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Abstract

Exploring individual differences and looking beyond averaged parameters of early numeracy in young children with mild intellectual disabilities has become an area of interest to many researchers worldwide. This study aimed to identify the different profiles of early numeracy skills in young children with mild intellectual disabilities. For this purpose, we assessed early numeracy through Utrecht early numeracy test and learning aptitude through Detroit Test, in a sample of 135 children diagnosed with intellectual disabilities. The mean of their mental age was 5:09 (years:months). Two-step cluster analysis identified four homogenous groups of children with distinct early numeracy profiles as follows:C1 were fluent in relational and numerical skills up to 20, C2 were fluent in relational skills and numerical skills up to 10, C3 had basic knowledge of relational skills and inconsistent numerical skills up to 10 and C4 had inconsistent relational skills and numerical skills. Results are discussed with reference to their educational implications.

Keywords

profiles early numeracy Intellectual Disabilities cluster analysis

Introduction

Over the past decades, the importance of early childhood education has gradually gained recognition, especially for young children with intellectual disabilities (Charitaki, et al., 2021; Hojnoski et al., 2018). Since, they are faced with significant deficits in intellectual functioning and adaptive behaviour (American Psychiatric Association, 2013; Schalock et al., 2010), they need systematic instruction to achieve a later independent living and social inclusion (Storey & Miner, 2017). Admittedly, individual differences in gaining early numeracy skills could be considered as a crucial aspect in a macro-level for their social inclusion (Atkinson et al., 2002; Duncan et al., 2007; Geary et al., 2013; Ritchie & Bates, 2013; Trilling & Fadel, 2009), but also in a micro-level for school inclusion (Skwarchuk, 2020). Contrary to the growing recognition of the importance the early numeracy skills bear in the field, early numeracy profiles in young children with intellectual disabilities is an underrepresented area of research (Sermier Dessemontet, et al., 2020; Schnepel, et al., 2020).

According to recent societal trends, exploring individual differences and looking beyond averaged parameters of early numeracy in young children with mild intellectual disabilities has become an area of interest to many researchers worldwide (Porter & Campbell, 2020). Nevertheless, research evidence seems to be scarce in terms of early numeracy profiles in young children with intellectual disabilities (Sermier Dessemontet et al., 2020; Schnepel et al., 2020). Existing published research in the field has focused on exploring early numeracy in young children with intellectual disabilities (Brodeur et al., 2013; Brigstocke et al., 2008; Camos, 2009; Charitaki et al., 2014a, 2014b, 2015; Clarke & Faragher, 2014; Herrera et al., 2010; King et al., 2017; Lanfranchi et al., 2015; Noda & Bruno, 2017; Porter, 2019, 2018, 1999) through a variable-centered approach, assuming that all participants from the sample are represented only by a single population through estimated “averaged” parameters. However, a drawback of this approach is the possibility that the sample might include multiple subpopulations characterized by different sets of parameters (Howard, & Hoffman, 2018; Morin, et al., 2016). Consequently, uniqueness and variability of the sample are disregarded. On the other hand, the existing person-centered published research is limited to cases of typically developing children (Hannula-Sormunen et al., 2017; Hellstrand, 2021; Newton & Penner-Wilger, 2015;) and children with autism spectrum disorder (McDougal et al., 2020). There is only one study (Sermier Dessemontet et al., 2020) exploring profiles of early numeracy in young children with intellectual disabilities. Though, they employed a small sample of 57 participants and they assessed only early numeracy parameters. Such a sample size (n=57) could be appropriate for variable-centered analyses, but not for person-centered analyses since convergence problems and difficulty in identification of smaller profiles will possibly emerge (Vargha et al., 2016).

Early numeracy

It is clear that early numeracy consists of important skills assisting later achievement in elementary school mathematics (Aunio & Niemivirta, 2010; Braeuning et al., 2020; Charitaki et al., 2020; Clarke & Faragher, 2014; Desoete et al., 2009; Gersten et al.,2015; Krajewski & Schneider, 2009; Zhang et al., 2014), the achievement in high school mathematics (NRC, 2009), and the academic achievement in general (Jordan et al., 2007). Despite the significance of early numeracy, research on the individual differences in terms of early numeracy acquisition in young children with intellectual disabilities is in need of expansion (Kleemans et al., 2012). Additionally, there is a well-established belief in placing numerical skills under the definition of early numeracy (Kleemans et al., 2012; Porter, 2019).

Early numeracy consists of a range of different skills, such as relational skills, counting skills and operations (O) (Aunio et al., 2009; Purpura & Lonigan, 2013; Van de Rijt et al., 2003). With respect to relational skills, their definition includes a synthesis of areas such as classification, seriation, comparison, and correspondence (Piaget, 2013). The ability of creating and identifying homogenous groups of items, in terms of one or more perceptual features (e.g., green books), is assessed through classification tasks. Comparison tasks involve the identification of non-equivalent items (e.g., the shorter girl), while seriation tasks include repetitive tasks of comparison in order to create a series of items. Finally, correspondence tasks assess the ability of one-to-one pairing an item from set A with one and only one item from set B.

The definition of counting skills describes the child’s ability to perceive quantity, to know and understand the rules and processes of the counting sequence (Purpura & Lonigan, 2013). Finally, the definition of the term operations describes child’s skills of composition and decomposition of sets of objects (Clements & Sarama, 2007; Jordan et al., 2006; Purpura & Lonigan, 2013; Starkey et al., 2004).

Relations between early numeracy and cognitive functions-learning aptitude

Several studies on early numeracy either in typical, or in populations of young children with intellectual disabilities, support the basis of individual differences and the existence of significant variations in early numeracy attainments even before the typical school entry. Prior knowledge could be considered the best predictor for later academic achievements (Aunola et al., 2004). Consequently, it is important to begin with a thorough investigation of learning aptitude. With respect to learning aptitude, cognitive functions in areas of language, attention and motor abilities (Figure 1) seem to share significant variance in predicting early numeracy skills and could be considered predictors (Newton & Penner-Wilger, 2015).

Figure 1.

Cognitive functions in areas of learning aptitude (Hammill & Bryant, 2005).

