Introduction

Executive functioning (EF) has been defined and described in many ways since the inception of the term. However, for the purpose of the present research, the authors have adopted the following definition: “[t]he functions an organism employs to act independently in its own best interest as a whole, at any point in time, for the purpose of survival” (Koziol & Lutz, 2013, p.1). The definition implies that adapting to the environment, indeed, seems to be the central purpose of the development of EF. From an evolutionary perspective, early humans who were better at planning and organizing an approach to finding food, building shelter, communicating with others, and overpowering rivals would be the ones who were more likely to survive, and that simply reacting to threats—rather than planning ahead and anticipating outcomes of one’s own as well as others’ behaviors—likely would have been much less effective for survival (Adornetti, 2016). Evolutionarily, executive functions, such as sustained attention, behavioral inhibition, working memory, and planning and organization, would have been essential for survival, and therefore reproductive fitness. Koziol and Lutz (2013) posited that EF relies on procedural learning and automatically executed behaviors to allow for humans’ adaptation to changing environmental demands, and that the human functional brain structure evolved to facilitate efficient executive functioning. Automaticity of procedural memories facilitates adaptation to the environment because higher cortical reasoning is freed and made nimble in real time as lower procedures are not required to be consciously directed; they are automatic. Although the extent and aim of what is considered adaptive in modern times have steadily become more focused on academic performance—and thereby future income potential (Maner & Menzel, 2012)—intact and efficient EF skills remain advantageous to survival.

Rather than describing EF in the traditional manner as a serial order process focused on thinking and cognitive processes (Chidekel & Budding, 2010; Koziol et al., 2014a; Njiokiktjien, 2010), extant research posits the necessity of describing EF as simultaneous processing and automation as it relates to academic learning, which are functions often associated with the cerebellum (Koziol & Lutz, 2013). In Koziol and Lutz (2013), the authors proposed an interactive model of EF that emphasizes action and control of actions, as well as coupling of actions for automatic expression, as the primary mediator of behavior and cognition. According to the model, EF developed from the necessity of early human ancestors to make quick survival decisions relying more predominantly on motor skills. This appears to be consistent with the fact that motor skills are the first to appear in prenatal and neonatal development, and they are the first skills to become automated (Koziol et al., 2014b, 2016; Malina, 2004; Njiokiktjien, 2010). This is not to say that motor skills are in and of themselves EF, but rather that executive functions allow for the automatic expression of motor skills, which were initially necessary for survival. Serial order processing requires sensorimotor feedback, time for that feedback to be processed, transmission of motor commands to muscles, and thinking and organizing of motor responses—a process which can be quite slow in comparison to automated responses. Evolutionarily, survival responses needed to be quick, accurate, and automatic. Practically speaking, considering the necessary skills of modern humans, basic motor skills also need to be habituated and automatic; otherwise, people would be walking around actively thinking about the process of walking (“Right foot, left foot, right foot, left foot…”), writing, typing, turning a page in a book, among others. Automation of responses is key, and it relies on the anticipation of outcomes. Both processes are dependent on large-scale brain systems of which a key feature is cerebellar function (Koziol & Lutz, 2013). Anticipation of outcomes allows humans to alternate between reactive/automatic processing and proactive/serial order processing (Koziol & Lutz, 2013). Through repetition of identical and similar activities, the brain begins to learn (with assistance of the cerebellum) to anticipate certain outcomes and to then adjust motor responses accordingly. The cerebellum then coordinates this anticipation of outcomes with the prefrontal cortex (PFC) in an effort to “teach” the PFC to anticipate those outcomes and to store the most efficient routines and automatic responses. Executive functioning as mediated by cerebellar processes, therefore, is the foundation of many procedural skills.

Under the same evolutionary paradigm, the adaptive expression of efficient performance of math operations relies on automation of mathematic facts and procedures in regards to motor and cognitive skills (Ashkenazi et al., 2013; Blythe, 2009; Fuchs & Fuchs, 2003). When considering how this process may relate to mathematics, consider a thought experiment. In order to express quick and accurate simple one-step math calculation, one must automate answers to basic mathematical facts across the operations of addition, subtraction, multiplication, and division. Only once each skill is individually habituated and consolidated can they be integrated into higher-order mathematics, such as algebraic calculations. Thus, failure to automate such procedural responses results in inefficient and disordered mathematic performance, which by extension ultimately impedes the smooth development of higher order mathematical ability.

