Parental influences on the development of single and co-occurring difficulties in reading and arithmetic fluency

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Parental influences on the development of single and co-occurring difficulties in reading and arithmetic fluency © 2023 the Authors Published version Khanolainen, Daria; Koponen, Tuire; Eklund, Kenneth; Gerike, Georgia; Psyridou, Maria; Lerkkanen, Marja-Kristiina; Aro, Mikko; Torppa, Minna Khanolainen, D., Koponen, T., Eklund, K., Gerike, G., Psyridou, M., Lerkkanen, M.-K., Aro, M., & Torppa, M. (2023). Parental influences on the development of single and co-occurring difficulties in reading and arithmetic fluency. Learning and Individual Differences, 105, Article 102321. https://doi.org/10.1016/j.lindif.2023.102321 2023 Learning and Individual Differences 105 (2023) 102321 Available online 28 June 2023 1041-6080/© 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Parental influences on the development of single and co-occurring difficulties in reading and arithmetic fluency Daria Khanolainen a , * , Tuire Koponen a , Kenneth Eklund b , Georgia Gerike b , c , d , Maria Psyridou b , Marja-Kristiina Lerkkanen a , Mikko Aro e , Minna Torppa a a Department of Teacher Education, University of Jyv¨ askyl¨ a, Finland b Department of Psychology, University of Jyv¨ askyl¨ a, Finland c Centre for Interdisciplinary Brain Research, University of Jyv¨ askyl¨ a, Finland d Niilo M¨ aki Institute, Finland e Department of Education, University of Jyv¨ askyl¨ a, Finland ARTICLE INFO Keywords: Reading difficulties Mathematical difficulties Home learning environment Familial risk Comorbidity ABSTRACT This study investigated how single and co-occurring difficulties in reading and arithmetic fluency developed among Finnish children across Grades 1–9 (N =2151). Latent profile analysis among 391 children who had reading and/or arithmetic fluency difficulties in Grade 9 revealed profiles that followed three distinct patterns: reading difficulties (N =121), mathematical difficulties (N =94), and comorbid difficulties (N =176). The profiles and typical performers were compared on parental reading and mathematical difficulties, parental education, the early home learning environment, and parental assistance with school homework across Grades 1–9. Results showed that although parents whose children had difficulties provided them with domain-specific support across all grades, the amount of support gradually declined and the performance gap between the profiles increased. 1. Introduction The end of comprehensive school is a critical time point—this is when adolescents face important choices regarding their future educational pathways. Unfortunately, these choices can be negatively affected by reading and mathematical difficulties (RD and MD, respectively), as poor foundational academic skills are a known risk factor for later lower academic motivation (Klauda & Guthrie, 2015), higher levels of school burnout and dropout after compulsory education (Korhonen et al., 2014), which can lead to unemployment and mental health problems in adulthood (Aro et al., 2019). International assessments show that many teenagers struggle with reading and mathematical tasks that are well below their grade level (Schleicher, 2018); nevertheless, longitudinal research on reading and mathematical skill development mostly focuses on early childhood and primary school education, rarely extending into education during adolescence. Moreover, reading and mathematical skills are interrelated (Cirino et al., 2018) and difficulties in these domains often co-occur (Moll et al., 2019), placing individuals at even higher risk for the negative outcomes. Nevertheless, most previous studies examining the comorbidity of RD and MD are cross-sectional and long-term developmental patterns leading to RD, MD, and comorbid difficulties remain to be identified and examined. During the last decade, an increasing amount of research has examined the cognitive factors related to the co-variance of reading and mathematical skills (Cirino et al., 2018) and the comorbidity of difficulties in these domains (Landerl et al., 2009; Van Daal et al., 2012). However, notably less attention has been paid to the related environmental factors. Although numerous studies have shown positive correlations between home learning activities and children's reading (Dong et al., 2020) and mathematical skills (Dunst et al., 2017), whether the existing differences in the characteristics of the home learning environment could be related to divergent outcomes in adolescence is still unclear. Studying differential pathways to adolescent performance and identifying the environmental factors that predict them can elucidate the risk and protective factors operating in children's everyday life. The main objective of this study is to gain new insights into the developmental patterns that result in RD and MD among adolescents. To this end, we identify latent profiles of reading and mathematical skill development among Finnish schoolchildren who demonstrate low performance in reading and arithmetic fluency at the end of comprehensive * Corresponding author at: University of Jyv¨ askyl¨ a, Huone RUU B325, Ruusupuisto, Alvar Aallon katu 9, 40600 Jyv¨ askyl¨ a, Finland. E-mail address: [email protected] (D. Khanolainen). Contents lists available at ScienceDirect Learning and Individual Differences journal homepage: www.elsevier.com/locate/lindif https://doi.org/10.1016/j.lindif.2023.102321 Received 29 July 2022; Received in revised form 17 April 2023; Accepted 22 June 2023 Learning and Individual Differences 105 (2023) 102321 2 school (Grade 9, age 16). We then compare the skill levels of the low performing profiles to the skill levels of their typically performing peers to see how the pace of their development differs. In addition, we examine the role of various family factors in the development of foundational academic skills. We include family risk status (parental RD and MD), parental education, and learning activities at home, as these all are linked with children's reading and mathematical skill development (Dong et al., 2020; Dunst et al., 2017; Esmaeeli et al., 2019; Van Bergen et al., 2014). To