5.8 THE CONCEPT OF THE DERIVATIVE IN MODELLING AND APPLICATIONS Gerrit Roorda, Pauline Vos, Martin Goedhart University of Groningen, The Netherlands Abstract−The question addressed in this paper is how to measure students’ transfer skills with respect to the concept of the derivative in modelling and applications. We adapted a framework developed by Zandieh (2000) for analysing students’ understanding of the concept of the derivative. Nine grade-11 students made a test and a task-based interview, which consisted of mathematical and application problems. The task-based interview included tasks about the application of the derivative in a physics and economics context. The analysis of interview data showed some difficulties emerging with the framework. With respect to modelling and applications, we found that sometimes students cannot apply knowledge of calculus that they have, and that translations between representations, and especially between a context set in the real world and mathematics, and backwards, cannot be represented in the framework. Our analyses gave directions for further development of a framework for understanding derivatives in relation to applications. 1. INTRODUCTION In the Dutch mathematics curriculum for secondary schools, the role of modelling and applications increased in the past fifteen years. When students in grades 10-12 learn the concept of the derivative, most textbooks avoid the use of the formal limit definition (or only mention it on one page). Instead, textbooks provide students with opportunities to learn the concept in different representations (e.g. symbolical and graphical) and in applications (e.g. physical and economical). Many application problems in mathematics textbooks describe a real situation but often the mathematical model is simultaneously offered by a formula. In terms of the modelling circle described by Blum (2005), the real world and the mathematical world are both present in many Dutch textbook exercises. Most exercises are set within the real world and offer a problem context. This context has to be understood (step 1) and structured (step 2). The ensuing step, to mathematise the problem (step 3) into a mathematical model, is already provided for by the textbook authors. What remains is that students have the opportunity to make the last steps in the modelling circle such as ‘work mathematically’, ‘interpret’ and ‘validate’ the model (steps 4, 5 and 6). In tasks involving derivatives, this means that students can use their mathematical knowledge to solve an application problem and compare their answer with the situation in the real world.