The purpose of this study was to explore problem solving with kindergarten students. This line of inquiry is highly significant given that Common Core State Standards emphasize deep, conceptual understanding in mathematics as well as problem solving in kindergarten. However, there is little research on problem solving with kindergarten students. This study is one of a few to explore problem solving with kindergarten students. This study explored problem solving with kindergarten children using pencil paper. The study also presents a preliminary scale for assessing students' representation and problem solving. It explored the relationships between students' levels of conservation, representation and problem solving. The data confirmed previous findings that stated kindergarten students can solve a wide variety of problems, including multiplication and division problems and that most students represent number in writing at or below their level of conservation. New findings from this data also suggest that there are levels of representation and problem solving through which students progress. However, problem solving is not a lockstep process through which students progress, rather, it is a complex individual process that combines factors related to the student, the problem and the problem context. This new way of conceptualizing problem solving has several implications that change the way problem solving is frequently taught. Approaching problem solving as a complex interwoven process between these three factors changes the focus to each individual student's construction of logico-mathematical knowledge rather than the transfer of social knowledge from teacher to student. The data suggests that including a conference, like the one used in this study, between all students and the teacher is a crucial part of building the knowledge (mental structures) for all types of students. The instructional method of solving a wide variety of authentic and relevant mathematical word problems using pencil and paper utilized in this study is a viable means of improving students' problem solving proficiency. This study also presents the idea that assessing problem solving is possible using a scale. The preliminary scales created by this study are presented with the invitation that further study will be undertaken to confirm them. The hope is that future studies will continue to document the highly complex and sophisticated nature of problem solving skills that kindergarten students demonstrate when allowed to solve problems using their own strategies. Scales and taxonomies can be used to document these natural processes in the current climate of accountability. [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://www.proquest.com/en-US/products/dissertations/individuals.shtml.]