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IJEC (2012) 44:91–114 DOI 10.1007/s13158-011-0050-x ORIGINAL ARTICLE

The Mathematical Competencies of Toddlers Expressed in Their Play and Daily Life Activities in Norwegian Kindergartens Elin Reikera˚s • Inger Kristine Løge Ann-Mari Knivsberg



Published online: 13 January 2012  Springer Science+Business Media B.V. 2012

Abstract Research on toddlers’ mathematical knowledge is sparse. Studies on children’s mathematical competencies before school age have mostly focused on older children. Few of the previous studies have included large groups of toddlers, few have been conducted in natural settings, and few have been directed at a broad field of mathematical knowledge. The objective of this study was to investigate which mathematical competencies a large group of toddlers’ in Norwegian kindergartens expressed through play and daily life activities. A total of 1,003 children participated. Their competencies were registered when they were between 30 and 33 months. The assessment material consisted of 36 items, divided into three main areas: number and counting, geometry and problem solving. The information was collected through authentic assessment; the staff in the kindergartens observed the toddlers’ competencies in play and daily life activities. The competencies were registered as mastered, partly mastered or mastering not yet observed. The toddlers showed mathematical competencies in all areas observed. A wide dispersion was found; both for the total score and the subareas’ scores. The largest variance was found in number and counting. Our participants displayed lower levels of competencies in using number words and reciting number sequences than reported from previous studies and higher competencies in puzzle-making and following instructions on spatial words. The results indicate that the assessment material may be a valuable tool for the preschool teachers in identifying the variety of competencies mastered by the children in kindergarten. The need for future research is highlighted and discussed. E. Reikera˚s (&)  A.-M. Knivsberg Faculty of Arts and Education, National Centre for Reading Education and Research, University of Stavanger, Hulda Garborg’s House, 4036 Stavanger, Norway e-mail: [email protected] I. K. Løge Faculty of Arts and Education, Centre for Behavioural Research, University of Stavanger, Hulda Garborg’s House, 4036 Stavanger, Norway

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Keywords Toddlers’ mathematics  Authentic assessment  Natural settings in Norwegian kindergartens Re´sume´ Les recherches sur les connaissances mathe´matiques des jeunes enfants sont rares. Les e´tudes re´alise´es sur les compe´tences mathe´matiques des enfants avant l’aˆge scolaire ont surtout porte´ sur des enfants plus aˆge´s. Peu d’entre elles ont inclus de grands groupes de jeunes enfants, ont e´te´ mene´es en milieu naturel et se sont inte´resse´es a` un domaine de connaissances mathe´matiques large. L’objectif de la pre´sente e´tude e´tait de savoir quelles compe´tences mathe´matiques e´taient exprime´es dans le jeu et les activite´s de la vie courante par un grand groupe de jeunes enfants des classes maternelles de Norve`ge. Au total, 1003 enfants ont participe´. Leurs compe´tences ont e´te´ enregistre´es lorsqu’ils avaient entre 30 et 33 mois. Le mate´riel d’e´valuation se composait de 36 items, divise´s en trois sphe`res principales: les chiffres et la nume´ration, la ge´ome´trie et la re´solution de proble`mes. Les informations ont e´te´ recueillies par des e´valuations authentiques; l’e´quipe pre´sente dans les classes maternelles a observe´ les compe´tences des enfants durant les jeux et les activite´s de la vie courante. Celles-ci e´taient enregistre´es comme maıˆtrise´es, partiellement maıˆtrise´es ou maıˆtrise pas encore observe´e. Les enfants ont montre´ des compe´tences mathe´matiques dans toutes les sphe`res e´tudie´es. Une grande dispersion a e´te´ constate´e, tant au score total qu’aux scores des sous sphe`res. La plus grande variance se trouve a` la sphe`re des chiffres et de la nume´ration. En comparaison avec des e´tudes ante´rieures, nos participants ont des niveaux infe´rieurs de compe´tences dans l’utilisation de mots lie´s aux nombres et dans la re´citation de se´quences de chiffres, mais ils ont des compe´tences supe´rieures pour faire des casseteˆte et suivre des consignes sur des mots lie´s a` l’espace. Les re´sultats indiquent que le mate´riel d’e´valuation pourrait s’ave´rer un outil valable pour les enseignants de maternelle dans l’identification des diffe´rentes compe´tences maıˆtrise´es par les jeunes enfants de leur classe. Le besoin de poursuivre les recherches est souligne´ et examine´. Mots cle´s Mathe´matiques chez les jeunes enfants  E´valuation authentique  Cadre naturel dans des classes maternelles en Norve`ge Resumen La investigacio´n en cuanto al conocimiento de matema´ticas de nin˜os pequen˜os es poco abundante. Los estudios de la competencia matema´tica de nin˜os preescolares se han enfocado, en su mayorı´a, en nin˜os mayores. Muy pocos de los estudios realizados anteriormente han incluido grandes grupos de nin˜os pequen˜os, pocos han sido llevados a cabo en entornos naturales, y pocos han estado dirigidos a un amplio campo de conocimiento matema´tico. El objetivo del presente estudio fue el de determinar que´ competencia matema´tica demostro´ un grupo grande de nin˜os pequen˜os de jardines de infancia noruegos a trave´s del juego y actividades diarias. Un total de 1,003 nin˜os participaron. Sus competencias fueron registradas cuando tenı´an entre 30 y 33 meses de edad. El material de evaluacio´n consistio´ de 36 artı´culos, divididos en tres a´reas principales: nu´meros y conteo, geometrı´a y resolucio´n de problemas. La informacio´n fue recabada a trave´s de una evaluacio´n aute´ntica; el personal de los jardines de infancia observo´ las competencias de los