our knowledge, this is the first study with such an objective. This study builds on our previous research (Khanolainen et al., 2020) where we examined the effects of parental difficulties and the early home learning environment on children's reading and arithmetic skills. In that study we showed that parental difficulties had small predictive effects on children's skills in the general population sample. The overall group level analyses are informative but assume that developmental patterns and associations are similar for all participants. This study is different as we focus on the group of children with learning difficulties and test if there are differential developmental paths leading to learning difficulties at the end of Grade 9. This approach allows us to investigate the possibility of heterogeneous long-term developmental pathways (including pathways with single and comorbid difficulties). This is an important extension of the previous research as we know that reading and arithmetic difficulties are often comorbid (Moll et al., 2019). Furthermore, in the previous study we specifically examined the role of the early home environment (measured when children were in kindergarten) in subsequent skill development. In this study, however, we incorporate parental academic assistance from Grades 1 to 9. 1.1. Developmental pathways of reading and arithmetic fluency development Reading fluency is most often defined as the skill that allows reading with speed and accuracy. It forms the foundation for developing more complex skills, such as reading comprehension (Florit & Cain, 2011; Pikulski & Chard, 2005). Similarly, arithmetic fluency can be understood as the skill needed for speed and accuracy in simple mathematical calculations (additions, subtractions, multiplications, and divisions). Strong arithmetic fluency is critical for further mathematical development and difficulties in arithmetic fluency are an important precursor of difficulties in higher-order mathematical skills (Cowan et al., 2011; Jordan et al., 2003). Throughout this study we will refrain from using the terms “dyslexia” and “dyscalculia”, instead opting for “reading difficulties” and “arithmetic difficulties”. Our participants completed only skill assessments that are sufficient for identifying reading and arithmetic difficulties but are not sufficient for diagnosing either dyslexia or dyscalculia. Conducting extensive diagnostic assessments required for a diagnosis was beyond our study's objectives. In addition, identifying reading and mathematical difficulties (rather than diagnosing dyslexia and dyscalculia) is in line with the support system provided within Finnish education where support is made available based on teachers' identification of learning difficulties. No official diagnoses of dyslexia or dyscalculia are needed for special needs support. At the same time, it is likely that a sizable proportion of children who demonstrated reading and arithmetic difficulties in our sample had in fact dyslexia, dyscalculia, or both. Existing research on adolescent reading development suggests that those who slightly lag in their reading fluency development during early grades often experience a significantly more pronounced lag in academic performance immediately after elementary school (ages 11–12), as the demands of the curriculum become increasingly rigorous and much of secondary school teaching starts taking place outside their zone of proximal development (Blanton et al., 2007; Deshler & Hock, 2007). Similarly in mathematics, high rates of acceleration in development have been noted among high performers compared with low performers, contributing to an increasing variance in skills over time and a gradually widening skill gap between low performers and high performers during the early grades (Aunola et al., 2004); however, this process has not been traced to adolescence. Although longitudinal research is still lacking, existing evidence shows that both reading and arithmetic fluency have high interindividual stability, suggesting that even though fluency develops over time, the individual rank order in skill is established in early years and remains fairly time-invariant (Hulslander et al., 2010). For this reason, early skills are strong predictors of later skills. For example, Aunola et al. (2004) reported the correlation between mathematical skills (arithmetic fluency tasks were included in the assessments) in Grades 1 and 2 to be 0.79, while Eklund et al. (2015) found that the correlation between reading fluency scores in Grades 2 and 8 was 0.78. Despite such a high stability of reading and arithmetic fluency, large variances that are unexplained still remain, allowing room for change. This means that less predictable developmental patterns are possible, and they are usually studied within RD and MD research. For example, RD do not always demonstrate longitudinal stability (Torppa et al., 2015), even after controlling for measurement error and using a simulation-based analysis with buffer zones to counter the effects of arbitrary cut-offs (Psyridou et al., 2020). Torppa et al. (2015) found that only around 40 % of all children with RD in their sample had persistent difficulties identifiable in both elementary and lower secondary school (at ages 8 and 14). A similar longitudinal instability was recently observed in the identification of MD—only about 50 % of learners with an early diagnosis retained clear difficulties over the first two years of elementary school (ages 7 and 8) (Chan & Wong, 2020). These studies, however, focused on the stability of either RD or MD without testing for their possible comorbidity and its impact on stability. In contrast, working with the same data set as we used in the present study Koponen et al. (2018) examined the stability of RD and MD as well as comorbid difficulties across Grades 1–4 and found lower stability in Grade 1 and higher stability thereafter. In addition, starting from Grade 2, comorbid difficulties were stable and more so than the single difficulties—68 % of second graders with comorbid difficulties demonstrated persistent difficulties in both domains and confirmed their status in Grade 4, whereas only 46 % and 39 % of those with single RD and MD, respectively, remained in the same developmental group. Interestingly, only 1 %, went from typical performance in both skills to comorbid difficulties over time; however, note that the study ended at Grade 4. The fact that children frequently display comorbid difficulties and may transition from one deficit group to another over time is best explained by the multiple deficit model (Pennington, 2006), a theoretical framework that accounts for the emergence and dynamic nature of RD and MD by the complex interactions between multiple shared risk factors that are associated with the two types of difficulties probabilistically (rather than deterministically). However, further research focusing on RD and MD along with their comorbidity and longitudinal stability is needed to gain a better understanding of the possible factors that shape different developmental patterns from a long-term perspective. 