### Browsing by Author "Craig, Tracy S"

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- ItemOpen AccessAddressing dualism in mathematical abstraction: An argument for the role of Construal Level Theory in mathematics education(School of Computing, Engineering and Mathematics, University of Western Sydney, 2013-11) Torr, Stuart; Craig, Tracy SLearners of mathematics often struggle to balance the apparently conflicting demands for abstract thinking as well as (often simultaneous) concrete cognitive engagement. Conflicting demands of successful mathematical engagement have been addressed in the literature pertaining to procedural versus conceptual approaches to mathematical learning as well as in the literature on cognitive and meta-cognitive mathematical demands. Construal Level Theory offers an opportunity to understand both these dualities as aspects of the same psychological response to contextual priming. In addition, Construal Level Theory can be understood to illuminate student difficulties with heuristic strategies in mathematical problem-solving. The focus of Construal Level Theory on abstract and concrete cognitive construals as a consequence of psychological distance provides a useful lens for teaching and learning opportunities. We argue that Construal Level Theory offers an opportunity to draw together several strands of mathematics education theory and to help educators address learning challenges in the classroom.
- ItemOpen AccessCategorisation and analysis of explanatory writing in mathematics(Taylor & Francis, 2011) Craig, Tracy SThe aim of this paper is to present a scheme for coding and categorising students’ written explanations of mathematical problem-solving activities. The scheme was used successfully within a study project carried out to determine whether student problem-solving behaviour could be positively affected by requiring the writing of explanatory strategies to mathematical problem-solving processes. The rationale for the study was the recognised importance of mathematical problem-solving, the widely acknowledged challenge of teaching problem-solving skills directly and the evidence in the literature that writing in mathematics provides a tool for learning. The study was carried out in a first-year mathematics course at the University of Cape Town, South Africa. Students’ written submissions were categorised and analysed through use of an adaptation of a journal entry classification scheme. The scheme successfully observed positive changes over the experimental period in students’ level of engagement with the mathematical material and with their stance towards knowledge.
- ItemOpen AccessConceptions of mathematics and student identity: implications for engineering education(Taylor & Francis, 2013) Craig, Tracy SLecturers of first-year mathematics often have reason to believe that students enter university studies with naive conceptions of mathematics and that more mature conceptions need to be developed in the classroom. Students' conceptions of the nature and role of mathematics in current and future studies as well as future career are pedagogically important as they can impact on student learning and have the potential to influence how and what we teach. As part of ongoing longitudinal research into the experience of a cohort of students registered at the author's institution, students' conceptions of mathematics were determined using a coding scheme developed elsewhere. In this article, I discuss how the cohort of students choosing to study engineering exhibits a view of mathematics as conceptual skill and as problem-solving, coherent with an accurate understanding of the role of mathematics in engineering. Parallel investigation shows, however, that the students do not embody designated identities as engineers.
- ItemOpen AccessConstrual level theory and mathematics education(2013) Torr, Stuart; Craig, Tracy SA common complaint of mathematics students is that mathematics is highly abstract. Students often find it difficult to attach meaning to the mathematical concepts they are expected to master. In addition to coming to grips with the abstract nature of the subject, mathematical proficiency requires engagement at a more concrete level. Students must be able to perform step by step algorithmic procedures, detailed algebraic manipulations and master new symbol systems. Mathematical competence often requires thinking at high and low levels of abstraction almost simultaneously and this creates a tension which lies at the core of mathematics education. This tension has been addressed in the literature on procedural versus conceptual approaches to mathematics education and in the literature on cognitive and metacognitive mathematical demands. Construal level theory, and to a lesser extent dual process theory, are theories in cognitive and social psychology which provide a lens through which the difficulties of reasoning at multiple levels of abstraction can be viewed. Construal level theory posits that thinking about psychologically distant objects influences the extent to which we view possibly unrelated objects abstractly or concretely. Psychological distance and abstract thought are cognitively linked together and make up Far Mode thinking. Psychological proximity and concrete thinking are intrinsically linked together to form Near Mode thinking. It is argued that construal level theory forms a useful framework for interpreting much mathematics education research as well as helping to explain the difficulties students experience in implementing problem solving heuristic strategies. Evidence is presented suggesting that priming mathematics students to adopt either a Near or Far mental mode has an impact on their performance in solving conceptually challenging mathematical problems.