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nin˜os durante sus juegos y actividades de vida diaria. Las actividades fueron registradas como dominadas, parcialmente dominadas o domino au´n no observado. Los pequen˜os nin˜os demostraron competencia matema´tica en todas las a´reas observadas. Se encontro´ una amplia dispersio´n, tanto en las puntaciones finales como en las puntuaciones de las sub-a´reas. Las variaciones ma´s grandes se encontraron en nu´meros y conteo. Nuestros participantes mostraron niveles ma´s bajos de competencia al utilizar los nu´meros en palabras y al recitar las secuencias de nu´meros en comparacio´n con aquellos reportados en estudios anteriores en donde habı´a mayor competencia en el armar rompecabezas y el seguir instrucciones en palabras espaciadas. Los resultados indican que el material de evaluacio´n puede ser una valiosa herramienta para los maestros de preescolar para identificar la variedad de competencias que son dominadas por los nin˜os en el jardı´n de infancia. Se resalta y argumenta la necesidad de futurasinvestigaciones. Palabras clave Matema´tica de nin˜os pequen˜os  evaluacio´n aute´ntica  entorno natural en jardines de infancia noruegos

Introduction Children acquire many mathematical competencies during their first years, and findings support that early mathematical development is important for later mathematical achievement (Aunola et al. 2004; Claessens et al. 2006; Hannula and Lehtinen 2005; Prentice Starkey et al. 2004). Research in this field has increased during the past 10 years, but more knowledge is needed (Baroody et al. 2006). Few studies have surveyed mathematical knowledge in children younger than 5 years (Sarama and Clements 2009). Research is particularly scarce on early mathematics observed in natural settings (English 2004; Tudge et al. 2008). These factors are taken into account in this study, which focuses on mathematical competencies observed in play and the daily life activities in a large group of toddlers in Norwegian kindergartens. Toddler age here refers to the age between 1 and 3 years. The toddlers’ behaviour will reflect their acquired skills and knowledge, their competencies. As background, we shall first present some of the research that has been carried out on children’s early mathematical development. Children’s Early Mathematics Children seem to be born with a driving force to explore their surroundings, to make sense of them and set things in order (Bjo¨rklund 2007; Geist 2009). Infants a few hours old, for instance, classify by discriminating their mothers from strangers (Alan and Paul 2001; Bushnell 2001). Infants also pay attention to numerical changes (Cordes and Brannon 2009; Wood and Spelke 2005; Xu et al. 2005; Xu and Spelke 2000), and choose between different problem-solving strategies (Wellman et al. 1985). The child’s exploration of and interaction with his or her surroundings involves many mathematical concepts that give the child valuable mathematical experience that he or she learns from. When reaching toddler age, the child has

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knowledge in a broad field of mathematics, for example, in number and quantitative thinking, geometry and problem solving (e.g. Baroody et al. 2006; Bjo¨rklund 2008; Solem and Reikera˚s 2008). Toddler’s Mathematics Number and quantitative thinking is often said to be the most important mathematical area in early childhood development, and is also the area in which most research has been conducted (Sarama and Clements 2009). Although infants can observe numerical changes (Cordes and Brannon 2009; Wood and Spelke 2005; Xu et al. 2005; Xu and Spelke 2000), children need a number of experiences to develop more conscious relation to quantities (Mix et al. 2002; Sophian 2008). Through the toddler age the child develops important quantitative skills such as making judgments of ‘‘more’’ (Brannon et al. 2004), to distinguish between one and many (Spelke and Kinzler 2007), knowledge of number words (Durkin et al. 1986) and making bijections (Geist 2009). The understanding of the role of units in the numerical representation of quantities is, however, still not functional until several years later (Sophian and Kailihiwa 1998). The children for example have to learn all five principles central to counting which are outlined by Gelman and Gallistel (1978). The different quantitative components seem to develop as separate skills before they are combined, and the development of quantitative skills lasts through the entire childhood period (Mix et al. 2002). Geometrical experiences and concepts about space, shape, pattern, and order are also central in early mathematical development (Sarama and Clements 2009). The development of spatial competence begins early, and many visual and other perceptual experiences as well as the child’s motor development contribute to this (Acredolo 1990; Clearfield 2004). By the age of 16 months, children begin to show the hierarchical combinations that characterise adult spatial coding (Newcombe and Huttenlocher 2000). As toddlers, they are capable of orientating in their nearest surroundings and use their spatial knowledge when playing with blocks (Geist 2009). Recognition of shapes is the tool the children use to establish order of their surroundings by classifying and naming objects (Clements and Sarama 2009; Oakes and Madole 2003). Through infant age, toddler age and in preschool age children gain important experience and develop concepts through tactile and visual senses of different shapes as they recognise, distinguish and classify (Geist 2009). Puzzle-making gives the children practise in recognition of parts of shapes and putting the parts together (Montford and Readdick 2008). Recognition and analyses of patterns are also essential components in mathematical development, for geometry, arithmetic and algebraic thinking (Sarama and Clements 2009). Through toddler and preschool age the children gain experience with patterns and produce their own (Seo and Ginsburg 2004). The children’s problem solving and logical reasoning processes are found to be powerful facilitators of early learning, even more than specific items of mathematical knowledge (Perry and Dockett 2002), and the children use many strategies (Chen and Siegler 2000). They can reason and draw conclusions if the task is motivating and the context familiar (DeLoache et al. 1985; English 2004; Goswami 2001).