1.2. Family risk for RD and MD The reasons behind the differences in the patterns of reading and arithmetic fluency development can be multiple, e.g. children's cognitive skills, motivation-related factors, family factors (both parental reading and mathematical skills and the home learning environment), etc. The present study focuses on a variety of family factors that might influence how children's skills develop. In reading, parental RD (family risk) are one of the best early predictors of children's reading skills (Esmaeeli et al., 2019; Puolakanaho, 2007; Van Bergen et al., 2014) while family risk studies are still rare in mathematics-related research (Shalev et al., 2001). Nevertheless twin, molecular genetic, and adoption studies indicate a high heritability of different mathematical skills D. Khanolainen et al. Learning and Individual Differences 105 (2023) 102321 3 (Borriello et al., 2020; Docherty et al., 2010; Kovas et al., 2007), including skills such as arithmetic fluency, suggesting that parental MD could be a strong predictor of children's general mathematical skills over time. Existing research on the etiology of comorbid difficulties in reading and mathematics-related skills reported that the two types of difficulties stem largely from the same genetic factors (Daucourt et al., 2020). Moreover, reading and arithmetic fluency share considerable genetic overlap not only with one another but also with general cognitive ability (Hart et al., 2009). Nevertheless, there are few family risk studies that focus on the comorbidity of RD and MD (Nguyen et al., 2022). Our present study offers novel insights into how different parental learning difficulties (family risk for both RD and MD) influence children's reading and arithmetic fluency development. 1.3. Home learning environment and parental academic assistance Children's skills develop under the influence of not only genetic but also environmental factors which has been understood through studying the home learning environment. In research, the home learning environment is commonly divided into the home literacy environment (HLE) and home numeracy environment (HNE) which refer to at-home interactions between parents and their children, learning materials, and parental attitudes related to literacy and numeracy, respectively. Multiple studies have produced compelling evidence indicating significant positive associations between the early home learning environment and both reading and mathematical development (Dong et al., 2020; Dunst et al., 2017). However these studies were conducted with young children who were not yet enrolled in formal schooling (Dong et al., 2020; Dunst et al., 2017). Studies with children of school age looking into parental academic assistance and involvement in homework are still quite rare and have provided mixed evidence. Some studies with general population samples of school-age children suggest that parental academic assistance is beneficial (Dumont et al., 2012; Patall et al., 2008), whereas other studies report a negative association between parental involvement and children's academic performance (Hill & Tyson, 2009; Pomerantz & Eaton, 2001). This negative association does not necessarily mean that parental involvement itself is detrimental for academic skill development but rather that children's lower academic achievement likely evokes more parental academic assistance (Silinskas et al., 2010; the researchers used the same data set as we did but only the data from early grades was available at that time). Contradicting and inconsistent findings could also be attributable to the use of different research measures. For example, Dumont et al. (2014) highlighted that some studies collect data on the quantity of all academic assistance activities whereas others differentiate between qualitatively different types of activities and show that only some of these activities help children learn. Additionally, some researchers have pointed out that parental learning difficulties could be an important confounding factor that needs to be investigated in research on HLE and HNE (Puglisi et al., 2017; Van Bergen et al., 2014). Indeed, parents with learning difficulties could be organizing fewer learning activities at home but it is not necessary the reason why their children demonstrate lower academic skills, the real reason could be that these children have inherited parental learning difficulties. Therefore, the inclusion of both home environmental factors and parental skill measures is important. 