Tracking a multitude of abilities as they develop.

Recently, the Urnings algorithm (Bolsinova et al., 2022, J. R. Stat. Soc. Ser. C Appl. Statistics, 71, 91) has been proposed that allows for tracking the development of abilities of the learners and the difficulties of the items in adaptive learning systems. It is a simple and scalable algorithm which is suited for large-scale applications in which large streams of data are coming into the system and on-the-fly updating is needed. Compared to alternatives like the Elo rating system and its extensions, the Urnings rating system allows the uncertainty of the ratings to be evaluated and accounts for adaptive item selection which, if not corrected for, may distort the ratings. In this paper we extend the Urnings algorithm to allow for both between-item and within-item multidimensionality. This allows for tracking the development of interrelated abilities both at the individual and the population level. We present formal derivations of the multidimensional Urnings algorithm, illustrate its properties in simulations, and present an application to data from an adaptive learning system for primary school mathematics called Math Garden.

Control Theory Forecasts of Optimal Training Dosage to Facilitate Children's Arithmetic Learning in a Digital Educational Application.

Education can be viewed as a control theory problem in which students seek ongoing exogenous input-either through traditional classroom teaching or other alternative training resources-to minimize the discrepancies between their actual and target (reference) performance levels. Using illustrative data from [Formula: see text] Dutch elementary school students as measured using the Math Garden, a web-based computer adaptive practice and monitoring system, we simulate and evaluate the outcomes of using off-line and finite memory linear quadratic controllers with constraintsto forecast students' optimal training durations. By integrating population standards with each student's own latent change information, we demonstrate that adoption of the control theory-guided, person- and time-specific training dosages could yield increased training benefits at reduced costs compared to students' actual observed training durations, and a fixed-duration training scheme. The control theory approach also outperforms a linear scheme that provides training recommendations based on observed scores under noisy and the presence of missing data. Design-related issues such as ways to determine the penalty cost of input administration and the size of the control horizon window are addressed through a series of illustrative and empirically (Math Garden) motivated simulations.

Tracking with (Un)Certainty.

One of the highest ambitions in educational technology is the move towards personalized learning. To this end, computerized adaptive learning (CAL) systems are developed. A popular method to track the development of student ability and item difficulty, in CAL systems, is the Elo Rating System (ERS). The ERS allows for dynamic model parameters by updating key parameters after every response. However, drawbacks of the ERS are that it does not provide standard errors and that it results in rating variance inflation. We identify three statistical issues responsible for both of these drawbacks. To solve these issues we introduce a new tracking system based on urns, where every person and item is represented by an urn filled with a combination of green and red marbles. Urns are updated, by an exchange of marbles after each response, such that the proportions of green marbles represent estimates of person ability or item difficulty. A main advantage of this approach is that the standard errors are known, hence the method allows for statistical inference, such as testing for learning effects. We highlight features of the Urnings algorithm and compare it to the popular ERS in a simulation study and in an empirical data example from a large-scale CAL application.

Formal Modeling of the Resistance to Peer Influence Questionnaire: A Comparison of Adolescent Boys and Girls With and Without Mild-to-Borderline Intellectual Disability.

Items of the Resistance to Peer Influence Questionnaire (RPIQ) have a tree-based structure. On each item, individuals first choose whether a less versus more peer-resistant group best describes them; they then indicate whether it is "Really true" versus "Sort of true" that they belong to the chosen group. Using tree-based item response theory, we show that RPIQ items tap three dimensions: A Resistance to Peer Influence (RPI) dimension and two Response Polarization dimensions. We then reveal subgroup differences on these dimensions. That is, adolescents with mild-to-borderline intellectual disability, compared with typically developing adolescents, are less RPI and more polarized in their responses. Also, girls, compared with boys, are more RPI, and, when high RPI, more polarized in their responses. Together, these results indicate that a tree-based modeling approach yields a more sensitive measure of individuals' RPI as well as their tendency to respond more or less extremely.

Mutualistic Coupling Between Vocabulary and Reasoning in Young Children: A Replication and Extension of the Study by Kievit et al. (2017).

Recent work suggests that the positive manifold of individual differences may arise, or be amplified, by a mechanism called mutualism. Kievit et al. (2017) showed that a latent change score implementation of the mutualism model outperformed alternative models, demonstrating positive reciprocal interactions between vocabulary and reasoning during development. Here, we replicated these findings in a cohort of children (N = 227, 6-8 years old) and expanded the findings in three directions. First, a third wave of data was included, and the findings were robust to alternative model specifications. Second, a simulation demonstrated that data sets of similar magnitude and distributional properties could have, in principle, favored alternative models with close to 100% power. Third, we found support for the hypothesis that mutualistic-coupling effects are stronger and self-feedback parameters weaker in younger children. Together, these findings replicated the work of Kievit et al. (2017) and further support the hypothesis that mutualism supports cognitive development.