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The toddlers’ logical reasoning and problem solving are mainly expressed through action, and they are not yet fully able to express themselves verbally (Bjo¨rklund 2007). The children are surrounded by adults and other children who use language to explain sizes, shapes, numbers, relations, directions, quantities, etc. This interactive and linguistic context is important for the children to learn the language of mathematics (Bjo¨rklund 2008; Durkin et al. 1986). Mathematical language is classified in some taxonomies in the problem-solving area together with logical reasoning (Magne 2003) even though this language develops through interaction with acquisition of skills in most other mathematical areas. Verbal counting is, for instance, central in the quantitative development (Mix et al. 2002), and relational language is found to be a powerful contribution to spatial learning (Gentner 2008). In the same way, quantitative development affects geometrical development and not only arithmetical development (Mix et al. 2002). Geometrical development also supports the learning of numbers and later arithmetic (Arcavi 2003). Rationale for Performing the Study The different parts of mathematics are, as described, not separate. Most studies of children’s early mathematics have, however, dealt mainly with arithmetic; few studies have focused on other aspects of children’s mathematical development such as shape, space, measurement, etc. (Dowker 2005). Research that covers more than one area in the child’s mathematical development is needed to obtain knowledge on how overall mathematical performance develops (Aunola et al. 2004). This study takes this into account. Children younger than 3 years have not been included in many studies on mathematical development (Bjo¨rklund 2008). Studies on toddlers’ mathematical competencies are, in other words, few and knowledge in this area is consequently limited. Most research on small children’s mathematical competencies has additionally been performed in cultures with a more academic approach to early childhood education than the Nordic, which has a dominating social pedagogical tradition with a holistic approach (Jensen 2009; OECD 2006). Differences in mathematics in early age have been reported from cross-cultural studies (Aunio et al. 2008; Geary et al. 1993; Stevenson 1987; Yuzawa et al. 1999), which indicate that existing research results cannot be attributed directly to Norwegian children. Observations of the toddlers’ interaction with their surroundings are important to obtain information on small children’s mathematical competencies (Bjo¨rklund 2008; Piaget and Inhelder 2002; Sa¨ljo¨ 2001; Vygotskij and Davydov 1997), but little research has so far been done in children’s natural environment (English 2004; Tudge et al. 2008). In a Norwegian kindergarten context where play is the main pedagogical tool (Ministry of Education and Research 2006a), research in a natural environment means identifying children’s competencies as they are expressed through play and daily life activities. Data obtained in this way is needed, as much of our current knowledge is based on standardised assessment or surveys on reported behaviour (Downer et al. 2010).

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Most studies report only mastering or no mastering and thereby lose important information about the period when support is still needed to develop competencies. Competencies develop gradually, and a child may initially need much help and support to partly master a task. With experience the need for support will gradually diminish until the child can automatically conduct the task by him or herself as described in the theory of the Zone of Proximal Development by Vygotskij (1978). In this study, information about this stage between no mastering and mastering was included. Mathematics was included for the first time in the latest framework plan for Norwegian kindergartens and was defined as a central learning area (Ministry of Education and Research 2006a). In the 3-year long preschool teacher training mathematics in kindergarten is a compulsory subject (Ministry of Education and Research 2003). Mathematics is, however, reported to be under-investigated in Norwegian kindergartens (Borg et al. 2008), and it has been underlined that quantitative and longitudinal studies in kindergarten children’s competencies are needed (Borg et al. 2008; Greve and Solheim 2010). Information on toddlers’ mathematical competencies enables the preschool teachers to arrange the kindergarten environment in a way that can give the toddlers the possibility to develop their potentials. Based on the information presented, the following research question was formulated: •

Which mathematical competencies do a large group of toddlers express through play and daily life activities in Norwegian kindergartens?

Method Participants The participants were 1,003 toddlers, 513 males and 490 females. They were enrolled in pedagogical programmes in kindergartens. A total of 16.5% of the toddlers were multi-linguistic. Data were collected when they were between 30- and 33-month-old. All parents in these kindergartens with children in appropriate age were invited to let their children participate. The toddlers were in 185 groups in the kindergartens, most of them in toddler groups with children aged 1–3 years. These groups usually consist of 9–10 toddlers and a staff of 3 adults. In Norway, the majority of children are enrolled in kindergartens between the age of 1 and 2 years: 87% of the 2-year-old and 95% of the 3-year-old children are in kindergartens1. This study was conducted as part of a longitudinal, multidisciplinary project, which is carried out in close cooperation between the National Centre for Reading Education and Research at the University of Stavanger and the municipality of Stavanger. 1

http://www.ssb.no/english/subjects/04/02/10/barnehager_en/tab-2011-03-15-04-en.html.

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The objective in the longitudinal project is to generate new knowledge on children’s development, from the age of 2‘ to 10 years. Focus in the preschool period is on knowledge within and between the areas of language, mathematics, movement and social skills, and in the early school period on and between reading, spelling and arithmetic. The preschool children’s skills and competences will be related to the same children’s later skills in reading, spelling and arithmetic. Assessment is carried out four times, when the children are 33 and 57 months, and when they are 7‘- and 9‘-year-old. The aim is to identify early developmental factors that may enhance or inhibit the acquisition of reading, spelling and arithmetic. The aim is furthermore to generate knowledge important for early identification and stimulating of children that may struggle to acquire these basic skills. Competence enhancement for professionals working with the children in the municipal services is also a central objective. All kindergartens in the municipality were invited to take part in the larger project, and all municipality owned kindergartens and a number of kindergartens owned by groups of parents, churches, private enterprises, etc. participated. They all worked according to The Framework Plan for the kindergartens (Ministry of Education and Research 2006a). This is obligatory for all kindergartens in Norway regardless of ownership, and they all receive funding from the municipality. The data on mathematics presented here are from the first assessment carried out in the larger project. Material The observation material used for assessment is called ‘‘The Mathematics, the Individual, and the Environment’’, and consists of a handbook describing the observations to be done (Davidsen et al. 2008b) and a registration form for each child whose skills and competencies are to be observed (Davidsen et al. 2008a). The Norwegian abbreviation of the material is MIO (Matematikken, Individet og Omgivelsene). In the handbook, every observation item is described with three examples (Reikera˚s 2008), see a translated example in Appendix 1. The material has been developed in Norway for the age band from 2 to 5 years, and is constructed for use in kindergartens. The registration form has 36 items, divided into three main areas: number and counting, geometry and problem solving. The main areas are divided into two subareas. Table 1 describes what is to be observed. The registration form is constructed as a circle with three rings of items, displayed in Fig. 1. The items are of increasing difficulty with the easiest items in the inner circle, and the most demanding in the outer. The circle has 36 areas, one for each of the items to be observed. The area for an item is to be marked completely with a pen or a pencil when the child shows competence in all situations. When the child is beginning to show competence, in some situations, or with help from staff, the area is filled with stripes. When competence has not yet been observed by the staff in the kindergarten, the area is not filled in.