1.4. The present study Our review of previous research suggests that further investigation of different long-term patterns within skill development leading to RD and MD at the end of compulsory schooling is important. While research focusing on individuals with resolving difficulties is valuable because it can help identify protective and promotive factors, it is important to recognize that research with a specific focus on individuals with below grade level outcomes is also valuable because it can help establish and better understand specific risk factors. The heterogeneity of learning difficulties is multi-layered, as distinct groups of difficulties can be identified based on their stability, time of emergence, and co-occurrence with other difficulties. In view of this, the present study aims to address two main research questions. The first is, “What patterns of developmental progress can be identified among those leaving comprehensive school with lower foundational skills (reading and mathematical difficulties)?” To identify the patterns of developmental progress, we used latent profile analysis (LPA), which is currently one of the most common scientific approaches to retrieve homogeneous subgroups (profiles) from heterogeneous populations. Based on previous findings about the prevalence of comorbid RD and MD (Moll et al., 2019), we expected to identify distinct groups of learners with RD, MD, and comorbid difficulties. Moreover, based on previous research on developmental changes in the domain of reading (Catts et al., 2012; Torppa et al., 2015) and mathematics (Chan & Wong, 2020), we expected to find persistent (emerging during early grades) and late-emerging (emerging only after Grade 3) difficulty profiles. Our second research question focused on parental influences: “Do profiles of low performers differ from one another and from typical performers based on the family risk status (parental RD and MD), parental education, or home learning activities (the early home learning environment, assessed when children were in kindergarten, as well as parental academic assistance, repeatedly measured when children were in school—in Grades 1–9)?” Taking into account previous studies, we expected to find the following significant predictors of children's profiles: family risk as a negative predictor (Esmaeeli et al., 2019; Shalev et al., 2001; Van Bergen et al., 2014), the home learning environment as a positive predictor (Dong et al., 2020; Dunst et al., 2017; Van Bergen et al., 2017), and parental academic assistance as either a positive (Dumont et al., 2012) or a negative predictor (Hill & Tyson, 2009). To answer the second research question, we compared the low performers and typical performers using one-way ANOVAs. We additionally tested if any of the family factors predicted the low performing profiles using the three-step approach in our LPA (Asparouhov & Muth´ en, 2014). 2. Methods 2.1. Participants and procedure This study is part of the First Steps Study (Lerkkanen et al., 2006) that followed children from kindergarten (aged 6–7 years) to Grade 9 (aged 15–16 years), the end of comprehensive schooling. The sample includes 2614 children. The study ensured balanced sampling of participants from rural, urban, and mixed areas in western, central, and eastern Finland. Marital statuses and educational levels of participating parents were very close to the national distribution. Overall, the sample can be considered representative of the Finnish population in terms of ethnic composition, family structure and educational levels (Statistics Finland, 2007). The current study complied with the guidelines of the Finnish National Board on Research Integrity (TENK, 2019). The Ethical Committee of the University of Jyv¨ askyl¨ a reviewed the study and provided an ethical evaluation statement on June 6th, 2006. Throughout the whole study research was conducted in accordance with the ethical guidelines for research with human subjects. Around 83 % of all contacted families participated in the study and provided informed consent prior to participation. 2.2. Measures In this study, we utilized data from eight available time points (kindergarten and Grades 1, 2, 3, 4, 6, 7, and 9). Children's assessments were conducted in schools, where trained researchers administered tests for reading and arithmetic fluency in classrooms. Parental questionnaires were administered at all time points when children's skills were D. Khanolainen et al. Learning and Individual Differences 105 (2023) 102321 4 assessed, starting at kindergarten. The children's fathers were less likely to report their home activities than mothers (e.g., in Grade 1, 3 % of mothers' replies were missing, whereas for fathers, this number was 33 %). Therefore, only mothers' self-reports were analyzed (except for the family risk questionnaire, explained in more detail in the section “Familial risk for RD and MD” below). 2.2.1. Reading fluency The measure of reading fluency comprised three standard groupadministered tests. The first test was an 80-item word-reading task that is part of the nationally standardized reading test (ALLU; Lindeman, 2000). Each item offered a picture along with four phonologically similar written words. The task was to read the words silently and select the one that semantically matched the picture. Participants were allotted 2 min to complete this task, and their score was the sum of all correct answers. The pictures and words used in this test were simple and familiar to children. The second reading fluency test was a word chain task comprising 10-word chains, each with 4–6 words presented in a row without any spaces (Nevala & Lyytinen, 2000). Participants needed to read the chains silently and provide boundary lines between all words they could identify. This task was also time-limited (1.25 min in Grades 1 and 2, 1.20 min in Grade 3, 1.05 min in Grade 4, 1 min in Grades 6 and 7, and 1.30 min in Grade 9), and each participant's score was calculated as the sum of all correct answers. The third reading fluency test was a sentence reading task. In Grades 1–4, the Finnish version of the Test of Silent Reading Efficiency and Comprehension (TOSREC; Wagner et al., 2010; Finnish version by Lerkkanen & Poikkeus, 2009) was used. This task comprised 60 sentences, and the duration to complete the task was 3 min. Participants were asked to read each sentence and decide if it was true or not (e.g., apples are blue). In Grade 6, a similar task was administered—the Finnish adaptation of the Salzburg Sentence Reading Test (Pichler & Wimmer, 2006). Participants were asked to verify the truthfulness of 69 sentences in 2 min. In Grades 7 and 9, this test was replaced with a similar 3-min assessment—the standardized Finnish reading test for lower secondary school sentence reading (YK¨ A; Lerkkanen et al., 2018). This test had the same instructions, but the items were designed for older children. The final sum of scores was also based on the number of correct answers. The mean of the three standardized fluency measures was used as the score. Cronbach's alpha reliability coefficients for the composite ranged in different grades between 0.78 and 0.84. The score in each grade was standardized before proceeding with analysis. 