Procedural Consolidation Deficit

Procedural learning can be defined as “gradual, incremental learning of skills and habits that can be demonstrated through improvements in task performance, but do not require conscious memorization or recollection (e.g., riding a bike, tennis swing, driving)” (Gonzalez et al., 2008, p. 776). Procedural learning is heavily involved in all aspects of daily life, including skills in motor, communication, adaptive, social, academic, and vocational functioning (Chidekel & Budding, 2010; Gilmore et al., 2017; Lum et al., 2013; Nicolson & Fawcett, 2007). For example, automated procedures are crucial for efficient written communication skills (Ashkenazi et al., 2013; Lum et al., 2013; Gabay et al., 2012; Nicolson & Fawcett, 2007; Ullman & Pierpont, 2005), word reading recognition skills, and mathematical ability. The attainment of educational automation can have a long-standing impact on academic success (Ashkenazi et al., 2013; Chidekel & Budding, 2010; Rubinsten & Henik, 2005; Swanson & Saez, 2003). Thus, procedural learning may be one of the most important forms of learning as it lends to automaticity (Chidekel & Budding, 2010; Grafton et al., 1992; Koziol et al., 2013; Ullman & Pierpont, 2005). One crucial outcome of developing habituated procedural learning skills is the reallocation of cognition to problem solve in real time as opposed to the necessity to overburden cognitive capacity (Ashkenazi et al., 2013; Feifer, 2017).

Research on the neural basis of procedural learning indicates involvement of fronto-striatal-thalamic circuitry, hippocampal connections, basal ganglia connections, and cerebellar inputs and pathways, with additional pathways and structures involved depending upon the specific skill or academic domain (Evans & Ullman, 2016; Gabay et al., 2012; Grafton et al., 1992; Koziol et al., 2014a, b; Lum et al., 2013; Molinari et al., 1997; Ullman & Pierpont, 2005). Neurodevelopmental difficulty with, or acquired injury to, one or more of these diffuse areas or pathways can interfere with the development of procedural learning. When occurring in the absence of a memory deficit, there is a resulting failure to automate a procedural skill through repetition—termed a procedural consolidation deficit (PCD). Given the necessity of intact procedural learning ability for efficient academic learning, the accurate identification of PCD may provide a theoretical underpinning of various learning disorders, including MLD (Evans & Ullman, 2016). Thus, one aim of the present study is to determine the extent to which performance across neuropsychological tasks assessing procedural learning skills correlate with and predict MLD.

Math Learning Disorder

The majority of research conducted on learning disorders primarily focuses on reading learning disorders, with a limited focus on MLD (Fuchs & Fuchs, 2003; Gabay et al., 2012; Lum et al., 2013; Miller et al., 2003; Nicolson & Fawcett, 2019). The few number of extant studies on MLD primarily explore mathematical concepts (Ashkenazi, 2013; Butterworth et al., 2011; Geary, 2003; Piazza et al., 2010). As such, a large gap remains in the understanding of the process by which procedural learning affects math learning. Failure to consolidate repeatedly practiced skills due to PCD may be linked to the failure to consolidate basic mathematical operations—herein referred to as MLD. While one may conceptually understand mathematical operations, the failure to fluently recall math facts is the crux of MLD of the PCD-type (Ashkenazi et al., 2013; Chidekel & Budding, 2010; Feifer, 2017; Geary, 2003, 2004; Rubinsten & Henik, 2005). Although a deficit in one’s ability to efficiently recall mathematical facts is described as being a component of verbal dyscalculia (Feifer & De Fina, 2005), poor math fact automation appears to be overlooked in much of the available literature. Indeed, the association of poor math fact automation with poor procedural automation in general also appears to be overlooked in much of the available literature.

Aims and Hypotheses

Extant research indicates those with MLD may have more generalized procedural learning deficits than those without MLD (Evans & Ullman, 2016; Uittenhove et al., 2016; Ullman, 2004). Thus, the present study aims to identify specific procedural learning deficits related to MLD. A secondary aim is to develop a clinically useful regression model of MLD utilizing performance on procedural learning tasks. Hypotheses include:

  1. 1.

    There will be a significant correlation between performance on neuropsychological tasks requiring procedural learning and a diagnosis of MLD.

  2. 2.

    There will be an algorithmic logistic regression model with sufficient sensitivity and specificity for accurately predicting a classification of MLD given an individual’s performance on procedural learning tasks.

Materials and Methods

Participants

The researchers extracted archival data from the neuropsychological reports of a private neuropsychology practice. Demographic, assessment, and diagnostic data were used in the present study. Due to the administration requirements of the selected tests in accordance with their published norms, inclusion criteria included participant age. Across the tests used, the sample was restricted to participants between the ages of 6 and 15 years–11 months. Owing to diagnostic guidelines of the DSM-5, only participants with a derived FSIQ on the Wechsler Intelligence Scale for Children, 5th Edition (WISC-V; Wechsler, 2014) above the threshold for intellectual disability (a standard score of 70) were included in the current sample.

