Children’s mathematical skills in kindergarten are important for later mathematical achievement (e.g., Jordan et al., 2009). Using sequential process growth curve models across five time points from first through third grade, Jordan et al. (2009) found that kindergarten number sense predicted later mathematics achievement while accounting for child age, gender, and family income. However, even before that, preschool skills predict later mathematics outcomes throughout childhood and adolescence. For example, Watts et al. (2014) demonstrated that preschool mathematics predicted achievement in second grade, fifth grade, and at age 15, even after controlling for a broad set of child and family characteristics, including early reading skills, vocabulary, attention, general cognitive skills, behavior ratings, demographic factors, and home environment quality. Taken together, these findings suggest that early mathematics skills follow a developmental trajectory, meaning that early competencies provide a foundation that supports subsequent skill acquisition over time. Although prior work has examined this trajectory longitudinally, given children’s rapid skill development, more assessments over a shorter time frame are needed to better understand these periods (e.g., the beginning of preschool and the end of kindergarten).

One essential question yet to be addressed by prior research is whether classifiers of high or low mathematical performance at the end of kindergarten (6 – 8 years old) change from the beginning of preschool (3 – 5 years old), end of preschool (4 – 6 years old), or beginning of kindergarten (5 – 7 years old)? Addressing these questions can provide teachers and practitioners with more information about which types of activities or instruction may be most appropriate for a specific developmental period. Thus, an important contribution of including multiple developmental time points across multiple years would be to provide more precise information regarding appropriate developmental instruction. For example, Purpura, Day, et al. (2017) found that children’s mathematical language skills at the beginning of preschool were an important predictor of low mathematics performance at the end of preschool. However, a different cognitive skill (other than mathematical language) at the end of preschool may serve as an important classifier for mathematics at the end of kindergarten. Therefore, examining multiple developmental time points will be informative about which skills to focus on and when.

Given that early childhood mathematics performance is predictive of later college attainment (e.g., Davis-Kean et al., 2022) and income (Rose & Betts, 2004), it is important to identify best practices for promoting positive mathematical development early on in a child’s life. Research on children’s reading skills has used information to specifically target important predictors of later reading in early childhood (Carlisle, 2003; Tyler et al., 2002). However, unlike reading research, less is known about the specific skills that inhibit or advance children from succeeding in mathematics. Not necessarily the skills that predict mathematical giftedness or disabilities, but the skills that may hinder or advance children on a typical trajectory. With information about domain-specific and domain-general skills that contribute to mathematics, we can better specify which skills are most important at particular developmental periods, informing learning trajectories that support children who may struggle. Understanding these predictors is especially important in the context of socioeconomic differences in early mathematics, as children from lower-income backgrounds tend to show lower average performance (Purpura & Reid, 2016; Starkey et al., 2004) but substantial heterogeneity in developmental pathways (Jordan et al., 2009), highlighting the need to identify skills that may serve as targets for early intervention and support.

Although much of the literature focuses on predictors of low mathematics achievement (Jordan et al., 2002; Purpura, Day, et al., 2017), evidence also suggests that high mathematics performance is associated with early domain-general (e.g., executive function; Ribner et al., 2023), domain-specific (e.g., numeracy; Nguyen et al., 2016), and overlapping skills (e.g., mathematical language; Purpura, Day, et al., 2017). Importantly, predictors of high performance may reflect both areas for intervention and the presence of qualitatively distinct strengths, emphasizing the need for analytic approaches that identify classifiers across the full distribution of mathematics achievement.

Classification and regression tree (CART) analyses present an ideal modeling approach for identifying children at the high and low ends of mathematics performance. CART is a technique that uses recursive partitioning to create more alike groups of children from a larger sample using a range of skills as predictor variables (Gruenewald et al., 2008). Unlike multivariate regression, CART models create data-driven split points (or nodes) based on a continuous outcome variable to partition it into categories such as “high risk” and “low risk.” These models are inherently non-parametric; thus, they can include any number of possible predictors (Lewis, 2000).

