The Elusive Slope

Thomas Lingefj?rd 0 Djamshid Farahani 0 0 University of Gothenburg , Gothenburg , Sweden In understanding upper secondary school students' interpretations of information in graphical representations of a distance-time graph and an ECG graph, little attention has been paid to the analysis of the condition of the conceptual development related to their utterances. Understanding this better can help improve the teaching of interpretations of information in graphical representations of different situations. This paper integrates results from 2 studies and 3 theoretical perspectives: Tall and Vinner's theoretical perspectives on learning, Chi's ontological perspectives on conceptual development and Friel's theoretical perspectives on interpretation of graphical information. The findings provide evidence to support the conjecture that iconic interpretations could be stimulated and generated as a result of student categorisation of a distance-time graph as an event, when in fact the graph is being used to describe and communicate a process. The outcome further indicates that students found a resemblance between the ECG diagram and the periodic function of f(x) = sin(x). Conceptual development; Graphical representations; Interpretations; Intuitive ideas; Slope - Mathematical representations such as diagrams, histograms, functions, graphs, tables and symbols facilitate understanding and communication of abstract mathematical concepts or other situations described in mathematical terms (Elby, 2000; Leinhardt, Zaslavsky, & Stein, 1990). Nevertheless, humans of today are facing a world that is shaped by increasingly complex, dynamic and powerful systems of information through various media. Being able to interpret, understand and work with graphical representations involves mathematical processes the student needs to appreciate, comprehend and be able to address when facing interpretation challenges (Friel, Curcio & Bright, 2001). For mathematics education in an elementary, middle, lower secondary and upper secondary perspective, teachers use different representations in order to make it possible for students to gradually understand more and more complex mathematical objects and concepts. Geometrical constructions, graphs of functions and a variety of diagrams of different kinds are used to introduce new concepts and to study relations, dependency and change (Trigueros & Mart?nez-Planell, 2010). Mathematical representations, structures and constructions are also used in different scientific branches, such as biology, chemistry, physics or social science. It is of major importance that students learn how to interpret graphical representations in a scientific and successful way. Understanding a graphical representation of a situation requires different concepts be incorporated in the specific representation. The critical problem of transition between and within representations has been addressed in several studies (Breidenbach, Hawks, Nichols & Dubinsky, 1992; Janvier, 1987; Sfard, 1992). They claim that bridging the gap between algebraic and graphic representations depends highly on how students encapsulated relevant concepts involved in the representation. We acknowledge that slope is a universal topic in every country?s mathematics curricula. It is usually introduced with linear functions. It is central for describing the behaviour of a curve and has an essential role in the development of calculus (Lobato & Thanheiser, 2002; Stump, 1997, 1999, 2001a, b; Zaslavsky, Sela & Leron, 2002). These investigations on the understanding of slope have made valuable contributions to understanding what makes this concept so complex to learn. How do students interpret and understand graphical representations of a situation? How do students use their interpretation in order to investigate special features of the situation at hand? A good deal of research has been conducted to investigate student?s alternative conception about scientific concepts (diSessa, Hammer, Sherin & Kolpakowski, 1991; diSessa, 1993; Elby, 2000; Hammer, 2000; McDermott & Schaffer, 2005). In our study, we use the term alternative conception instead of misconception, which is associated with inaccuracy and mistakes. diSessa (1993) claimed that humans gradually acquire an elaborate sense of mechanism?a sense of how things work in dealing with the physical world, what sorts of events are necessary, likely, possible or impossible. Control of the physical world is one function for the sense of mechanism and, in addition, of humans being capable of taking actions with appropriate consequences. diSessa et al. (1991) refer to the present view of the sense of mechanism as Bknowledge in pieces^. This view of physics understanding and physics learning is knowledge-based. It assumes only a few very simple cognitive mechanisms, although the resulting knowledge system is conjectured to be large and complex. The central focus in addressing these issues is a hypotheti (...truncated)