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Table 1 The items in the MIO registration form Number and counting Counting and series of numbers C1. Distinguishes between one and many C2. Uses number words (e.g. ‘‘I have a thousand cars’’) C3. Has started pointing and at the same time using number words (not necessarily matching objects and number words) C4. Perceives number of objects up to three without having to count them (e.g. dots on a dice) C5. Counting to five while correctly pointing at objects. C6. Can recite the number sequence up to ten Enumeration E1. Fetches two objects on request E2. Can hand out one item to each person (e.g. one spade to each child in the sandpit) E3. Fetches three objects on request E4. Shows with the fingers how old he or she is E5. Sets the table for five persons E6. Can answer how many there are after having counted five objects Geometry Shape and space SS1. Can point at different parts of the body (at least four: arm, foot, eye etc.,) SS2. Shows that he or she distinguishes between different shapes (e.g. fit the pieces into a peg board or use a sorting box) SS3. Puzzles a jigsaw with 3–4 pieces into a picture SS4. Can by request go to a fixed place in the room (e.g. ‘‘The queen commands: go to the doll corner’’) SS5. Draws a human body (at least a head with eyes inside and legs, a ‘‘tadpole’’ man) SS6. Copies simple figures (e.g. on paper, in the sand) Pattern and order PO1. Places a picture on an identical picture (e.g. when playing a lotto game) PO2. Is interested in rhythms and movement (demonstrated e.g. by ‘‘dancing’’, clapping hands) PO3. Has knowledge about the daily routines (e.g. ‘‘When we have finished the meal, we shall go outside.’’) PO4. Puts objects in a line according to size (e.g. cars, dolls) PO5. Makes his or her own patterns (e.g. with beads, in drawings, while jumping) PO6. Sorts objects according to one characteristic (e.g. shape, size or colour) Problem solving Mathematical language ML1. Distinguishes between the concepts large and small ML2. Knows what is up and what is down (e.g. ‘‘Sit down on the floor’’) ML3. Uses words that describe toys (e.g. the soft bear, the red car) ML4. Follows instructions on placing (e.g. over the table, under the bench, through the tunnel) ML5. Uses words describing relationship between sizes (e.g. the balloon is lighter than the stone, ‘‘I have longer hair than you’’) ML6. Shows what is in the middle (e.g. cars lined up in a row) Logical reasoning LR1. Knows what outdoor clothing to put on when it is raining

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Table 1 continued LR2. Tidies toys and puts them in the right places LR3. Can share equally with a friend (e.g. four or six crayons) LR4. Fetches objects needed for the activity (e.g. hammer in the carpenter storage, doll in the doll corner) LR5. Reasons out what comes first and last when dressing LR6. Knows the difference between what has happened and what is going to happen

Fig. 1 MIO circle

The assessment material can be used in specific periods during a year or as an instrument to register whenever it has been observed that the child has acquired or nearly acquired a new skill. Natural situations and children’s free play activities are recommended as the main observational arena (Davidsen et al. 2008b). The material required for observing the items (such as puzzles, toys with different sizes and shapes, etc.) should be easily accessible to the children in the kindergarten during most of their free playtime. Only if it is difficult to observe the child’s competencies thoroughly, the adults should organise play-based activities to focus on the items’ content. Examples of such activities are given in the handbook (Davidsen et al. 2008b). The instrument has been developed in accordance with earlier research on children’s mathematical development. The items were discussed and tried out in close cooperation with preschool teachers, and the registration form was piloted with more than 1,000 children in kindergartens throughout Norway. The description

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of how the material was developed is only published in Norwegian (Løge and Lunde 2008), and we shall therefore provide information on the development of the material here. The first version had 9–15 items for each of the subareas. This was tried out for 411 children in the age range 2–5 years (97 aged 24–35 months, 156 aged 36–47 months, and 158 aged 48–60 months) in eight Norwegian municipalities. Each municipality had a contact, a preschool teacher trained to use the material and who was responsible for training the staff in the kindergartens where the observations were made. After observations had been carried out, the staff returned the registrations and their judgement of the content of the material to the group developing the MIO. The overall aim with the material was that it should identify children at risk of getting difficulties. It was consequently decided that the items should be adapted to age groups in following way: *70% of children aged 24–35 months should master the items in the inner circle, and 20% of them partly master these items. In the same way, the middle circle was adapted to 70% mastering and 20% partly mastering for the children aged 36–47 months, and this was also the case for the outer circle in the older age group. This, along with scientific consideration, contributed to a revised material with 36 items. For some of these items smaller linguistic changes were made in line with suggestions from the preschool teachers. The revised material was tried out in 12 new municipalities. Five hundred and sixty-two children participated: 109 aged 24–35 months, 207 aged 36–47 months, and 246 aged 48–60 months. All of the preschool teachers making the observations were trained in how to conduct them by members of the group that developed the material. The way in which different observers influenced the results was tested when 90 children were observed by two independent observers. The Wilcoxon signed rank test showed good inter-rater reliability for most items with values above 0.3; only three items had values between 0.05 and 0.10. Significant differences (\0.05) were found for three items. For these items, some smaller linguistic adjustments were made before the material was published. Also, the examples in the handbook were rewritten to make them more precise. Computation of Cronbach’s alpha showed a reliability of 0.96 indicating a good internal consistency. No other Norwegian material is available for assessment of toddlers’ mathematical competence. Validation towards other material could therefore not be carried out. Procedure The longitudinal project, of which this study is part, has been approved by the Norwegian Social Science Data Services. Participation is based on the parents’ voluntary and written consent. They can withdraw their children from the project at any time. Inclusion criteria were that the child, before the age of 30 months, was enrolled in a kindergarten participating in the project, and that the child was born in the period between July 1st 2005 and December 31st 2007. No exclusion criteria were set up. A total of 1,364 children are registered as participants, and data from 1,003 were available when the current study was conducted.

