What Do We Measure When We Measure Magnitudes?

Using nonsymbolic representations of magnitudes (eg, arrays of items) is a common way to study the processing of the number of items in a set. This ability is considered to be very basic, innate, and automatic. However, arrays of items always include noncountable continuous magnitudes such as the density of the items, their total surface area, and so forth. These continuous magnitudes can affect performance when comparing numerosities (eg, where are more dots?) and when comparing proportions (eg, where are more blue dots compared with yellow dots?). Numerosities and continuous magnitudes are so inherently correlated that it is impossible to be certain that the required task indeed measured what we intended to measure (ie, numerosity or proportion processing). Researchers in the field are aware of this problem and employ different methods to reduce possible influences that continuous magnitudes might have on performance. In this chapter we will introduce such methods and discuss the different and often contradictory conclusions they lead to. We will also suggest-in light of recent studies-that great benefit can come from directly studying (1) how continuous magnitudes affect comparison of numerosity and proportions, and (2) the reciprocal interaction between numerosities and continuous magnitudes. Studying these issues during different stages of development and in typically and atypically developed adults can shed new light on the most elementary building blocks of our mathematical abilities.