Scope and method of study: The purpose of this study was to unpack the understandings pre-service elementary teachers have pertaining to the ideas of pattern and function. The intent was to bring insight into how mathematics teacher educators can use patterning activities to prepare pre-service elementary teachers to support the development of algebraic thinking in their future students. The assumption that learning in mathematics is a hermeneutical process indicated that access to pre-service elementary teachers' understanding of pattern and function could be obtained by analyzing the explanations offered while they were engaged in pattern-finding activities. Based on this assumption, the texts produced by six pre-service elementary teachers while they were engaged in the process of generalizing linear patterns were collected and analyzed. The methodological framework of hermeneutic phenomenology guided the research design and the analysis of the textual accounts collected for this study. Findings and conclusions: Four overlapping ideas of function emerged through the analysis of the conversations and written texts associated with the task of pattern-finding: the ideas of function as a pattern, as a relation, as a rule, and as a process. The region where all four ideas overlapped represented the kernel of algebraic thinking, the idea of function as generalization. When these pre-service elementary teachers combined these four ideas in an integrated approach to pattern-finding, they were able to generate a way to describe the general term in a linear pattern. The omission of one of these four ideas, particularly the ideas of function as a pattern or as a relation, presented a roadblock to generalization. In conclusion, the results of this study pointed to a four-stage process to understanding linear functions. The first stage begins with the identification of the pattern of change. This was followed by a stage of discrimination, in which the individual noted both the variable and invariant features of a linear model. The act of discrimination often led to the ability to generalize the pattern. Ultimately, a few of the participants began to notice those properties that all linear patterns had in common. This final stage of synthesis resulted in a more complete understanding of the concepts of pattern and function. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]