Paola Iannone

University students’ perceptions of summative assessment: The role of context

Abstract We report on a mixed-method study that compared students’ perceptions of summative assessment across two distinct disciplines – education and mathematics, at two research-intensive institutions in the UK. The disciplines chosen represent opposing positions in Biglan’s classification of academic disciplines, as well as having very different assessment practices. Results suggest that these education students prefer to be assessed by methods they perceive to discriminate on the basis of academic abilities. Moreover, they perceive the traditional closed-book examination as inadequate to assess the capabilities which are key to being successful in their subject, which fits some but not all of the general findings in the literature. However, comparing these results with those of an identical study with mathematics students, we find that the perceptions of summative assessment are very different. We account for that difference by suggesting that students’ epistemic beliefs play a role in shaping these perceptions and conclude that, in designing summative assessment in higher education, generalised and centralised forces for change need to be tempered by contextual and disciplinary factors.

Housing mix, school mix: barriers to success

Recent UK policy has emphasised both the development of socially mixed communities and the creation of balanced school intakes. In this paper, we use a case study of an area of mixed tenure in eastern England to explore policy in practice and the extent to which mechanisms of segregation impact on both the creation of socially mixed neighbourhoods and socially mixed schools. We draw on parent and pupil views of schools, combined with local authority data and geo-demographic classification, to explore the background to school destinations. Overall, we find different expectations underlying the two strands of policy and show how a more detailed understanding of context helps those supporting or judging schools.

“If you can count to ten you can count to infinity really”: Fostering conceptual mathematical thinking in the first year of primary school

In this paper we investigate how teachers can foster conceptual mathematical thinking in five- and six-year-old pupils in a classroom situation. In this context we define conceptual mathematical thinking as instances where pupils verbalise mathematical thinking linked to abstraction and generalisation. We focus on one case study to explore the impact of teachers’ epistemology of mathematics in introducing two sociomathematical norms in the classroom: what counts as a mathematical answer and how such an answer is incorporated in the mathematical discourse of the teacher and pupils in the classroom. We then link the introduction of these sociomathematical norms to the presence of instances of conceptual mathematical thinking as observed in the classroom. We conclude that pupils are more consistently engaged in conceptual mathematical thinking in the classrooms of teachers who view mathematics as a web of interconnected ideas, and perceive it as being about general structure and patterns.

Students’ views of oral performance assessment in mathematics: straddling the ‘assessment of’ and ‘assessment for’ learning divide

This paper explores the views of a group of students who took an oral performance assessment in a first-year mathematics module. Such assessments are unusual for most subjects in the UK, but particularly within the generally homogenous assessment diet of undergraduate mathematics. The evidence presented here resonates with some, but not all, of the existing literature on oral assessment and suggests that, despite concerns about anxiety and fairness, students see oral assessments as encouraging a focus on understanding, being relatively authentic and reactive to their needs. We argue that, suitably implemented, oral assessment may be a viable assessment method for straddling the ‘assessment for’ and ‘assessment of’ learning divide in higher education.

Educators and the Teacher Training Context

There is increasing recognition in the United States of the need for mathematics departments and mathematicians to become involved in training mathematics teachers for primary and secondary schools. Certainly, issues that are valued by both mathematicians and mathematics educators could promote collaboration. For example, the Mathematical Sciences Research Institute, in Berkeley, California, has initiated a series of workshops entitled “Critical Issues in Mathematics Education,” which aims “to provide opportunities for mathematicians to cooperate with experts from other communities on the improvement of mathematics teaching and learning” (Thames, 2006, p. iii; see also, for example, Conference Board on Mathematical Sciences, 2001; McCallum, 2003). However, this area of the 15th study conference received no papers. In this article, we discuss ideas that might be of interest to both mathematicians and mathematics educators, and we give examples of some projects that could represent ways to begin collaborations. The ideas that we discuss include the concept of the Knowledge Quartet (Rowland et al. 2005a, 2005b), the notion of a mathematical habit of mind, and the comparison of ways in which mathematicians present fractions to future teachers (McCrory, 2006). Finally, we look at preliminary results from a small international survey to consider whether and how mathematicians and mathematics educators might collaborate on content and pedagogy courses for intending teachers. It is important to find common grounds for discussion between mathematicians and mathematics educators on topics valued by both groups: when the communication between the two groups goes astray, difficulties such as the socalled “Math Wars” in the United States (Ralston, 2003) can result in the waste of much valuable energy.

