5. Analysis of task characteristics of questions in the two disciplines

Before the empirical results are discussed, the characteristics of the tasks set in the mathematics and kinematics contexts were analysed on the level of the participants. This implies that this analysis may differ for more or less advanced participants. For example, more experienced participants may distinguish characteristic features of graphs by visual decoding only and consequently may not need to explicitly perform judgement.

As indicated in Table 1, each task (e.g., reading coordinates, etc.) requires different mathematical and kinematics contextual knowledge, although similar visual decoding and judgement are to be performed in both the contexts. The first task, reading coordinates, is the simplest and requires only contextual knowledge and visual decoding. The other graph tasks require contextual knowledge, visual decoding and judgement.

It is important to note that the kinematics tasks can only be done if the mathematics contextual knowledge is transferred and integrated with kinematics knowledge. In the first task (reading coordinates), participants should have contextual mathematics knowledge of Cartesian coordinates and integrate it with kinematics knowledge about the variables of position (s), velocity (v), acceleration (a) and time (t). Conventionally, the independent variable t is placed on the x-axis and the dependent (s, v or a) on the y-axis. In the questionnaire items, participants needed to connect the proper dependent variable (function value) to a given independent variable, using visual decoding.

The second task (called connecting representations) requires mathematical knowledge of the graphical representation and formula of straight-line, parabolic and hyperbolic functions. In the kinematics questions, participants needed to recognise the mathematical formats and graph forms of the given expressions containing kinematics variables, instead of mathematical symbols. Proper understanding further requires insight that the given kinematics equations and graphs represent functions of time. Without having and integrating this contextual mathematics and kinematics knowledge, the participants will not know which visual decoding and judgement tasks to perform.

In order to accomplish "area quantitative" and "area qualitative" tasks (tasks 3a and 3b in Table 1) on kinematics, participants must recall the kinematics relation <sup>s</sup> <sup>¼</sup> <sup>Ð</sup> vdt. Then they should know from mathematics that the integral is determined from the area under a line graph. Blending these kinematics and mathematics knowledge elements should result in understanding that displacement in interval dt is <sup>s</sup> <sup>¼</sup> <sup>Ð</sup> vdt = area under v-t graph. Only then can the participants perform the expected visual decoding and judgement tasks.

Requirements for successful execution of the qualitative and quantitative tasks on slopes (tasks 4a and 4b) are similar to those for area. From mathematics, participants should know the meaning and formula for calculating the slope of a graph and be able to attach the kinematics meaning to it, that is, v ¼ ds=dt = gradient of s-t graph at time t. Thereafter, the visual decoding and judgement required by the different questions can be performed.

For all tasks, the discipline characteristics of the question thus determine what visual decoding and judgement tasks have to be done. Inability to perform the correct contextual tasks is expected to prohibit execution of correct visual decoding and judgement.
