Children's understanding of density is riddled with misconceptions-or so it seems. Yet even preschoolers at times appear to understand density. This article seeks to reconcile these conflicting outcomes by investigating the nature of constraints available in different experimental protocols. Protocols that report misconceptions about density used stimulus arrangements that make differences in mass and volume more salient than differences in density. In contrast, protocols that report successful performance used stimulus arrangements that might have increased the salience of density. To test this hypothesis, the present experiments manipulate the salience of object density. Children between 2 and 9 years of age and adults responded whether an object would sink or float when placed in water. Results indicated that children's performance on exactly the same objects differed as a function of the saliency of the dimension of density, relative to the dimensions of mass and volume. These results support the idea that constraints--rather than stable knowledge--drive performance, with implications for teaching children about nonobvious concepts such as density.

This chapter argues that connectionist and dynamic systems models complement each other and collectively move toward a unified theory of development if they subscribe to the second view of mechanism—one that treats behavior as soft assembled in the immediate context. The chapter is organized as follows. It first addresses why models aimed at reducing behavior to cognitive components cannot make clear headway. The argument is that a reduction of behavior requires human performance to be relatively context free. However, it is shown with the example of balance task performance that human performance is highly context dependent, even in the sterile laboratory context of balance experiments. The chapter then elaborates on what such context dependence could mean. The chapter reviews more pointed evidence for soft assembly and discusses why models that take soft assembly seriously—connectionist or dynamic systems—anticipate the unified theory.

Students rarely practice math outside of school requirements, which we refer to as the “math-practice gap”. This gap might be the reason why students struggle with math, making it urgent to develop means by which to address it. In the current paper, we propose that math apps offer a viable solution to the math-practice gap: Online apps can provide access to a large number of problems, tied to immediate feedback, and delivered in an engaging way. To substantiate this conversation, we looked at whether tablets are sufficiently engaging to motivate children’s informal practice math. Our approach was to partner with education agencies via a community-based participatory research design. The three participating education agencies serve elementary-school students from low-SES communities, allowing us to look at tablet use by children who are unlikely to have extensive access to online math enrichment programs. At the same time, the agencies differed in several structural details, including whether our intervention took place during school time, after school, or during the summer. This allowed us to shed light on tablet feasibility under different organizational constraints. Our findings show that tablet-based math practice is engaging for young children, independent of the setting, the student’s age, or the math concept that was tackled. At the same time, we found that student engagement was a function of the presence of caring adults to facilitate their online math practice.

In this paper, we argue that beliefs share common properties with the self-sustaining networks of complex systems. Matching experiences are said to couple with each other into a mutually reinforcing network. The goal of the current paper is to spell out and develop these ideas, using our understanding of ecosystems as a guide. In Part 1 of the paper, we provide theoretical considerations relevant to this new conceptualization of beliefs, including the theoretical overlap between energy and meaning. In Part 2, we discuss the implications of this new conceptualization on our understanding of belief emergence and belief change. Finally, in Part 3, we provide an analytical mapping between beliefs and the self-sustaining networks of ecosystems, namely by applying to behavioral data a measure developed for ecosystem networks. Specifically, average accuracies were subjected to analyses of uncertainty (H) and average mutual information (AMI). The ratio between these two values yields degree of order, a measure of how organized the self-sustained network is. Degree of order was tracked over time and compared to the amount of explained variance returned by a categorical nonlinear principal components analysis (CATPCA). Finding high correspondence between the two measures of order, together with the theoretical groundwork discussed in Parts 1 and 2, lends preliminary validity to our theory that beliefs have important similarities to the structural characteristics of self-sustaining networks.

Mistaken beliefs pose a barrier to science learning. For this reason, it is important to understand the circumstances in which they emerge and change. In the current paper, we apply complexity theory to shed light on the nature of mistaken beliefs. The strength of this approach lies in conceptualizing beliefs as dynamic stabilities, a well-defined construct that can be indexed precisely. For example, Recurrence Quantification Analysis (RQA) can determine the presence of dynamic stabilities by analyzing variability in time-series data. We applied this analytical tool to probe for mistaken beliefs in a beam-balancing task, a task that is known to elicit mistaken beliefs in preschoolers. Using a case-study design with four preschoolers, we tracked children’s hand position with motion sensors as they balanced various beams. The resulting time series of hand position was submitted to RQA, yielding two important results: First, we found that consistent mistakes in trying to balance the beams were not always accompanied by dynamic stability. This undermines the common assumption that overt consistency in task performance is sufficient to conclude the presence of beliefs. Second, we found strong individual differences over time, as children explored the balance beams. Applications to science education are discussed.