This study looks at conceptual explanations given in a reform-oriented mathematics class for preservice secondary mathematics teachers and extends Toulmin's argumentation framework to account for some of the complexities of the explanations given by these students. This study explains the complexities that arose in applying Toulmin's framework to explanations and extends the framework by accounting for the features of conceptual explanations. The complexities of these explanations are that they are made up of multiple arguments that build on each other to reach a final conclusion and that they are also dependant upon the social aspects of the class in which they are situated. Recognizing that some statements have dual purposes in the explanation and that there are varying levels of justification used in the explanations helped to account for the first complexity of explanations. The classification of class conventions helps to account for the social influences on explanations. This study differs from other studies that use Toulmin's framework to analyze formal proofs or to identify taken-as-shared understanding in a classroom. This study instead focuses on using the framework to analyze the components of explanations and to provide insight into the structure of conceptually oriented explanations. This study contributes to the existing research by extending Toulmin's argumentation framework to account for how social influences help determine the appropriate components of an explanation.