The resultant sample included 299 participants, with 173 males (58% of the sample) and 126 females (42% of the sample). Participants had a mean age of 10 years–0 months (SD = 2.8 years). The sample had a mean FSIQ standard score of 105 (SD = 9), as measured by the WISC-V. The sample consisted of 150 individuals (50% of the sample) identified with MLD, in accordance with diagnostic criteria in the DSM-5 as well as observed difficulty with the consolidation of mathematics fundamentals on standardized academic achievement tests. Likewise, 42% of the sample had a prior diagnosis of Specific Learning Disorder with Impairment in Reading (Dyslexia), and 36% had a prior diagnosis of Specific Learning disorder with Impairment in Written Expression (Dysgraphia). Approximately 65% of the sample had a prior diagnosis of Attention-Deficit/Hyperactivity Disorder (ADHD).

Materials

The records consisted of normed test data from the following psychometric tests: Wechsler Individual Achievement Test, 3rd Edition (WIAT-III; Wecshler, 2009); Wechsler Intelligence Scale for Children, 5th Edition (WISC-V) Symbol Search and Coding subtests (Wechsler, 2014); A Developmental Neuropsychological Assessment (NEPSY-II) Inhibition subtests (Korkman et al., 2007); Trail Making Test for Children Parts A and B (Spreen & Strauss, 1998).

Design and Procedure

Given theoretical models posited in extant research, the present study utilized an exploratory correlational and regression analysis to further develop a model for predicting a classification of MLD using normed scores on standardized neuropsychological tasks measuring procedural learning. The Coding and Symbol Search subtests of the WISC-V, the NEPSY-II Inhibition subtest, and a serial administration of TMT A and B were used as measures of procedural learning. While the Processing Speed tasks—including WISC-V Coding and Symbol Search subtests—are traditionally used to measure processing speed, the manner in which a participant repeatedly completes a finite number of tasks in a serially administered manner suggests they are also measures of procedural learning. A biserial correlation was conducted using SPSS to determine the relationship between participants’ performance on WISC-V Coding and Symbol Search subtests, NEPSY-II Inhibition subtests, Trail Making Test A, and Trail Making Test B, and a dichotomous classification of MLD compared to a non-classification of MLD. Additionally, a stepwise binary logistic regression model was developed to provide the clinical framework from which to predict a classification of MLD given an individual’s performance on tests measuring procedural learning. The stepwise binary logistic regression methodology was utilized as it would produce an accurate and parsimonious set of predictors (measures of procedural learning) by elucidating relationships between predictors that have not yet been explored in extant research in order to develop a more efficient algorithmic model for clinical use. A follow-up validation procedure was conducted to establish classification accuracy.

Results

There was no statistically significant difference in age between the MLD and non-MLD groups (p>.05). There was a statistically significant difference in FSIQ between the MLD and non-MLD groups in which those with MLD scored lower (mean = 100; SD = 10) than the non-MLD group (mean = 109; SD = 9), which would be expected as noted in extant research such that individuals with ADHD tend to underperform on measures of intellect (Kuntsi et al., 2004). Lastly, there was a statistically significant difference in rates of ADHD diagnosis between the MLD and non-MLD groups, insomuch that those with MLD were more often comorbidly diagnosed with ADHD (p<.01).

A point-biserial correlation revealed a significant correlation between a prior diagnosis of MLD and scores on the following procedural learning tasks (Table 1).

Table 1 Table of means for participants
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A stepwise binary logistic regression model using scores on procedural learning tasks (Table 2) as predictor variables and a classification of MLD as the predicted variable was highly significant (p<.0001). The model was found to have goodness of fit (nagelkerke r square = .686; Hosmer-Lemeshow = .690), indicating that of the above-mentioned procedural learning tests (see Table 1) those most predictive of a classification of MLD included: WISC-V Coding, TMT B1, TMT B1-5 slope. There was no significant collinearity between predictor variables.

Table 2 Correlation between procedural learning tasks and a diagnosis of specific learning disorder with impairment in mathematics
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The vectored relationship between the three tests and a subsequent predicted diagnosis of MLD is such that a lower score on each task results in a higher odds-ratio of being classified with MLD. Thus, lower scores on WISC-V Coding, Trails B and the slope of one’s performance across five serial administrations of Trails B significantly predicted a classification of MLD (Table 3).

Table 3 Outcome of binary logistic regression
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The predictive model demonstrated a classification accuracy of 87.4% of those with MLD and those without MLD, with an ROC sensitivity of 84.7% and specificity of 89.3%, for an overall AUC of 0.870. Therefore, the newly established predictive model of scores on procedural learning tasks demonstrates an 87% accuracy in identifying those with MLD.