Through learning aptitude assessment, a significant number of individual cognitive functions are being simultaneously assessed. More specifically, conceptual matching assesses two cognitive functions. The first one is related to verbal reduced domain of language and non-verbal comprehension, while the other one is the correlation of concepts. Design reproduction assesses four cognitive functions, summarized to attention, motor abilities, direct memory, and spatial relations. Digit sequences assess memory recall, and short-term memory. Motor directions assess reduced attention and motor abilities. Object sequences assess short-term memory and visual correlation. Picture identification assesses long-term memory and concept comprehension. Sentence limitation assesses language enhanced domain and more specifically, the comprehension and use of oral speech. Symbolic relations assess non-verbal symbolic reasoning, perceptual integration, and long-term memory. Word sequences assess memory recall and short-term memory (Hammill & Bryant, 2005).

Limited ability in the above-mentioned specific skills, could explain deficits in early numeracy acquisition as well. Research findings indicate the significant effect of linguistic factors to procedural aspects of early numeracy, since they share a common basis in terms of learning rules (LeFevre et al., 2010). Moreover, they share a common basis on abstract hierarchical representations (Kleemans et al., 2012). Besides language, attention and motor factors are strongly related to early numeracy (Abdelhameed, 2007; Brueggemann & Gable, 2018).

Person-centered investigation of early numeracy

Uniqueness is a commonly admitted characteristic of human beings. Despite the similarities, each human differs from one another, in terms of strengths and weaknesses. Heterogeneity is a significant factor which can be observed within the same human in different times. Though, it is important to mention that both heterogeneity and homogeneity are significant. Admittedly, it seems impossible to find even two humans that are 100% identical or 100% different in every way (Porter & Campbell, 2020). Despite, the fact that children with intellectual disabilities are typically clustered, there is a considerable heterogeneity-diversity in terms of their attainments, which still remains under investigation.

Admittedly, all children (with and without intellectual disabilities) have different strengths and weaknesses (Burack et al., 2011, 2012, 2020). Recent findings describing the children’s individual characteristics regarding early numeracy, could facilitate teachers’ efforts to implement effective interventions. This is the main reason why the researchers have been turned to person-centered approaches (Hannula-Sormunen et al., 2017; Hellstrand, 2021; McDougal et al., 2020; Newton and Penner-Wilger, 2015; Schnepel et al., 2020; Sermier Dessemontet et al., 2020).

The critical synthesis of these limited research efforts provides us with two basic conclusions: a) there is a significant lack of research findings in the field and b) significant differences and controversies exist within the available research findings. Additionally, these significant variations can be attributed to differences in research questions, research tools and data analyses. The only common basis in the proposed early numeracy profiles, within all populations (intellectual disabilities, autism spectrum disorder and typical development), includes the suggestion of two distinct profiles (the lowest achieving and the highest achieving) (Hannula-Sormunen et al., 2017; Hellstrand, 2021; McDougal et al., 2020; Newton & Penner-Wilger, 2015; Sermier Dessemontet et al., 2020). Moreover, we should highlight that only Sermier Dessemontet et al. (2020) worked with a sample of children with intellectual disabilities.

For example, McDougal et al. (2020), working with typically developing children, suggested a three-profile solution which emphasized on the attention (Pr_1: Good attention – higher achieving, Pr_2: Average attention – average achieving and Pr_3: Poor attention – lower achieving). Despite the fact that they assessed all domains of early numeracy including numerical operations and mathematical reasoning, they didn’t present a detailed description of diversity within early numeracy domains. As a result, no significant evidence regarding early numeracy variability can be gained through this study.

Similarly, Hellstrand (2021), suggested a three-profile solution emphasizing on overall performance, while Newton & Penner-Wilger (2015) suggested a profile solution which emphasized the visuospatial working memory. Both of them worked with typically developing children. In the same way, no significant evidence regarding early numeracy variability can be gained through these studies, too.

The only study that was exclusively oriented to the potential profiles of early numeracy in a sample of young children with intellectual disabilities was this of Sermier Dessemontet et al. (2020). More specifically, Sermier Dessemontet et al. (2020) suggested a four-profile solution (Pr_1: children with inconsistent basic numerical skills with numbers up to 10, Pr_2: children with basic numerical skills with numbers up to 10, Pr_3: children with numerical skills with numbers up to 20 and Pr_4: children with numerical skills with numbers up to 100 and arithmetic skills), which provided sufficient evidence for children’s individual early numeracy attainments and could promote individual interventions. Nevertheless, their findings were limited to the domain of counting skills. Although they assessed relational skills through a number of tasks, no further analysis was provided regarding conceptual aspects of early numeracy, thus focusing only on the procedural aspects of early numeracy. Moreover, children’s characteristics related to learning aptitude (language, attention and motor abilities) were not taken into account.

As previously mentioned, it is obvious that there is a significant gap regarding variability of early numeracy skills of young children with intellectual disabilities. Researching their profiles would enable us to reveal their strengths and weaknesses and promote individualized interventions. Thus, the aim of the current study is to evaluate whether procedural and conceptual aspects of early numeracy in young children with intellectual disabilities can be better described through specific and discrete profiles. Moreover, the effect of learning aptitude (language, attention and motor abilities) on early numeracy individual differences in children with intellectual disabilities will be evaluated. Finally, whether early numeracy profiles demonstrate significant variations across relational and counting skills will be examined.

Method

Variable-centered approaches are commonly used in educational research and support the assumption that all the sample participants are representative of the population, which can be adequately described by averaged parameters. Contrary to those, person-oriented approaches reinforce the possibility that there is heterogeneity in the sample, which enables the identification of multiple subpopulations with different characteristics (Howard, & Hoffman, 2018). Person-oriented approaches reinforce the possibility that there is heterogeneity in the sample, which enables the identification of multiple subpopulations with different characteristics (Howard, & Hoffman, 2018). This type of approach was also employed in the present study to identify the different profiles of early numeracy skills in young children with mild intellectual disabilities.

