This study concentrates on the relationship between the students’ thinking styles (Stenberg, 1999) and the strategies (Kirby, 1988) the students employ when dealing with exam-type questions in mathematics. Thinking styles are the “preferred way[s] of using the ability one has” (Sternberg, 1999, p. 8) and are conceptualised to be relatively stable over time and context. A strategy is the “combination of tactics, or a choice of tactics, that forms a coherent plan to solve a problem” (Kirby, 1988, p. 230-231). The students’ attainment, the nature of task and the students’ views are also considered in this study. A three-phase study including both quantitative and qualitative techniques was designed with the aim of delineating this relationship. The study was conducted with 2nd year students (N=99) following a BSc in Mathematics in the Department of Mathematics of the University of Athens, although, for methodological reasons, additional data were collected from a broader group of undergraduates (NUG=224). The students’ thinking styles were identified through a version of the Sternberg-Wagner Thinking Styles Inventory (Sternberg, 1999), translated into Greek. Two main Style Cores were identified: Core I (creative, original, critical and non-prioritised thinking) and Core II (procedural, already tested and prioritised thinking). Based on these cores, the students were assigned to two clusters: Cluster 1C2C (High Core I/Low Core II) and Cluster 3C4C (High Core II/Low Core I). In order to identify the students’ strategies, the A-B-Δ strategy classification was introduced, expanding on Weber’s (2005) semantic, syntactic and procedural strategies. The AB-Δ strategies were grouped in three Strategy Types depending on their links with truth,memory and flexibility, respectively identified as: α-type, β-type and δ-type. Students assigned to Cluster 1C2C appeared to prefer more α-type and less β-type Initial Strategies than those assigned to Cluster 3C4C. The nature of the task appeared to affect this link. On the other hand, in the context of Back-Up Strategies, stylistic preferences and ‘high’ attainment appeared to regulate a link between the nature of the task and a Back-Up Strategy, rather than forming a style-strategy link (as in the case of Initial Strategy). Drawing from Skemp’s (1979) views about reality (inner and social) and survival (respectively, internal consistency and social survival), it is argued that the students choose different strategies, because they essentially perceive the given task in qualitatively different ways. The students’ different stylistic preferences indicate differences in their inner reality, thus affecting their choice of an ascertaining argument, which in turn determines their selection of Initial Strategy. The failure of the students’ Initial Strategy leads them to re-evaluate the task itself, thus resulting in a change of the reality in which the students have to survive and this, in turn, determines the students’ Back-Up Strategies.

Although there has been considerable research into real-world contexts in elementary mathematics, little work has been done at a more advanced, post-16 level. This thesis explores the origin, function and effect of real-world contextual framing (RWCF)in GCEA/AS mathematics examinations. The study develops an evaluation framework (ARTA)based on the notions of accessibility, realism and task authenticity, derived from assessment theory, and considers ‘context’ in relation to theoretical ideas such as Realistic Mathematics Education, construct validity and construct-irrelevant variance. The function and effect of RWCF are investigated using the ARTA framework on samples of A/AS questions. Its effect is explored using sequence questions with the same solutions with and without real-world context, set to a sample of nearly 600 students, together with a questionnaire that surveys students’ attitudes to RWCF. Quantitative differences in the use of RWCF are established and traced to early project syllabuses such as SMP and MEI. The study finds that RWCF in general adds to the difficulty of questions, unless they can be solved by ‘thinking within the context’. The accessibility of questions with RWCF is a function of comprehensibility of language, and the explicitness of the match between context and mathematical model. The study distinguishes between natural and synthetic contexts, according to the extent to which the context matches reality, or reality is configured to match the mathematics. Natural contexts are more realistic; but synthetic contexts can serve the purpose of reifying abstract mathematical ideas. At best, RWCF in examination questions require solvers to engage in pseudo-modelling: they cannot test aspects of the modelling cycle such as discussing assumptions, refining, and critical reading of longer arguments. There is, moreover, a gender difference in students’ attitudes to RWCF, with boys in general expressing more favourable views about its use in pure mathematics questions. These findings have the following implications for A/AS assessment. Current examination questions are not able to satisfy current QCDA (Qualifications and Curriculum Authority, 2002) assessment objectives on mathematical modelling. Questions with RWCF need to be authentic, and require careful construction to ensure that language is precise and unambiguous. Longer questions, which present and invite comparison of more than one model, are desirable, in order that students appreciate the relationship between reality and mathematical models.