Synthesis

Theoretical work in statistics has deepened our understanding of what learning in the discipline requires of a student. Studies conducted have used quantitative methods, in the main, to generate an extensive body of evidence of the methods that broadly support learning in statistics. Details gleaned from this body of scholarship matches the central idea of a liminal transition in learning and has, to varying degrees, highlighted the importance of pre-conceptions, conceptual change, troublesomeness, identity aspects and meta-learning capacity in disciplinary learning as aspects of learning that are emphasised in a Threshold Concepts Framework orientation.

Reflecting on the extant statistics education literature, guidelines informing best pedagogical practice in the discipline entailed the use of real-world application incorporating the use of real data, increased opportunities engendering active learning through cooperative learning exercises and the use of technology to foster deep approaches to learning. The threshold concepts framework approach agrees with these disciplinary pedagogical guidelines to learning in statistics (Bulmer et al., 2007; MacDougall, 2010). In South Africa, statistics education research has focused predominantly on quantitative studies of performance determinants and (to a much

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lesser extent) on teaching innovations (de Wet, 1998; Galagedera, 1998; Galagedera et al., 2000). However, becoming proficient in disciplinary ways of thinking and doing is a dynamic process, requiring the complete enculturation of the student into the discipline – this involves interaction of the student with the nature of the discipline, teaching approaches, course content and broader contextual features. Given the idiosyncrasies of individual learner journeys and the notoriety of statistics as being difficult amongst students, learners’ emotions are bound to play a significant role in their learning journey. The significant absence of extant studies in the statistics education literature focused on an entanglement of student learning and self-identity is lamentable.

In contrast to this gross neglect in statistics education literature, the Threshold Concepts Framework approach offers an encapsulation of the two strands of the ‘DNA’ that characterises statistical learning, viz. the cognitive and the affective. The threshold concepts framework to learning offers a holistic view of learning initially identifying sources of difficulty, then interrogating cognitive and affective elements to assist students in crossing the liminal space, and finally describing the epistemological and ontological metamorphosis of the learner. While this learning perspective is well-established in research into learning internationally, it has not been deeply explored in statistics as a discipline. In South Africa there appears to have only been a study conducted by Dunne et al. (2003) on identifying threshold concepts in statistics.

In this study, I set out to explore how (and why) statistics students learn in a threshold concepts- enriched tutorial programme. In using a threshold concepts orientation here, I hoped to address both conceptual and contextual gaps in the extant statistics literature. The study focuses on processes and experiences of students’ learning, and indicates possible ways to support and facilitate learning transitions and sheds further light on the less understood, liminal aspects of statistics disciplinary learning in terms of threshold concepts theory.

Reflecting on Chapter 2, the threshold concepts view to learning speaks to current debates in statistics higher education research arising from concerns with the quality of learning and calls for more student-centeredness to foster deep approaches and active learning. Threshold concepts orientation also reconciles the apparent poles of cognitive and affective aspects of learning by highlighting their united impact on student’s learning within the liminal space (Rattray, 2014,

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Attempts at augmentation of the threshold concepts framework includes a proposed framework for the design of university curricula in the form of the Threshold Capability Integrated Theoretical Framework (TCITF) (Baillie et al., 2012). The TCITF is an amalgamation of Threshold Concepts Framework and Capability Theory30 _{(Bowden, 2004). Capability Theory} draws upon phenomenography and variation theory,31 and is concerned with the development of knowledge capability (Baillie et al., 2012). The thinking behind linking the two approaches is the dynamic progression from understanding threshold concepts to developing threshold capabilities, the ability to use threshold concepts in context, to knowledge capability (for example, enabling a student to think and practice as a statistician). For example, in statistics, understanding confidence intervals is considered to be a threshold concept (Khan, 2014), but being able to discern the implications of altering the sample size, say, on the confidence interval, requires the student to have threshold capabilities.

The TCITF offers a new perspective on disciplinary learning, wherein the emphasis is placed on preparing and equipping students to apply their understanding to new, previously unseen situations, thus enabling students to achieve a longer-term professional goal of gaining knowledge capability (Land, 2014). The TCITF contributions to knowledge acquisition is achieved through: (i) the capability to handle previously unseen situations, where a common requirement in professional life is developed through multiple experiences of dealing with new situations, and (ii) discernment of key aspects is assisted by experiencing variation, i.e. experiencing a range of situations in which the key aspects vary (Bowden and Marten, as cited in Baillie et al., 2012).

The essential elements of the learning process within a TCITF orientation to the statistics discipline is represented in the schematic below and explained in the subsequent paragraph.

