The purpose of this study was to understand children's operations on fragmenting schemes while they are engaged with three different visual models, including circle, rectangle, and length model. This study was a sequential explanatory mixed method, which included two phases that happened sequentially with the dominant use of the quantitative approach. A fragmenting survey was given to 132 first and second grade students from three different schools located in the Midwestern U.S. Out of 132 participants who completed the test, 11 participants were purposefully selected for one-on-one clinical interviews. Results from the descriptive statistics showed that almost half of the students are at Level 1 of fragmenting. It implies that students are able to partition a model into a given number of parts but are not able to make the parts equal or exhaust the whole. Analysis of the Friedman test and clinical interviews showed that students performed better when partitioning models into even numbers than into odd numbers. Further, findings from comparing the circle, rectangle, and length models and their relation to children's fragmenting scheme suggest that children's fragmenting knowledge is consistent across the three visual models. However, the type of model operated on children's successful partitioning when they work with odd numbers of parts. The findings of the study suggested circle models can add an extra layer of difficulty for equal partitioning into third and fifth. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://bibliotheek.ehb.be:2222/en-US/products/dissertations/individuals.shtml.]