Abstract
This study sought to explore whether access to definitions and general representations influences the construction of general direct arguments. Data was collected in college mathematics courses for prospective elementary school teachers. Participant arguments were analyzed along two variables: the generality of the representations and the viability of the conceptual insight they included. Participants were given one of three proving tasks. One task included no definitions, one task included definitions in the conceptual register, and the third task included definitions in the symbolic register. A randomized block design was used to explore the relationship between the definitions and the two variables. Qualitative methods were used to explore how participants intended their arguments. This study found that: 1) the inclusion of definitions on proving tasks does not have a substantial influence on the generality of the arguments or the viability of the conceptual insight used and 2) examples and algebraic representations were used as placeholders to demonstrate a procedure and to stand for the domain as a class of objects. The findings also indicate that the manner in which students are generalizing about the domain of the claim determines the structure of the domain they attend to, the conceptual insights available to them, and thus their ability to construct a viable argument. Future research is needed to connect the existing understanding of actions that support student generalization to supporting students in developing viable general direct proof.