof the first year of school when the same components of number sense were assessedbut at a more advanced level with tasks involving numbers between 10 and 20 ratherthan below 10. Mathematics performance, which was assessed at the end of the firstand second years of school, was measured using the Applied Problems and Quantita-tive Concepts sub-tests of the Woodcock-Johnson III Tests of Achievement (Australianedition, 2001) (Woodcock, McGrew, & Mather, 2001). These sub-tests comprise theMath Reasoning cluster of the mathematics assessments. The Peabody Picture

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


Vocabulary Test – PPVT-III (Dunn & Dunn, 1997) was also administered at the endof preschool in order to provide a rough measure of cognitive ability and a measure ofreceptive vocabulary, both of which have the potential to influence learning.

Unfortunately, because the numbers were small, subject attrition was great and therange of ability across subjects was skewed towards the more able children (meanPPVT age for the preschool children was 73.4 months), there were no conclusiveresults in relation to the predictive nature of any of the components of number senseincluded in these assessments. The only result that appeared to have any significancewas the correlation between the number of items passed at the preschool level andreceptive vocabulary as measured by the PPVT (r = 0.452; p < 0.02). The pilot studydid, however, enable the researchers to fine-tune the assessment items and to standar-dise their implementation.

The research reported in this paper is part of a broader study in which the research-ers sought to examine possible links between components of early number senseassessed using procedures identified in the number sense literature or a modificationof these and mathematics performance at the end of preschool and after one year offormal schooling. The purpose of the current paper is to report on receptive vocabu-lary and mathematics skills, including early number sense skills, demonstrated by abroad range of children assessed just prior to starting school. Receptive vocabularywas included not just as an indication of cognitive ability but also because of thecorrelation found between this measure and the number of number sense itemsdemonstrated in the preschool phase of the pilot study. In discussing the data, acomparison is made of the skills demonstrated by children assessed before schoolentry and skills that were highlighted in two Delphi studies as being essential compo-nents of number sense. Also discussed are the skills demonstrated by children thatwere not expected of children prior to school entry according to the data collected inthe Delphi studies but which do appear in the early number sense assessment litera-ture. Further, the implications of the data are discussed in relation to the teaching ofearly number skills prior to school entry and in the early years of school. Specifically,the research questions are as follows:

(1) What receptive vocabulary and mathematical skills, including skills identifiedin the literature as potential components of early number sense predictive ofmathematics difficulties, do 176 children attending preschools and childcarecentres in a socially inclusive urban area of Sydney, Australia, demonstrate?

(2) What percentage of the children demonstrates skills identified in previousstudies, including two Delphi studies, as essential components of early numbersense?

(3) Are there differences in receptive vocabulary and mathematics skills accord-ing to type of early childhood setting or gender?

Method

Participants

A total of 180 children participated in the study. The data for 172 of these childrenwere complete. The data for a further four children were complete but for a single itemthat was inadvertently omitted from the assessment. Children not included in thisreport were those who were not assessed on all three measures and one child whoseparents did not provide a date of birth. This gave a total of 176 children (89 boys and

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


87 girls). Of these, 93 (52.84%) from 17 different childcare centres and 83 (47.16%)from eight different preschools were included in the final analysis. The ages of the 176children (89 boys and 87 girls), assessed from October to December in the year beforeschool, ranged from 50 months to 68 months, the mean age being 59.9 months. Theparticipants were chosen from one local government area that included the full rangeof socio-economic (socio-economic status [SES]) backgrounds as determined by theAustralian Bureau of Statistics (2007).

Measures

Components of early number sense

The 18 components included for assessment related to counting (11), number princi-ples (three) and number magnitude (four). Details for each of the components and thetasks used to assess these across the three areas are presented in Tables 1–3. Rotecounting appeared as the first item, and the child’s counting range then determined thenumbers used to assess other components of number sense. No item that requirednumeral knowledge was included in the assessment.

The number of items chosen to assess each component was based on a judgementas to whether or not performance on a single presentation was capable of indicating

Table 1. Tasks to measure counting components.

Component Task description

Rote counting Allow the child three opportunities to count aloud. Stop counting at 30.

Count from a given number Ask the child to count on from three numbers (other than 1) within their rote counting range.

Counting backwards Ask the child to count backwards from three numbers within their counting range (excluding 10).

1:1 correspondence Ask the child to count a random display of 7 items and 9 items. If the child cannot do this, ask the child to count 7 items and 9 items displayed in a horizontal line.

Cardinal value In the task above, the child is asked to count the sets and say how many there are.

Counting out a set to match a spoken number

Ask the child to count out 6 and 8 counters from a group of counters.

Number after Say, ‘I’m going to say a number and I want you to tell me the number which comes next. 5 – What number comes next?’ Repeat with 2 and 7.

Number before Say, ‘I’m going to say a number and I want you to tell me the number which comes before it. What number comes before 10?’ Repeat with 6 and 3.

Ordinal value Ask the child to point to the third of 10 apples, sixth of 10 swans and ninth of 15 bugs shown on cards in a horizontal line.

Addition story (addition to 5) Explain that the child can use counters if they like. Say, ‘If you have 3 sweets and I give you 2 more how many will you have?’

Subtraction story (subtraction from 4)

Explain that the child can use counters if they like. Say, ‘If you have 4 sweets and you eat 3 sweets how many will you have left?’