2.2.2. Arithmetic fluency The measure of arithmetic fluency allocated 3 min for completion and included one standardized group-administered subtest of the arithmetic test developed by Aunola and R¨ as¨ anen (2007). In Grades 1–3, the measure comprised 14 addition (e.g., 2 +4 =__, 5 +3 +7 =__) and 14 subtraction tasks (e.g., 8–2 =__, 18–5 −4 =__). In Grade 4, the measure slightly changed and offered 25 addition and subtraction tasks (e.g., 117–9 +13 =__; 485–42 =__; 1635 +576 =__) as well as 1 multiplication and 2 division tasks (e.g., 40:8–3 =__, 240:80 =__, 12 ⋅ 28 =__). In Grade 6, the measure included 23 addition and subtraction tasks, 3 division tasks, 1 multiplication task, and 1 task with decimal numbers (e.g., 106.2–30.04 =__). Finally, in Grades 7 and 9, the measure included 19 addition and subtraction tasks, 3 division tasks, 3 multiplication tasks, and 3 tasks with decimal numbers. The score on this measure reflected both the speed and accuracy of foundational mathematical calculations, allowing to assess children's arithmetic fluency. Cronbach's alphas varied in different grades between 0.68 and 0.94. The score in each grade was standardized before proceeding with analysis. 2.2.3. Familial risk for RD and MD When children were in kindergarten, mothers and fathers were asked if they themselves or their spouse had experienced learning difficulties in reading or mathematics. Responses were measured on a three-point scale: 1 (no difficulties), 2 (some difficulties), and 3 (clear or serious difficulties). Self-reports were given priority, whereas spouse reports were used to fill in missingness. The children were considered to be at family risk if they had at least one parent with some or clear difficulties. Although measuring parental RD and MD with a single item for each difficulty type does not capture all aspects of familial risk, previous large-scale research has shown that even a single familial risk item can be an important predictor of children's skills (Esmaeeli et al., 2019). 2.2.4. Parental education The parents were asked about their education level as well as that of the other parent using a seven-point scale: 1 (no vocational education) (5.1 % of mothers and 1.8 % of fathers), 2 (vocational courses) (3.1 % of mothers and 1.7 % of fathers), 3 (vocational school degree) (30.8 % of mothers and 14.3 % of fathers), 4 (vocational college degree) (23.2 % of mothers and 10.1 % of fathers), 5 (polytechnic degree or bachelor's degree) (9.7 % of mothers and 4.2 % of fathers), 6 (master's degree) (23.7 % of mothers and 8.0 % of fathers), and 7 (licentiate or doctoral degree) (4.4 % of mothers and 2.7 % of fathers). The information about parental education was collected when children were in kindergarten and the sum score was computed as an average of both parents' individual scores. 2.2.5. Home learning environment For kindergarteners, parents completed a questionnaire about the learning activities they organized at home. The questionnaire was based on items developed by S´ en´ echal et al. (1998) and S´ en´ echal (2006), which have been used successfully in the Finnish context (Silinskas et al., 2020). It included four questions about the frequency of home teaching activities (teaching letters, teaching reading, teaching numbers, and teaching arithmetic skills). In addition, the questionnaire had an item about shared reading: “How often do you read books to your child or together with your child?” All answers were given on a fivepoint Likert-type scale: 1 (less than once a week), 2 (1–3 times a week), 3 (4–6 times a week), 4 (once a day), and 5 (more than once a day). The sum scores for the three HLE and two HNE items were calculated by adding the individual scores of activities related to each domain. Cronbach's alphas for HLE and HNE were 0.79 and 0.86, respectively. 2.2.6. Parental academic assistance with literacy tasks In each grade, parents were asked to indicate the frequency of different literacy-related activities organized at home using a five-point scale ranging from 1 (not at all) to 5 (on a daily basis). In Grade 1, the questionnaire had two items about reading (“How often do you teach your child to read?” and “How often do you encourage your child to read independently?”). In Grades 2 and 3, the questionnaire included four items—two were the same as those in Grade 1 and two were about writing (“How often do you teach your child to write?” and “How often do you encourage your child to write independently?”). In Grade 4, in addition to the items in the previous grades, two items about parental assistance were included (“How often do you help your child with reading homework?” and “How often do you help your child with writing homework?”). In Grades 6, 7, and 9, to ensure that the questionnaire is age-appropriate in relation to school subjects, the items about reading and writing were replaced with equivalent items about Finnish language tasks. At these time points, the questionnaire included three items (“How often do you teach your child to do Finnish language tasks?”, “How often do you help your child with Finnish language home assignments?”, and “How often do you encourage your child to do Finnish language tasks independently?”). Similar items have been successfully used in earlier studies (e.g., Edwards, 2014; Haney & Hill, 2004; Silinskas, Kiuru, et al., 2013). Cronbach's alpha coefficients for the parental literacy assistance measure were 0.55, 0.80, 0.80, 0.89, 0.66, 0.62, and 0.63 in Grades 1, 2, 3, 4, 6, 7, and 9, respectively. 