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: 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: 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: 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.

Description: In this editorial we explain the background to the research papers reported in this special issue, and to some extent how each paper relates to this body of work. In particular we outline the research projects and research teams that worked together between 2006 and 2014 on projects that related to the theme of transition, and we provide the knowledge base on which these papers build.

Description: This article builds on previous results of the Transmaths studies concerning transmissionist teaching practices—and especially adds the significance of students’ perceptions of these practices—in their association with students’ declining dispositions for studying mathematics. It addresses a gap in this work, and the literature in general, regarding the relationship between teachers’ and students’ perceptions of pedagogy. Drawing on data analyses from a recent, large survey of teaching and learning mathematics in secondary schools, the article: (a) demonstrates and validates two new measures of perceptions of transmissionist practices, as experienced from students’ and teachers’ perspectives, (b) investigates the comparability of these two measures, and (c) identifies their associations with students’ dispositions to study mathematics. Analysis draws on measures of students in Years 7 to 11 (involving 13,000+ students) and from 132 of their mathematics teachers, and shows low correlation at class level and negligible correlation at student level. Results of regression analysis confirm previous work with older students, i.e. that teachers’ self-reported transmissionism is negatively associated with learners’ dispositions, but adds that students’ perceptions of transmissionism are much more strongly negatively associated with these dispositions, and largely mediate the effect of teachers’ (self-reported) transmissionism. Further, the differences between year groups and gender show how girls and older learners suffer significantly larger negative effects. The article concludes with a brief discussion of these complexities and some implications for students’ trajectories and transitions into (and out of) mathematics.