University of Chester
Drawing on research that informs transformative teacher education, this paper will report on an ongoing study that develops mathematics teachers’ knowledge and practice collaboratively. This paper accounts the experiences of a group of Welsh secondary school educators participating in collaborative classroom enquiry designed to develop GCSE students’ understanding of linear and quadratic algebraic expressions. The paper identifies the potential to disturb and improve learning through the use of enactive and iconic representations of algebraic concepts, whilst identifying tensions that arise in the act of changing the context for learning in a secondary school classroom.
University of Bristol
Taking an enactivist perspective, when a teacher talks about what has happened in their classroom their speaking, learning and doing is a function of their structural coupling with the people and contexts of past interactions. With each interaction, they are changed and they change the world that emerges. Drawing on existing models for what can be inferred about teachers’ epistemological framing from what they say about their practice, particularly in response to video clips of the classroom, this report considers one stimulated recall interview with a teacher of mathematics. Their reflections provide a new interaction, building on, yet separate from, what they noticed in the original classroom event. In this context, I consider how the teacher frames accounts of the lesson, giving particular attention to accounts of what was noticed during the lesson and what this might indicate in terms of post-hoc noticing and future classroom observations.
Clelia Cascella, Maria Pampaka, Julian Williams
University of Manchester
Research on gender differences in mathematics has received a lot of attention but has failed to explain important patterns across cultures and geographies. Most of the comparative research on this topic has been carried out at national level. In this paper, Italian gender gap is explored at intra-national level to understand if the national differences appear also at local (i.e. province, regional, and macro-area) levels and variations within the country are measured. A multilevel model of mathematics achievement tests administered in Italy in 2017 with grade 10 students (n=38120) confirmed the importance of the regional levels in the hierarchical data structure and revealed a strong interaction between local clustering and gender.
University of Bradford
This paper presents findings from a study, exploring experiences of numerate graduates who self-identify as struggling, or having struggled, with mathematics. Eight graduates participated, all of whom were working as experienced professionals in fields that require evidence of mathematics or numeracy skills. Creative qualitative methods were used. There were two in-depth interviews with each participant, focused around a personal timeline. Subsequently, Interview Story-Diagrams were created and shared with the participants for verification. Finally, thematic analysis was undertaken, and two global thematic networks were developed entitled ‘Purpose’ and ‘Identity’. Common experiences among the graduates were noted and consequently, with reference to literature, it is argued that dis-empowering and inequitable tendencies within mathematics education are ‘sticky’ in nature and seem to be replicated, or preserved, over many years and in many contexts. The role of graduates as stakeholders in mathematics education is highlighted and recommendations for practice and further research are made.
University of Reading
Programmes of teacher training almost universally follow a model of the allocation of an experienced teacher as a mentor in a school placement. This arrangement is intended to help the trainee to understand her experiences, develop her in the skills of teaching, integrate her with the school and ensure the expectations of stakeholders such as the school leadership, government, students and parents, are met. These are complex and sometimes conflicting requirements, meaning the role of mentor is demanding and difficult. The literature commonly refers to the mentor-trainee relationship as being particularly problematic. This study reports on an attempt to identify the most important elements of mentoring by surveying PGCE secondary maths teacher trainees. The results indicate remarkable consistency of the importance of all mentor practices despite highly varied trainee background, knowledge and experience, with no strong skew regarding relationships. The implications of this consistency are considered.
Dr Helen Drury1 and Laurie Jacques2
1Mathematics Mastery, 2University of London
This piece of work was commissioned by Mathematics Mastery – a UK-based non-profit organisation using and advocating a pedagogical approach that seeks to enable all pupils to experience deep mathematical learning. This literature review contributes to the knowledge about deep mathematical learning. The review explores three mathematics education papers that contain ‘deep’ in their titles to consider the authors’ uses of ‘deep’ in the context of mathematical learning and any analytical or theoretical frameworks upon which they drew. We reveal that ‘deep’ was not always clearly defined in the research. Rather, the authors’ interpretations were left implicit in the terminology they used. A word frequency analysis revealed common words used across the articles. These data were used to offer a ‘lexicon for deep mathematical learning’ to assist teachers to describe the quality of pupils’ mathematical understanding.