2.2.7. Parental academic assistance with numeracy tasks A similar five-point measure ranging from 1 (not at all) to 5 (on a daily D. Khanolainen et al. Learning and Individual Differences 105 (2023) 102321 5 basis) was used to collect information about the frequency of numeracyrelated activities. In Grades 1–3, the questionnaire had two items about mathematics (“How often do you teach your child to do calculations?” and “How often do you encourage your child to do calculations independently?”). In Grade 4, one more item was added that asked about parental assistance (“How often do you help your child with calculation tasks?”). In Grades 6, 7, and 9, the items about calculations were replaced with equivalent items about mathematical tasks (“How often do you teach your child to do mathematical tasks?”, “How often do you help your child with mathematical home assignments?”, and “How often do you encourage your child to do mathematical tasks independently?”). The majority of these items were based on the literacy assistance items listed above (e.g., Edwards, 2014; Haney & Hill, 2004; Silinskas, Kiuru, et al., 2013) and have been used previously by Silinskas et al., 2010. Cronbach's alpha coefficients for the parental numeracy assistance measure were 0.67, 0.76, 0.72, 0.81, 0.73, 0.70, and 0.69 in Grades 1, 2, 3, 4, 6, 7, and 9, respectively. 2.3. Statistical analysis A preliminary step was data preparation: the whole sample was checked for entry errors and outliers. Using Mahalanobis distance test, we identified and deleted 13 multivariate outliers. We then examined the patterns of missing data. Little's test of missing completely at random (which included all questionnaire items of literacyand numeracyrelated activities organized at home) confirmed that mothers' home activities reports were missing at random ( χ 2 (5068) =5051.034, p = .564), indicating that all mothers were equally likely to submit selfreports at different time points. Another Little's MCAR test was conducted (which included all reading and arithmetic fluency assessments from Grades 1 to 9) to determine whether children's skill performance was associated with the likeliness of data missingness. Results showed that children who performed lower in reading ( χ 2 (165) =314.477, p < .001) and in math ( χ 2 (178) =339.301, p <.001) were more likely to not be included in each wave of the study. More details on missing values can be found in Appendix 1. Next, to answer the first research question and to examine the patterns of developmental progress that preceded students' graduation from a comprehensive school with below grade level foundational academic skills (reading and arithmetic fluency difficulties in grade 9), we ran a type of mixture model (LPA; Oberski, 2016) (Fig. 1). For this type of analysis, we decided not to use the whole sample (N =2151) because of the large variability in reading and arithmetic fluency present in a general population sample. This large variability can potentially prevent LPA from identifying distinct profiles that might exist in the data (see Huijsmans et al. (2020), who provided an example of such problem occurring in LPA). In view of this, we started our analysis by separating low performers from the rest of the sample to ensure that LPA could retrieve distinct profiles from the population of interest—that is, the participants with learning difficulties at the end of comprehensive school. Composite scores for reading fluency in Grade 9 and arithmetic fluency in Grade 9 were calculated and everyone who performed at least one standard deviation below the mean (the mean was calculated based on the whole sample) in either reading fluency or arithmetic fluency were considered to be a low-performing adolescent. In total, the scores of 391 adolescents were below the cut-off for reading fluency, arithmetic fluency, or both. Table 1 presents the descriptive statistics for this group. Once the population of interest was selected (those graduating from school with low foundational academic skills), LPA was conducted for these 391 participants. Based on their performance on all reading and arithmetic fluency tasks (using continuous standardized variables) across all seven time points (Grades 1–9), we examined whether distinct profiles existed (Muthen, 2001) using Mplus version 7.3. Seven indicators for reading fluency and seven indicators for arithmetic fluency (one for each time point) were entered into our mixture model as indicators. The number of indicators were deemed appropriate for this research questions based on the findings of Wurpts and Geiser (2014), which established that adding more indicators in mixture models improves their performance and can compensate for small sample sizes. Our mixture model performed well without running into any problems Fig. 1. Latent profile analysis model for the reading and arithmetic fluency measures. Note. C represents the latent profiles, Level RF and Level AF represent the initial level of reading fluency (RF) and arithmetic fluency (AF). Numbers next to RF and AF indicate the assessment time point (grade). D. Khanolainen et al. Learning and Individual Differences 105 (2023) 102321 6 Table 1 Descriptive statistics for all variables across time. Whole sample Low performers only N Minimum Maximum Mean SD Skewness (std. error) Kurtosis (std. error) N Minimum Maximum Mean SD Skewness (std. error) Kurtosis (std. error) Reading fluency (z-scores) Grade 1 2037 −2.44 4.02 0.00 1.00 0.62 (0.05) 0.44 (0.11) 276 −1.98 2.03 −0.54 0.68 0.55 (0.15) 0.14 (0.29) Grade 2 1991 −2.89 3.89 0.00 1.00 0.26 (0.05) 0.23 (0.11) 280 −2.16 2.69 −0.58 0.74 0.47 (0.15) 1.39 (0.29) Grade 3 1980 −4.42 3.19 0.00 1.00 −0.04 (0.05) 0.43 (0.11) 287 −2.72 2.46 −0.59 0.83 0.47 (0.14) 0.66 (0.29) Grade 4 1939 −4.62 2.76 0.00 1.00 −0.17 (0.05) −0.30 (0.11) 286 −2.64 2.16 −0.62 0.84 0.34 (0.14) 0.16 (0.29) Grade 6 1807 −3.58 3.28 0.00 1.00 0.10 (0.05) −0.11 (0.11) 365 −2.92 2.61 −0.75 0.90 0.58 (0.13) 0.76 (0.25) Grade 7 1755 −4.20 3.04 0.00 1.00 −0.07 (0.05) −0.00 (0.12) 369 −4.13 2.11 −0.87 0.90 0.25 (0.13) 0.47 (0.25) Grade 9 1706 −2.98 2.99 0.00 1.00 −0.09 (0.05) −0.14 (0.12) 391 −2.98 2.88 −1.02 0.93 0.82 (0.12) 1.12 (0.25) Arithmetic fluency (z-scores) Grade 1 2035 −2.55 4.25 0.00 1.00 0.33 (0.05) 0.26 (0.11) 275 −2.55 2.07 −0.44 0.78 0.25 (0.15) −0.16 (0.29) Grade 2 1986 −3.28 2.44 0.00 1.00 −0.09 (0.05) −0.46 (0.11) 278 −2.47 1.83 −0.53 0.86 0.27 (0.15) −0.32 (0.29) Grade 3 1979 −4.25 1.82 0.00 1.00 −0.64 (0.05) 0.45 (0.11) 287 −3.39 1.60 −0.56 0.96 −0.07 (0.14) −0.25 (0.29) Grade 4 1938 −4.18 2.44 0.00 1.00 −0.63 (0.06) 0.80 (0.11) 286 −3.20 1.71 −0.65 0.89 −0.33 (0.14) 0.35 (0.29) Grade 6 1802 −4.14 2.63 0.00 1.00 −0.27 (0.06) 0.17 (0.11) 