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


an understanding of the number sense component. Another factor influencing thenumber of items used to assess a single component was the time taken to implementeach item. With these factors in mind, a single item was included for the assessmentof the commutative law and for simple word problems of addition and subtraction.Only one item assessing counting on from a number greater than 10 and counting backfrom a number greater than 10 was included.

Interrater reliability was calculated for 43 of the number sense assessments. Thisrepresented 24% of the 176 assessments. Mean interrater reliability for scoring thechildren’s responses to the tasks included in the number sense assessments was 98.8%(range 93.2–100%).

Vocabulary knowledge

The Peabody Picture Vocabulary Test – third edition – PPVT-III (Dunn & Dunn,1997) is a tool commonly used in research to provide a rough measure of cognitive

Table 2. Tasks to measure number principle components.


Order irrelevance The child observes a puppet counting and says whether the count is right or wrong. The puppet counts correctly from left to right starting with the first item, correctly but starting from the third item and incorrectly counting the third item twice.

Inversion Establish with the child that two groups of counters are ‘equal’ (each group has four counters). The two groups are each hidden under a card. Two counters are added to one side of one group, and then two counters are removed from the other side of that group. The child says whether the two groups are the same or different. The process is repeated by adding and removing one counter from one group. In the third trial, one counter is added to a group, but two counters are removed.

Commutative addition Establish that the containers with the same coloured sweets have the same number of sweets. The assessor shares out the containers of sweets to two puppets. The puppets receive containers of coloured sweets in different order. The child says whether the assessor has been fair and given the puppets the same number of sweets.

Table 3. Tasks to measure number magnitude components.


Larger group Show two cards and ask the child to point to the group with more. Present groups of 2 and 5, 8 and 4.

Larger number The child says which of the two numbers spoken by the assessor is the biggest. Spoken numbers 4 and 2, 3 and 5, 2 and 1.

Ordering groupsOrdering two groupsOrdering three groups

After a demonstration of the task, the child places two cards, each displaying a group of dots in order on a ladder with the small group low on the ladder and big group high on the ladder. The task is repeated with three cards.

Subitising The child is told that a card will be quickly shown and that they are to say how many dots they see. A card with random dot pattern of 2, 4 and 3 is flashed.

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


ability. PPVT-III Form A is reported to have a split half reliability coefficient of 0.95for children of five to six years of age and a test–retest reliability of 0.92 for childrenof two to six years of age (Dunn & Dunn, 1997, pp. 50–51). Scores on the PPVT-IIIwere examined to see whether there was any relationship between verbal ability andnumber sense as measured in the current study. A relationship between scores on thePPVT-III and number sense may have been a reflection of the role of language in earlymathematics development.

Mathematics achievement

Mathematics achievement was measured using the Quantitative Concepts and AppliedProblems sub-tests of the Woodcock-Johnson III Tests of Achievement (Australianedition, 2001) (Woodcock et al., 2001), which comprise a standardised battery of testswith American norms. Although norms for the Australian version of the assessmentwere not available at the time this paper was written, this version of the test doesprovide a linguistically and culturally appropriate version of the assessment. Each ofthe tests is administered orally on an individual basis.

The Quantitative Concepts and Applied Problems tests comprise the cluster scorefor Math Reasoning (Woodcock et al., 2001, p. 18), which was selected because it isless likely to assess achievement that is the result of formal teaching. The view wastaken that, as a measure, Math Reasoning would be more closely aligned withnumber sense than a measure of computational skill, which is likely to be influencedby mathematics instruction. The median reliability for Math Reasoning is reported tobe 0.95 for individuals of 5–19 years of age (Woodcock et al., 2001, p. 18). TheApplied Problems test measures an individual’s ability to analyse and solve mathe-matics problems. The Applied Problems test is reported to have a median reliabilityof 0.92 for individuals of 5–19 years of age (Woodcock et al., 2001, p. 14). TheQuantitative Concepts test includes the sub-tests of concepts and number series and isreported to have a median reliability of 0.90 for individuals of 5–19 years of age(Woodcock et al., 2001, p. 15).

Procedure

The local government body (the Council), which had responsibility for all participat-ing childcare centres and preschools, had already been collaborating with the secondauthor on an unrelated research project. Separate ethics approval for the current studywas obtained from the university and from the participating Council. Following aninformation session presented to all directors of childcare centres and preschoolsoperated by the local Council, each director was requested to distribute permissionnotes and an information sheet to the parents/carers of all children in their centre orpreschool who were to commence kindergarten (the first year of formal schooling inNew South Wales, Australia) at the beginning of the following year. Informedconsent was obtained from the parents of all participating children prior to thecommencement of the study. Parents were also asked to indicate their willingness tobe contacted for follow-up assessment of their child after a year of formal schooling.Although children were encouraged to participate in the assessment, it was deter-mined in advance that a refusal to cooperate would be accepted as a withdrawal ofconsent. All children indicated their willingness to participate in the assessmentprocess. Financial and time restraints meant that the research team visited centres on

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


prescheduled days. Children for whom a permission note had been signed andreturned and who were at the centre on the day/s the research team was present wereincluded in the study.

