A growing body of research focuses on computationally detecting controversial topics and understanding the stances people hold on them. Yet gaps remain in our theoretical and practical understanding of how to define controversy, how it manifests, and how to measure it. In this paper, we introduce a novel measure we call "contention", defined with respect to a topic and a population. We model contention from a mathematical standpoint. We validate our model by examining a diverse set of sources: real-world polling data sets, actual voter data, and Twitter coverage on several topics. In our publicly-released Twitter data set of nearly 100M tweets, we examine several topics such as Brexit, the 2016 U.S. Elections, and "The Dress", and cross-reference them with other sources. We demonstrate that the contention measure holds explanatory power for a wide variety of observed phenomena, such as controversies over climate change and other topics that are well within scientific consensus. Finally, we re-examine the notion of controversy, and present a theoretical framework that defines it in terms of population. We present preliminary evidence suggesting that contention is one dimension of controversy, along with others, such as "importance". Our new contention measure, along with the hypothesized model of controversy, suggest several avenues for future work in this emerging interdisciplinary research area.