Abe D. Hofman

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.

Distinguishing fast and slow processes in accuracy : Response time data

We investigate the relation between speed and accuracy within problem solving in its simplest non-trivial form. We consider tests with only two items and code the item responses in two binary variables: one indicating the response accuracy, and one indicating the response speed. Despite being a very basic setup, it enables us to study item pairs stemming from a broad range of domains such as basic arithmetic, first language learning, intelligence-related problems, and chess, with large numbers of observations for every pair of problems under consideration. We carry out a survey over a large number of such item pairs and compare three types of psychometric accuracy-response time models present in the literature: two ‘one-process’ models, the first of which models accuracy and response time as conditionally independent and the second of which models accuracy and response time as conditionally dependent, and a ‘two-process’ model which models accuracy contingent on response time. We find that the data clearly violates the restrictions imposed by both one-process models and requires additional complexity which is parsimoniously provided by the two-process model. We supplement our survey with an analysis of the erroneous responses for an example item pair and demonstrate that there are very significant differences between the types of errors in fast and slow responses.

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.

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 = 12.6yr.) 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.

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.

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.