Having heard "a pimwit", English-speakers assume that "the pimwit" is also possible. This type of productivity is attributed to syntactic categories such as NOUN and DETERMINER, but the key question is "how" do humans become endowed with these categories in the first place. We propose a novel approach that combines corpus analysis with computational modeling to analyze the productivity of DETERMINER+NOUN constructions in child-produced utterances. Our experiments on two corpora of child-adult interactions using two different methods of quantifying linguistic productivity show that children do not display productivity at early stages. Using a model trained on child-directed utterances, we simulate children's developmental trajectory with great precision, suggesting that the emergence of productivity in human language can be explained without the need to postulate a "priori" access to syntactic categories. [This paper was published in: "Proceedings of the 13th International Joint Conference on Natural Language Processing and the 3rd Conference of the Asia-Pacific Chapter of the Association for Computational Linguistics (Volume 1: Long Papers)," Association for Computational Linguistics, 2023, pp.330-343.]