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Data was collected through structured observation of the children’s competencies during play and daily life activities in the kindergartens. The observations and registrations of competencies were made by the staff in the kindergartens. The method is gentle and non-intrusive for the child whose competencies were observed in natural surroundings by familiar adults and adults who knew the child. According to Bagnato (2007), this method gives ecologically valid data, it reveals information on the child’s functional behaviour, and is therefore named authentic assessment. The method is in line with the social pedagogical tradition in Norwegian kindergartens. In this tradition children’s own play and daily life activities are more central than formal teacher-driven activities (OECD 2006). It is also in line with the national framework plan for the kindergartens (Ministry of Education and Research 2006a). Norwegian trained preschool teachers have solid training and experience in observing children. This is an essential part of their education (Ministry of Education and Research 2003). The importance of observation as a pedagogical tool is also underlined in the Framework plan for the kindergartens (Ministry of Education and Research 2006a). The method did, however, make a close cooperation with staff in the kindergartens both necessary and natural because the staff needed to be familiar with the assessment material and how to use it. The material was first presented to preschool teachers from each of the kindergartens at a dialogue seminar. They discussed, in groups, how to carry out the observations in the best possible way during daily life in the kindergartens. Before the observation started, up-dated information on young children’s mathematical development and training in how to use the material were given in courses especially developed for this purpose. Also special in-depth courses in early mathematics were given at the request of the preschool teachers. The necessary observation material for each participating child and the handbook with detailed descriptions of the items were forwarded to each participating kindergarten units. The data to be registered were the competencies demonstrated by the children in the age period from 30 to 33 months. For each item several observations of the child’s competence were done. To obtain as good data as possible the following procedures were used in addition to those described in the handbook: The child had to master each item in different situations over time. The registrations were done when at least two of the staff in the kindergarten had, independently, observed that the child had mastered or partly mastered the various items. The preschool teachers had the main responsibility for assuring the quality of requisite observations for each item. The material was returned to the researchers for statistical purposes when the observations had been carried out. When the necessary registration in the database had been conducted, the material was sent back to the kindergartens. There it could be used in planning and implementing pedagogical programmes, and for sharing information with and cooperating with parents. When coding the results three levels of achievement were set: 2 points for mastering, 1 for partly mastering and 0 for not yet mastering. This gives a total of 72 points for all items, 24 points in each main area and 12 points in the subareas.

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102 Table 2 The toddlers results in the different areas (N = 1003)

Mean

SD

Variance

Min

Max

6.67

0

12

Numbers and counting (N&C) Counting and Series of numbers

4.31

2.58

Enumeration

4.70

2.23

4.99

0

12

Sum all items (N&C)

9.01

4.36

19.03

0

23

Shape and space

5.95

2.02

4.07

0

12

Pattern and order

5.14

1.87

3.51

0

12

Sum all items (G)

11.10

3.54

12.53

1

22

Geometry (G)

Problem solving (P) Mathematical language

5.58

2.16

4.65

0

12

Logical reasoning

5.14

2.11

4.44

0

12

Sum all items (P)

10.72

3.95

15.62

0

24

30.82

10.80

116.54

2

65

Total sum all items all areas

Results and Discussion The main aim of this study was to investigate which competencies, in a broad field of mathematics, a large group of toddlers expressed through play and daily life activities in Norwegian kindergartens. The results for the three main areas and the six subareas are presented in Table 2. Table 2 illustrates a large dispersion for each of the subareas and also for the total score, the latter varying from 2 to 65. This was expected, due to the large number of participants and the fact that no exclusion criteria for participation had been set up. The Kolmogorov–Smirnov test of normality revealed non-normal distribution for the sum score of all items (p \ .01). This is quite normal for large samples where graphical inspection often gives a better picture (Pallant 2007). The normal Q–Q plot showed a reasonably straight line, and we assumed a normal distribution. This was also the case for the six subareas. When measuring the inner consistency of MIO, Cronbach’s alpha coefficient was 0.91, and according to Coolican (2004) considered to be good. Results on item level in each of the three areas are presented in the following. Results showing mastering or that mastering had not yet been observed in the kindergarten are compared with previous reports on children’s mathematics. For the evaluation of this comparison, it should be noted that most of the previous studies on toddler’s mathematics have been conducted with far fewer participants than this study. The studies we refer to using large samples of children have older participants, for example, Aunio et al. (2006, 2008) in which children from 4 years were included. In addition, information about the results of partly mastered items has not been included in most earlier studies; therefore the results on partly mastering can only be commented and reflected on.

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The Mathematical Competencies of Toddlers Table 3 Percentage performance score in the area Numbers and counting