The data collection took place over the last three months of the year prior to thechildren starting school. Each child was assessed on the components of number sense,The Peabody Picture Vocabulary Test – third edition – PPVT-III (Dunn & Dunn,1997) and two sub-tests of the Woodcock-Johnson III Tests of Achievement 2001(Australian edition) (Woodcock et al., 2001). The assessments took place in thechild’s preschool or childcare centre in a separate room or space set-up, with a tableand chairs for each assessor and child. After spending a few minutes in informaldiscussion with the child, each assessment was implemented individually by one ofthe four researchers experienced in the implementation of the assessment.

Recording responses to number sense tasks

A form was developed for recording children’s responses to each of the number sensetasks. For each component of number sense, the recording sheet provided a briefdescription of the component, instructions for the task and a column for recording thechild’s performance. In the performance column, assessors recorded rote countingrange, ticked correct responses, circled a (P) when a prompt was given and recordednumbers used in tasks when these had to be adjusted to match a child’s countingrange.

Scoring assessment data

Number sense. The rote counting range of each student was recorded to a maximumof 30. A score of 2 was given for counting back from 5 or 8, a score of 3 was givenfor counting back from 5 and 8, and a score of 4 was given for counting back from 5,8 and 15. A score of 2 was given for correctly ordering two dot cards on a ladder anda score of 3 was given for correctly ordering three dot cards. For all other skills, chil-dren were given a score of 1 if they did not demonstrate the skill and a score of 2 ifthey did demonstrate the skill. Where the task included more than one item to assessa skill, the child had to get each item correct to score 2.

Test of receptive vocabulary. A precise PPVT age below 21 months is not provided inthe assessment manual. Instead, when a child scores 22 or below, an age equivalent ofless than 21 months is given. For the purpose of calculating mean PPVT age, an ageequivalent of 20 months was recorded for children scoring PPVT ages of less than 21months. Only three of the 176 children scored below 21 months, the lowest scoreprovided in the assessment manual.

Test of math reasoning. The Applied Problems and Quantitative Concepts sub-testsof Woodcock et al.’s (2001) assessment provide age equivalents for each raw score.The mean age was calculated for the Applied Problems and the Quantitative Conceptssub-tests. The raw scores of each of these sub-tests are combined to give a clusterscore for Math Reasoning. No age equivalent is provided in the manual for thiscluster score. To calculate an age equivalent for Math Reasoning, the mean of the ageequivalents for the sub-tests Applied Problems and Quantitative Concepts wascalculated.

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


Data analysis

Data are presented descriptively in the first instance, as the purpose of the researchpresented in this paper was to document the mathematical performance of childrenimmediately prior to school entry. In order to determine whether there were any differ-ences in performance on the norm-referenced tests between boys and girls or childrenin preschools and childcare centres, independent sample t-tests were used. Calculationof mean differences in performance between these groups on the components ofnumber sense required the use of non-parametric tests of mean difference becausethese data were either dichotomous or rank order data. A conservative alpha level ofless than 0.01 was adopted to avoid the threat of a Type 1 error when multiple statis-tical tests are used.

Results

Norm-referenced tests

The mean scores in months on the PPVT and Woodcock et al.’s (2001) test arepresented in Table 4. The children’s receptive vocabulary, as measured by the PPVT,spanned at least five years 10 months. There was also a range of four years sevenmonths in the ability to solve mathematical problems, as measured by the WoodcockApplied Problems sub-test, and a range of six years 10 months in mathematicalconcepts, as measured by the Woodcock Quantitative Concepts sub-test.

While the overall mean differences between the boys and girls on all measureswere small (raw score range = 0.28–0.75; range in months = 0.49–0.98), the differencewas significant in favour of the boys for the raw score on the measure of QuantitativeConcepts (t (174) = 567; p < 0.01), and the mean difference for age in months on theQuantitative Concepts measure was approaching significance (t (174) = 0.484; p =0.01). There were no significant differences between the performances of children inchildcare centres and preschools for any of the measures.

Components of number sense

The number and percentage of children who demonstrated each component of numbersense are presented in Tables 5–7 (Table 5 for counting skills, Table 6 for numberprinciples and Table 7 for number magnitude). The rote counting range of childrenvaried from 2 to 30+ (see Table 5 for the number of children able to rote count to a

Table 4. Mean performance on norm-referenced tests.

Norm-referenced tests Boys (N = 89) Girls (N = 87) Total (N = 176)

Mean PPVT age in months 60.25 (SD 14.85)Range 20–90

61.03 (SD 13.55)Range 20–88

60.64 (SD 14.18)Range 20–90

Mean applied problems age in months

62.89 (SD 10.15)Range 30–87

62.28 (SD 6.92)Range 43–78

62.57 (SD 8.68)Range 30–87

Mean quantitative concepts age in months

66.08 (SD 11.50)Range 26–108

65.59 (SD 7.85)Range 40–83

65.84 (SD 9.84)Range 26–108

Mean maths reasoning age in months*

64.43 (SD 9.98)Range 28–94.50

63.92 (SD 6.72)Range 44–78.50

64.20 (SD 8.51)Range 28–94.50

Note: *This is reported as the mean of Applied Problems and Quantitative Concepts.

