Issues in longitudinal research on motivation

This paper discusses two methodological issues regarding the analysis of longitudinal data using structural equation modeling that emerged during the reconsideration of the analysis of a recent study on the relationship between academic motivation and language achievement in elementary education [Stoel R.D., Peetsma, T.T.D. and Roeleveld, J. (2003). Relations between the development of school investment, self-confidence and language achievement in elementary education: a multivariate latent growth curve approach. Learning and individual differences, 13, 313–333]. The issues are related to the factorial structure of the repeatedly measured variables, and to the explanation of interindividual difference by means of covariates [see Stoel, R.D., Van den Wittenboer, G. and Hox, J.J. (2004a). Including time-invariant covariates in the latent growth curve model. Structural Equation Modeling, 11, 155–167, Stoel, R.D., Van den Wittenboer, G. and Hox, J.J. (2004b). Methodological issues in the application of the latent growth curve model. In K. van Montfort, H. Oud, and A. Satorra (Eds.). Recent developments on structural equation modeling: Theory and applications. (pp. 241–262). Amsterdam: Kluwer Academic Press. It is illustrated that standard modeling practices may sometimes lead to incorrect conclusions regarding the concepts under investigation, and that ideally alternative modeling possibilities should be considered in order to check the adequacy of the standard practice.     

The relationship between science process skill and formal thinking abilities

The purpose of this study is to examine integrated process skill and formal thinking abilities of middle and high school students and determine the relationship, if any, between the two. A relationship was thought to exist since both sets of skills strongly emphasize conducting fair experiments as well as other abilities. Pencil and paper measures of formal operational and integrated process skill achievement were given to almost 500 grade 7–12 students. Resulting correlations showed a strong relationship between achievement on the two meansures (r = 0.73) and all subtests of both measures. Factor analysis data corroborate the correlational evidence. One potential inference to be drawn from these results is that process skill teaching might influence formal thinking ability. A follow-up experimental study will determine this.     

Modeling the Growth of Decoding Skills in First-Grade Children

The purpose of this study was to demonstrate the usefulness of combining curriculum-based measurement and hierarchical linear modeling procedures to identify the characteristics of first-grade children that predict growth rates in the acquisition of decoding skills (as assessed through measures of isolated word and nonword reading). This study examined the relative importance of both static (initial levels of performance) and dynamic (rate of growth) measures of cognitive-processing abilities (i.e., phonemic awareness and rapid naming speed) and emergent print knowledge (i.e., letter name, letter sound, more advanced graphophoneme knowledge, and orthographic awareness) as predictors of decoding growth in a sample of 75 first-grade children. Over the course of an academic year, a set of parallel word and nonword reading tasks, constructed using curriculum-base measurement techniques and administered on a monthly basis, were capable of demonstrating individual change in decoding skill. Furthermore, results indicate that growth in cognitive-processing abilities and general knowledge about print could likewise be measured and adequately modeled. In this sample of children, rate of growth in word and nonword reading was predicted by a different combination of static and dynamic variables representing both cognitive-processing abilities and print knowledge. Results suggest that in the very earliest stages of word reading development there may be a strong association between the rate of growth in cognitive processing, print knowledge, and decoding skills.     

Modeling Latent Growth Curves With Incomplete Data Using Different Types of Structural Equation Modeling and Multilevel Software

This article offers different examples of how to fit latent growth curve (LGC) models to longitudinal data using a variety of different software programs (i.e., LISREL, Mx, Mplus, AMOS, SAS). The article shows how the same model can be fitted using both structural equation modeling and multilevel software, with nearly identical results, even in the case of models of latent growth fitted to incomplete data. The general purpose of this article is to provide a demonstration that integrates programming features from different software. The most immediate goal is to help researchers implement these LGC models as a useful way to test hypotheses of growth.     

Growth curve analysis of ability means and variances in measures of fluid intelligence of older adults

This article is concerned with an application of growth curve analysis to modeling test performance changes demonstrated by older adults following cognitive training or based on experience with repeatedly presented fluid intelligence measures. Group differences and invariance in change over time in ability means and indices of interindividual differences at the ability level in a 2‐group cognitive intervention study (Baltes, Dittmann‐Kohli, & Kliegl, 1986) are examined using 3 latent curve analysis (Meredith & Tisak, 1990) models. The results indicate (a) group equivalence in the pattern of temporal development of ability variances and (b) training effects in the experimental group that are stronger than the practice/experience effects in the control group, whereby both types of effects are maintained over the 6‐month testing period. The findings provide further suggestive evidence for plasticity in fluid intelligence of elderly adults (cf. Baltes et al., 1986; Raykov, 1995, 1996).     