- ItemOpen AccessDevelopment of an engineering identity: Personal discovery of classroom mathematics in “real engineering”(Taylor & Francis / Unisa Press, 2010) Craig, Tracy SThis article reports on an activity in a first-year engineering mathematics class designed to strengthen students’ personal identities as novice engineers. The literature on identity suggests that students are more likely to be retained in an engineering degree programme if they develop an identity as an engineer and that development of such an identity is encouraged and supported if students can see the relevance of their studies to future studies or their future career. The mathematics encountered at a first-year level is often of an unrealistic nature due to its largely algebraic content as well as to the fact that real-world engineering problems are often intractable without using more advanced mathematics than is accessible in a first-year course. The activity described in this article endeavoured to build empirically on the theoretical issues raised in the literature by asking “Can students identify the presence of classroom mathematics in real-world engineering texts and does this recognition encourage the development of identity as a novice engineer?” Sixty-six students studying first-year engineering mathematics in an academic development programme at a South African university took part in the activity. Data consisted of the students’ written assignments and their responses to a Likert-style questionnaire. The written assignments were graded on the strength of their alignment with the task’s mathematical requirements. Specifically within the course topic of Applications of Differentiation, the students were required to use resources from the library and the internet to find examples in real-world engineering where differentiation is used for practical purposes. The examples that the students investigated were necessarily expressed in the discourse of engineering, yet drew on mathematics the students had recently encountered in the classroom. This evident trajectory of knowledge from pure classroom practice to real-world engineering use allowed the students ready access to the discourse of engineering and ideally fostered development of identity as an active novice participant in the world of real engineering. A minority of students did not succeed in the task requirements, but the bulk of the students found the task interesting and informative. Several students expressed surprise and pleasure that they were able to understand what they were reading, revealing to them that they were already participants in the engineering community with some fluency in the discourse.
- ItemOpen AccessEnabling Capabilities in an Engineering Extended Curriculum Programme(Bloomsbury Press, 2017-09) Craig, Tracy S; Bangeni, Bongi; Kapp, Rochelle
- ItemRestrictedLanguage and communication in mathematics education: an overview of research in the field(Springer, 2014-10) Morgan, Candia; Craig, Tracy S; Schütte, Marcus; Wagner, DavidWithin the field of mathematics education, the central role language plays in the learning, teaching, and doing of mathematics is increasingly recognized, but there is not agreement about what this role (or these roles) might be or even about what the term ‘language’ itself encompasses. In this issue of ZDM we have compiled a collection of scholarship on language in mathematics education research, representing a range of approaches to the topic. In this introduction we outline a categorisation of ways of conceiving of language and its relevance to mathematics education, the theoretical resources drawn upon to systematise these conceptions, and the methodological approaches employed by researchers. We will also identify some outstanding issues and questions and suggest some ways of building upon the diversity in order to strengthen the coherence of the field and the utility of its outcomes.
- ItemOpen AccessLearning as acquiring a discursive identity through participation in a community: improving student learning in engineering education(Taylor & Francis, 2009) Allie, Saalih; Armien, Mogamat Noor; Burgoyne, Nicolette; Case, Jennifer M; Collier-Reed, Brandon I; Craig, Tracy S; Deacon, Andrew; Fraser, Duncan M; Geyer, Zulpha; Jacobs, Cecilia; Jawitz, Jeff; Kloot, Bruce; Kotta, Linda; Langdon, GenevIn this paper, we propose that learning in engineering involves taking on the discourse of an engineering community, which is intimately bound up with the identity of being a member of that community. This leads to the notion of discursive identity, which emphasises that students' identities are constituted through engaging in discourse. This view of learning implies that success in engineering studies needs to be defined with particular reference to the sorts of identities that students develop and how these relate to identities in the world of work. In order to achieve successful learning in engineering, we need to recognise the multiple identities held by our students, provide an authentic range of engineering-related activities through which students can develop engineering identities and make more explicit key aspects of the discourse of engineering of which lecturers are tacitly aware. We include three vignettes to illustrate how some of the authors of this paper (from across three different institutions) have applied this perspective of learning in their teaching practice.
