# The unfolding of third graders’ conceptualization of a function: a collective case study

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The goal of this study was to understand elementary students’ meaning-making process of early algebra representations in the context of number patterns about even and odd numbers within their natural classroom setting. A particular emphasis was placed on understanding the role of the function as a way to generalize a relationship between inputs and outputs where every input has exactly one output, i.e. an even number can be defined as f(n)= 2n for an integer n and an odd integer could be defined as f(n)= 2n+1 for an integer n. Through a collective case study of individual students in a third-grade classroom, I qualitatively analyzed and characterized individual noticing patterns when given a function rule table and moreover, concluded that third-graders are able to conceptualize a function rule in the context of generating even integers and were able to act upon a function rule when generating odd integers. The study suggests that the benefit of integrating function tables into the early elementary instruction will support formal functional understanding in later grades.