Item

103

Mastering

Partly mastering

No mastering

Counting and series of numbers C1

86.9

5.3

7.8

C2

50.5

15.1

34.4

C3

32.8

10.8

56.4

C4

8.5

8.8

82.7

C5

7.2

2.4

90.4

C6

7.3

2.3

90.4

E1

82.9

7.1

10.0

E2

82.2

7.6

10.2

E3

21.1

9.1

69.8

E4

29.3

7.5

63.2

E5

1.1

1.1

97.8

E6

1.8

0.5

97.7

Enumeration

Number and Counting Table 2 shows that the variance is largest in the area number and counting. Counting and series of numbers (C) is the subarea where the toddler’s competencies differed most. The toddlers’ mean performance on each item is shown in Table 3. The results presented in Table 3 illustrate that the items have varying degrees of difficulty. The majority of the participants could distinguish between one and many (Item C1), in line with other findings (e.g. Wynn 1992). Half of the group used number words (Item C2), a much lower result than Fuson (1988) reported. It should also be noted that for more than a third of the group mastering number words was not observed at all. This may mirror a cultural difference; in interaction and communication with toddlers in a Norwegian context number words may be less emphasised than in other cultures. A third of the toddlers had started pointing at objects when using number words (Item C3), but less than 10% carried this out correctly (Item C5). This is in line with results reported by Mix et al. (2002). In a study by Fluck (1995), most toddlers could recite the number sequence up to ten or beyond, while our participants expressed lower competence at this item (Item C6). Wynn (1992) found that most 3-year-old children could recite number words at least up to six. We wonder if we would have registered more mastering in this study, if we had observed the recitation of the number sequence up to five. In the studies cited there were much fewer participants than in our study. For instance, Wynn (1992) had 20 toddlers and Fluck (1995) had 13. Our study may reflect a wider variety of toddlers. The majority of the toddlers could fetch two objects on request (Item E1) while fetching three was apparently much harder (Item E3). One-fifth of the group mastered this, and nearly a tenth was about to gain this competence. Wynn (1992) found that children learn the meaning of two about 9 months before the meaning of three and this learning has been characterised as a developmental step. Whether

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104 Table 4 Percentage performance score in the area Geometry

Item

Mastering

Partly mastering

No mastering

Shape and space SS1

97.1

0.9

SS2

89.4

7.6

2.0 3.0

SS3

41.0

10.1

48.9

SS4

54.4

6.0

39.6

SS5

0.8

1.1

98.1

SS6

1.1

2.0

96.9 7.8

Pattern and order PO1

83.9

8.3

PO2

89.6

7.6

2.8

PO3

44.2

13.0

42.8

PO4

14.8

11.8

73.4

PO5

0.7

0.4

98.9

PO6

1.9

2.4

95.7

subitizing, perceiving small quantities without counting, is inborn and whether it is used to enumerate small sets has been extensively discussed (e.g. Mix et al. 2002). Our findings show that less than 10% had mastered recognition of numbers of objects up to three without counting (Item C4) supporting the step model. Making bijections, such as handing out one item to each person (Item E2), is an essential prerequisite for counting. The majority of the toddlers in this study had gained this competence. This has also previously been reported (Geist 2009; Sarama and Clements 2009). Nearly, a third of the children could also show with their fingers how old they were (Item E4). Only 1% had mastered and another 1% were about to master setting a table for five (Item E5), a more complicated task combining several components. Very few, nearly 2%, were likewise capable of answering a question on how many there were after having counted five objects (Item E6). These results reflect earlier findings on quantitative development (Aunio et al. 2006; Mix et al. 2002). Geometry The results for the items in geometry are shown in Table 4. Also in this area some of the tasks were easy for most toddlers, whereas others were difficult. Almost every toddler could point out the different body parts (Item SS1), while the related item, drawing a human (‘‘a tadpole man’’, Item SS5) was difficult for most of them. The transference of three-dimensional placing related to own body into two-dimensions is rather complex, and mastered late in preschool age (Golomb 1981, 2004). The majority of the participants distinguished between different shapes (Item SS2). This was also the case for shape recognition when placing a picture on an identical picture (Item PO1). This corresponds with findings reporting that differentiating between and recognising shapes are important tools that infants

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The Mathematical Competencies of Toddlers

105

and toddlers use to make order of their surroundings (Clements and Sarama 2009; Oakes and Madole 2003). Copying simple figures (Item SS6) was much more difficult for the young participants; 1% mastered this. This could be expected as it is a competence that according to Piaget and Inhelder (2002) is mastered late in preschool age. Approximately, 40% managed to make a picture out of a jigsaw with 3–4 pieces (Item SS3). Such puzzle-making challenges involve recognition of parts of shapes and are reported as more commonly mastered around the age of 4 years (Montford and Readdick 2008), so a higher than expected number of the toddlers in our study mastered this task. The majority of the participants were interested in rhythm and movement (Item PO2). Such recognition of order is also found in a study by Geist and Geist (2008). Although nearly half of the group also recognised the pattern in the daily routines (Item PO3), making their own patterns (Item PO5) was more difficult, mastered by less than 1%. Sorting is difficult for children under 3 years (Sarama and Clements 2009), and in our study less than 2% managed to sort objects according to one characteristic (Item PO6). Seriation is reported in several studies to be only partly mastered before the age of 5 years (Sarama and Clements 2009). It is therefore not surprising that only about 15% of the toddlers in our study could put objects in size order in a row (Item PO4). Problem Solving The percentage scores for each of the items in problem solving are presented in Table 5.