Participants

A total number of n = 155 young children with intellectual disabilities were initially enrolled in the study. An examination of outliers revealed their significant effect on Type I error rates. More specifically, we observed that the outliers showed up as a separate cluster and also caused other clusters to merge. Moreover, the assessment of cluster stability suggested that clustering was not efficient when outliers remained in data set. (explanation) Consequently, n = 20 children were excluded from the initial sample. All of them had a full-scale intelligence quotient ranging from 63 to 65 and their mean chronological age was 9:00 years old. Their full-scale intelligence quotient was estimated with Wechsler Intelligence Scale for Children-Third Edition (WISC-III)). Their mean mental age was 5 years and 09 months (min = 5:02, max = 6:08, sd = 0.678), while their chronological age was 8 years and 11 months (min = 7:02, max = 9:06, sd = 0.713). 23.7% of them were diagnosed with Down’s syndrome, while all the others were diagnosed with intellectual disabilities without any other etiology (participants with autism spectrum disorder or other disabilities were excluded from the sample). All children were enrolled in 14 primary special school settings in Greece (Attica). The sample can be considered as representative of the population of Greek children with intellectual disabilities, since 40% of them live in Attica. There was also an attempt for an equal distribution of participants by gender with 42.5% (n = 66) being girls. Consent letters were signed by the parents of all children included in the sample.

Measures

Utrecht early numeracy test

For the aims of the study, Utrecht early numeracy test (Van Luit et al., 1994) was employed. It is a standardized criterion and the specific age range of the Greek standardization sample was formed at 4.00 – 7.05 (years: months) (Barbas et al., 2008). More specifically, ENT assesses a set of 40 independent items. Eight equal-multitude of item sub-scales assess aspects of early numeracy, namely comparison, classification, correspondence, seriation, using number words, structured counting, resultative counting and general number knowledge-problem solving). The initial construct of ENT was for the unidimensional assessment (Van Luit et al., 1994) of the three domains of early numeracy (relational skills, counting skills and operations). Contrary to the initial claims, our previous research findings (Charitaki, Soulis, & Alevriadou, 2021) in a sample of young children with Intellectual disabilities suggest a two-factor model for early numeracy (relational skills, counting skills + operations), which was employed for the interpretation of the Utrecht early numeracy test assessments in the current study.

Detroit Learning Aptitude Test (DTLA-P:3)

Moreover, Detroit learning aptitude test (DTLA-P:3) (Hammill & Bryant, 2005) was employed. It is a standardized criterion and the specific age range of the Greek standardization sample was formed at 4.00 – 7.11 (years: months) (Tzouriadou et al., 2008). More specifically, DTLA-P:3 assesses nine sub-scales of learning aptitude, such as word sequences, digit sequences, object sequences, symbolic relations, sentence imitation, picture identification, motor directions, conceptual matching and design reproduction. The test is used for the assessment of language, attention and motor abilities.

Procedure

Data collection took place in public special primary school settings. The Ministry of Education and Religious Affairs approved the implementation of the study. The number of the special primary schools participated in the study, comprised 90% of the existing special schools in Attica, Athens. The administration of ENT and DTLA-P:3 were implemented by the Garyfalia Charitaki with the assistance of a school psychologist. A one-to-one approach was used for each child’s assessment, away from disruptive factors. An average of 12 separate sessions (less than 8 minutes) took place for the entire assessment of each child. Avoidance behaviors, commonly presented in intellectual disabilities populations (Patel et al., 2018), determined the abovementioned model of assessment in the sample, in terms of the number of sessions needed to administer the assessment in each child.

Data Analysis

An initial identification of potential clusters was implemented with a two-step cluster analysis. Segmentation analysis indicated a 4-cluster solution. A further investigation for potential cluster solutions was made. Ratios of sizes (larger cluster to smallest cluster) and silhouette measures of cohesion and separation from the 1-cluster to 5-cluster solutions were assessed. Moreover, the values for the ratio of distance measures and the Schwarz’s Bayesian Criterion were estimated. Further, Multivariate Analysis of Variance (MANOVAs) was used to assess the effect of gender on the clusters. In addition, the effect of learning aptitude on the clusters was assessed (cluster membership was the independent variable, while all domains of learning aptitude were the dependent variables). In order to achieve a significant decrease of Type I error, level of significance in Bonferroni adjustments was a = .001. Next, a post-hoc analysis was performed with the use of Scheffe's method. Finally, the stability of cluster membership was assessed by performing the hierarchical four-cluster solution and comparing the stability of cluster membership across two-step and hierarchical solutions.

Results

Cluster profiles

Segmentation analysis through two-step cluster technique revealed a four interpretable clusters’ solution for early numeracy in young children with mild intellectual disabilities. Silhouette measure of cohesion and separation showed a “good” cluster quality for the four-cluster solution, while the evaluation of all the other potential solutions resulted in worse cluster quality. Moreover, ratios of sizes [largest cluster (n = 42 – 31.1%) to smallest cluster (n = 29 – 21.5%)] = 1.45< 3 was acceptable and lower than all the other potential solutions, which were greater than 3. We used Figure 2 in order to interpret the profiles of the four clusters.

Figure 2.

Early Numeracy Profiles for the four clusters.

Cluster 1: Fluent in relational and numerical skills up to 20

A total number of n = 29 young students with intellectual disabilities were enrolled in Cluster 1. These students were labeled as “Fluent in relational and numerical skills up to 20”. More specifically, the students in this cluster outperformed the other participants in all domains of early numeracy (Table 1, Table 2, Figure 2). Moreover, they outperformed in domains of learning aptitude such as conceptual matching, design reproduction, digit sequences, motor directions, object sequences, picture identification, sentence imitation, symbolic relations and word sequences (Table 2). The percentage of girls in the cluster was 38%.

Table 1.

Cluster means and standard deviations by areas of early numeracy.

	Students’ Clusters
	Cluster 1: Fluent in relational and numerical skills up to 20 (n=29)		Cluster 2: Fluent in relational and numerical skills up to 10 (n=31)		Cluster 3: Basic knowledge of relational skills and inconsistent numerical skills up to 10 (n=33)		Cluster 4: Inconsistent relational and numerical skills up to 10 (n=42)		ANOVAs
	M	SD	M	SD	M	SD	M	SD	F (3, 151)	η_p2
Comparison	4.44	1.014	3.67	1.528	2.14	1.726	1.83	1.722	16.834*	.134
Classification	3.11	1.167	2.33	1.505	.82	.858	.33	.516	43.520*	.377
Correspondence	4.22	0.667	3.33	1.437	1.73	.947	1.33	.816	43.093*	.345
Seriation	2.00	1.414	1.00	1.188	.36	.658	1.17	.983	17.354*	.133
Using Number Words	2.33	.707	1.33	1.089	.18	.385	.67	1.211	122.603*	.854
Structured Counting	2.33	1.323	1.67	1.499	.14	.419	.33	.516	54.683*	.473
Resultative Counting	1.11	1.691	1.33	1.069	.05	.213	.00	.000	17.037*	.151
General Number Knowledge – Problem Solving	2.00	1.502	2.67	1.155	1.27	.935	1.00	1.549	3.665**	.068

Note: M = Mean, SD = standard deviation, *: p = .000, **: p = .014<.05

Table 2.