Executive function facilitates learning from math instruction in kindergarten: Evidence from the ECLS-K

Extensive evidence has suggested mathematical skill in early childhood is a robust predictor of children's later academic skills and eventual labor market outcomes; however, there is substantial heterogeneity in the degree to which different students learn from the same instructional contexts. Using data from N = 12,082 children enrolled in the Early Childhood Longitudinal Study-Kindergarten Cohort, this paper employs a latent piecewise growth curve modeling approach to investigate the role of classroom math instruction and executive function and approaches to learning in the development of mathematical skills in kindergarten, first, and second grade. Findings suggest that overall instructional frequency relates to math development in kindergarten through second, and that this is driven by exposure to advanced content in kindergarten. Further, executive function moderates children's learning in kindergarten, such that children with higher levels of executive function benefit more from instruction than do those with lower levels.     

Application of Latent Growth Curve Analysis With Categorical Responses in Social Behavioral Research

Latent growth modeling allows social behavioral researchers to investigate within-person change and between-person differences in within-person change. Typically, conventional latent growth curve models are applied to continuous variables, where the residuals are assumed to be normally distributed, whereas categorical variables (i.e., binary and ordinal variables), which do not hold to normal distribution assumptions, have rarely been used. This article describes the latent growth curve model with categorical variables, and illustrates applications using Mplus software that are applicable to social behavioral research. The illustrations use marital instability data from the Iowa Youth and Family Project. We close with recommendations for the specification and parameterization of growth models that use both logit and probit link functions.     

Training Interventions to Foster Skill and Will of Argumentative Thinking

Argumentative thinking requires not only the skill to apply argumentative strategies such as supporting theories with evidence but also the will to apply these strategies by considering argumentative thinking to be both reasonable and worthwhile. Focusing on direct instruction for the initial acquisition of both skill and will, we developed a new short-term computer-based training intervention. We tested its effects on learning processes and learning outcomes in an experimental study with 147 German high school students. Our intervention fostered facets of both skill (i.e. a declarative knowledge about argumentation) and will (i.e., epistemic orientation, intellectual values, and epistemic knowledge). We gained additional insights into learning mechanisms, such as the mediating effects of the learners' self-explanation activities and the advantage of addressing will before skill.     

Changes in acceptance of dating violence and physical dating violence victimization in a longitudinal study with teens

Teen dating violence is a pervasive issue in adolescence and has been linked to maladjustment (Temple, Shorey, Fite et al., 2013). Physical dating violence is a particularly significant problem with one in five adolescents reporting experiencing physical teen dating violence (TDV; Wincentak et al., 2017). Acceptance of violence has been suggested to increase the risk of TDV; however, most studies to date have been cross-sectional. The purpose of the current study is to examine patterns of acceptance of dating violence and TDV victimization across time. Participants were ethnically diverse teenagers (N = 1042; ages 13–18) who were followed over a four-year period. Multivariate latent growth curve modeling techniques were used to determine trajectories of physical TDV victimization and attitudes accepting of dating violence. Results showed two trajectories for physical TDV victimization, linear and quadratic, and two trajectories for acceptance of dating violence, non-linear and quadratic. Parallel models investigating the interplay between TDV victimization and acceptance demonstrated two possible trends; however, we did not find any evidence for a longitudinal relationship between the two variables, suggesting that change in acceptance was not related to change in physical TDV victimization. Instead, our results suggest a significant amount of heterogeneity in these trajectories. These findings suggest studies are still needed to further explore longitudinal patterns of TDV to better understand how to reduce the risk of teen dating violence.     

Is three‐dimensional videography the cutting edge of surgical skill acquisition?

The process of learning new surgical technical skills is vital to the career of a surgeon. The acquisition of these new skills is influenced greatly by visual‐spatial ability (VSA) and may be difficult for some learners to rapidly assimilate. In many cases, the role of VSA on the acquisition of a novel technical skill has been explored; however, none have probed the impact of a three‐dimensional (3D) video learning module on the acquisition of new surgical skills. The first aim of this study is to capture spatially complex surgical translational flaps using 3D videography and incorporate the footage into a self‐contained e‐learning module designed in line with the principles of cognitive load theory. The second aim is to assess the efficacy of 3D video as a medium to support the acquisition of complex surgical skills in novice surgeons as evaluated using a global ratings scale. It is hypothesized that the addition of depth in 3D viewing will augment the learner's innate visual spatial abilities, thereby enhancing skill acquisition compared to two‐dimensional viewing of the same procedure. Despite growing literature suggesting that 3D correlates directly to enhanced skill acquisition, this study did not differentiate significant results contributing to increased surgical performance. This topic will continue to be explored using more sensitive scales of measurement and more complex “open procedures” capitalizing on the importance of depth perception in surgical manipulation. Anat Sci Educ. © 2012 American Association of Anatomists.     