CERME7 Working Group 14: University mathematics education

The WG14 papers presented at CERME7 provide ample evidence of growth in this area of research. The nutshell descriptions of WG14 papers that follow are structured around the main themes of the Group’s Call for Papers (such as the teaching and learning of particular topics, pedagogical and curricular issues at university level, the transition from school to university mathematics) and the observation that, beyond staple references to classic constructs from the AMT era, several works employ approaches such as the Anthropological Theory of Didactics (ATD: Chevallard 1985), and discursive approaches (e.g. Sfard 2008). Xhonneux and Henry employed the ATD framework to distinguish between mathematical and didactic praxeologies in the context of teaching and learning of Lagrange’s Theorem in calculus courses to mathematics and economics students. Gyöngyösi, Solovej and Winslow was another: in it a part of a transitional course in Analysis was taught with a combination of Maple and paper-based techniques, resulting in mixed reception and performance by students. A third was Barquero, Bosch and Gascón: from its analyses ‘applicationism’ emerges as the prevailing epistemology of mathematics in science departments, potentially hindering the teaching of mathematical modelling to science students. Several papers employed a discursive approach. Jaworski and Matthews used this to trace university mathematicians’ pedagogical discourse, and suggested links to their ontological and epistemological perspectives. Biza and Giraldo described how computational inscriptions have potentialities and limitations that can be helpful in students’ exploration of newly introduced mathematical concepts. Three papers made use of Sfard’s commognitive framework. Viirman employed this framework to trace the variation in the pedagogical discourses of mathematics lecturers in the course of their introducing the concept of function. Stadler described students’ experiences of the transition from school to university mathematics as an often perplexing re-visiting of content and ways of working that seem simultaneously familiar and novel (for example in the case of solving equations). Nardi outlined interviewed mathematicians’ perspectives on their newly arriving students’

Students' affective responses to the inability to visualise cosets

In this report we investigate the relationship between visualisation and the development of Year 2, undergraduate students’ affective responses during their first encounter with abstract algebra. We also consider the impact that the reinforcement of such affective responses has on student engagement, focusing on the concept of coset. For the purposes of this study, we predominantly use one of the eight sources of data (Ioannou & Nardi 2009), which was collected during a compulsory, 10 week, abstract algebra module, taught in a university mathematics department in the UK. The module consisted of 20 lectures and three seminars. 78 students attended, and 13 of these participated voluntarily in the three one-to-one interviews in weeks 2, 6 and 10. In the interviews, they were asked about their experience with this module and given some tasks related to the module content. We use Goldin’s (2000) theory of affect as generalised by Weber (2008). Students may follow two possible affective pathways (Weber 2008): favourable (encouragement, pleasure, elation and satisfaction) and unfavourable (frustration, anxiety and fear/ despair). According to Weber (2008), these affective pathways may be selfstrengthening if their duration is long. A repeated emotional experience may cause stable attitudes and beliefs that may be related to particular cognitive responses. Interviews were fully transcribed, and analysed in the spirit of data-grounded theory (Strauss and Corbin 1998). Amongst the themes emerging from the interviews were students’ difficulties in visualising abstract concepts, often paired with their diminishing engagement as the module was progressing. Considering cosets, one of the tasks given in the second interview was to produce their own image of a coset. Only two of the 13 students interviewed produced an image, and only one of these was correct. One aspect of the relationship that emerges from the interview data relates to students’ expressed need for visualisation, and their inability to use any visual imagery offered by the lecturer (e.g. image of group, coset, equivalence classes and homomorphism). The lecturer’s coset image is shown in Figure 1. Additionally, the majority of students interviewed expressed their need to have mental images of the concepts studied. Surprisingly, when students were asked to produce simple proofs involving cosets, they were concerned about their lack of visual imagery, and the anxiety and despair this task had caused was apparent in

Special issue on summative assessment

This is an Accepted Manuscript of an article published by Taylor & Francis in Research in Mathematics Education on 01 Aug 2017, available online: https://doi.org/10.1080/14794802.2017.1334578

Students’ experiences of teaching at secondary school and university: sharing responsibility for classroom engagement

Abstract Recently much research has focused on student engagement, both at school and at university. This attention is motivated by the role that engagement plays in student learning and in the student experience. Acknowledging that student engagement is a multifaceted construct we focus on the contribution that teaching and teacher traits make to the quality of student engagement, from the student’s perspective. In this small scale study, we adopt a qualitative methodology to investigate students’ perceptions of what factors impact on their engagement and what role the students themselves have in fostering such engagement. Focus group and one-to-one interviews with students in the last year of school and at university were analysed to reveal four overarching themes related to engagement in classroom life including the importance of active listening on the part of both students and teachers. The resulting Refined Quality Teaching Initiatives Framework outlines how dual engagement and active listening can be viable pedagogical strategies both at school and university. The framework also brings to the fore the active role and responsibilities that students have, in their own perceptions, for engagement in the classroom. We conclude with a reflection on the implications of our findings for teaching and teacher training.

Mathematics & mathematics education: searching for common ground

This is an Accepted Manuscript of an article published by Taylor & Francis in Research in Mathematics Education on 22nd Aug 2014, available online: http://dx.doi.org/10.1080/14794802.2014.937354