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


range of numbers), but few children (7.39%) could not count to 10. Within theircounting range, most (89.20%) could count with 1:1 correspondence. There was greatvariation across the components in the percentage of children who successfullydemonstrated skills. For example, 163 (92.61%) of the 176 children assessed couldrote count up to 10, while only 8.52% could demonstrate ordinal number. Of the 176children in the study, 71.02% could count from a number other than 1 when asked tocount from a number below 10, 53.98% were able to count backwards from 5, 40.34%were able to count as far as 20 and 19.89% were able to count to at least 30. An under-standing of the order irrelevance principle was demonstrated by 23.86%, and anunderstanding of the inversion principle was demonstrated by 9.66%. The percentageof children demonstrating number magnitude ranged from 45.45% for ordering threegroups to 95.45% for identifying the larger group. This indicates that the ability to usesome form of mental number line to make magnitude judgements about numbersbelow 5 was demonstrated by 64.20% of the children. The area of number principlesappeared to be the area in which fewer of the children performed well.

Analysis of each of the number sense components was carried out to determinewhether there was a significant difference between boys and girls on any componentand whether there was a significant difference on any component for children attend-ing a preschool versus children attending a childcare centre. A Fisher exact test wasused for each dichotomous number sense component and a chi-square test was used

Table 5. Components of number sense demonstrated – counting skills.

Male (N = 89) Female (N = 87) Total (N = 176)

Num

ber

suc

cess

ful

Per

cent

age

succ

essf

ul

Num

ber

suc

cess

ful

Per

cent

age

succ

essf

ul

Num

ber

succ

essf

ul

Per

cent

age

succ

essf

ul

Rote count to:2 89 100 87 100 176 1005 87 97.75 86 98.85 173 98.3010 82 92.13 81 93.10 163 92.6115 54 60.67 51 58.62 105 59.6620 35 39.32 36 41.38 71 40.3430+ 18 20.22 17 19.54 35 19.89

Count from a number:<10 68 76.40 57 65.51 125 71.02>10 24 26.97 15 17.24 39 22.15

Count 1:1 83 93.26 74 85.05 157 89.20Cardinal value 78 87.64 74 85.05 152 86.36Next number 51 57.30 45 51.72 96 54.54Number before 27 30.34 19 21.83 46 26.13Count backwards from

at least 551 57.30 44 50.57 95 53.98

Count out* 42 47.19 45 51.72 87 49.43Ordinal number 10 11.24 5 5.75 15 8.52Addition to 5 26 29.21 22 25.28 48 27.27Subtraction from 4 24 26.97 16 18.39 40 22.73

Note: *Data missing for one male and three female participants.

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


for counting back, which had three ratings. The only significant difference betweenboys and girls, which favoured the girls, was for subitising, χ2 = 8.567; p < 0.01. Onecomponent, simple addition, which was assessed by a single item, was approachingsignificance in favour of children attending preschools, χ2 = 4.655; p = 0.42.

Discussion

Mathematics and language skills demonstrated prior to school entry

The children included in the current study demonstrated a range of mathematics andlanguage skills, including skills identified in the literature as potential components ofearly number sense predictive of mathematics difficulties. The PPVT scores indicatedthat these children had a range of cognitive ability, with some scores indicatingdisability and severe disadvantage and others indicating that the children wereperforming significantly better than what would be expected of children of their age.Similarly, the scores and mathematical age equivalents of the children on the two sub-tests of Woodcock et al. (2001) indicated that the children included in the currentstudy had a wide range of mathematical ability.

The children demonstrated varying levels of success with the components ofnumber sense included for assessment. While the majority of children in the currentstudy demonstrated competence with many of the basic counting and number magni-tude skills, a smaller percentage of children demonstrated the more difficult skills inthese areas. Of all the areas assessed, the children seemed to have most difficulty withan understanding of number principles.

Table 6. Components of number sense demonstrated – number principles.


Num

ber

succ

essf

ul

Per

cent

age

succ

essf

ul

Num

ber

succ

essf

ul

Per

cent

age

succ

essf

ul

Num

ber

suc

cess

ful

Per

cent

age

succ

essf

ul

Order irrelevance 26 29.21 16 18.39 42 23.86Inversion 12 13.48 5 5.75 17 9.66Commutative addition 42 47.19 36 41.38 78 44.32

Table 7. Components of number sense demonstrated – number magnitude.


Num

ber

succ

essf

ul

Per

cent

age

succ

essf

ul

Num

ber

succ

essf

ul

Per

cent

age

succ

essf

ul

Num

ber

succ

essf

ul

Per

cent

age

succ

essf

ul

Larger group 82 92.13 86 98.85 168 95.45Larger number 56 62.92 57 65.51 113 64.20Ordering groups:

Two groups 74 83.15 78 89.66 152 86.36Three groups 42 47.19 38 43.68 80 45.45

Subitising to 4 50 56.18 67 77.01 117 66.48

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


Demonstration of components of early number sense identified in previous research including the two Delphi studies

Rote counting, 1:1 correspondence, cardinal value and comparison of quantity wereall identified as essential measures of number sense at school entry in the both theAustralasian and International Delphi studies. Each of these components also appearsin the recent assessment literature and is described by some as having predictive value.In the current study, number sense was assessed with quantities of up to 10, with theexception of rote counting which was assessed up to 30. Tasks were adapted to usenumbers within the counting range of children who could not rote count to 10, withthis adjustment being needed for less than 10% of children. The findings in the currentstudy that more than 92% of children could count to 10 suggest that at school entry, itis appropriate to assess children’s number sense using tasks that involve numbers upto 10.