Fitting Nonlinear Latent Growth Curve Models With Individually Varying Time Points

Individual growth trajectories of psychological phenomena are often theorized to be nonlinear. Additionally, individuals’ measurement schedules might be unique. In a structural equation framework, latent growth curve model (LGM) applications typically have either (a) modeled nonlinearity assuming some degree of balance in measurement schedules, or (b) accommodated truly individually varying time points, assuming linear growth. This article describes how to fit 4 popular nonlinear LGMs (polynomial, shape-factor, piecewise, and structured latent curve) with truly individually varying time points, via a definition variable approach. The extension is straightforward for certain nonlinear LGMs (e.g., polynomial and structured latent curve) but in the case of shape-factor LGMs requires a reexpression of the model, and in the case of piecewise LGMs requires introduction of a general framework for imparting piecewise structure, along with tools for its automation. All 4 nonlinear LGMs with individually varying time scores are demonstrated using an empirical example on infant weight, and software syntax is provided. The discussion highlights some advantages of modeling nonlinear growth within structural equation versus multilevel frameworks, when time scores individually vary.     

Numerical magnitude processing in abacus-trained children with superior mathematical ability: an EEG study

Distance effect has been regarded as the best established marker of basic numerical magnitude processes and is related to individual mathematical abilities. A larger behavioral distance effect is suggested to be concomitant with lower mathematical achievement in children. However, the relationship between distance effect and superior mathematical abilities is unclear. One could get superior mathematical abilities by acquiring the skill of abacus-based mental calculation (AMC), which can be used to solve calculation problems with exceptional speed and high accuracy. In the current study, we explore the relationship between distance effect and superior mathematical abilities by examining whether and how the AMC training modifies numerical magnitude processing. Thus, mathematical competencies were tested in 18 abacus-trained children (who accepted the AMC training) and 18 non-trained children. Electroencephalography (EEG) waveforms were recorded when these children executed numerical comparison tasks in both Arabic digit and dot array forms. We found that: (a) the abacus-trained group had superior mathematical abilities than their peers; (b) distance effects were found both in behavioral results and on EEG waveforms; (c) the distance effect size of the average amplitude on the late negative-going component was different between groups in the digit task, with a larger effect size for abacus-trained children; (d) both the behavioral and EEG distance effects were modulated by the notation. These results revealed that the neural substrates of magnitude processing were modified by AMC training, and suggested that the mechanism of the representation of numerical magnitude for children with superior mathematical abilities was different from their peers. In addition, the results provide evidence for a view of non-abstract numerical representation.     

Basic books for a fraternity house library

The purpose of this article is to demonstrate how recent methodological developments in the analysis of individual growth can inform important problems in education policy. Specifically, this article focuses on a method referred to as growth mixture modeling. Growth mixture modeling is a relatively new procedure for the analysis of longitudinal data that relaxes many of the assumptions associated with conventional growth curve modeling. In particular, growth mixture modeling tests for the existence of unique growth trajectory classes through a combination of latent class analysis and standard growth curve modeling. Antecedent predictors of the latent classes can be incorporated as well as relations from the latent classes to specific outcomes. This article applies growth mixture modeling to data from the Early Childhood Longitudinal Study-Kindergarten class of 1998-1999. The specific policy question posed in this article focuses on the estimation of latent growth trajectory classes in reading proficiency and the effects of full-day or part-day kindergarten programs on growth within reading trajectory classes. Results identify a 3-class solution corresponding to slow-developing, normal-developing, and fast-developing reading growth in children. The results further show that full-day kindergarten attendance benefits children in the slow-reading development class relative to the normal and fast-reading development class, but the effect is lessened when holding constant socioeconomic status and age of entry into kindergarten. The implications of the method for quantitative education policy analysis are also discussed.     