Table 5 Percentage performance score in the area Problem solving

Item

Mastering

Partly mastering

No mastering

Mathematical language ML1

92.5

3.4

4.1

ML2

92.0

3.9

4.1

ML3

27.5

13.1

59.4

ML4

45.9

11.0

43.1

ML5

1.6

1.3

97.1

ML6

2.4

1.8

95.8 11.7

Logical reasoning LR1

79.1

9.2

LR2

91.1

7.0

1.9

LR3

11.7

11.5

76.8

LR4

46.9

9.2

43.9

LR5

3.9

3.0

93.1

LR6

2.4

3.4

94.2

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The first two items in mathematical language (ML1 and ML2), distinguishing between large and small, and knowledge about up and down, were easy for most of the toddlers. Approximately half of them also managed to follow instructions with spatial words (Item ML4). This requires navigation abilities not common before later preschool age (Sarama and Clements 2009). Our results are therefore better than expected. Nearly a third of the toddlers naturally used words that describe toys (Item ML3), but very few used words for relations between sizes (Item ML5), which was also the case when they were asked to show what was in the middle (Item ML6). This corresponds with earlier research reporting that although toddlers show problem-solving strategies, they are not yet fully able to express themselves verbally (Bjo¨rklund 2007). Since cross-cultural language differences are related to learning mathematical language (Choi and Bowerman 1991), more research on how Norwegian toddlers’ acquire mathematical language is needed. The easiest item in logical reasoning was tidying toys and putting them in the right places (Item LR2), only 2% had not mastered this at all. Nearly half of the toddlers could also fetch objects they needed in their activity (Item LR4). Little research has been conducted on these types of competencies. The Norwegian tradition, which emphasises free play and activities initiated by the children versus other cultures with more academic tradition, may cause cultural differences that should be further explored. Only a tenth of the toddlers could share equally with a friend (Item LR3), even though the majority of the group, in the related Item E2, could hand out one to each. Nearly 80% of them knew what outdoor clothing they needed when it was raining (Item LR1), but determining what came first and last in dressing (Item LR5) was much more difficult. Likewise, very few knew the difference between what had happened/what was going to happen (Item LR6). Both these items include seriating. As previously stated, in several studies seriating is reported to be only partly mastered before the age of 5 years (Sarama and Clements 2009). Overall Information and Reflections The level of difficulty of the items varies a lot. This was reflected in the results, where 97.1% of the toddlers mastered the easiest item (Item SS1) in MIO, while only 0.7% of them mastered the most difficult one (Item PO5). The same wide variety was registered for not mastering the items. The category ‘‘partly mastering’’ varied within a smaller range, from 15.1 to 0.4%. The small number of observations in this category was somewhat surprising in light of Vygotskij (1978) theory of the Zone of Proximal Development. This may be due to lack of confidence in observation and registration of partly mastering. Most assessment material is based on mastered/not mastered. Information on partly mastering is also important for daily work in the kindergarten. The caretakers can then provide opportunities for the child to gain more experience in the particular area. Regarding the subareas, these are, as previously described, not strictly separated, and analyses of correlations show strong relations: Pearson product moment correlations revealed that all coefficients were above .5 between the six subareas (Table 6).

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107

Table 6 Intercorrelations among main areas and subareas Total

N&C

G

P

i

ii

iii

iv

Total score all items



N&C

.91



G

.89

.68



P

.93

.76

.79

i. Counting and series of numbers

.82

.92

.61

.69



ii. Enumeration

.82

.89

.63

.69

.64



iii. Shape and space

.83

.63

.92

.74

.56

.59



iv. Pattern and order

.80

.61

.90

.69

.54

.57

.66

v. Mathematical language

.86

.71

.71

.93

.65

.64

.70

.61

vi. Logical reasoning

.87

.70

.74

.93

.63

.65

.68

.67

v

vi



– – .72



Spearman rank order correlations of the items, used due to ordinal level, also showed many correlations of .3 and above. When examining the pattern of factors measured with the material, we found that the material measured the same phenomenon, mathematical competence, at three levels of difficulty. Details can be found in Appendix 2. Even though all the participants in our study came from the same Norwegian municipality, they represented a diversity of children enrolled in 185 different units in kindergartens. The results illustrate the competencies from this group of toddlers, not from a representative sample of toddlers in Norway. However, both the main and subareas had reasonably straight Q–Q plot lines, indicating normal distribution of the results. Together with the large number of participants, this makes the results statistically robust (Tabachnick and Fidell 2006). With the limited research on the age group in focus, and the limited number of studies with a large group of participants, we assume the results might be of interest and importance to researchers both nationally and internationally. When evaluating the results, it is important to reflect on and evaluate the assessment method for collecting data on the toddlers’ mathematical competencies. Assessment methods might influence the results (Tudge et al. 2008). It has been assumed that toddlers in play and daily life activity show more competence than they do in clinical tasks (Baroody et al. 2006; Mix et al. 2002). This was not found in this study, most of the results were at the same or lower level than results from clinical studies. The assumptions of better results in a natural setting, might have been due to the supposition that clinical tasks were more stressful than tasks in natural settings, or that observations in natural settings could easily be biased; the observer noting more or less what he or she wanted to see. The differences in results do highlight the need for further research to be done. Assessment done in clinics and natural settings is, however, different and consequently comparisons should be treated and interpreted with care. In our study, for example, the toddlers had to display mastering the items in at least two different settings. In addition, this mastering should have been observed independently by two of the kindergartens’

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staff. We consider this procedure strengthens our results. According to Bjo¨rklund (2008), toddlers need to encounter a phenomenon with a mathematical content in many settings before it is mastered. This was taken into account when the procedure for registration of competencies was chosen. In this way, we think our study has contributed to a realistic picture of the toddlers’ mathematical competencies. This can only be confirmed through future studies exploring the same areas of competencies ideally though authentic assessment in a population based sample. The difference between the results in this study and earlier research may also reflect cultural aspects. Perhaps the weak results for using number words and reciting number sequences may reflect that we in Norway do not emphasise counting and numbers in the same way as in countries from which previous research on toddlers has been reported. The Nordic countries have often been said to have the same kindergarten culture (Jensen 2009; OECD 2006). Cross-cultural research on children aged 4–5 years showed that Finish children were outperformed by Beijing children overall in early numeracy performance (Aunio et al. 2008). There is a need for more research to investigate whether our results in the numerical area are a Norwegian challenge or a Nordic one. It may be that children’s motor skills have a higher focus in Norway since the results for puzzle-making and following instructions with spatial words were higher than compared to earlier findings by, for example, Montford and Readdick (2008) and Sarama and Clements (2009). Our results support the need for more research on Norwegian toddlers’ competencies from a cross-cultural perspective. It should also be noted that especially earlier research in geometry and problem solving in general is limited (Dowker 2005; Sarama and Clements 2009). The numerical area is more studied in infant and later preschool age than in toddler age, where there are few studies on quantity development (Mix et al. 2002). Differences in mathematical development between boys and girls in preschool age has previously been reported (Levine et al. 1999), but has not been examined in the present study. Studies on eventual sex differences in mathematical competencies in toddler age are needed and planned in our longitudinal project. Very large differences in gained competence were observed in the toddlers in our study. This was to be expected as all children in Norway are included in kindergartens in line with Norwegian educational policy (Ministry of Education and Research 2006b). There are very few special institutions for children with special educational needs in Norway. The results, however, illustrate and underline the kindergartens’ challenges; every child is to be given opportunities to prosper. The results also naturally raise the research question on low performance in toddler age as an eventual predictor of future difficulties in arithmetic. Earlier research has reported the numerical area to be critical for arithmetical development (Locuniak and Jordan 2008; Mazzocco and Thompson 2005). In our study, the dispersion was largest in the number and counting area, but also in the other areas the children’s mastering varied widely. The importance of early mastering in geometry and problem solving for children’s mathematical development is still unknown (Gersten et al. 2005). Research in these areas is needed. Our larger project may in a longer perspective contribute interesting

