Toward a description of how engineering students think mathematically

The purpose of this study was to build a descriptive model for how individuals add mathematical structure to a problem setting. Blum and Leiss's (2007) mathematical modeling cycle was adopted as a research framework. Data was collected using task-based clinical interviews with four engineering students enrolled in differential equations. Analysis led to the creation of three theoretical constructs which together describe the process of structurally enriching (Schwarzkopf, 2007) a nonmathematical context: mathematical framing, the pseudo-empirical setting, and intertwining. The students in this study made sense of the modeling tasks by assuming that the solution had a certain mathematical structure (the mathematical framing) and then verifying that choice by making sure all the necessary information was present (comparing the information in the task to the pseudo-empirical setting). This process of matching up the variables and operations in the mathematical framing with the quantities and relationships in the pseudo-empirical setting was called intertwining. To move forward in the modeling task, the students then externalized a mathematical representation and analyzed it.;Validating activity was observed throughout this process and analysis confirmed five distinct types of validating activity: (i) to check alignment of the mathematical representation with the individual's interpretation of the context (checking mathematical representation against the real model); (ii) confirming alignment between the mathematical framing and the individual's interpretation of the context (checking the mathematical representation against the situation model); (iii) to check alignment between the results of analysis and the individual's interpretation of the context (checking the real results against the real model); (iv) to check the analysis itself (checking mathematical results against the mathematical representation); (v) to check agreement between the results of the analysis and the information available from the real world (checking real results against the situation model).;These findings were used to build a theoretical model of the mathematical modeling process which deviates from previous theoretical and research frameworks used to study mathematical modeling.