Developmental trajectories of children's spatial skills: Influencing variables and associations with later mathematical thinking

Few studies have examined the long-term relations between children's early spatial skills and their later mathematical abilities. In the current study, we investigated children's developmental trajectories of spatial skills across four waves from age 3–7 years and their association with children's later mathematical understanding. We assessed children's development in a large, heterogeneous sample of children (N = 586) from diverse cultural backgrounds and mostly low-income homes. Spatial and mathematical skills were measured using standardized assessments. Children's starting points and rate of growth in spatial skills were investigated using latent growth curve models. We explored the influence of various covariates on spatial skill development and found that socioeconomic status, language skills, and sex, but not migration background predicted children's spatial development. Furthermore, our findings showed that children's initial spatial skills––but not their rate of growth––predicted later mathematical understanding, indicating that early spatial reasoning may play a crucial role for learning mathematics.     

Developmental pathways of early numerical skills during the preschool to school transition

Most longitudinal evidence explores the average level of development, suggesting that the relationships between a limited number of variables applies to all learners in the same way. This is the first longitudinal study that investigates multiple component numeric skills within a preschool population using a person-centered approach (i.e., a latent transition analysis), thus allowing for an investigation of different subgroup learning pathways of mathematical skills over time. 128 children aged 43–54 months (at Time 1) were tracked at three time points over 8 months encompassing the transition from preschool through to their first year of primary education. Findings suggest that there are five developmental pathways of mathematical learning with some groups of children making more rapid progress on entry to school than other groups. Those children in the low number skill pathway have a lower rate of growth than more advanced pathways, possibly due to a lack of understanding in cardinality. Findings highlighted the potential importance of language and working memory abilities on mathematical skills development over time.     

Predictions of Individual Change Recovered With Latent Class or Random Coefficient Growth Models

Popular longitudinal models allow for prediction of growth trajectories in alternative ways. In latent class growth models (LCGMs), person-level covariates predict membership in discrete latent classes that each holistically define an entire trajectory of change (e.g., a high-stable class vs. late-onset class vs. moderate-desisting class). In random coefficient growth models (RCGMs, also known as latent curve models), however, person-level covariates separately predict continuously distributed latent growth factors (e.g., an intercept vs. slope factor). This article first explains how complex and nonlinear interactions between predictors and time are recovered in different ways via LCGM versus RCGM specifications. Then a simulation comparison illustrates that, aside from some modest efficiency differences, such predictor relationships can be recovered approximately equally well by either model—regardless of which model generated the data. Our results also provide an empirical rationale for integrating findings about prediction of individual change across LCGMs and RCGMs in practice.     

Longitudinal Analysis of Early Mathematics Learning

The trend in mathematics achievement from preschool to kindergarten is studied with a longitudinal growth item response theory model. The three measurement occasions included the spring of preschool and the spring and fall of kindergarten. The growth trend was nonlinear, with a steep drop between spring of preschool and fall of kindergarten. The modeling results provide validation for the argument that a classroom assessment in mathematics can be used to assess developmental skill levels that are consistent with a theory of early mathematics acquisition. The statistical model employed enables an effective illustration of overall gains and individual variability. Implications of the summer loss are discussed as well as model limitations.     

Evidence for referential understanding in the emotions domain at twelve and eighteen months

Infants as young as 12 months readily modulate their behavior toward novel, ambiguous objects based on emotional responses that others display. Such social-referencing skill offers powerful benefits to infants' knowledge acquisition, but the magnitude of these benefits depends on whether they appreciate the referential quality of others' emotional messages, and are skilled at using cues to reference (e.g., gaze direction, body posture) to guide their interpretation of such messages. Two studies demonstrated referential understanding in 12- and 18-month-olds' responses to another's emotional outburst. Infants relied on the presence versus absence of referential cues to determine whether an emotional message should be linked with a salient, novel object in the first study (N = 48), and they actively consulted referential cues to disambiguate the intended target of an affective display in the second study (N = 32). These findings provide the first experimental evidence of such sophisticated referential abilities in 12-month-olds, as well as the first evidence that infant social referencing at any age actually trades on referential understanding.     

An Illustration of Second-Order Latent Growth Models

Methods of latent curve analysis (latent growth modeling) have recently emerged as a versatile tool for investigating longitudinal change in measured variables. This article, using higher order factor models as suggested by McArdle (1988) and Tisak and Meredith (1990), illustrates latent curve analysis for the purpose of modeling longitudinal change directly in a latent construct. The construct of interest is assumed to be indicated by several measured variables, all of which are observed at the same multiple time points. Examples with simultaneous estimation of covariance and mean structures are provided for both a single group and a two-group scenario.