When the formula survives but the explanation for it does not: how prospective teachers justify the formula for the area of a rectangle with rational sides

A commonly documented phenomenon involves properties from an earlier encountered set being overextended to a later encountered superset (e.g., the natural number bias). This work focuses on a related but separate phenomenon, where a formula does carry over from a set to its superset, but the justification for why the formula holds does not. Specifically, justifications of the area formula for rectangles, which is typically only justified in the natural number case, are explored. Prospective teachers were asked to justify this formula in the case of rational side lengths. The data indicated three distinct types of justifications: counting with fractional units, counting with a new unit, and expanding the rectangle. The first two of these categories are otherwise viable arguments, which inadvertently took the formula working in the rational number case for granted. We discuss the varied ways prospective teachers’ approaches take advantage of or modify the known justification for the formula in the integer case.