My research broadly demonstrates how quantitative approaches can be effectively leveraged for developmental research. In this dissertation, I show one quantitatively precise way to identify the nature of developing mental representations in a variety of domains; my approach utilizes the connection between a learners input, creation of a potential mental representation from that input, and evaluation with respect to the learners output. More specifically, the quantitative approach I use leverages both realistic input data and realistic output data as part of the model design and evaluation. Using modeling, we have the opportunity to concretely evaluate representational options that we would not otherwise be able to disambiguate. I demonstrate this quantitative approach with three case studies in language development: (I) the development of adjective ordering preferences, where I find that the representations that adults use to talk to children are different than the ones used to talk to other, adults, (II) immature individual syntactic category representations, where I identify precisely which immature category representation young children are likely to be using, and (III) the development of adult productive syntactic category representations, where I identify when adult category knowledge emerges in typically and atypically developing populations.