In cognitive development, learning more than the input provides is a central challenge. This challenge is especially evident in learning the meaning of numbers. Integers – and the quantities they denote – are potentially infinite, as are the fractional values between every integer. Yet children’s experiences of numbers are necessarily finite. Analogy is a powerful learning mechanism for children to learn novel, abstract concepts from only limited input. However, retrieving proper analogy requires cognitive supports. In this talk, I seek to propose and examine number lines as a mathematical schema of the number system to facilitate both the development of rational number understanding and analogical reasoning. To examine these hypotheses, I will present a series of educational intervention studies with third-to-fifth graders. Results showed that a short, unsupervised intervention of spatial alignment between integers and fractions on number lines produced broad and durable gains in fractional magnitudes. Additionally, training on conceptual knowledge of fractions – that fractions denote magnitude and can be placed on number lines – facilitates explicit analogical reasoning. Together, these studies indicate that analogies can play an important role in rational number learning with the help of number lines as schemas. These studies shed light on helpful practices in STEM education curricula and instructions.

Extensive research on science teaching has shown the effectiveness of analogies as a didactic tool which, when appropriately and effectively used, facilitates the learning process of abstract concepts. This seminar does not contradict the efficacy of such a didactic use of analogies in this seminar but switches attention and interest on their heuristic use in approaching and understanding of what previously unknown. Such a use of analogies derives from research with 10 to 17 year-olds, who, when asked to make predictions in novel situations and to then provide explanations about these predictions, they self-generated analogies and used them by reasoning on their basis. This heuristic use of analogies can be used in science teaching in revealing how students approach situations they have not considered before as well as the sources they draw upon in doing so.

Abstract concepts are a powerful tool to store information efficiently and to make wide-ranging predictions in new situations based on sparse data. Whereas looking-time studies point towards an early emergence of this ability in human infancy, other paradigms like the relational match to sample task often show a failure to detect abstract concepts like same and different until the late preschool years. Similarly, non-human animals have difficulties solving those tasks and often succeed only after long training regimes. Given the huge influence of small task modifications, there is an ongoing debate about the conclusiveness of these findings for the development and phylogenetic distribution of abstract reasoning abilities. Here, we applied the concept of “overhypotheses” which is well known in the infant and cognitive modeling literature to study the capabilities of 3 to 5-year-old children, chimpanzees, and capuchin monkeys in a unified and more ecologically valid task design. In a series of studies, participants themselves sampled reward items from multiple containers or witnessed the sampling process. Only when they detected the abstract pattern governing the reward distributions within and across containers, they could optimally guide their behavior and maximize the reward outcome in a novel test situation. We compared each species’ performance to the predictions of a probabilistic hierarchical Bayesian model capable of forming overhypotheses at a first and second level of abstraction and adapted to their species-specific reward preferences.

People from cultures around the world tend to borrow from the domain of space to represent abstract concepts. For example, in the domain on time, we use spatial metaphors (e.g., describing the future as being in front and the past behind), accompany our speech with spatial gestures (e.g., gesturing to the left to refer to a past event), and use external tools that project time onto a spatial reference frame (e.g., calendars). Importantly, these associations are also present in the way we think and reason about time, suggesting that space and time are also linked in the mind. In this talk, I will explore the developmental origins and functional implications of these types of cross-dimensional associations. To start, I will discuss the roles that language and culture play in shaping how children in the US and India represent time. Next, I will use word learning and memory as test cases for exploring why cross-dimensional associations may be cognitively advantageous. Finally, I will talk about future directions and the practical implications for this line of work, with a focus on how encouraging spatial representations of abstract concepts could improve learning outcomes.