Cluster means and standard deviations by EN domains.

	Students’ Clusters
	Cluster 1: Fluent in relational and numerical skills up to 20 (n=29)		Cluster 2: Fluent in relational and numerical skills up to 10 (n=31)		Cluster 3: Basic knowledge of relational skills and inconsistent numerical skills up to 10 (n=33)		Cluster 4: Inconsistent relational and numerical skills up to 10 (n=42)		ANOVAs
	M	SD	M	SD	M	SD	M	SD	F (3, 151)	η_p2
Relational Skills (RS)	13.77	4.262	10.33	5.658	5.05	4.189	4.33	4.037	53.501*	.464
Counting Skills & Operations (CS+O)	7.77	5.223	7.00	4.812	2.64	1.952	2.00	3.276	49.398*	.412
Counting Skills (CS)	5.77	3.721	4.33	3.657	0.37	1.017	1.00	1.727	67.515*	.579
General number knowledge – Operations (O)	2.00	1.502	2.67	1.155	1.27	.935	1.00	1.549	4.926*	.168

Note: M = Mean, SD = standard deviation, *: p = .000, **: p = .014<.05

Cluster 2: Fluent relational skills and numerical skills up to 10

A total number of n = 31 young students with intellectual disabilities were enrolled in Cluster 2. These students were labeled as “Fluent relational skills and numerical skills up to 10”. More specifically, the students in this cluster outperformed the participants of Cluster 3 and 4 in all domains of early numeracy (Table 1, Figure 2). Significant difficulties were observed in seriation and counting tasks, while their performance in problem solving tasks is almost the same with the participants of Cluster 1. Their performance in tasks related to learning aptitude domains such as design reproduction, digit sequences, symbolic relations and word sequences was lower compared to the performance of the participants of Cluster 1 (Table 2). The percentage of girls in the cluster was 51.2%.

Cluster 3: Basic knowledge of relational skills and inconsistent numerical skills up to 10

A total number of n = 33 young students with intellectual disabilities were enrolled in Cluster 3. These students were labeled as “Basic knowledge of relational and inconsistent numerical skills up to 10”. Outperformance is observed in the first three domains of relational skills (Table 1, Figure 2). Moreover, participants of Cluster 3 underperformed in domains of learning aptitude such as conceptual matching, design reproduction, digit sequences, motor directions, object sequences, picture identification, sentence imitation, symbolic relations and word sequences compared to participants of Cluster 2. (Table 2). The percentage of girls in the cluster was 37.2%.

Cluster 4: Inconsistent relational skills and numerical skills up to 10

A total number of n = 42 young students with intellectual disabilities were enrolled in Cluster 4. These students were labeled as “Inconsistent relational skills and numerical skills up to 10”. More specifically, the students in this cluster underperform in all domains of early numeracy compared to the other participants (Table 1, Figure 1). Moreover, they underperformed in all domains of learning aptitude such as conceptual matching, design reproduction, digit sequences, motor directions, object sequences, picture identification, sentence imitation, symbolic relations and word sequences (Table 2). The percentage of girls in the cluster was 24.1%.

Cluster differences in students’ learning aptitude domains

Cluster differences in students’ learning aptitude domains were examined through MANOVA. Cluster membership was set as an independent variable, while all domains of learning aptitude were the dependent variables. Significant differences were observed across the extracted four clusters in terms of their learning aptitude. A post-hoc analysis through Scheffe's method was conducted (Table 3). Analysis indicated significant differences (p<.05) of row means between the clusters and no differences within the same cluster. Students from Cluster 1 had the best performance in conceptual matching, digit sequences and symbolic relations. Students from Cluster 2 and Cluster 3 had a similar performance and were better than those from Cluster 4 in the above domains. An opposite pattern was observed for design reproduction, with students from Cluster 4 outperforming those from Cluster 1 and Cluster 2, which is similar to Cluster 3.

Table 3.

Cluster means and standard deviations by learning aptitude domains.

	Students’ Clusters
	Cluster 1: Fluent in relational and numerical skills up to 20 (n=29)		Cluster 2: Fluent in relational and numerical skills up to 10 (n=41)		Cluster 3: Fluent in relational skills and inconsistent numerical skills up to 10 (n=43)		Cluster 4: Inconsistent relational skills and numerical skills up to 10 (n=42)		ANOVAs		Post-hoc Tests
Early Numeracy Domains	M	SD	M	SD	M	SD	M	SD	F (3, 151)	_ηp 2	Post-hoc Tests
Conceptual matching	3.451	1.233	3.275	1.393	3.217	.939	2.285	1.181	16.321^*	.245	C1^>C2^=C3^>C4^
Design reproduction	2.541	1.305	2.075	.916	2.150	.659	3.234	1.805	36.516^*	.420	C4^>C1^>C2^=C3^
Digit sequences	5.458	.988	4.675	.888	4.294	1.013	3.023	.075	22.012^*	.304	C1^>C2^=C3^>C4^
Motor directions	5.471	1.628	5.350	1.657	3.493	1.191	3.714	.460	45.301^*	.474	C1^=C2^>C3^=C4^
Object sequences	5.735	1.217	5.725	1.339	4.547	.501	3.016	.050	80.080^*	.614	C1^=C2^>C3^>C4^
Picture identification	6.174	1.339	5.800	.822	4.342	.767	4.128	1.317	69.870^*	.581	C1^>C2^>C3^=C4^
Sentence imitation	5.464	1.239	5.325	1.347	4.164	.645	3.971	.093	30.302^*	.376	C1^=C2^>C3^=C4^
Symbolic relations	4.922	2.359	3.950	2.551	3.797	1.460	2.113	.473	55.561^*	.525	C1^>C2^=C3^>C4^
Word sequences	4.277	1.286	3.150	1.311	2.863	.961	2.571	.503	40.988^*	.449	C1^>C2^>C3^=C4^

Note: M = Mean, SD = standard deviation

A similar performance was observed in domains of motor directions and sentence imitation for students from Cluster 1 and Cluster 2, while they outperformed students from Cluster 3 and 4, who had similar performance. Moreover, students from Cluster 1 had similar performance in the object sequences with those of Cluster 2 and outperformed students from both Cluster 3 and Cluster 4. Lastly, students from Cluster 1 had the best performance in the picture identification and word sequences domains. Students in Cluster 2 had better performance than those in Cluster 3 and Cluster 4, who had similar performance.