Importantly, CART is designed for prediction and classification rather than modeling relations among constructs (Strobl et al., 2009). This approach allows for the identification of complex interactions and nonlinear relations among predictors without requiring these to be specified a priori, which would be necessary in approaches such as structural equation modeling. Because CART is data-driven, it identifies combinations of predictors that best classify outcomes based on observed data and detects meaningful cut points that may not be captured using traditional continuous modeling approaches.

Finally, CART also facilitates communication through its tree format (see Figure 1), allowing teachers and practitioners to better translate results into practice and determine which skills are more important than others for classifying math ability. In Figure 1, the full sample of 140 children was first split based on Bracken. Children with a score of 41 or lower had the lowest average mathematics score (Terminal Node 1), whereas children with a score above 58 on Bracken and above 32 on HTKS had the highest average mathematics score (Terminal Node 5).

Given the benefits of CART as a methodological approach for identifying risk classifiers, it is not surprising that prior work has leveraged this technique. Specifically, CART has been used in both mathematics and reading. Some studies have used CART analyses to determine classifiers of reading difficulties (Compton et al., 2006; Koon et al., 2014). Notably, Purpura, Day, et al. (2017) used CART to determine predictors of high and low mathematics performance between fall and spring of preschool. Finally, other studies have found that CART results are consistent with logistic regression results when more potential predictor variables are considered, and fewer assumptions are made about the model framework (Koon & Petscher, 2015).

The current study extends Purpura, Day, et al. (2017) by examining predictors across multiple developmental time points spanning preschool through kindergarten in an independent sample and builds upon previous research examining classifiers of later low and high mathematics performance in early childhood. Given that prior research has been limited by developmental timing (Jordan et al., 2002; Purpura, Day, et al., 2017), this study specifically extends prior work by using four (rather than two) developmental time points across years to identify classifiers of low and high mathematics performance. Notably, these four time points span the early years of children’s educational experiences in the United States and capture the critical transition from preschool to kindergarten: fall of preschool (Time 1), spring of preschool (Time 2), fall of kindergarten (Time 3), and spring of kindergarten (Time 4). All children were assessed on the same battery of measures at each time point (with math performance as the outcome at Time 4), but the developmental timing of these assessments allows us to examine how predictors operate differently across educational transitions. To do this, we examined three CART analyses that differed in the range of time points. The first assessed classifiers of initial school entry skills (using data from Time 1 to Time 4). The second assessed classifiers of kindergarten readiness skills (Time 2 to Time 4). The final assessed classifiers of kindergarten entry skills (Time 3 to Time 4). Based on results from Purpura, Day, et al. (2017), we expected children’s previous mathematics achievement and mathematical language skills (domain-specific and overlapping) to be consistent classifiers of later low mathematical skills across all time points; however, we did not explicitly expect this to differ from high mathematical skills. Further, given results from Purpura, Day, et al. (2017) suggesting that classifiers differ by children’s age, we also expected domain-general classifiers to differ across time points for both high and low mathematical skills.

Data came from a larger study evaluating a state-funded preschool program that was approved by the Institutional Review Board [#1807020801] and represents a quasi-experimental study in which a state-developed preschool program was designed to offer high-quality early education to children from families with low incomes. However, there were no significant differences in outcomes between the treatment and control groups, so we combined them into a single sample in this study. The original study consisted of N = 684 children; however, CART methods require complete case analysis, resulting in listwise deletion of all variables included in the models up to the final time point. As reported in studies using the same dataset, the data met the criteria for missing at random (Ehrman et al., 2025). Thus, the present sample included 322 children (50.6% female) from three cohorts (i.e., year of entry into preschool 2015, 2016, and 2018, respectively). To participate in the original study, all children’s families met an income-eligibility requirement of a family income at or below 127% of the federal poverty level. At the first time point (fall of preschool), children ranged in age from 3.74 to 5.45 years (M_age = 4.81 years; SD = 0.30). Per the parent report, the sample was 41% Black or African American, 35.1% White, 12.4% Hispanic or Latine, 10.5% multiracial, and 1% Asian. Further, children had no known diagnoses of developmental, neurological, or sensory differences.