King’s College London
Teachers in the UK are required to teach a diverse range of students, with increasing inclusion of those with Special Educational Needs and Disabilities (SEN/D) in mainstream classrooms. This exploratory research investigates the initial conceptions of a Secondary Mathematics PGCE cohort regarding inclusive education of students with SEN/D, and the ways in which these developed over the course of the academic year. Many participants were willing to share their views and discuss their experiences. The preliminary findings shared here indicate a variety of initial conceptions and knowledge of SEN/D and attitudes to inclusion.
UCL Institute of Education, London
The 2014 English national curriculum for mathematics suggests that students from 14 to 16 should be taught one of two freshly-aspirational curricula, at ‘Foundation’ or ‘Higher’ tier. Related student learning is assessed by two overlapping sets of papers. I draw on two longitudinal studies that included classroom observations and teacher and student interviews involving >500 GCSE students and >60 GCSE teachers, together with teacher and student surveys, in a representative sample of about 40 schools. Data suggest enactment of the curriculum as intended is unusual. Instead, and for a variety of reasons, teachers at both tiers commonly adopt approaches that include teaching a range of content superficially. For students with average prior attainment, this includes coaching for routine questions at Higher tier rather than supporting a broad and deep experience of the Foundation curriculum. I consider apparent and reported implications.
09 The influence of context and numerical complexity on the tendency to focus on scalar relations when solving missing-value ratio items. A study involving lower secondary school pupils and PGCE mathematics students.
In this paper I discuss responses to three pairs of missing-value ratio items presented in the form of parallel one-page written tests. The items varied in numerical complexity (involving ratios that ranged from 1:3 and 1:4 to 2:9 and 3:10). They also varied in the contexts used (recipe, geometric enlargement and currency exchange). Within each pair of items, the given numbers were the same but ‘flipped’, so what was a scalar relation of 1:3, say, in one version of the item, became a functional relation of 1:3 in the other version. I use these notions of scalar and functional, derived from Vergnaud, to classify responses, and to examine the influence of context and numerical complexity on the frequency of such responses and on overall item facility. From a teaching perspective, the findings suggest that it is not helpful to think of ratio concepts as forming a simple hierarchy.
UCL Institute of Education
Teaching is both a cognitive and emotional undertaking, but we know less about how emotions are used by teachers in the classroom. This paper draws from observations and interviews with mathematics teachers in the UK. I attend to what is meant by teacher modelling and provide the context of the wider study. Using two examples from experienced teachers’ practices and a classification system, this paper considers one form of modelling, which I call ‘stepping back’. I explore some affective associations of this form of modelling, including that modelling is affectively driven. Some possible next steps for exploring modelling are suggested.
Ka Hei Lei and Maria Pampaka
University of Manchester
In this report, we draw on data from a study of UK secondary school students with a focus on mathematics self-efficacy (MSE) and its relation with other variables. We are particularly interested in parental support and perceptions of teaching practice, and to further explore if these relationships are affected when considering student characteristics, such as gender and year group. Our methodological approach includes a validation stage with the use of the Rasch model and a modelling stage with linear regression. Some indicative results with the sample of 13643 students are discussed in relation to their implications for practice and also in the context of an upcoming (comparative) study in Macau.
Salvador Llinares1 and Laurinda Brown2
1University of Alicante, Spain, 2University of Bristol, UK
Laurinda and Salvador both work in Mathematics Teacher Education (MTE) in different institutional contexts and with different perspectives. We are working, together with colleagues in our universities, dialoguing about how the different aspects of enactivism and professional noticing approaches can help us learn as mathematics teacher educators about prospective teachers’ learning. We consider how activities in mathematics teacher education shape the way in which prospective teachers reason about teaching situations.
