Recent educational policy documents have encouraged engaging students in mathematical argumentation via discussion and writing. Most recently in the U.S., the Common Core State Standards recommend that children "construct viable arguments and critique the reasoning of others." One often advocated means of engaging students in this mathematical practice is mathematical writing. This requires students to develop mathematical writing that demonstrates careful analysis, a command of sequence, and a level of detail considered fundamental for constructing effective argumentative, persuasive and informative mathematical explanations. However, there is currently little to no research examining how mathematical writing develops in elementary grades. The present study examined K-3 students' mathematical writing using modified Piagetian tasks. Incorporating elements of Toulmin's argumentation scheme, a set of classifications for mathematical writing emerged from K-3 student samples. Further, these classifications are sequential, with strong statistical correlations associated with children's grade levels. The findings indicate a potentially useful set of classification schemes for identifying children's writing and examining how such writing develops in early grades.