A component of counting identified as an essential component of number sensein the international Delphi study, also identified in the early number sense assess-ment literature, is counting backwards. Over half of the children in the current studydemonstrated some ability to count backwards. The results of this study suggest thatthe majority of children begin formal schooling with well-developed counting skillsbeyond those expected by some of the academics who participated in the two Delphistudies. Counting skills alone, however, are not sufficient for mathematics success(Baroody & Wilkins, 1999; Briars & Siegler, 1984; Cowan et al., 1996; Fuson,1988).

Number principles assessed included order irrelevance, inversion and the commu-tative law of addition. Each of these principles was identified as an essentialcomponent of number sense in the international Delphi study, and each appears insome early mathematics assessments. None was identified as a component of numbersense expected at school entry in the Australasian Delphi study. In comparison withcounting skills, markedly fewer children demonstrated an understanding of thenumber principles ‘order irrelevance’, ‘inversion’ and the commutative law of addi-tion. The skills demonstrated by children in the current study support the view thatbasic counting skills develop prior to and independently of counting principles.

Number magnitude is a recurring theme in the early number sense literature(Griffin, 2003; Griffin et al., 1994; Laski & Siegler, 2007; Malofeeva et al., 2004;Ramani & Siegler, 2008) as is the mental number line often measured by the abilityto count on from a given number and the ability to provide the number both after andbefore a given number (Chard et al., 2005; Griffin, 2003; Griffin et al., 1994). Whilenumber magnitude was not identified in the Australasian Delphi study as a componentof number sense expected of children at school entry, in the current study, the vastmajority of children could identify which of the two groups to 10 is ‘more’ and couldorder two quantities to 10 on a ladder, with the lesser quantity placed lower on theladder. Clearly, the majority of children commence school with some understandingof number magnitude as it applies to the size and order of groups.

Gender differences and differences across type of setting

Despite the fact that boys did perform better on the test of Quantitative Concepts, over-all, there was no difference in performance between the boys and the girls on most ofthe components of number sense as measured in the present study. Subitising, whichis a component of number sense appearing in much of the literature, was the only

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


component for which there was a significant difference between boys and girls, withthe difference being in favour of girls. In the current study, subitising required imme-diate recognition of a random display of dots on a card, not immediate recognition ofdot patterns as they appear on dice or tens frames. Given that immediate attention wasrequired for a correct response to the subitising task, perhaps, it is not surprising thatgirls outperformed boys. The better performance of boys on the test of QuantitativeConcepts is consistent with the findings of Jordan, Kaplan, Olah, and Locuniak (2006),which suggest that at a kindergarten level, boys outperform girls on some measures ofnumber sense, non-verbal calculation and estimation. A substantial part of the test ofQuantitative Concepts involved recognising number sequences, and number sense, asmeasured by Jordan et al. (2006), included missing number. Performance in thiscomponent does not rely on high levels of language, and this may have contributedpartially to this result. The finding in the present study, that girls outperformed boyson the subitising task, warrants further investigation, as it appears to contradict thefindings of Jordan et al.

There were no significant differences in performance on number sense tasksbetween children attending preschools as opposed to childcare centres. Likewise,there were no significant differences between the performances of children inchildcare centres and preschools for any of the standardised measures. While simpleaddition was found to be approaching significance in favour of preschools, careshould be taken in interpreting this result as there was only one item assessing thiscomponent. In New South Wales, Australia, where the current study took place, it isa requirement that childcare centres employ a qualified early childhood teacher whenthey enrol more than 29 children (New South Wales Government Department ofCommunity Services, 2004). The childcare centres in the current study all employedat least one qualified early childhood teacher. Given that there were university-qualified early childhood teachers in both preschools and childcare centres includedin this study, it is probably not surprising that there were no significant differences inperformance across type of setting. This may not apply in other Australian states, orindeed in other countries, where this level of staffing for early childcare settings maynot apply.

Implications for practice

The data from the preschool number sense and standardised mathematics and recep-tive vocabulary assessments used in the current study provide valuable informationrelating to the range of early number skills and level of receptive vocabulary evidentin young children prior to school entry. For children in this study, who all had a chro-nological age of four or five years, there was indeed a considerable range in receptivevocabulary and mathematical ability. For example, some children would begin theirformal schooling with receptive vocabulary poorer than that of a typical two-year-old,and mathematics skills typical of children aged two years two months. On the otherhand, some children demonstrated the mathematics skills of children aged nine years.There was also a broad range of skills as demonstrated by the children’s performanceon the assessment of number sense components. Even so, the vast majority of the chil-dren were able to count to at least 10 and many were able to count as far as 20. Thismeans that the counting tasks traditionally included in early kindergarten mathematicsprogrammes may have already been mastered by the majority of children prior toschool entry. On the other hand, there is a small minority of children with very poorly

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


developed early counting and other number-related skills who, despite exposure tonumber-related activities in prior-to-school settings, have made very slow progress inmathematics in their initial early education settings. Such children may not respond tothe tasks that are included in many early kindergarten programmes. Some of thesechildren will have global delays while others may be identified with specific problemsin the area of mathematics.