- ItemOpen AccessMathematical, cognitive and didactic elements of the multiplicative conceptual field investigated within a Rasch assessment and measurement framework(2011) Long, Margaret Caroline; Dunne, Tim; Craig, Tracy SHow may the essential elements of a framework including mathematical, cognitive, and didactic elements, and applied in the multiplicative conceptual field, address the challenges in mathematics education, and inform the curriculum and the validity of assessment processes.
- ItemOpen AccessMeeting the requirements of both classroom-based and systemic assessment of mathematics proficiency: the potential of Rasch measurement theory(AOSIS, 2012) Dunne, Tim; Long, Caroline; Craig, Tracy S; Venter, ElsieThe challenges inherent in assessing mathematical proficiency depend on a number of factors, amongst which are an explicit view of what constitutes mathematical proficiency, an understanding of how children learn and the purpose and function of teaching. All of these factors impact on the choice of approach to assessment. In this article we distinguish between two broad types of assessment, classroom-based and systemic assessment. We argue that the process of assessment informed by Rasch measurement theory (RMT) can potentially support the demands of both classroom-based and systemic assessment, particularly if a developmental approach to learning is adopted, and an underlying model of developing mathematical proficiency is explicit in the assessment instruments and their supporting material. An example of a mathematics instrument and its analysis which illustrates this approach, is presented. We note that the role of assessment in the 21st century is potentially powerful. This influential role can only be justified if the assessments are of high quality and can be selected to match suitable moments in learning progress and the teaching process. Users of assessment data must have sufficient knowledge and insight to interpret the resulting numbers validly, and have sufficient discernment to make considered educational inferences from the data for teaching and learning responses.
- ItemOpen AccessA new selection model for the academic development programme for engineering at UCT(SASEE, 2018-02-16) Campbell, Anita; Craig, Tracy S; le Roux, PierreThe Academic Support Programme for Engineering at the University of Cape Town (ASPECT) has operated under a slowly evolving model since its inception in 1989. Different models of access and curriculum are frequently under consideration and in 2014 we had the opportunity to put into practice a new model, involving self-selection and delayed transition into ASPECT driven by first term assessment. In this paper we present a historical overview, reflect on the 2014 experiences of students and staff in light of relevant theory and conclude with an argument in favour of the delayed transition model.
- ItemOpen AccessObservations and Conclusions of Dynamics Students’ Mathematical Fluency(South African Society for Engineering Education, 2015-07-29) Craig, Tracy S; Cloete, Trevor JThe course Dynamics I in mechanical engineering is a challenging course for many reasons, one of them being its mathematical demands. A collaboration between the first author (a mathematics lecturer and mathematics education researcher) and the second author (a mechanical engineer and the Dynamics I lecturer) sought to answer the question “What specific and identifiable mathematical difficulties are experienced by the Dynamics I students?” The observational results of this, in essence, ethnographic case study suggest that there are two levels of mathematical challenge, namely specific symbolic and computational difficulties as well as the need for well-developed problem-solving processes. We discuss our observations and provide pedagogic advice for lecturers of mathematics to help ease the transition to Dynamics I.
- ItemMetadata onlyProficiency in the multiplicative conceptual field: using Rasch measurement to identify levels of competence(Taylor & Francis, 2010) Long, Caroline; Dunne, Tim; Craig, Tracy SIn the transition years, Grades 7 to 9, the shift from natural numbers to rational numbers and the associated multiplicative concepts prove challenging for many learners. The new concepts, operations and notation must be mastered if the student is to thereafter rise to meet the challenges of algebra and more advanced and powerful mathematics. The multiplicative conceptual field (MCF) groups together such concepts as fraction, ratio, rate, percentage and proportion, all of which are related yet subtly distinct from one another, each with its own challenges. Rasch analysis allows us to compare the difficulty of mathematical problems located within the MCF while, on the same scale, locating the degree to which individual learners have mastered the necessary skill set. Such location of problems and learners on the same unidimensional scale allows for fine-grained analysis of which aspects of the problems being analysed make one problem more difficult than another. Simultaneously the scale gives the teacher clear evidence of which students have mastered which concepts and skills and which have not, thereby allowing more targeted assistance to the class and individual learners. This paper illustrates the process involved in such analysis by reporting on results located within a larger study. It is suggested that implementing Rasch analysis within the school classroom on appropriately designed assessment instruments would provide clarity for the teacher on the locations of difficulty within the problems used in the assessment and the relative degree to which individual learners are achieving success at mastering the targeted concepts.
