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.

Eyes abound in the animal kingdom. Some are large as basketballs and others are just fractions of a millimetre. Eyes also come in many different types, such as the compound eyes of insects, the mirror eyes of scallopsor our own camera-like eyes. Common to all animal eyes is that they serve the same fundamental role of collecting external information for guidingthe animal’s behaviour. But behaviours vary tremendously across the animal kingdom, and it turns outthis is the key to understand how eyes evolved. The lecture will take a tour from the first animals that could only sense the presence of light, to those that saw the first crude image of the world and finally to animals that use acute vision for interacting with otheranimals. Amazingly, all these stages of eye evolution still exist in animals living today, and this is how we can unravel the evolution of behaviours that has been the driving force behind eye evolution