The positive manifold of intelligence has fascinated generations of scholars in human ability. In the past century, various formal explanations have been proposed, including the dominant g factor, the revived sampling theory, and the recent multiplier effect model and mutualism model. In this article, we propose a novel idiographic explanation. We formally conceptualize intelligence as evolving networks in which new facts and procedures are wired together during development. The static model, an extension of the Fortuin–Kasteleyn model, provides a parsimonious explanation of the positive manifold and intelligence’s hierarchical factor structure. We show how it can explain the Matthew effect across developmental stages. Finally, we introduce a method for studying growth dynamics. Our truly idiographic approach offers a new view on a century-old construct and ultimately allows the fields of human ability and human learning to coalesce.

The positive manifold of intelligence has fascinated generations of scholars in human ability. In the past century, various formal explanations have been proposed, including the dominant g-factor, the revived sampling theory, and the recent multiplier effect model and mutualism model. In this article we propose a novel idiographic explanation. We formally conceptualize intelligence as evolving networks, in which new facts and procedures are wired together during development. The static model, an extension of the Fortuin-Kasteleyn model, provides a parsimonious explanation of the positive manifold and intelligence's hierarchical factor structure. We show how it can explain the Matthew effect across developmental stages. Finally, we introduce a method for studying growth dynamics. Our truly idiographic approach offers a new view on a century-old construct, and ultimately allows the fields of human ability and human learning to coalesce.

With the advent of computers in education, and the ample availability of online learning and practice environments, enormous amounts of data on learning become available. The purpose of this paper is to present a decade of experience with analyzing and improving an online practice environment for math, which has thus far recorded over a billion responses. We present the methods we use to both steer and analyze this system in real-time, using scoring rules on accuracy and response times, a tailored rating system to provide both learners and items with current ability and difficulty ratings, and an adaptive engine that matches learners to items. Moreover, we explore the quality of fit by means of prediction accuracy and parallel item reliability. Limitations and pitfalls are discussed by diagnosing sources of misfit, like violations of unidimensionality and unforeseen dynamics. Finally, directions for development are discussed, including embedded learning analytics and a focus on online experimentation to evaluate both the system itself and the users' learning gains. Though many challenges remain open, we believe that large steps have been made in providing methods to efficiently manage and research educational big data from a massive online learning system. Notes for Practice• We analyzed an online adaptive practice environment for arithmetic, actively used by over 400,000 primary school children in the Netherlands.• Adaptive practice is achieved by continuously tracking both student abilities and item difficulties, and matching students to items.• A unidimensional adaptive algorithm, separately employed within each domain (e.g., multiplication), takes care of tracking abilities and difficulties.• We show that the obtained unidimensional ability and difficulty estimates are, to a large extent, reliable and accurate.• Moreover, we explore the many sources of misfit, or violations of the unidimensionality assumption, including differences in response processes (fast and slow responders) and response strategies (erroneous strategies that work for clusters of items).• To advance the field of learning analytics, we call for active analytics such as exemplified in this paper. Learning analytics must actively help direct a student towards his or her educational objective by means of embedded analytics that not only analyze the student, but also shape their learning path (such as the discussed adaptive algorithm) and includes experiments that ensure changes to the system have the desired effect.

Use and benefits of the possibility to choose a success rate are studied in a math-practice application that is used by a considerable percentage of Dutch primary school children. Study 1 uses data that were collected with the application, using children's practice data (N = 40,329; grades 1-6). Children differed in their preference for a high, medium, or low success rate. Preferences were associated with gender, age, and ability, matching expectations that follow from the literature. Study 2 is an experimental study with 192 children from grades 3-6, using a pretest, training phase, and posttest. The possibility to choose a success rate was manipulated. Unexpectedly, beneficial effects for math practice, improvement of math skills, and self-belief concerning math were absent. Results suggest an appreciation of the possibility to choose, although beneficial effects of choosing were not observed for motivation to practice, skill improvement, and self-belief concerning math.

Pairwise correlations are currently a popular way to estimate a large-scale network (> 1000 nodes) from functional magnetic resonance imaging data. However, this approach generally results in a poor representation of the true underlying network. The reason is that pairwise correlations cannot distinguish between direct and indirect connectivity. As a result, pairwise correlation networks can lead to fallacious conclusions; for example, one may conclude that a network is a small-world when it is not. In a simulation study and an application to resting-state fMRI data, we compare the performance of pairwise correlations in large-scale networks (2000 nodes) against three other methods that are designed to filter out indirect connections. Recovery methods are evaluated in four simulated network topologies (small world or not, scale-free or not) in scenarios where the number of observations is very small compared to the number of nodes. Simulations clearly show that pairwise correlation networks are fragmented into separate unconnected components with excessive connectedness within components. This often leads to erroneous estimates of network metrics, like small-world structures or low betweenness centrality, and produces too many low-degree nodes. We conclude that using partial correlations, informed by a sparseness penalty, results in more accurate networks and corresponding metrics than pairwise correlation networks. However, even with these methods, the presence of hubs in the generating network can be problematic if the number of observations is too small. Additionally, we show for resting-state fMRI that partial correlations are more robust than correlations to different parcellation sets and to different lengths of time-series.