Categorisation and analysis of explanatory writing in mathematics

The 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

The role of expository writing in mathematical problem solving

Mathematical 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

Conceptions of mathematics and student identity: implications for engineering education

This is an Accepted Manuscript of an article published by Taylor & Francis in International Journal of Mathematical Education in Science and Technology on 8 March 2013, available online: http://www.tandfonline.com/10.1080/0020739X.2013.823521.Lecturers 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

Observations and Conclusions of Dynamics Students’ Mathematical Fluency

The 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

Simple rule, hidden meaning: the scalar product in engineering mathematics

Engineering 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

Development of an engineering identity: Personal discovery of classroom mathematics in “real engineering”

This 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

Meeting the requirements of both classroom-based and systemic assessment of mathematics proficiency: the potential of Rasch measurement theory

The 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