Children were assessed at four time points: the fall and spring of preschool and kindergarten. The time between assessments during the school year was approximately 6 months (M = 5.56 months; SD = 0.79 months). Executive function, global school readiness, early literacy, receptive vocabulary, mathematical language, and early numeracy variables were each assessed at the first three time points following the same procedure at each time point. Early mathematics was assessed at all four time points following the same procedure at each time point. Assessments were administered one-on-one by a trained research assistant in a quiet area at the child’s school.

We used CART analyses in IBM SPSS 28.0 to identify predictors of high and low mathematics performance using a data-driven approach. Sum scores of children’s executive function, global school readiness, early literacy, receptive vocabulary, mathematical language, and early numeracy, as well as sex and age were used to predict low and high performance in mathematics in the spring of kindergarten. CART analyses consist of both trees and forests (Gruenewald et al., 2008). A tree represents a series of hierarchical decision rules in which the sample is repeatedly split into more homogeneous subgroups based on predictor variables, resulting in terminal nodes that represent groups of children with similar mathematics performance. Each split is selected to maximize differences between groups while minimizing differences within groups. A forest is a collection of trees trained on the same set of predictors but with different random subsamples of the data, allowing for the evaluation of the stability and replicability of classification patterns (Gruenewald et al., 2008). Rather than relying on a single tree, examining a forest of trees reduces the likelihood that results are driven by sample-specific variation. Given the non-parametric nature of CART, a traditional power analysis is not appropriate, as it relies on hypothesis testing. However, SPSS automatically adjusts for multiple comparisons using a built-in Bonferroni correction, as indicated in our syntax (ADJUST = BONFERRONI), which we have made publicly available (https://osf.io/t6dp9/files/52xmu). Accordingly, significance values for both merging and splitting criteria were adjusted using the Bonferroni method.

We generated three forests of 25 trees each: a) predicting mathematics performance in the spring of kindergarten using predictors from the fall of preschool, b) predicting mathematics performance in the spring of kindergarten using predictors from the spring of preschool, and c) predicting mathematics performance in the spring of kindergarten using predictors from the fall of kindergarten. The use of 25 trees per forest follows prior developmental CART applications and provides a balance between model stability and interpretability (e.g., Purpura, Day, et al., 2017). We used split-sample validation by growing a training tree on a randomly selected 60% of participants and replicating it with a test tree on the remaining 40%.

After generating forests of 25 trees for each time point of predictors, we used three criteria to choose the final representative trees for each forest (Day & Dotterer, 2018). First, we determined whether any trees had a terminal node (i.e., the children with the lowest mathematics scores) that contained less than 10% of the sample. Such trees were cut, as they were determined to be non-replicable. Next, we compared the mean mathematics score in spring kindergarten for children in the lowest terminal node across training and test tree pairs. Those with significantly different mean scores were excluded because they were non-replicable. Finally, we calculated the proportion of variance in mathematics achievement in the spring of kindergarten explained by each test tree. We excluded any that explained a substantially low proportion of variance (20%). For more information about CART, see Breiman and Ihaka (1984) and Purpura, Day, et al. (2017) for a closer example of using CART to classify low mathematics performance.

Descriptive statistics are presented in Table 1. All variables demonstrated adequate variability, with no evidence of floor or ceiling effects. Low mathematical performance was determined by focusing on the terminal node with the lowest mean mathematical score at the end of kindergarten in each individual tree; high mathematical performance was determined by focusing on the terminal node with the highest mean mathematical score. Full trees across all time points are shown in the appendix. Given the data-driven nature of CART analyses, it is best not to rely upon single trees, but rather to consider the entire forest to identify consistent classifiers. In this instance, after the exclusions listed above, there were 23 trees in the forest at the beginning of preschool classifications, 24 at the end of preschool classifications, and 19 at the beginning of kindergarten classifications. See (Ellis, 2026S) for syntax, output, and supplemental materials.

