In this dissertation, I report on a study with seven pre-service teachers and their construction and interpretation of formulas through reasoning with dynamic geometric objects. Historically, individuals have used formulas to evaluate quantities and to represent precise relationships between quantities, and many mathematics curricula include instruction on formulas, particularly area formulas, as early as elementary school and into postsecondary education. Nevertheless, researchers report on students various non-quantitative conceptions of symbols within formulas and express students difficulties with constructing symbols that represent variables.In this study, I designed a semester-long teaching experiment to support and understand pre-service teachers quantitative construction and interpretation of formulas based on a conceptual analysis of productive meanings for formulas I proposed as a result on experimental teaching I did with four pre-service teachers. Specifically, I designed tasks in which pre-service teachers reasoned about various pairs of quantities and their covariation within dynamic geometric objects (e.g., triangle, rectangle, parallelogram) to construct and relate formulas that represented those relationships. They did so throughtransformations of the geometric objects and the construction of symbols as constants, parameters, and variables.As a result of my analysis of my experimental teaching and my teaching experiment with two pre-service teachers, I identify various ways of reasoning the students exhibited including implicative reasoning, engaging in cross-quantity comparisons, states-based reasoning, numerical reasoning, and reasoning with extreme cases. I identify key developmental mental operations involved in the construction of formulas: constructing a multiplicative object, reasoning with quantities as magnitudes, constructing a formula as a standard format, using transformations to identify invariant quantitative relationships, and considering the role of units in the measurement process. I compare and contrast these key developmental operations with two pre-service teachers from the teaching experiment. The findings have important implications for teaching, curriculum development, and research on students learning and application of formulas.