365 −4.14 2.09 −0.80 0.88 −0.21 (0.13) 0.53 (0.25) Grade 7 1734 −3.61 3.50 0.00 1.00 −0.16 (0.06) 0.35 (0.12) 367 −3.61 1.39 −0.81 0.86 −0.06 (0.13) 0.22 (0.25) Grade 9 1690 −3.56 3.09 0.00 1.00 −0.11 (0.06) 0.02 (0.12) 391 −3.56 1.56 −1.03 0.88 0.41 (0.12) 0.43 (0.25) Parental academic assistance with literacy tasks (mean composites of items) Grade 1 1474 1 5 2.94 0.91 0.19 (0.06) −0.39 (0.13) 203 1 5 3.24 0.91 0.28 (0.17) −0.29 (0.34) Grade 2 1430 1 5 2.29 1.05 0.69 (0.06) 0.24 (0.13) 200 1 5 2.60 0.83 0.44 (0.17) −0.15 (0.34) Grade 3 1360 1 5 2.06 0.95 0.79 (0.07) 0.76 (0.13) 198 1 4.50 2.31 0.72 0.64 (0.17) 0.52 (0.34) Grade 4 1269 1 5 1.85 0.97 0.92 (0.07) 1.19 (0.14) 187 1 4.50 2.05 0.67 0.55 (0.18) 0.59 (0.35) Grade 6 999 1 4 1.95 0.59 0.29 (0.08) −0.08 (0.15) 182 1 4 2.19 0.53 0.22 (0.18) 0.73 (0.36) Grade 7 768 1 3.67 1.83 0.57 0.30 (0.08) −0.33 (0.18) 141 1 3.67 2.03 0.57 0.11 (0.20) −0.08 (0.41) Grade 9 892 1 4 1.73 0.54 0.45 (0.08) 0.02 (0.16) 169 1 3.33 1.87 0.53 0.10 (0.19) −0.44 (0.37) Parental academic assistance with numeracy tasks (mean composites of items) Grade 1 1470 1 5 2.93 0.89 0.13 (0.06) −0.47 (0.13) 202 1 5 3.18 0.93 0.10 (0.17) −0.53 (0.34) Grade 2 1440 1 5 2.45 0.91 0.44 (0.06) −0.23 (0.13) 203 1 5 2.77 0.96 0.23 (0.17) −0.53 (0.34) Grade 3 1362 1 5 2.31 0.82 0.49 (0.07) 0.22 (0.13) 197 1 4.50 2.55 0.77 0.18 (0.17) −0.60 (0.34) Grade 4 1280 1 5 2.16 0.76 0.68 (0.07) 0.50 (0.14) 188 1 5 2.42 0.78 0.54 (0.18) 0.45 (0.35) Grade 6 987 1 4.67 2.07 0.66 0.39 (0.08) 0.20 (0.16) 180 1 4 2.33 0.66 0.41 (0.18) 0.26 (0.36) Grade 7 765 1 5 1.90 0.65 0.57 (0.09) 0.50 (0.18) 140 1 3.57 2.09 0.66 0.10 (0.20) −0.59 (0.41) Grade 9 890 1 4 1.71 0.61 0.73 (0.08) 0.45 (0.16) 169 1 4 1.93 0.60 0.37 (0.19) 0.12 (0.37) D. Khanolainen et al. Learning and Individual Differences 105 (2023) 102321 7 leading us to conclude that we had an adequate balance between the sample size and model indicators. Maximum likelihood with robust standard errors was used to estimate model parameters. Moreover, missing data was handled using full information maximum likelihood estimation (FIML). Mixture models do not have one commonly accepted criterion for deciding the number of classes (profiles); therefore, we relied on several statistical information criteria as well as on the interpretability of the final solution and graphic presentations of all possible solutions to decide the number of classes (profiles indicated by the model) (Yu & Park, 2014). Note that theory and past findings play an important role in the decision (Berlin et al., 2014; Geiser, 2012). Next, we validated the classification by conducting repeated measures analysis of variance (ANOVA) on children's skills. To determine whether children's RD and MD were associated with family-related variables, we conducted chi-square tests and ANOVAs. This second part of the analysis was conducted in SPSS Statistics 26. Finally, using the “three-step approach” we added all family-related factors as predictors to our mixture model (Asparouhov & Muth´ en, 2014). This statistical approach allows covariates to be tested as predictors of latent profiles in a multinomial logistic regression by using the Bolck-Croon-Hagernaars (BCH) method (Asparouhov & Muth´ en, 2014; Bakk et al., 2016). The BCH method uses weights based on the posterior probabilities to adjust for classification error. To analyze the relative contribution of each predictor to the identified latent profiles, we conducted hierarchical regression analyses in a structural equation modeling (SEM) framework by applying a Cholesky model (De Jong, 1999). Two separate Cholesky models were used, one model examined the relative contribution of the factors related to literacy (parental reading difficulties, teaching literacy at home when children were in kindergarten, parental assistance with literacy tasks in Grades 1–9, parental education) and the other model examined the relative contribution of the factors related to numeracy (parental math difficulties, teaching numeracy at home when children were in kindergarten, parental assistance with numeracy tasks in Grades 1–9, parental education). Parental education was treated as a general control measure and thus entered in both models. Maximum likelihood estimation with robust standard errors (MLR) was used as estimator for the analysis. The second half of our analysis that included family-related factors (using chi-square tests, ANOVAs, and the three-step approach) was performed to answer the second research question. All these analytical procedures were performed with the same goal in mind, but they had important differences. Compared to ANOVAs and chi-square tests the three-step approach is a more reliable method to identify factors that are significantly associated with latent profiles, however in the present study the three-step approach could not include typical performers for comparison (this was only possible in ANOVAs). Thus, only the combination of different statistical approaches allowed us to answer the second research question comprehensively. 3. Results 3.1. Descriptive statistics and group comparisons Table 1 presents the descriptive statistics for children's skills and parental academic assistance measures for all participants. All of the measure distributions were close to normal distribution. 