Six of the eleven possible variables appeared in at least one pathway when predicting children’s low mathematical performance, and eight appeared for high mathematical performance from the beginning of preschool to the end of kindergarten (see Table 2). The most common predictors across all trees were prior mathematics scores (WJAP and PENS) and the school readiness assessment (Bracken). Trees assessing classifiers from the beginning of preschool accounted for 19.98-41.47% of the variance in the end of kindergarten mathematics scores. Tree 5 accounted for the most variance (41.47%) in later mathematics performance (Figure 1). In this tree, a score below 41 on the school readiness assessment was associated with a low mathematics score (14.09), and a score above 58 on the school readiness assessment and above 32 on behavioral self-regulation (HTKS) was associated with a high mathematics score (20.97) at the end of kindergarten. These results suggest that the combination of low levels of prior mathematics achievement and low levels of school readiness skills is a strong classifier at the beginning of preschool for low math performance at the end of kindergarten.

At the end of preschool, seven of the eleven possible variables appeared in at least one pathway when predicting children’s low and high mathematical performance at the end of kindergarten (see Table 3). All the same variables from the fall of preschool were present except the reading measure (GRTR), and instead, children’s behavioral self-regulation (HTKS) and inhibitory control (DNS) skills were included. This time, the most common predictors across all trees were prior mathematics scores (mainly PENS) and low scores on the behavioral self-regulation assessment (HTKS). A consistent classifier of only low mathematics performance at the end of kindergarten was behavioral self-regulation skills (HTKS) at the end of preschool. Trees assessing classifiers at the end of preschool accounted for 29.22-59.37% of the variance in the end of kindergarten mathematics scores. Tree 24 accounted for the most variance (59.37%) in later mathematics performance (Figure 2). In this tree, a score below or equal to 10 on prior mathematics scores (WJAP) in combination with a score below or equal to 10 on early numeracy skills (PENS) was associated with a low mathematics score (12.67), and a score of greater than 15 on early numeracy skills (PENS) and a score above 13 on math language (MLA) was associated with a high mathematics score (21.04) at the end of kindergarten. These results suggest that the combination of low levels of prior mathematics achievement and low levels of numeracy knowledge are strong classifiers at the end of preschool of low math performance, and high mathematics achievement, numeracy knowledge, and math language are strong classifiers at the end of preschool of high math performance at the end of kindergarten.

Finally, at the beginning of kindergarten, seven of the eleven possible variables appeared in at least one pathway when predicting children’s low and high mathematical performance at the end of kindergarten (see Table 4).

The most common predictors across all of the trees were prior mathematics scores (WJAP and PENS). A consistent classifier of only high mathematics performance at the end of kindergarten was reading skills (GRTR) at the beginning of kindergarten. Trees assessing classifiers from the beginning of kindergarten accounted for 36.48-52.81% of the variance in the end of kindergarten mathematics scores. Tree 8 accounted for the most variance (52.81%) in later mathematics performance (Figure 3). In this tree, a score below or equal to 13 on prior mathematics scores (WJAP) in combination with a score below or equal to 16 on early numeracy skills (PENS) was associated with a low mathematics score (13.56) at the end of kindergarten. A score above 19 on prior mathematics scores (WJAP) in combination with a score above 23 on reading skills (GRTR) was associated with a high mathematics score (21.00) at the end of kindergarten. These results suggest that the combination of low levels of prior mathematics achievement and low levels of numeracy knowledge is a strong classifier at the beginning of kindergarten of low math performance, and high levels of numeracy knowledge and high levels of reading skills are strong classifiers of high math performance.