Discussion

This is the first study to date examining the theoretical and clinical relationship between PCD and MLD. The results of the present study provide statistical support for the theoretical model positing a relationship between PCD and MLD. Procedural learning tasks that correlated with MLD included the following: WISC-V Symbol Search; WISC-V Coding; several NEPSY-II Inhibition tasks; Trail Making Test A; Trail Making Test B.

More generally, results of the present study corroborate extant research suggesting procedural learning underpins math ability. While some researchers have posited standard clinical practice for identifying MLD is via tests of math concepts, the present study supports the conception of MLD as a deficit of procedural consolidation aside from conceptual understanding of mathematics. Therefore, MLD can be characterized as a deficit in consolidating the incremental complexity of mathematical concepts, insomuch that those with such a learning disorder struggle to express procedural automaticity through repetition of lower order mathematical fundamentals from which they are able to blend higher order mathematic concepts to demonstrate novel learning (Christensen et al., 2014; Mirsky et al., 1999; Poljac et al., 2009; Swanson & Saez, 2003).

In addition, the present study demonstrated that the performance on procedural learning tasks could be used to predict a clinical diagnosis of MLD using a regressive algorithm. The best overall predictors of MLD were performance on the following procedural learning tasks, in hierarchically ranked order: whether one increases by at least 10% in time across each of five serial administrations of Trail Making Test B, WISC-V Coding, and the first administration of Trail Making Test B. The results suggest that PCD is a strong predictor of MLD, and that such may serve as an addendum in clinical diagnostics. Overall, these findings reveal that procedural learning was identified as a highly effective algorithmic underpinning useful for clinicians to screen for children and adolescents who are likely to develop MLD.

A potential limitation of the present study included the demographics of the available sample. As the participants were sampled from an outpatient pediatric neuropsychology practice, many participants had comorbid diagnoses. In total, nearly 65% of the sample had prior diagnoses of Attention-Deficit/Hyperactivity Disorder (ADHD), which constituted the most frequent comorbidity across the sample. Likewise, 42% of the sample had a prior diagnosis of Specific Learning Disorder with Impairment in Reading (Dyslexia), and 36% were diagnosed with a Specific Learning Disorder with Impairment in Written Expression (Dysgraphia). However, such rates of Learning Disorders (LD) are commensurate with what would be expected given the observed rate of ADHD in an outpatient clinical setting (DuPaul et al., 2013; Wadsworth et al., 2015).

An additional potential limitation of the study was due to the theoretical nature of the concept of PCD as it relates to available neuropsychological tests. Many of the tests herein utilized were not originally developed as pure measures of procedural learning. For example, some of the measures assess procedural learning in concert with working memory. However, PCD should be considered separate from working memory, which is a single trial phenomenon. The development of procedural memory is based on motor repetition. Thus, as available neuropsychological tests are not pure measures of procedural learning at the present, the modeling process utilized in the current study should be classified as exploratory. For example, consider the Trail Making Test Part B, which has been classified as a measure of working memory when administered on a single trial basis. The present use of Trail Making Test Part B, administered five times serially, is a theoretically informed exploratory method. Due to such, further neurological and neuropsychological research is needed to strengthen the conceptual validity of such a neuropsychological phenomenon of procedural learning as measured by available neuropsychological tests, as well as the development of more pure tests of procedural learning. However, the scope of the present study served to introduce such topics for further study.

The present study provides theoretical and clinical results which warrant future research in more fully refining the relationship between the constructs of PCD and MLD, and between PCD and other learning disorders. These findings may be applied to other learning disorders that require repetition to consolidate the academic skill (Ashkenazi et al., 2013; Chidekel & Budding, 2010; Geary & Hoard, 2001), such as orthographic dyslexia and dysgraphia. Orthographic dyslexia and dysgraphia are two such disorders that heavily rely on procedural learning (Evans & Ullman, 2016). Orthographic dyslexia involves the failure to habituate common/high frequency words for spelling or reading despite much practice (Wang et al., 2014; Feifer & De Fina, 2000). The child must sound out words that should be easily recognized. Dysgraphia is not only poor handwriting but also the incomplete consolidation of automating grammar, case, and punctuation to write expressively (Chidekel & Budding, 2010; Feifer & De Fina, 2002). Future research should thus focus on adding additional literature on the theoretical and clinical applicability of using pediatric neuropsychological assessments of procedural learning for identifying MLD. Likewise, comparison of PCD in reading, writing, and math disorders would provide valuable information regarding shared and unique variance of PCD among these clinical groups.