Because there is such a broad range of number skills in children entering theschool system, a range of teaching practices needs to be employed in any one kinder-garten classroom if all children are to receive instruction at a level that reflects theirmathematical understanding and number sense at school entry. Also, some children inthe current study were to begin their formal schooling with a receptive vocabularylower than that of a typical two-year-old, while others demonstrated the receptivevocabulary knowledge of children over seven years. The wide range in receptivevocabulary must surely have implications for the current pedagogy that emphasisespeer discourse as instrumental in mathematics learning. Further research is needed todetermine whether the findings that boys outperformed girls on tasks involvingnumber sequences and that girls outperformed boys on subitising have implicationsfor differentiating kindergarten instruction for boys and girls.

The data reported in this study clearly indicate that many children begin schoolwith an intuitive understanding of number magnitude as it applies to quantities to 10and confirm that many children do start school with some form of mental numberline by which they make judgements about numbers. Questions worthy of furtherexamination include whether children, who begin school without any level ofmental number line are given adequate opportunities to develop one and whetherthere is an optimum time frame in which children should acquire some facility witha mental number line if they are to develop mental strategies for simple additionand subtraction.

The results of the current study highlight the fact that most children begin schoolwithout understanding counting principles such as order irrelevance, inversion or thecommutative law of addition. This raises questions as to whether early mathematicsprogrammes place sufficient emphasis on counting principles as opposed to countingskills. It also has implications for teaching practices in preschool settings in countrieslike Australia where there are moves towards providing more structured curriculum atthe preschool level (see The early years learning framework, Australian GovernmentDepartment of Education, Employment and Workplace Relations, 2009). An analysisof the problem-solving tasks provided to children in kindergarten may reveal thatmany tasks merely require counting rather than an understanding of inversion, orderirrelevance and the commutative law of addition. It could certainly be argued that chil-dren who begin school with an understanding of these three principles are well placedto develop the mental strategies for early addition and subtraction that are expected ofthem within the constructivist pedagogy underpinning the mathematics curriculum incountries such as the USA and Australia. Further research is needed to determinewhether children without such understandings are given adequate opportunities todevelop them. The extent to which children without an understanding of these princi-ples can benefit from the explanations of mental strategies provided by their peers isalso worthy of exploration. Likewise, an investigation of whether or not understandingof these principles is linked to the development of strategies, such as the ‘min’ strategyfor addition or part-whole strategies for solving two-digit addition and subtractionproblems, is warranted. It could be that in the early years, some children learn to

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


execute mental strategies, such as ‘counting on’, as rote procedures without a deepunderstanding of the principles that the strategies reflect and, as such, are not devel-oping the number sense crucial to higher-level skills.

The data from the assessment of components of early number sense and from thestandardised mathematics assessment used in this study provide valuable informationrelating to the range of early number skills evident in young children prior to schoolentry. The results highlight the fact that most children begin school without under-standing counting principles, such as order irrelevance, inversion or the commutativelaw of addition, and prompt the question as to whether early mathematics programmesadequately address these principles.

Limitations of the study

All children participating in the current study attended an early childhood facility andspoke English as their first language. It may be that children who have not receivedany out-of-home early care or education or who speak a first language other thanEnglish would demonstrate different skills to those reported in the current study. Thefact that the children attended centres located in a range of socio-economic areas, thatthe number of boys and girls participating in the research was roughly equal, and thatthe children’s mean age on the PPVT (III) and their mean chronological age were verysimilar support the assumption that this was a representative sample of children at theend of the year prior to school entry. With many of the components of number sensebeing assessed by a single item, it could well be argued that too few items wereincluded to provide a convincing assessment of each of the number sense components.The purpose of the assessment, however, was not to provide a comprehensive assess-ment of each of the individual components but to isolate potential components ofnumber sense that might be predictive of future mathematics difficulties and thatwould, therefore, warrant further investigation. The results reported in this study,nonetheless, provide a snapshot of mathematics skills demonstrated by children in thefinal months of the year prior to school.

Conclusion

Any proposed causal relationship between the construct of number sense and mathe-matics performance is still very much at an investigatory stage. The components ofnumber sense assessed in this study reflect components of number sense that appearin the broad number sense literature and components that were identified in the twoDelphi studies, but not all of them are included in any one number sense assessmentreported in the literature. There is a long history of research into young children’sdevelopment of number knowledge prior to formal schooling, and counting skills havelong been recognised as essential building blocks of mathematical understanding. Theresults reported in this paper confirm that the majority of children in Australia arelikely to commence school with basic counting skills to at least 10. The poorerperformance on number sense components reflecting number principles and numbermagnitude, demonstrated by children in the current study, raises the question as towhether or not these components are essential for the development of later mathemat-ical competence.

The results of the current study provide important information about number sensecomponents demonstrated by children at the end of preschool worthy of further

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


investigation. The results also reveal the range of language and mathematics skills thatmay exist in any kindergarten classroom and that need to be taken into considerationby teachers as they strive to provide teaching and learning opportunities that respondto the mathematics learning needs of individual children. This is particularly relevantto the classrooms of the early twenty-first century, which place high expectations onchildren to construct their own mathematical understanding through dialogue withtheir peers. Because of the broad range of ability and environmental backgrounds ofthe children participating in this research, the data reported in this study have thepotential to provide useful information about components of number sense believed tobe important for later mathematical learning. The findings may also have relevance tothe development of an assessment for screening for MD at school entry and for thedevelopment of early mathematics programmes for young children at risk of MD. Atthe very least, the data confirm that kindergarten teachers need to adopt a wide rangeof teaching strategies and activities to cater for the diverse mathematical understand-ing of children as they begin formal schooling.