Internal Validity

127 students, comprising 94% of the initial sample remained in the same cluster across clustering techniques. A random selection of 70% of the initial sample was used for the re-estimation of the cluster solution. The solution indicated stability of the clusters, since, 88% of the students remained in the same cluster.

Discussion

The aim of this study was to explore diversity in early numeracy attainment of young children with intellectual disabilities and, more specifically, to suggest specific profiles in terms of early numeracy. Segmentation analysis through two-step cluster technique revealed a four interpretable clusters’ solution for early numeracy in young children with Intellectual disabilities, labeled as follows: C1 were fluent in relational and numerical skills up to 20, C2 were fluent in relational skills and numerical skills up to 10, C3 had basic knowledge of relational skills and inconsistent numerical skills up to 10 and C4 had inconsistent relational skills and numerical skills up to 10. Our findings are in line with those of Sermier Dessemontet et al. (2020), who assessed specific profiles of early numeracy in children with Intellectual disabilities and explain the diversity of the early numeracy attainments in young children with Intellectual disabilities.

Nonetheless, the existing diversity related to teaching methods, family background and support, may influence the early numeracy skills acquisition. A significant number of parameters-characteristics may possibly have a direct effect on child’s early numeracy attainments and should be carefully considered in terms of educational implications. Within the child’s zone of proximal development, teachers could support each child individually, by taking into consideration the children’s individual characteristics as emerged in the findings of the present study. A systematic and intensive teaching of early numeracy concepts (Hardy & Hemmeter, 2019; Jimenez & Besaw, 2020), based on children’s early numeracy profiles will possibly ameliorate their performance in the domain. It is worth noticing that a link between the developmental approach and early numeracy attainment can be observed (Burack et al., 2011, 2012, 2020), since, there is clear evidence that children with intellectual disabilities follow the developmental pathways of their aged-matched typically developing children.

Conceptual understanding vs procedural fluency

According to our findings, relational skills are better developed than counting skills in all early numeracy profiles apart from C4 and conceptual knowledge of early numeracy domains is grounded on prior procedural knowledge of early numeracy domains, similarly to typically developing children (Gersten et al., 2015; Kolkman et al., 2013; Missall et al., 2012; Toll et al., 2016). It can be observed that as children’s performance decreases in conceptual tasks from Cluster 1 to Cluster 4, the same applies to the procedural tasks in which children significantly underperformed. A possible explanation may lie on cognitive functions, such as language skills, memory and non-verbal reasoning skills which support counting skills (Koponen, Eklund, & Salmi, 2018). In young children with intellectual disabilities, all the above-mentioned cognitive functions are impaired (Charitaki et al., 2015). At the same time, estimations and magnitude comparisons promoting conceptual learning of early numeracy seem to be better established in young children with intellectual disabilities, similarly to typically developing preschoolers (Bernabini et al., 2020).

The effect of Learning Aptitude

The specific domains of learning aptitude assessed were: conceptual matching, design reproduction, digit sequences, motor directions, object sequences, picture identification, sentence imitation, symbolic relations and word sequences. Results indicated that all domains of learning aptitude, apart from picture identification and word sequences domains, have a direct and strong effect on early numeracy of young children with intellectual disabilities.

A possible interpretation of the results may lie on the fact that visualization has a direct effect on early numeracy attainments and offsets short-term memory problems in young children with intellectual disabilities (Charitaki et al., 2015). Commonly, children with intellectual disabilities demonstrate impaired short-term memory, which is closely related to their counting skills. A potential explanation of the phenomenon may be related to the fact that spoken instruction rather than visual presentations are used (Abdelhameed, 2007). Their short-term memory deficits compromise the development of early numeracy skills (Kolkman et al., 2013; Kroesbergen & Van Dijk, 2015; Toll et al., 2016). Consequently, they should not be underestimated. Apart from the limited short-term memory capacity, children with intellectual disabilities have severe deficits in their expressive language and difficulty in using rehearsal strategy (Abdelhameed, 2007). They also have significant problems with their long-term memory (Henry & Winfield, 2010; Schuchardt, Gebhardt, & Mäehler, 2010). As a result, children who enrolled in cluster 4 could employ the visual support that was provided and outperformed in tasks which assessed long-term or short-term memory. Results indicate that children in cluster 4 seem to overuse their visual short-term memory to solve tasks.

Practical Implications

It is generally admitted that all children with intellectual disabilities need systematic instruction. Significant variations within the group of young children with intellectual disabilities suggest systematic research of their profiles, which could be considered as a common basis for educational practice. More specifically, a significant effort for the identification of students whose profile is closer to Cluster 3 and Cluster 4 is needed. Teachers should be aware of these considerable variations of their early numeracy skills. Their precise and comprehensive assessment could support the smooth delivery of an effective intervention programme, especially in terms of setting learning goals for each student and planning instruction.

Mapping early numerical ability and its subcomponents in young children with intellectual disabilities will provide insights of their individual differences in terms of early numeracy acquisition, which in turn could support teachers to plan evidence-based and personalized instructions. Moreover, such an effort could promote the understanding of the learning mechanisms that children with intellectual disabilities utilize to perceive, represent, learn, and manipulate early numerical concepts (Desoete & Warreyn, 2020).

Moreover, universal screening with reliable and valid measures assessing all domains of early numeracy is necessary to be established in early-childhood educational settings, allowing teachers to identify children with intellectual disabilities, who enter typical elementary education, and are not able to count to 20 or even have difficulties with relational skills. At the same time, there are children with intellectual disabilities whose attainments in terms of early numeracy should not be underestimated. Participants of Cluster 1, who demonstrated fluent relational and numerical skills up to 20 were formed at 21.5%. Consequently, special education teachers should not be biased against their students’ early numeracy achievements.