A Solution to the Measurement Problem in the Idiographic Approach Using Computer Adaptive Practicing.

Molenaar's manifesto on psychology as idiographic science (Molenaar, 2004) brought the N = 1 times series perspective firmly to the attention of developmental scientists. The rich intraindividual variation in complex developmental processes requires the study of these processes at the level of the individual. Yet, the idiographic approach is all but easy in practical research. One major limitation is the collection of short interval times series of high quality data on developmental processes. In this paper, we present a novel measurement approach to this problem. We developed an online practice and monitoring system which is now used by thousands of Dutch primary school children on a daily or weekly basis, providing a new window on cognitive development. We will introduce the origin of this new instrument, called Math Garden, explain its setup, and present and discuss ways to analyze children's individual developmental pathways.

Bayesian evaluation of constrained hypotheses on variances of multiple independent groups.

Research has shown that independent groups often differ not only in their means, but also in their variances. Comparing and testing variances is therefore of crucial importance to understand the effect of a grouping variable on an outcome variable. Researchers may have specific expectations concerning the relations between the variances of multiple groups. Such expectations can be translated into hypotheses with inequality and/or equality constraints on the group variances. Currently, however, no methods are available for testing (in)equality constrained hypotheses on variances. This article proposes a novel Bayesian approach to this challenging testing problem. Our approach has the following useful properties: First, it can be used to simultaneously test multiple (non)nested hypotheses with equality as well as inequality constraints on the variances. Second, our approach is fully automatic in the sense that no subjective prior specification is needed. Only the hypotheses need to be provided. Third, a user-friendly software application is included that can be used to perform this Bayesian test in an easy manner. (PsycINFO Database Record

TRACING THE DEVELOPMENT OF TYPEWRITING SKILLS IN AN ADAPTIVE E-LEARNING ENVIRONMENT.

Typewriting studies which compare novice and expert typists have suggested that highly trained typing skills involve cognitive process with an inner and outer loop, which regulate keystrokes and words, respectively. The present study investigates these loops longitudinally, using multi-level modeling of 1,091,707 keystroke latencies from 62 children (M age=12.6 yr.) following an online typing course. Using finger movement repetition as indicator of the inner loop and words typed as indicator of the outer loop, practicing keystroke latencies resulted in different developmental curves for each loop. Moreover, based on plateaus in the developmental curves, the inner loop seemed to require less practice to develop than the outer loop.

The Balance-Scale Task Revisited: A Comparison of Statistical Models for Rule-Based and Information-Integration Theories of Proportional Reasoning.

We propose and test three statistical models for the analysis of children's responses to the balance scale task, a seminal task to study proportional reasoning. We use a latent class modelling approach to formulate a rule-based latent class model (RB LCM) following from a rule-based perspective on proportional reasoning and a new statistical model, the Weighted Sum Model, following from an information-integration approach. Moreover, a hybrid LCM using item covariates is proposed, combining aspects of both a rule-based and information-integration perspective. These models are applied to two different datasets, a standard paper-and-pencil test dataset (N = 779), and a dataset collected within an online learning environment that included direct feedback, time-pressure, and a reward system (N = 808). For the paper-and-pencil dataset the RB LCM resulted in the best fit, whereas for the online dataset the hybrid LCM provided the best fit. The standard paper-and-pencil dataset yielded more evidence for distinct solution rules than the online data set in which quantitative item characteristics are more prominent in determining responses. These results shed new light on the discussion on sequential rule-based and information-integration perspectives of cognitive development.

The role of pattern recognition in children's exact enumeration of small numbers.

Enumeration can be accomplished by subitizing, counting, estimation, and combinations of these processes. We investigated whether the dissociation between subitizing and counting can be observed in 4- to 6-year-olds and studied whether the maximum number of elements that can be subitized changes with age. To detect a dissociation between subitizing and counting, it is tested whether task manipulations have different effects in the subitizing than in the counting range. Task manipulations concerned duration of presentation of elements (limited, unlimited) and configuration of elements (random, line, dice). In Study 1, forty-nine 4- and 5-year-olds were tested with a computerized enumeration task. Study 2 concerned data from 4-, 5-, and 6-year-olds, collected with Math Garden, a computer-adaptive application to practice math. Both task manipulations affected performance in the counting, but not the subitizing range, supporting the conclusion that children use two distinct enumeration processes in the two ranges. In all age groups, the maximum number of elements that could be subitized was three. The strong effect of configuration of elements suggests that subitizing might be based on a general ability of pattern recognition.