ReferencesAunio, P., Hautamaki, J., & Van Luit, J.E.H. (2005). Mathematical thinking intervention

programmes for preschool children with normal and low number sense. European Journalof Special Needs Education, 20(2), 131–146.

Australian Bureau of Statistics. (2007). 2006 census community profile series. RetrievedDecember, 2007, from http://www.censusdata.abs.gov.au

Australian Government Department of Education, Employment and Workplace Relations.(2009). The early years learning framework. Retrieved March 15, 2010, from http://www.deewr.gov.au/EarlyChildhood/Policy_Agenda/Quality/Pages/EarlyYearsLearningFramework.aspx

Baroody, A.J., & Wilkins, J.L.M. (Eds.). (1999). The development of informal counting, number,and arithmetic skills and concepts. Reston, VA: The National Teachers of Mathematics.

Berch, D.B. (2005). Making sense of number sense: Implications for children with mathemat-ical disabilities. Journal of Learning Disabilities, 38(4), 333–339.

Briars, D., & Siegler, R.S. (1984). A featural analysis of preschoolers’ counting knowledge.Journal of Experimental Psychology, 20(4), 607–618.

Bryant, D.P. (2005). Commentary on early intervention for students with mathematicsdifficulties. Journal of Learning Disabilities, 38(4), 340–345.

Bryant, D.P., Bryant, B.R., Gersten, R., Scammacca, N., & Chavez, M.M. (2008). Mathemat-ics intervention for first- and second-grade students with mathematics difficulties. Theeffects of Tier 2 intervention delivered as booster lessons. Remedial and SpecialEducation, 29(1), 20–32.

Butterworth, B. (2005). The development of arithmetical abilities. Journal of Child Psychologyand Psychiatry, 46(1), 3–18.

Chard, D.J., Clarke, B., Baker, S., Otterstedt, J., Braun, D., & Katz, R. (2005). Usingmeasures of number sense to screen for difficulties in mathematics: Preliminary findings.Assessment for Effective Intervention, 30(2), 3–14.

Chiappe, P. (2005). How reading research can inform mathematical difficulties. Journal ofLearning Disabilities, 38(4), 313–317.

Clarke, B., & Shinn, M.R. (2004). A preliminary investigation into the identification anddevelopment of early mathematics curriculum-based measurement. School PsychologyReview, 33(2), 234–248.

Clements, D.H., & Sarama, J. (2008). Experimental evaluation of the effects of a research-based preschool mathematics curriculum. American Educational Research Journal, 45(2),443–494.

Clements, D.H., Sarama, J.H., & Liu, X.H. (2008). Development of a measure of early math-ematics achievement using the Rasch model: The research-based early maths assessment.Educational Psychology, 28(4), 457–482.

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


Cobb, P., Wood, T., & Yackel, E. (1991). A constructivist approach to second grade mathe-matics. In E. von Glaserfeld (Ed.), Radical constructivism in mathematics education(pp. 157–176). Dordrecht: Kluwer.

Cowan, R., Dowker, A., Christakis, A., & Bailey, S. (1996). Even more precisely assessingchildren’s understanding of the order-irrelevance principle. Journal of Experimental ChildPsychology, 62(1), 84–101.

Doig, B., McCrae, B., & Rowe, K. (2003). A good start to numeracy. Commonwealth ofAustralia. Retrieved November 14, 2009, from http://www.dest.gov.au/common _topics/publications _resources/All_Publications_AtoZ.htm

Dowker, A. (2005). Early identification and intervention for students with mathematicsdifficulties. Journal of Learning Disabilities, 38, 324–332.

Dunn, L.M., & Dunn, L.M. (1997). Examiners manual. Peabody picture vocabulary test (3rded.). Circle Pines, MN: American Guidance Service.

Fuchs, L.S., Fuchs, D., Compton, D.L., Bryant, J.D., Hamlett, C.L., & Seethaler, P.M. (2007).Mathematics screening and progress monitoring at first grade: Implications for respon-siveness to intervention. Exceptional Children, 73(3), 311–330.

Fuchs, L.S., Fuchs, D., & Karns, K. (2001). Enhancing kindergarteners’ mathematical devel-opment: Effects of peer-assisted learning strategies. Elementary School Journal, 101(5),495–510.

Fuson, K.C. (1988). Children’s counting and concepts of number. New York: Springer-Verlag.Gersten, R., & Chard, D. (1999). Number sense: Rethinking arithmetic instruction for students

with mathematic disabilities. Journal of Special Education, 33(1), 18–28.Gersten, R., Clarke, B.S., & Jordan, N.C. (2007). Screening for mathematics difficulties

in k-3 students (Report). Portsmouth, NH: RMC Research Corporation, Center onInstruction.

Gersten, R., Jordan, N.C., & Flojo, J.R. (2005). Early identification and interventionsfor students with mathematics difficulties. Journal of Learning Disabilities, 38(4),293–304.