Conclusion – Suggestions for Future Research

The area of early numeracy skills in young children with intellectual disabilities is underinvestigated. This study suggests a four-profile solution for the description of all domains of early numeracy. A further investigation in a longitudinal basis could possibly shed light to their developmental pathways in terms of early numeracy. Moreover, early numeracy skills should not be assessed separately but in conjunction with their learning aptitude.

Footnotes

Authors’ contributions

All authors contributed equally in writing the manuscript.

Availability of data and material

Moreover, we declare that data will be available upon request.

Consent to participate

Signed consent forms were obtained from all teachers and parents of the children that participated in the study.

Consent for publication

All the teachers and parents of the participants were informed for the purposes of the research and gave their consent for publication of its findings.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Ethics approval

The study was approved by the Ministry of Education and Religious Affairs of Greece.

ORCID iDs

Garyfalia Charitaki

Spyridon-Georgios Soulis

References

Abdelhameed

(2007). Do children with Down syndrome have difficulty in counting and why? International Journal of Special Education, 22(2), 129-139.

American Psychiatric Association . (2013). Diagnostic and statistical manual of mental disorders (DSM–5). Washington: Author.

Atkinson

Cantillon

Marlier

Nolan

(2002). Social indicators: The EU and social inclusion. Oxford: Oxford University Press.

Aunio

Hautamäki

Sajaniemi

Van Luit

J. E.

(2009). Early numeracy in low‐performing young children. British Educational Research Journal, 35(1), 25-46. doi:10.1080/01411920802041822.

Aunio

Niemivirta

(2010). Predicting children’s mathematical performance in grade one by EN. Learning and Individual Differences, 20(5), 427–435. doi:10.1016/j.lindif.2010.06.003.

Aunola

Leskinen

Lerkkanen

M. K.

Nurmi

J. E.

(2004). Developmental dynamics of math performance from preschool to grade 2. Journal of Educational Psychology, 96(4), 699–713. doi:10.1037/0022-0663.96.4.699.

Barbas

Vermeoulen

Kioseoglou

Menexes

(2008) Psychometric criterion of early mathematical competence of Utrecht (adaptation - standardization), In the framework of the project EPEAEK “Psychometric - differential assessment of children and adolescents with learning difficulties”, Thessaloniki.

Bernabini

Tobia

Guarini

Bonifacci

(2020). Predictors of Children’s Early numeracy: Environmental Variables, Intergenerational Pathways, and Children’s Cognitive, Linguistic, and Non-symbolic Number Skills. Frontiers in Psychology, 11.

Braeuning

Ribner

Moeller

Blair

(2020). The Multifactorial Nature of Early numeracy and Its Stability. Frontiers in Psychology, 11, 1-13. doi:10.3389/fpsyg.2020.518981

10.

Brigstocke

Hulme

Nye

(2008). Number and arithmetic skills in children with Down syndrome. Down syndrome Research and Practice. online. doi:10.3104/reviews/2070

11.

Brodeur

Trick

Flore

Marr

Burack

(2013). Multiple-object tracking among individuals with Down syndrome and typically developing children. Development and Psychopathology, 25(2), 545–553. doi:10.1017/S095457941200123X

12.

Brueggemann

Gable

(2018). Preschoolers’ selective sustained attention and numeracy skills and knowledge. Journal of experimental child psychology, 171, 138-147.

13.

Burack

J.A.

Evans

D.W.

Lai

Russo

Landry

Kovshoff

Goldman

K.J.

Iarocci

(2020). Edward Zigler's legacy in the study of persons with intellectual disability: The developmental approach and the advent of a more rigorous and compassionate science. Journal of Intellectual Disability Research, 64(1), 1-6.

14.

Burack

Hodapp

Iarroci

Zigler

(2012). On knowing more: Future issues for developmental approaches to understanding persons with intellectual disability. In Burack

Hodapp

Iarroci

Zigler

(Eds.), The Oxford handbook of intellectual disability and development (pp. 1-16). Oxford: Oxford University Press on line.

15.

Burack

Russo

Flores

Iarocci

Zigler

(2011). The more we know, the less we know, but that's OK: Developmental implications for theory, methodology, and interpretation. In Burack

J. A.

Hodapp

R. M.

Iarocci

Zigler

(Eds.), Handbook of intellectual disabilities and development (pp. 3-12). New York: Oxford University Press.

16.

Camos

(2009). Numerosity discrimination in children with Down syndrome. Developmental Neuropsychology, 34 (4), 435–447. doi:10.1080/87565640902964557.

17.

Charitaki

Baralis

Polychronopoulou

Lappas

Soulis

G. S.

(2015). Difficulty in learning to count or effect of short-term memory deficiency in mathematical abilities? International Journal of Innovation and Research in Educational Sciences, 2(2), 60-62. Retrieved from http://www.ijires.org/index.php/issues?view=publication&task=show&id=56

18.

Charitaki

Baralis

Polychronopoulou

Lappas

Soulis

G. S.

(2014a). Early numeracy in children with Down's syndrome in Greece. Psychology, 5(12), 1426-1432. doi:10.4236/psych.2014.512153.

19.

Charitaki

Baralis

Polychronopoulou

Lappas

Soulis

S.G.

(2014b). Factors Related to Numerical Ability of Children with Down's syndrome. The International Journal of Early Childhood Learning, 21(1), 1-17. Retrieved from http://ijlecl.cgpublisher.com/product/pub.256/prod.48

20.

Charitaki

Soulis

S.G.

Alevriadou

(2021). Factor Structure of Early numeracy: Evaluation of a measurement model in Greek-speaking children with Intellectual Disability. International Journal of Developmental Disabilities, 1-10. doi:10.1080/20473869.2021.1950496.

21.

Charitaki

Tzivinikou

Stefanou

Soulis

S.G.

(2020). A meta-analytic synthesis of early numeracy interventions for low-performing young children. SN Social Sciences, 1(5), 1-28.

22.

Clarke

Faragher

(2014). Developing early number concepts for children with Down syndrome. In Faragher

Clarke

(Eds.), Educating learners with ID (pp. 146–162). London: Routledge.

23.

Clements

D. H.

Sarama

(2007). Effects of a preschool mathematics curriculum: Summative research on the Building Blocks project. Journal for Research in Mathematics Education, 28(4), 136-163. doi:10.2307/30034954.

