Cognitive supports for analogical reasoning in rational number understanding
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.
Using Developmental Trajectories to Understand Change in Children’s Analogical Reasoning
Analogical reasoning is a complex ‘high-level’ cognitive process characterised by making inferences based on analogical comparisons. As with other high-level processes, development takes place over a protracted time period and believed to result from changes in multiple ‘lower-level’ systems. In the case of analogical reasoning, changes in systems responsible for conceptual knowledge, task knowledge, inhibition, and working memory have all been causally implicated in development. Whilst there is evidence that each of these systems contributes to development, what the relative contribution of each across development is, and how they interact with each, remain largely unanswered questions. In this presentation, I will describe how cross-sectional trajectory analysis can be used as a complementary method to shed light on these questions.