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information as the same children are followed and assessed four times between the ages of 2‘ to 10 years. The results from the study indicate that the observation material used may be a valuable tool for the preschool teachers in identifying the variety of competencies mastered by the children in kindergarten. It enables the preschool teachers to differentiate between levels of competencies, and thus from as early as toddler age can differentiate stimulation to obtain the various competencies, or seek help other professional advice when needed. The latter is confirmed by the leader of the municipality’s educational psychological service who in an interview reported an increased number of children referred for more in-depth evaluation of skills from the kindergartens. She also pointed out that the referrals were better written than they used to be, more precise in descriptions of what the children had mastered or not (Klokk 2010). In this study focus was directed only at the toddlers’ mathematics. It has, however, been reported that development in one area may influence development in other areas (Semrud-Clikeman 2007). Further studies on how, for example, language, movement and social–emotional development may influence the toddlers’ mastering of mathematics are therefore important. The present study is part of a longitudinal project with focus on some of these questions, which aim to contribute more knowledge about children’s mathematical development—both children with a normal development and those at risk of struggling with mathematics in school. The study conducted is, to our knowledge, the first study to include such a large number of toddlers at the same age level, and to explore the participants’ mathematical competencies in a broad field of mathematics through observation carried out in play and daily life activities. The study, as such, contributes to filling a gap in quantitative assessment of mathematics in children attending kindergartens. We hope and assume this may lay the foundation for further research on these young children’s competencies, which is needed in several areas. We have outlined some of these, but would like to highlight the fact that our participants displayed lower competence in using number words and reciting number sequences than participants in previous studies. It is also a fact that most of the results were at the same or lower levels than previously reported. The exceptions were puzzle-making and following instructions on spatial words. Future research may give information on whether the lower or higher results are due to different data collecting methods, clinical trials versus authentic assessment, or cultural differences reflecting different pedagogical traditions. Acknowledgments We want to express our gratitude to all the parents who let their children take part in this study. Without their consent, we could not have obtained this knowledge about the toddlers’ mathematics. Our sincere thanks also go to all the personnel in the kindergartens whose observations and registration during their daily work are the basis for our research. They have done a fantastic job. We also want to express gratitude to our municipality representatives who have supported our collaboration practically and economically since we first presented the idea for the project. We are grateful for funding received from the university, the faculty and our own centre. It would also have pleased us all to share the results from this study with our late colleague and friend, associate professor Synnøve Iversen, PhD, who actively contributed to the project and participated in the first part of it.

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Appendix 1 Two examples of observation items (Reikera˚s 2008).

Problems solving ML - Mathematical language

2-3 years

ML1. Distinguishes between the concepts large and small The child demonstrates, through action, that he/she distinguishes between sizes. The child does not need to use the words large and small Example

To be marked If the child demonstrates both in play and communication that he/she distinguishes between the concepts.

If the child sometimes distinguishes between the concepts large and small.

Finds a small and a large stone if asked to do so (e.g. in “Simon says”). Understands what it means when someone asks for a large block.

Chooses the largest cracker, but cannot, when asked to, find the smaller one.

If the child does not distinguish between the concepts large and small.

Number and counting E - Enumeration

3-4 years

E4. Shows with the fingers how old he or she is The child does not need to say with words how old he/she is To be marked If the child illustrates with his/her fingers how old he/she is If the child illustrates with his/her fingers how old he/she is, but with the incorrect number of raised fingers If the child does not illustrate with his/her fingers how old he/she is

123

“I am so many years!” (The three-year-old child holds up three fingers.) “I am so many years!” (The two-year-old child holds up five fingers.)

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Appendix 2 Components in the material To uncover an eventual pattern of factors measured with the material, the suitability of data for factor analysis was assessed. The correlations between the subareas and items are reported in Table 6. The Kasier—Meyer–Olkin Measure of Sampling Adequacy value was .91 and Bartlett’s test of Sphericity reached statistical significance, supporting the factorability of the correlation matrix (Pallant 2007). When subjected the six subareas in principal components analysis (PCA), the presence of only one component with eigenvalues exceeding 1 was revealed, explaining 73.9% of the variance. The one component solution explained 76.5% of the variance, and the loading on the six subareas were all over 0.65. Consequently, all the subareas seem to contribute to measure the same phenomenon. When all items were subjected, the PCA revealed the presence of eight components with eigenvalues exceeding 1, explaining 24.0, 10.0, 7.2, 3.8, 3.6, 3.2, 3.1, and 2.8% of the variance, respectively. An inspection of the screeplot revealed a clear break after the third component. Using Catell’s scree test, it was decided to use three components for further investigation. The three component model explained a total of 41.2% of the variance, with the components contributing, respectively, 24.0, 10.0, and 7.3%. In order to clarify interpretation of these three components, oblim rotations were performed. The rotated solutions revealed strong loadings of a number of each of the three components and the variables loading substantially on only one component. There were small to medium correlations between the three components (r = .11, 36, and .39). Component 1 consists of the items in the middle circle of the observation material, representing the items of middle degree of difficulty, Component 2 consist of 11 of the 12 items from the inner circle, the easiest one, and also Component 3 consists of 11 of the 12 items from the outer circle, the most difficult. The two items not included in this structure matrix were item PO2 and item PO5. The results from the PCA indicated that we have a material measuring different levels of performance of the same phenomenon.

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