24.

Desoete

Stock

Schepens

Baeyens

Rieyers

(2009). Classification, seriation, and counting in grades 1, 2, 3 as two-year longitudinal predictors for low achieving in numerical facility and arithmetical achievement? Journal of Psychoeducational Assessment, 27(3), 252–264. doi:10.1177/0734282908330588.

25.

Desoete

Warreyn

(2020). Introduction to the Special Issue: Mathematical abilities in developmental disabilities. Research in Developmental Disabilities, 107, 103805. doi: 10.1016/j.ridd.2020.103805.

26.

Duncan

G. J.

Dowsett

C. J.

Claessens

Magnuson

Huston

A. C.

Klebanov

Japel

(2007). School readiness and later achievement. Developmental Psychology, 43(6), 1428–1446.doi:10.1037/0012-1649.43.6.1428.

27.

Geary

D. C.

Hoard

M. K.

Nugent

Bailey

D. H.

Krueger

(2013). Adolescents’ functional numeracy is predicted by their school entry number system knowledge. PLoS ONE, 8(1), 1–8.

28.

Gersten

Rolfhus

Clarke

Decker

Wilkins

Dimino

(2015). Intervention for first graders with limited number knowledge: Large-scale replication of a randomized controlled trial. American Educational Research Journal, 52(3), 516–546. doi:10.3102/0002831214565787.

29.

Hammill

D. D.

Bryant

B. R.

(2005) Detroit tests of learning aptitude: Primary. Pro-ed.

30.

Hannula-Sormunen

M. M.

Nanu

C. E.

Laakkonen

Munck

Kiuru

Lehtonen

Pipari Study group . (2017). Early mathematical skill profiles of prematurely and full-term born children. Learning and Individual Differences, 55, 108-119. doi:10.1016/j.lindif.2017.03.004.

31.

Hardy

J. K.

Hemmeter

M. L.

(2019). Systematic instruction of early math skills for preschoolers at risk for math delays. Topics in Early Childhood Special Education, 38(4), 234-247.

32.

Hellstrand

(2021). Early numeracy Development: Identifying and Supporting Children at Risk for Mathematical Learning Difficulties. Turku: Åbo Akademi University.

33.

Henry

Winfield

(2010). Working memory and educational achievement in children with intellectual disabilities. Journal of Intellectual Disability Research, 54(4), 354–365.

34.

Herrera

A. N. A.

Bruno

González

Moreno

Sanabria

(2010). Addition and subtraction by students with Down syndrome. International Journal of Mathematical Education in Science and Technology, 42(1),13–35. doi:10.1080/0020739X.2010.500698.

35.

Hojnoski

R. L.

Caskie

G. I.

Miller Young

(2018). EN trajectories: Baseline performance levels and growth rates in young children by disability status. Topics in Early Childhood Special Education, 37(4), 206-218. doi:10.1177/0271121417735901.

36.

Howard

M. C.

Hoffman

M. E.

(2018). Variable-centered, person-centered, and person-specific approaches: where theory meets the method. Organizational Research Methods, 21(4), 846-876.

37.

Jimenez

B. A.

Besaw

(2020). Building early numeracy through virtual manipulatives for students with intellectual disability and autism. Education and Training in Autism and Developmental Disabilities, 55(1), 28-44.

38.

Jordan

N. C.

Kaplan

Nabors Oláh

Locuniak

M. N.

(2006). Number sense growth in kindergarten: A longitudinal investigation of children at risk for mathematics difficulties. Child Development, 77(1), 153-175. doi:10.1111/j.1467-8624.2006.00862.x.

39.

Jordan

N. C.

Kaplan

Locuniak

M. N.

Ramineni

(2007). Predicting first grade math achievement from developmental number sense trajectories. Learning Disabilities Research and Practice, 22, 36-46.

40.

King

S. A.

Powell

S. R.

Lemons

C. J.

DavIntellectual disabilitieson

K. A.

(2017). Comparison of mathematics performance of children and adolescents with and without Down syndrome. Education and Training in Autism and Developmental Disabilities, 52(2), 208-222.

41.

Kleemans

Peeters

Segers

Verhoeven

(2012). Child and home predictors of early numeracy skills in kindergarten. Early Childhood Research Quarterly, 27(3), 471-477. doi:10.1016/j.ecresq.2011.12.004.

42.

Kolkman

M. E.

Kroesbergen

E. H.

Leseman

P. P. M.

(2013). Early numerical development and the role of non-symbolic and symbolic skills. Learning and Instruction, 25(1), 95–103.

43.

Koponen

Eklund

Salmi

(2018). Cognitive predictors of counting skills. Journal of Numerical Cognition, 4(2), 410-428.

44.

Krajewski

Schneider

(2009). Exploring the impact of phonological awareness, visual-spatial working memory, and preschool quantity-number competencies on mathematics achievement in elementary school: Findings from a 3-year longitudinal study. Journal of Experimental Child Psychology,103(4), 516–531. doi:10.1016/j.jecp.2009.03.009.

45.

Kroesbergen

E. H.

Van Dijk

(2015). Working memory and number sense as predictors of mathematical (dis-)ability. Zeitschrift für Psychologie, 223, 102–109.

46.

Lanfranchi

Berteletti

Torrisis

Vianello

Zorzi

(2015). Numerical estimation in individuals with Down syndrome. Research in Developmental Disabilities, 36(1), 222–229. doi:10.1016/j.ridd.2014.10.010.

47.

LeFevre

J. A.

Fast

Skwarchuk

S. L.

Smith‐Chant

B. L.

Bisanz

Kamawar

Penner‐Wilger

(2010). Pathways to mathematics: Longitudinal predictors of performance. Child development, 81(6), 1753-1767.

48.

McDougal

Riby

D. M.

Hanley

(2020) Teacher insights into the barriers and facilitators of learning in autism. Research in Autism Spectrum Disorders 79: 101674.

49.

Missall

K. N.

Mercer

S. H.

Martínez

R. S.

Casebeer

(2012). Concurrent and longitudinal patterns and trends in performance on early numeracy curriculum-based measures in kindergarten through third grade. Assessment for Effective Intervention, 37(2), 95–106.

50.