30 _{Capabilities Theory or Capabilities Approach was introduced as an approach in welfare economics (Sen, 1979).} The core characteristic of this approach is a focus on what people are effectively able to do and be – in other words, their capabilities (Robeyns, 2003).

31 _{The Variation Theory of Learning (Marton, 2015; Marton & Tsui, 2004) arises out of the phenomenographic} research approach. Through pedagogical designs, students are introduced to variations in the illustrations of the critical aspects of a disciplinary concept, skill or practice. This variation is seen as an expansion in awareness of the different aspects of disciplinary concepts, and of the relationships between those aspects (Akerlind, 2015).

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Figure 3. A TCITF view of learning

(Based on an amalgamation of the descriptions of a TCITF and Threhold Concepts Framework view of learning taken from Baillie et al. (2012); HEReflections (2015)) This model positions disciplinary knowledge as the over-arching framework for a curriculum. Threshold concepts are the building blocks that support disciplinary knowledge (Meyer et al., 2010). Use of threshold concepts allows students to move deeper within their discipline to either specialise within coherent, particular spheres in their discipline or outwards towards a wider field of study. If disciplinary knowledge and threshold concepts are engaged with alone, then there is a deficit in the applied/practical use of the emerging learning. It is at the intersection of the three dimensions of disciplinary knowledge, concepts, and application, where we strive for learners to

Statistics Knowledge Application Threshold Concepts Theoretical understanding but limited practical ability Uncritical, ‘mechanistic application’ Emergent, critical understanding and application Weak application due to a lack of specific and wider statistical

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be. The journey to this level is a personal one for each student with different contexts, interests and applications driving learning (HEReflections, 2015).

The threshold concepts framework has been characterised as a “transactional curriculum inquiry”, in the sense that it is neither teacher-centred nor student-centred, but rather, a collaboration between subject specialists, researchers and learners (Cousin, 2008). Thus, in a threshold concepts view to learning, the pedagogical innovations and curriculum re-designs carried out by disciplinary teachers are inspired and motivated by their learners. In an attempt to bring this vast and expanding body of research knowledge to the threshold concepts community of scholars, the Integrated Threshold Concept Knowledge (ITCK) has been proposed (Meyer & Timmermans, 2016). The ITCK attempts to translate threshold concepts findings in a theoretically sound and actionable form in an unifying approach extending across disciplines, while “remaining non-prescriptive and adaptive to the various contexts in which threshold concept research and practice occur” (Meyer & Timmermans, 2016, p. 25). ITCK is described as being “socio-empirically constructed knowledge” constituted by different ‘types of knowledge’ (Meyer & Timmermans, 2016, p. 25), arising from: (i) the critical features of threshold concepts; (ii) learner variations with respect to the cognitive, affective and ontological aspects of experiencing threshold concepts in the liminal state; and (iii) pedagogical responses to engendering threshold concept crossing.

It is within these two broadly specified frameworks, that of the TCITF and ITCK, that the findings of this study fit. Firstly, the exploration of how (and why) statistics students learn in a threshold concepts-enriched tutorial programme in this study context will shed light on the following critical curriculum design features of the TCITF:

• The kinds of learning experiences and combinations thereof, which would best assist the learner to develop interim threshold capabilities, and ultimately build on, to develop the capability to handle an unknown future after graduation?

• How can the learning environment be best arranged to provide access to those optimal capability development experiences?

• How can the differing needs of individual students be catered for?

• What, specifically, is the role of teachers in supporting such learning by students? (Baillie et al., 2012, p. 237).

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Secondly, by foregrounding the students’ voices to answer the research questions, this study’s approach is also quite rare in statistics education research, which is almost exclusively from the perspective of the educator. Within threshold concepts scholarship, questions of student learning are central to the framework, and using student voice-data is increasingly recognised as an effective way of exploring cognitive, affective, and identity-related dimensions of learning. My research approach aims to elicit, deep, rich qualitative descriptions of the learning process in the participants’ voices, and was thus appropriate for capturing the details of cognitive, affective or ontological aspects of students’ encounters with learning statistics in threshold concepts-enriched tutorials and navigation of the liminal space of learning. The considerations made possible by such a close reading of statistics students’ learning may suggest ways in which the learning process may be supported or facilitated, which ultimately have implications for pedagogic and curricular responses such as those enumerated above in the ITCK.

This work also addresses a clear contextual gap in South Africa, where research in statistics higher education encompasses a narrower range of approaches and issues than those in evidence in the international literature, where little qualitative or conceptual enquiry has been undertaken, and an understanding of how students learn in statistics in South African higher education remains superficial. Furthermore, the setting of the case study within an introductory statistics course context could potentially generate possible insights into learners’ progress in crossing inter- disciplinary thresholds.

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