Using analogies to facilitate conceptual change in mathematics learning

The problem of adverse effects of prior knowledge in mathematics learning has been amply documented and theorized by mathematics educators as well as cognitive/developmental psychologists. This problem emerges when students' prior knowledge about a mathematical notion comes in contrast with new information coming from instruction, giving rise to systematic errors. Conceptual change perspectives on mathematics learning suggest that in such cases reorganization of students' prior knowledge is necessary. Analogical reasoning, in particular cross-domain mapping, is considered an important mechanism for conceptual restructuring. However, the use of analogies in instruction is often found ineffective, mainly because the structural similarity between two domains is obscure for students. To deal with this problem, John Clement and his colleagues developed the bridging strategy that uses multiple analogies to bring students to pay attention to the structural similarity that often goes unnoticed. This paper focuses on the cross-domain mapping between number and the (geometrical) line that has been instrumental in the development of the number concept. I summarize findings of a series of studies that investigated students' understandings of density in arithmetical and geometrical contexts from a conceptual change perspective; and I discuss how this research-based evidence was used to design an intervention study that used the analogy "numbers are points on the number line", and a bridging analogy ("the number line is like an imaginary rubber band that never breaks, no matter how much it is stressed") with the aim of bringing the notion of density within the grasp of secondary students.