To illustrate primary students' use of argumentation, this study reported on an analysis of elements in argumentation of 168 grade 5 students in Taiwan. This study adopted the van Hiele (vH) model as the main theoretical framework with relevant studies to show the development of argumentation. This study designed a geometric argumentation test (GAT) with four elements to evaluate students' justification in argumentation, including premise, conclusion, mathematical knowledge (MK) and reasoning. The GAT has three tasks and each task has four items to show students' use of argumentation. This study adopted the cluster analysis through four elements. The results showed that students performed better in the elements of premises and conclusions than in the elements of MK and reasoning, and these grade 5 students were grouped into three clusters. The names of three clusters from the basic to advanced levels were Naïve Argumentation, Initial Argumentation and Incomplete Empirical Argumentation. The significant finding was that students in this study had a similar trend with the results in TIMSS in 2011, 2015 and 2019. Students in the cluster of IEA were at Level 1, and students in the other two clusters were at Level 0 in the vH model. Students in each cluster had specific characteristics and the problems in each cluster were related to different factors. Finally, this study pointed out the limitations and addressed some suggestions for further researches.