Description: This report examines how students link continuity, differentiability and integrability concepts in their mind maps within the context of hierarchical thinking. A survey research design was used to obtain a large group of participants from three different mathematics departments in Turkey. While primary data acquired with the help of the concept map were analysed using descriptive statistics, secondary data acquired by way of interviews were analysed according to their content. The findings revealed that a great majority of the participants built wrong hierarchies between these concepts. The results also show that the students’ epistemological beliefs, or their sequential learning and instrumental understanding instead of relational understanding, hinder building correct hierarchies, and some suggestions show lecturers how to effectively use concept maps and counterexamples.

Description: The teaching of STEM subjects, and engineering and mathematics in particular, involves the use of a wide range of representational forms, including equations, diagrams, sketches and graphs, supported by speech and gestures. In the traditional face-to-face ‘board’ based classroom, the integration of writing, speech and gesture has been a key feature of pedagogical delivery approaches. The pen-enabled Tablet PC (penTPC), used in conjunction with a data projector, allows for the maintenance of a handwritten approach in teaching environments where traditional boards are unavailable or limited. However, it has been suggested that the use of the digital interface imposes restrictions on lecturer movement and gesture, compared to traditional board environments. This article examines the adaptations made by lecturers in using the penTPC in a classroom environment. The study suggests that the use of penTPC technology does not preclude the use of gesture, and that the augmented capability for annotation in conjunction with other digital representations can enhance teaching, particularly of STEM-based discipline subjects.

Description: Decision mathematics is at present the least established and arguably the most contested strand of applied mathematics within the British A-level mathematics and further mathematics qualifications. This article presents data from a study comprising 10 A-level further mathematics candidates who chose as a cohort to study a second module of decision mathematics. The students described their developing attitudes towards, and perceptions of decision mathematics in two sets of questionnaires and one round of follow-up interviews. Together they reported different levels of challenge, mostly positive affective profiles, and a high level of appreciation for the utility of decision mathematics. Using the concept of figured worlds, this article explores further how the students understood the activity contained within the decision mathematics modules, and the extent to which they positioned it as legitimate mathematical practice. The participants’ accounts show that the study of decision mathematics not only augmented the students’ knowledge of how mathematics might be applied in the world, but for some also worked to extend and reconfigure their perceptions of mathematics as a curriculum discipline.

Description: Linear algebra is one of the first abstract mathematics courses that students encounter at university. Research shows that many students find the dense presentation of definitions, theorems and proofs difficult to comprehend. Using a case study approach, we report on a teaching intervention based on Tall’s three worlds (embodied, symbolic and formal) of mathematical thinking, and use a framework combining these with Dubinsky’s Action, Process, Object and Schema (APOS) theory to analyse students’ resulting levels of understanding. Through interviews and analysis of test and examination scripts, we investigate students’ understanding of the basic concepts of linear algebra, their ability to use and explain these concepts and their relationship to definitional clarity. The results show that, while students tend not to learn definitions by rote and can be imprecise when expressing them in words, they seem to understand the concepts, can talk sensibly about them and are able to use their essential features in solving problems.

Description: Classroom flipping is the practice of moving new content instruction out of class time, usually packaging it as online videos and reading assignments for students to cover on their own, and devoting in-class time to interactive engagement activities. Flipping has garnered a large amount of hype from the popular education media and has been adopted in a variety of contexts. Despite this high amount of interest, few studies have evaluated the effectiveness of classroom flipping on student academic outcomes. Specifically, no rigorous studies of the effects of flipping a mathematics course on students’ mathematical understandings and achievement appear in the literature. This article reports results from a control group study of flipping a large ( N = 690), first-year university calculus course for life sciences students. Students in the flipped course sections on average outperformed their counterparts in the traditional sections on the final exam, though only by approximately 8%. A more detailed analysis reveals the true beneficiaries in a flipped classroom—those with high basic mathematical ability and low initial calculus knowledge. Gains for this group are considerable: approximately 10% on the final, with an effect size of d = 0.56, and comparable gains on an independent measure of calculus concept mastery. This study positions classroom flipping as an effective practice in undergraduate mathematics and calls for further research into the mechanisms behind its effectiveness.