- ItemOpen AccessPromoting understanding in mathematical problem-solving through writing : a Piagetian analysis(2007) Craig, Tracy S; Dunne, Tim; Webb, JohnIncludes bibliographical references (leaves 204-221).
- ItemOpen AccessThe role of expository writing in mathematical problem solving(Southern African Association for Research in Mathematics, Science and Technology Education, 2016-04) Craig, Tracy SMathematical problem-solving is notoriously difficult to teach in a standard university mathematics classroom. The project on which this article reports aimed to investigate the effect of the writing of explanatory strategies in the context of mathematical problem solving on problem-solving behaviour. This article serves to describe the effectiveness of using writing as a tool for deeper engagement with mathematical problems. Students’ claims about, and tutor observations of, problem-solving behaviour were analysed through the lens of Piaget’s theory of cognitive development. Examples of enhanced problem-solving behaviour are presented as well as reports from student interviews that writing “forces” deeper engagement. The analysis of students’ work and reflections indicated that writing about problem-solving processes potentially resulted in a cognitive perturbation when students were forced to confront their incomplete understanding (and hence their unstable knowledge structures) and therefore had to achieve a deeper level of understanding in order to adequately describe the solution process.
- ItemOpen AccessSimple rule, hidden meaning: the scalar product in engineering mathematics(Elephant Delta, 2017-09-13) Craig, Tracy S; Cloete, Trevor JEngineering is a highly mathematical field of study with different university courses requiring proficiency at different types of mathematics. Engineering dynamics requires the skilful use of vectors in various ways and proficiency at vector arithmetic, algebra and geometry is of vital importance to incoming students. This paper reports on findings from the administering of a vector proficiency assessment instrument across two semesters of a dynamics course. Findings suggest that problems requiring use of the scalar product embedded within a context are of the highest difficulty level. We argue that the geometric role of the scalar product is weakly understood by the majority of students, leading to poor performance at any problem requiring more than a basic calculation. We suggest that lecturers of engineering mathematics foreground the geometric role and that lecturers of engineering courses be aware of the level of challenge manifest in these problems.
- ItemOpen AccessStudent identity and the need to make classroom mathematics relevant to engineering practice(South African Society for Engineering Education, 2015-07-29) Craig, Tracy SCobb and Hodge’s (2005) identity theoretical framework suggests that learning is facilitated if normative identity (realised and co-constructed in the classroom by lecturer and student) is reconciled with core identity (the trajectory of who the student is and where he feels he is going). The cohort of students involved in the study discussed in this article largely embodies trajectories of social mobility, with a great willingness to study engineering for its role in providing a way out of poverty rather than for the sake of the discipline itself. The pedagogic implication is that teaching must proceed sensitive to the reality of the students which is that they potentially have little idea what engineering entails other than a route out of a disadvantaged background.
- ItemOpen AccessSuccessful students’ negotiation of township schooling in contemporary South Africa(University of the Free State, 2014-09) Kapp, Rochelle; Badenhorst, Elmi; Bangeni, Bongi; Craig, Tracy S; Janse van Rensburg, Vicki; le Roux, Kate; Prince, Robert; Pym, June; van Pletzen, ErmienThis article draws on data from a larger longitudinal qualitative case study which is tracking the progress of students over the course of their undergraduate degrees at a South African university. For this paper, we used background questionnaires and semi-structured interviews with 62 first-year students from working-class, township schools who were first registered for Extended Degree Programmes in 2009. The article draws on post-structuralist theory on learning and identity to describe and analyse the participants’ perspectives on how they negotiated their high school contexts. We analyse the subject positions in which participants invested, as well as how they negotiated their way through social networks and used resources. Our data illustrate the ways in which students had to carry the burden of negotiating their way through home, school and neighbourhood spaces that were generally not conducive to learning. Nevertheless, participants consciously positioned themselves as agents. They were resilient, motivated and took highly strategic adult decisions about their learning. We argue that a focus on how successful students negotiate their environments challenges the pathologising paradigm of “disadvantage” that characterises research and debates in higher education. It also offers an additional lens for admissions processes and for providing appropriate intervention strategies in the tertiary setting.