The purpose of this study was to explore the sources of mathematical ideas in terms of the relationships between meaning and formalism and their role in the transition between elementary mathematics and advanced mathematics. The two participants were high school mathematics teachers, who vary in their levels of experience. Two forms of data were collected to obtain more in-depth data about the transformations within among mathematical ideas: a questionnaire including 14 open-ended mathematical tasks and semi-structured interviews. Results indicated that individuals had different ways in constructing mathematical ideas and that their mathematical ideas were derived from the transition between meaning and formalism.