Ginsburg, H.P. (1997). Mathematics learning disabilities: A view from developmentalpsychology. Journal of Learning Disabilities, 30(1), 20–33.

Griffin, S. (2003). Laying the foundation for computational fluency in early childhood. TeachingChildren Mathematics, 9(6), 306–313.

Griffin, S.A., Case, R., & Siegler, R.S. (1994). Right start: Providing the central conceptualprerequisites for the first formal learning of arithmetic to students at risk for school fail-ure. In K. McGilly (Ed.), Classroom lessons: Integrating cognitive theory and classroompractice (pp. 25–49). Cambridge: MIT Press.

Howell, S., & Kemp, C. (2004). The role of number sense in the identification and preventionof mathematics disability: A consideration of the phonemic awareness/number senseanalogy. Australasian Journal of Special Education, 28(2), 65–78.

Howell, S., & Kemp, C. (2005). Defining early number sense: A participatory Australianstudy. Educational Psychology, 25(5), 555–571.

Howell, S., & Kemp, C. (2006). An international perspective of early number sense: Identify-ing components predictive of difficulties in early mathematics achievement. AustralianJournal of Learning Disabilities, 11(4), 197–207.

Howell, S., & Kemp, C. (2009). A participatory approach to the identification of measures ofnumber sense in children prior to school entry. International Journal of Early YearsEducation, 17(1), 47–65.

Jordan, N.C., Kaplan, D., Locuniak, M.N., & Ramineni, C. (2007). Predicting first-grade mathachievement from developmental number sense trajectories. Learning DisabilitiesResearch and Practice, 22(1), 36–46.

Jordan, N.C., Kaplan, D., Olah, L.N., & Locuniak, M.N. (2006). Number sense growth inkindergarten: A longitudinal investigation of children at risk for mathematics difficulties.Child Development, 77(1), 153–175.

Jordan, N.C., Kaplan, D., Ramineni, C., & Locuniak, M.N. (2009). Early math matters:Kindergarten number competence and later mathematics outcomes. DevelopmentalPsychology, 45(3), 850–867.

Landerl, K., Bevan, A., & Butterworth, B. (2004). Developmental dyscalculia and basicnumerical capacities: A study of 8–9 year-old students. Cognition, 93, 99–125.

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14


Laski, E.V., & Siegler, R.S. (2007). Is 27 a big number? Correlational and causal connectionsamong numerical categorization, number line estimation, and numerical magnitudecomparison. Child Development, 73(6), 1723–1743.

Locuniak, M.N., & Jordan, N.C. (2008). Using kindergarten number sense to predict calcula-tion fluency in second grade. Journal of Learning Disabilities, 41(5), 451–459.

Malofeeva, E., Day, J., Saco, X., Young, L., & Cianco, D. (2004). Construction and evalua-tion of a number sense test with head start children. Journal of Educational Psychology,96(4), 648–659.

Mazzocco, M.M.M. (2005). Challenges in identifying target skills for math disability screen-ing and intervention. Journal of Learning Disabilities, 38(4), 318–323.

Mulligan, J., Mitchelmore, M., & Prescott, A. (2005). Case studies of children’s developmentof structure in early mathematics: A two-year longitudinal study. In H.L. Chick & J.L.Vincent (Eds.), Proceedings of the 29th conference of the International Group for thePsychology of Mathematics Education (Vol. 4, pp. 1–8). Melbourne: PME.

National Council of Teachers of Mathematics Commission on Standards for School Mathe-matics. (1989). Curriculum and evaluation standards for school mathematics. Reston,VA: Author.

New South Wales Government Department of Community Services. (2004). Review of thechildren’s services regulation 2004: Discussion paper. Retrieved July 10, 2009, fromhttp://www.community.nsw.gov.au/for_agencies_that_work_with_us/childrens_services/regulations/review_of_the_regulation.html

Okamato, Y., & Case, R. (1996). Exploring the microstructure of children’s conceptual struc-tures in the domain of number. Monographs of the Society for Research in Child Develop-ment, 61, 27–59.

Ramani, G.B., & Siegler, R.S. (2008). Promoting broad and stable improvements in low-income children’s numerical knowledge through playing number board games. ChildDevelopment, 79(2), 375–394.

Steffe, L. (1991). The constructivist teaching experiment: Illustrations and implications. In E.von Glaserfeld (Ed.), Radical constructivism in mathematics education (pp. 177–194).Dordrecht: Kluwer.

Van Luit, J.E.H., & Schopman, E. (2000). Improving early numeracy of young children withspecial education needs. Remedial and Special Education, 21(1), 27–40.

von Glaserfeld, E. (1995). A constructivist approach to teaching. In L. Steffe & J. Gale (Eds.),Constructivism in education (pp. 3–16). Hillsdale, NJ: Erlbaum.

Woodcock, R.W., McGrew, K.S., & Mather, N. (2001). Woodcock-Johnson III tests ofachievement (Australian ed.). Itasca, IL: Riverside Publishing.

Dow

nloa

ded

by [

Am

s/G

iron

a*ba

rri L

ib]

at 0

4:57

14

Nov

embe

r 20

14

Documents

Assessing preschool number sense: skills demonstrated by children prior to school entry