Mathematics curriculum and policy documents for upper secondary schooling in many countries including Australia are placing increased focus on mathematical modelling and applications and forms of assessment that are more connected to real-world tasks. However, many fundamental theoretical issues relating to the location of mathematical tasks in meaningful contexts for both teaching and assessment purposes remain unresolved. The resolving of these issues becomes even more crucial in the context of a school based assessment system.
The general aim of the thesis is to develop an analysis system for the complexity encountered in application and modelling tasks in the senior secondary school. Such a system, being based on a sound theoretical framework, can then facilitate the framing of assessment tasks of known cognitive and contextual demand. Where assessment is school based, relying on the expertise of local teachers rather than a central examiner, such a system is particularly important when addressing questions of comparability of tasks set within different schools, districts and regions. Specific aims for this study are to:
• develop a framework for the analysis of applications/modelling tasks and
• a scheme for profiling tasks which
- identifies essential mathematical content and processes within the task,
- highlights the cognitive and contextual demand of the task, and
- investigates how task context, prior knowledge and experience, and mathematical content interact with performance on contextualised tasks.
Finally, a grounded theory of teacher and student construal of task difficulty and complexity in practice is presented.
A cognitive/metacognitive framework for analysing applications tasks was developed from a review of cognitive science literature. This was used with research tools based on the SOLO Taxonomy to develop a cognitive demand profile for applications tasks. A preliminary study was undertaken to examine and document current classroom practice through document analysis and an analysis, using the cognitive demand profiles, of scripts from the final two years of one secondary school's assessment program. This was followed by the main study employing Strauss and Corbin's (1990) reformulated grounded theory method. Individual task solving sessions were videotaped with 43 senior high school students from two schools. These sessions were immediately followed by clinical interviews using the videotapes as stimulus. In addition, eight teachers from the two schools were interviewed.
Intensive analyses of the task solving sessions revealed that students used a variety of cognitive strategies to take advantage of memory-related, perceptual and engagement conditions facilitating task access, and to overcome major conditions impeding task accessibility. Metacognitive strategies for monitoring, regulating and coordinating the use of these cognitive strategies were identified. A separation of task difficulty from task complexity was achieved with task difficulty determined by students' personal attributes and their interaction with the task, and complexity related to attributes of the task itself.
Student use of task context to keep their mathematics on track throughout a task did not appear to be related to the particular task but was student specific. Most students showed in the study that they would use the context to keep on track when the situation warranted it despite the claim of many that they used it only as a final check.
Prior knowledge of the task context was classified according to its source as being academic, encyclopaedic or episodic, a classification that proved useful in examining the nature of the prior knowledge reported which was task, rather than student, specific. The study highlights the largely unpredictable and differential nature of the effects of prior knowledge on student solutions. However, episodic prior knowledge was found to be more likely to have a positive influence than academic or encyclopaedic knowledge.
Moderate to high engagement with the task context of an application was not often associated with poor performance, which was more likely to be associated with nil to low engagement. However, high engagement was not a necessary condition for success as the degree of engagement necessary for success could be task specific. A sense of realism and having an objective to work towards were identified by students as facilitating their engagement.
Some students in the study who attempted tasks with similar mathematical structures in both abstract and contextualised formats had very little success in reproducing standard procedural knowledge in the contextualised setting. Such students believed that success on applications resulted from remembering methods and recognising when to apply them. Other students were much more successful in the contextualised settings. These students were able to successfully integrate task-defined information and constraints into clarifying diagrams. They either applied standard mathematical procedures which they recalled easily, or were able to select from their range of mathematical tools and adapt methods to fit novel aspects of an unfamiliar task. The successful students mostly appeared to have a more adaptable, relational understanding of mathematics and how it could be applied in contextualised settings.
Key differences in comparing in-school and out-of-school mathematics practices appear in relation to the flexibility of handling constraints. It is suggested that the salience of cues such as task constraints is mediated by how knowledge is acquired, with experential knowledge derived through observation proving less effective than that derived through action.
The construction of a grounded theory of how teachers and students construe complexity and difficulty of applications tasks, involved the identification of numerous attributes of a task that contribute to its overall complexity. These encompassed the mathematical, linguistic, intellectual, representational, conceptual and contextual complexities of the task and it was found that both teachers and students based their assessment of task complexity and difficulty on only a small subset of these possible cues.
The results of this study clearly demonstrate the complexities of teaching and using applications and modelling as a major teaching focus. They include four major contributions to the research in this area:
(1) the construction of a cognitive/metacognitive framework for analysing applications tasks;
(2) development of a new research tool, the cognitive demand profile;
(3) identification of significant insights into the role played by extra-mathematical knowledge in solving applications of different complexities, and
(4) construction of a grounded theory of teacher and student construal of task difficulty and task complexity in practice.
These outcomes make a substantial additional contribution to our understanding of the challenges involved in the successful teaching and assessment of applications tasks at the senior secondary level.