3.2. Identification of patterns within development leading to RD and MD in Grade 9 To examine the presence of differential patterns within skill development that lead to RD, MD, or both in Grade 9, we ran a series of LPA models. Fig. 1 depicts the LPA model, and Table 2 describes the LPA model outcomes for the first six profiling solutions. Models beyond six profiles became unstable and fitted the data poorly. Sixand five-profile models each had one very small profile (containing only seven people, which is <2 % of the sample). In the four-profile model, the average latent class probabilities declined below 0.80, suggesting greater uncertainty for this profile solution. In addition, BIC started increasing in the four-profile model, indicating a worsening fit, which continued through to the five and six profile models. The two-profile model had the highest entropy, and LMR and VLMR p-values suggested that two profiles are sufficient. However, the three-profile model had the lowest BIC value. We chose the three-profile model instead of the two-profile for two reasons. First, BIC has been reported to be the most efficient indicator for deciding the number of latent classes (profiles) (Yu & Park, 2014), especially when dealing with continuous variables (Fonseca & Cardoso, 2007). Second, the three-profile model was better fitted to theory, which is a strong argument in its favor (Geiser, 2012), because it included a distinct comorbid group whereas the two-profile model did not. The first profile (N =121) was named Reading Difficulties (RD), as the participants in this profile demonstrated low reading fluency but average arithmetic fluency. The second profile (N =176) was named Reading and Mathematical Difficulties (RD&MD), as it was characterized by low reading and arithmetic fluency. Finally, the third profile (N =94) was named Mathematical Difficulties (MD) in view of the participants having low arithmetic fluency but average reading fluency. Fig. 2 shows the reading and arithmetic fluency development in the low-performing profiles contrasted with typical performers. As can be seen in both Fig. 2 and Table 3, children with only RD significantly underperformed not only in reading fluency tasks but also in arithmetic Table 2 Fit indices for latent profile analyses (low performers only, N =391). Number of profiles BIC aBIC AIC Entropy p-Value of LMR p-Value of VLMR n in class 1 (ALCP) n in class 2 (ALCP) n in class 3 (ALCP) n class 4 (ALCP) n class 5 (ALCP) n class 6 (ALCP) 1 10,448.398 10,311.961 10,277.744 2 10,304.732 10,120.701 10,074.547 0.83 0.0010 0.0010 287 (0.964) 104 (0.925) 3 10,278.694 10,047.068 9988.978 0.73 0.3347 0.3303 176 (0.853) 121 (0.893) 94 (0.902) 4 10,293.652 10,014.433 9944.406 0.72 0.1931 0.1902 79 (0.809) 109 (0.776) 82 (0.908) 121 (0.875) 5 10,316.166 9989.352 9907.390 0.76 0.8263 0.8258 57 (0.89) 204 (0.84) 70 (0.80) 53 (0.84) 7 (0.96) 6 10,340.258 9965.85 9871.951 0.77 0.2548 0.2542 160 (0.81) 24 (0.88) 64 (0.82) 78 (0.88) 58 (0.83) 7 (0.97) Note. BIC =Bayesian Information Criterion; aBIC =Adjusted Bayesian Information Criterion; AIC =Akaike's Information Criterion; LMR =Lo-Mendell-Rubin Adjusted Likelihood Ratio Test; VLMR =Vuong-Lo-Mendell-Rubin Likelihood Ratio Test; ALCP =Average Latent Class Probabilities for Most Likely Latent Class Membership by Latent Class. D. Khanolainen et al. Learning and Individual Differences 105 (2023) 102321 8 fluency tasks compared with typical performers over all time points. However, RD gradually made more gains in arithmetic fluency than MD and RD&MD and progressed towards the skill level of typical performers by grade 9. Similarly, Fig. 2 suggests that children in early grades with only MD performed worse than typical performers in reading fluency tasks; however, the difference between these groups was not statistically significant (Table 3). Moreover, RD and MD gradually diverged in their skill gains (children with RD caught up with typical performers in arithmetic fluency, whereas children with MD only narrowed the gap with typical performers in reading fluency). RD&MD lagged increasingly on both skills over all time points. In view of the use of standardized reading and arithmetic scores, the downward patterns seen in Fig. 2 indicate a growing gap in grade level performance across the profiles, but they do not imply actual skill deterioration. 3.3. Profile differences in parental characteristics First, chi-square tests were performed to examine the relationship between parental difficulties (family risk) and profile membership, including typical performers (Table 4). Family risk for RD was not Fig. 2. Reading fluency (z-scores) and arithmetic fluency (z-scores) longitudinal pathways of different profiles across the seven time points. Note. RD =Reading Difficulty Profile; MD =Mathematical Difficulty Profile; RD&MD =Comorbidity Profile; TP =typical performers (added here for comparison but was not identified in LPA). Even though children's skills across all profiles were continuously developing over time, the graph shows some downward patterns. This is because standardized scores for age-appropriate measures were used for plotting this line graph, representing the relative performance compared to grade level peers. Table 3 Descriptive statistics and ANOVA comparisons for skill measures (z-scores) of different profiles. Measures Time point Typical performers (TP) RD RD&MD MD F Partial eta sq Significant pairwise differences between profiles (Bonferroni comparisons) N M (SD) N M (SD) N M (SD) N M (SD) Reading fluency Gr 1 1761 0.08 (1.01) 99 −0.69 (0.65) 109 −0.61 (0.60) 68 −0.23 (0.76) 36.43*** 0.05 RD, RD&MD <TP; RD <MD Gr 2 1711 0.09 (1.00) 100 −0.79 (0.68) 111 −0.69 (0.64) 69 −0.09 (0.75) 46.76*** 0.07 RD, RD&MD <TP, MD Gr 3 1693 0.10 (0.99) 103 −0.90 (0.67) 112 −0.72 (0.73) 72 −0.06 (0.84) 57.68*** 0.08 RD, RD&MD <TP, MD Gr 4 1653 0.10 (0.99) 103 −0.93 (0.71) 113 −0.83 (0.67) 70 0.18 (0.74) 69.33*** 0.10 RD, RD&MD <TP, MD Gr 6 1442 0.19 (0.93) 116 −1.13 (0.61) 158 −1.06 (0.68) 91 0.27 (0.76) 162.90*** 0.21 RD, RD&MD <TP, MD Gr 7 1386 0.23 (0.89) 117 −1.30 (0.67) 160 −1.12 (0.73) 92 0.12 (0.67) 214.86*** 0.27 RD, RD&MD <TP, MD Gr 9 1315 0.30 (0.80) 121 −1.55 (0.43) 176 −1.34 (0.59) 94 0.27 (0.68) 427.01*** 0.43 RD, RD&MD <TP, MD Arithmetic fluency Gr 1 1760 0.07 (1.01) 99 −0.33 (0.89) 108 −0.47 (0.74) 68 −0.55 (0.67) 21.87*** 0.03 RD, MD, RD&MD <TP Gr 2 1708 0.09 (0.99) 100 −0.31 (0.90) 109 −0.60 (0.89) 69 −0.74 (0.68) 34.60*** 0.05 RD, MD, RD&MD <TP; MD <RD Gr 3 1692 0.09 (0.97) 103 −0.19 (1.00) 112 −0.80 (0.93) 72 −0.74 (0.79) 46.32*** 0.07 RD, MD, RD&MD <TP; RD&MD, MD <RD Gr 4 1652 0.11 (0.97) 103 −0.23 (0.85) 113 −0.91 (0.87) 70 −0.84 (0.72) 61.76*** 0.09 RD, MD, RD&MD <TP; RD&MD, MD <RD Gr 6 1437 0.20 (0.92) 115 −0.18 (0.72) 159 −1.12 (0.87) 91 −1.01 (0.64) 147.67*** 0.20 RD, MD, RD&MD <TP; RD&MD, MD <RD Gr 7 1367 0.22 (0.92) 115 −0.07 (0.65) 160 −1.18 (0.73) 92 −1.08 (0.71) 170.45*** 0.23 RD, MD, RD&MD <TP; RD&MD, MD <RD Gr 9 1299 0.31 (0.81) 121 −0.05 (0.63) 176 −1.40 (0.63) 94 −1.59 (0.38) 405.09*** 0.42 RD, MD, RD&MD <TP; RD&MD, MD <RD Note. * p<.05, ** p<.01, *** p<.001. D. Khanolainen et al.