Both Thompson and the duo of Confrey and Smith describe how students might be taught to build "ways of thinking" about exponential behavior by coordinating the covariation of two changing quantities, however, these authors build exponential behavior from different meanings of covariation. Confrey and Smith advocate beginning with discrete additive and multiplicative changes, while Thompson advocates beginning with continuous variation. In light of these differences, this work investigates the questions of how students actually reason covariationally, and what the consequences of their reasoning are for mathematics involving exponential functions. This work describes a teaching experiment, consisting of a series of fifteen task-based exploratory teaching interviews with two high school student participants. The purpose of the experiment was to identify the operations of covariational reasoning that the students actually used, and the consequences of that reasoning for mathematics involving exponential growth. The tasks covered linear functions, compound interest, phase plane, exponential growth, and the logistic differential equation. In the conceptual analysis of the participants' mathematical work and spoken utterances, I identified two ways of thinking about change that differ from the discrete/continuous dichotomy above: thinking about "chunky" completed changes, or a "smooth" change in progress. With smooth and chunky as a basis, I also identify five different ways of understanding exponential growth: Geometric, compound, differential, harmonic, and stochastic. Lastly, I suggest that powerful understandings of exponential growth come not from the mastery of any one way of thinking, but from a rapid and fluent shifting amongst several ways of thinking. [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.]