From a broader developmental perspective, these findings align with transactional and systems-based theories of development, which emphasize that children’s skills emerge through dynamic interactions between individual characteristics and environmental contexts across time (Bronfenbrenner & Morris, 2007; Sameroff, 2010). The shifting importance of domain-specific, domain-general, and overlapping skills across developmental periods, as shown in this study, underscores the idea that early mathematics development is not driven by a single skill or pathway, but rather by multiple changing abilities and contexts that reflect children’s developmental stage and educational context. In particular, the consistent role of prior mathematics performance across time points is consistent with models of development (Duncan et al., 2007; Watts et al., 2014), whereas the time-specific emergence of school readiness, self-regulation, and reading skills highlights the importance of developmental timing and schooling experiences in shaping mathematics outcomes (Morrison et al., 2019). Together, these findings support a developmental systems view in which early competencies both scaffold and are reorganized by subsequent learning experiences, including formal instruction.

The current findings have implications for future research and practice. From a research perspective, identifying age-specific classifiers highlights the importance of examining developmental timing in studies of early mathematics achievement, rather than assuming the same predictors remain important across childhood. Future work should continue to examine how domain-specific and domain-general skills interact across developmental periods, particularly in more diverse and nationally representative samples, and whether similar classification patterns emerge when different instructional contexts or curricula are considered. From a practice perspective, these findings may suggest that early mathematics support may be most effective when aligned with children’s developmental age. For example, broad school readiness and self-regulation skills may be particularly important for identification in early preschool, whereas domain-specific numeracy skills may warrant greater emphasis as children approach and enter formal schooling. Although the current study does not test interventions directly, identifying developmentally sensitive classifiers provides a framework for informing differentiated instruction and early screening efforts to support children at risk for low mathematics performance.

A couple of limitations of the present study should be noted. First, CART is a data-driven analytic approach that may be susceptible to sample-specific biases. While many of our results are consistent with prior findings and theoretical evidence (Duncan et al., 2007; Purpura et al., 2011), there are limitations specific to our sample. For example, the analytic sample of this paper included children whose families met a specific income-eligibility requirement, and therefore, it does not capture the full range of possible educational and home experiences. Relatedly, although socioeconomic status (SES) is known to be associated with children’s language, literacy, and mathematics development, the SES variables in this dataset had substantial missing values. As CART requires complete cases, including SES would have reduced the analytic sample, making it infeasible to incorporate SES as a covariate in the present analysis. As a result, SES could not be directly examined in this study. However, one interesting future direction could be to use CART analyses with nationally representative data (i.e., Early Childhood Longitudinal Study-Kindergarten) to examine low and high mathematics performance across multiple time points and compare results.

Second, although the current study identifies classifiers of low and high mathematical performance, mathematical difficulties or giftedness could not be determined from these data. Math difficulty is often defined by low performance despite access to adequate instruction, resistance to intervention, or a specific math learning disability such as dyscalculia (Chan & Wong, 2020). Math giftedness has received more attention over recent years (Leikin, 2020), and although a precise definition remains lacking, it is often perceived as a person with high mathematical abilities and a strong inborn mathematical promise (Leikin, 2009). Importantly, including a broader range of learners may increase variability in math performance while also limiting conclusions drawn from more homogeneous samples. Despite this limitation, the current study can identify possible antecedents to children’s low and high performance represented in this sample. This information, used in conjunction with other important factors, can help better identify children who may experience difficulties or giftedness in math. Future research should also examine whether specific components of early numeracy (e.g., magnitude, cardinality, or symbolic knowledge) differentially classify later mathematics outcomes across developmental periods and levels of performance.

The current study assessed domain-specific, domain-general, and overlapping classifiers of low and high mathematics performance across different developmental periods. Findings from this study suggested that classifiers may differ depending on developmental period and performance level, such that domain-specific factors serve as classifiers across all developmental periods, whereas domain-general skills serve as important classification capabilities at different time points and performance levels. Moving forward, this methodological approach and understanding of domain-specific and domain-general skills should inform our research, policies, and practices to better identify and support children at risk of low mathematics performance.