Description: Teaching techniques of integration can be tedious and often uninspired. We present an obvious but underutilized approach for finding antiderivatives of various trigonometric functions using the complex exponential representation of the sine and cosine. The purpose goes beyond providing students an alternative approach to trigonometric integrals. It introduces a framework in which students can better understand more advanced mathematical ideas such as the inverse Laplace transform and also affords an opportunity to work with detailed algebraic manipulations involving the binomial expansion.

Description: This investigation explored pre-service teachers’ mathematics content knowledge (MCK) and beliefs associated with mathematics education practices. An Exploratory Factor Analysis, conducted on a beliefs and attitudes questionnaire, produced three common attitude factors associated with (1) inquiry-based teaching; (2) how mathematics knowledge is acquired; and (3) the applicability of mathematics. These factors were used in subsequent multivariate analyses to determine whether teachers’ mathematics competence influenced their personal mathematics viewpoints and perspectives. There was no difference between those students who had studied advanced and standard mathematics at school on the three belief and attitude measures, despite distinct differences in their MCK.

Description: A wide body of literature has highlighted how high achievement in mathematics in secondary school does not necessarily motivate students to both choose and succeed on mathematically demanding programmes at post-compulsory level. The recent Enterprising Science project (Archer et al. (2015, J. Res. Sci. Teach., 52, 922–948)) and before that, the ASPIRES project (Archer et al. (2013, London: Kings College)), have both highlighted that access to science capital is perhaps more important than prior achievement in shaping students’ aspirations and their future trajectories in Science, Technology, Engineering and Mathematics. In this article, we critically analyse the notion of science capital and its role in mediating students’ choice of and experience of studying mathematically demanding degree programmes at university. Drawing on data from the TransMaths project, we present two cases—Stacey and Elton—who are both enrolled on the same ‘Mathematics for Physics’ course at university. We show that although both discuss access to science capital in narrating their choice of degree, they do so in different ways and this invariably interplays with different forms of identification with ‘Mathematics for Physics’. We conclude that there is a need to re-conceptualize science capital so that the dialectic relationship between its exchange and use value is theorized more fully. Whilst some students may access science capital as a means to accumulate capital (e.g. qualifications) for its own sake (exchange value), others appear to recognize the ‘use value’ of science learning and knowledge and this produces different forms of engagement with science (and mathematics). We therefore argue that authoring oneself in the name of a STEM identity is crucial in mediating how one perceives science capital. Finally, we argue that mathematics should be a central part of this framework since it significantly contributes to the exchange value of science as a form of capital (especially Physics), but it also offers use value in scientific labour (e.g. in modelling scientific problems).

Description: This article is based on data from two major research projects that investigated students involved in mathematically demanding courses during their transition through college and into university. It explores the nature of modelling as a mathematical practice in this important transition phase for students. Theoretical considerations are informed by illustrative accounts of a college mathematical modelling lesson and engineering lecture exemplifying the complex nature of mathematical modelling in these two phases of post-compulsory education. This raises important issues concerning the teaching and learning of mathematical practices in relation to modelling and applications. The discussion presented here is seen through the lens of Cultural Historical Activity Theory that informed the project team’s analysis of the case studies developed of both institutions and individuals. In this article, data and earlier findings are reinterpreted to better understand how we might support students as they move from learning mathematics to learning to use mathematics effectively in pursuit of their other studies. The accounts of classroom and lecture activity illustrate how ‘doing mathematics’ is mediated in different ways ensuring that students experience modelling and applications as mathematical practices very differently in each. This leads me to explain why, but also infer that students are likely to experience difficulties in transition both ‘vertically’ in progression from one activity system to another over time (college to higher education) and ‘horizontally’ between activity systems in which they participate concurrently (maths and engineering classes in university).

Description: Drawing on large-scale survey data and interviews with students during their first year at university, and case studies in their institutions, we explore the problems faced by students taking mathematically demanding courses, e.g. physics and engineering. These students are often taught mathematics as a service subject by lecturers of mathematics. Analysis of students’ perceptions of transition suggests that ‘the lecture’ in Higher Education continues to pose problems. Thematic analysis of interview data shows that these problems relate to the way lectures involve ‘time pressure’ and ‘lack of dialogue/interaction’ which are practices that we associate with transmissionist pedagogy generally and can also create negative dispositions. A case study of one mathematics course for engineering that we argue made a difference is presented, and conclusions drawn for developing practice which are especially pertinent with the introduction of the Teaching Excellence Framework to monitor and assess teaching in universities.