Morin

A. J. S.

Gagne

Bujacz

(2016). Call for papers: Person-centered methodologies in the organizational sciences. Organizational Research Methods, 19, 8-9.

51.

National Research Council . (2009). Mathematics learning in early childhood: Paths toward excellence and equity. ( Cross

C. T.

Woods

T. A.

Schweingruber

, Eds.). Washington, DC: National Academy Press.

52.

Newton

A. T.

Penner-Wilger

(2015). The cognitive and mathematical profiles of children in early elementary school. Canadian Journal of Experimental Psychology/Revue Canadienne de Psychologie Expérimentale, 69(4), 348.

53.

Noda

Bruno

(2017). Assessment of the knowledge of the decimal number system exhibited by students with Down syndrome. Qualitative Research in Education, 6(1), 56–85. doi: 10.17583/qre.2017.2061.

54.

Piaget

(2013). Child's conception of number. Jean Piaget: Selected Works (Vol. 2). Oxon: Routledge.

55.

Porter

Campbell

(2020). Individuals difference in developmental disorders. Research in Developmental Disabilities, 108, 103814. doi:10.1016/j.ridd.2020.103814.

56.

Porter

(1999). Learning to count: A difficult task? Down Syndrome Research & Practice, 6(2), 85–94.

57.

Porter

(2018). Entering aladdin’s cave: Developing an app for children with Down syndrome. Journal of Computer Assisted Learning, 34(4), 429–439. doi: 10.1111/jcal.12246.

58.

Porter

(2019). Discriminating Quantity: New points for teaching children with Intellectual Disability about number? International Journal of Disability, Development and Education, 66(2), 133-150. doi:10.1080/1034912X.2019.1569208.

59.

Purpura

D. J.

Lonigan

C. J.

(2013). Informal numeracy skills: The structure and relations among numbering, relations, and arithmetic operations in preschool. American Educational Research Journal, 50(1), 178-209. doi: 10.3102/0002831212465332.

60.

Ritchie

S. J.

Bates

T. C.

(2013). Enduring links from childhood mathematics and reading achievement to adult socioeconomic status. Psychological Science, 24(7), 1301–1308.doi:10.1177/0956797612466268.

61.

Schalock

Borthwick-Duffy

S. A.

Bradley

V. J.

Bruntix

W. H. E.

Coulter

D. L.

Craig

E. M.

Yeager

M. H.

(2010). Intellectual disability: Definition, classification, and systems of supports. Washington: American Association on Intellectual and Developmental Disabilities.

62.

Schuchardt

Gebhardt

Mäehler

(2010). Working memory functions in children with different degrees of intellectual disability. Journal of Intellectual Disability Research, 54(4), 346–353.

63.

Sermier Dessemontet

Moser Opitz

Schnepel

(2020). The profiles and patterns of progress in numerical skills of elementary school students with mild and moderate intellectual disabilities. International Journal of Disability, Development and Education, 67(4), 409-423. doi:10.1080/1034912X.2019.1608915.

64.

Skwarchuk

S. L.

(2020). Factors to Consider while Teaching Early numeracy Skills in an Inclusive Education Setting. Oxford: Oxford Research Encyclopedia of Education.

65.

Schnepel

Krähenmann

Sermier Dessemontet

Opitz

E. M.

(2020). The mathematical progress of students with an intellectual disability in inclusive classrooms: results of a longitudinal study. Mathematics Education Research Journal, 32(1), 103-119. doi:10.1007/s13394-019-00295-w.

66.

Starkey

Klein

Wakeley

(2004). Enhancing young children’s mathematical knowledge through a pre-kindergarten mathematics intervention. Early Childhood Research Quarterly, 19(1), 99-120. doi:10.1016/j.ecresq.2004.01.002.

67.

Storey

Miner

(2017). Systematic instruction of functional skills for students and adults with disabilities. Springfield, Ilinois: Charles C Thomas Publisher.

68.

Toll

S. W. M.

Kroesbergen

E. H.

Van Luit

J. E. H.

(2016). Visual working memory and number sense: Testing the double deficit hypothesis in mathematics. British Journal of Educational Psychology, 86(3), 429–445.

69.

Trilling

Fadel

(2009). 21st century skills: Learning for life in our times. San Francisco: Jossey-Bass.

70.

Van de Rijt

Godfrey

Aubrey

van Luit

J. E.

GhesquiËre

Torbeyns

Tzouriadou

(2003). The development of Early numeracy in Europe. Journal of Early Childhood Research, 1(2), 155-180. doi:10.1177/1476718X030012002.

71.

Van Luit

J. E. H.

Van de Rijt

B. A. M.

Pennings

A. H.

(1994) Utrechtse Getalbegrip Toets (Utrecht Test of Number Sense) . Doetinchem, The Netherlands: Graviant.

72.

Vargha

Bergman

L. R.

Takacs

(2016). Performing cluster analysis within a person-oriented context: Some methods for evaluating the quality of cluster solutions. Journal of Person-Oriented Research, 2(1-2), 78-86.

73.

Zhang

Koponen

Räsänen

Aunola

Lerkkanen

M.-K.

Nurmi

J.-E.

(2014). Linguistic and spatial skills predict early arithmetic development via counting sequence knowledge. Child Development, 85(3), 1091–1107. doi:10.1111/cdev.12173.

Early numeracy profiles in young children with intellectual disabilities: The role of cognitive functions

Abstract

Keywords

Introduction

Early numeracy

Relations between early numeracy and cognitive functions-learning aptitude

Person-centered investigation of early numeracy

Method

Participants

Measures

Utrecht early numeracy test

Detroit Learning Aptitude Test (DTLA-P:3)

Procedure

Data Analysis

Results

Cluster profiles

Cluster 1: Fluent in relational and numerical skills up to 20

Cluster 2: Fluent relational skills and numerical skills up to 10

Cluster 3: Basic knowledge of relational skills and inconsistent numerical skills up to 10

Cluster 4: Inconsistent relational skills and numerical skills up to 10

Cluster differences in students’ learning aptitude domains

Internal Validity

Discussion

Conceptual understanding vs procedural fluency

The effect of Learning Aptitude

Practical Implications

Conclusion – Suggestions for Future Research

Footnotes

Authors’ contributions

Availability of data and material

Consent to participate

Consent for publication

Declaration of Conflicting Interests

Funding

Ethics approval

ORCID iDs

References