Natalia Molina, Anselm R. Strohmaier and Kristina M. Reiss
Technical University of Munich
Students’ solution process of mathematical word problems depends on the situational context, e.g. the school subject (Dewolf, Van Dooren & Verschaffel, 2011). We analysed approaches to “age of the captain problems” (ACP; Verschaffel, Greer & De Corte, 2000) that present a situation that makes no sense, but are nonetheless frequently “solved” by a majority of primary school students. 48 primary school students (age M = 9.4, 54% female) in a mathematics or a language classroom were given five ACP. Afterwards, classroom interviews were conducted in both groups. Quantitative analyses show a non-significant tendency that students in the mathematics class were more likely to provide an arithmetic response to ACP. Interviews revealed that students in both groups experienced a cognitive dissonance regarding the expectation to provide an arithmetic solution, but differed in their approach to resolve it. This suggests that sense-making in ACP is influenced by the classroom context.
Dominic Oakes1, Andrew Davies2, Marie Joubert1, Sofya Lyakhova1
1Swansea University, 2Eirias High School,
Many teachers of mathematics claim that they would like to teach in ways that promote understanding but that, because the curriculum is so crowded, they do not have time. One approach to freeing up time is known as the ‘Flipped Classroom’ approach (FCA), in which students learn new content at home, using resources such as videos and written explanations. This allows teachers to use time in the classroom to deepen understanding. This research is set in North Wales, and involves four teachers with A-level and Further Maths classes. Teachers inducted students into the ‘Flipped Classroom’ approach and followed this for their mathematics lessons for eight weeks. We investigated teachers’ experience of: pre-lesson resource production; effective pedagogic approaches in the lesson; differences from non-FCA pedagogic approach, materials etc.; what they might do differently another time; students’ responses. We also investigated students’ experiences of the flipped classroom approach.
Ahmet Şükrü Özdemir1 , Eyüp Sevimli2 , Emin Aydın1 , Gökhan Derin1
1Marmara University, 2Tokat Gaziosmanpaşa University
The aim of this paper, is to reveal pre-service mathematics teachers’ views on STEM education and coding activities. Research was conducted with 28 pre-service mathematics teachers studying at a mathematics education department in Turkey. There were two sources for data: the lesson plans analysed using the content analysis technique and the opinions of the participants which were collected through the questionnaire. The findings, were obtained through the views of the participants indicated that coding activities supported the students in terms of algorithmic thinking (n = 19) and that they provided the opportunity to learn mathematics with gaming (n = 14). In addition, participants stated that the workshops were needed to develop knowledge and awareness of technological-pedagogical content knowledge in order to increase the effectiveness of coding activities.
Ahmet Şükrü Özdemir and Volkan Yalçın
Marmara University, Turkey
The aim of this paper is to explore competencies of mathematics teachers who prepare students to mathematics olympiads. Competencies related to mathematics teachers and gifted students’ teachers were investigated in literature. Experts’ opinions were taken to determine related competencies. Competencies determined by experts were classified into three sub-competency groups as “Professional Knowledge”, “Professional Skill” and “Attitudes and Values” by using the Delphi method. After determination and classification of competencies, the order of importance of these competencies was calculated by using Analytic Hierarchy Process (AHP) which is a multiple criterion decision making method. Pairwise comparison of all competencies in the same groups were made and competencies were ordered according to their order of importance. The most important competencies are “he/she uses proof techniques such as induction, contrapositive and direct proof in order to demonstrate mathematical propositions”, “he/she constructs classroom environment which promotes mathematical thinking and reasoning” and “he/she renews himself/herself perpetually”.
Sheldon Phillips and Amanda McCrory
University College London, Institute of Education
A core component of mathematical knowledge acquisition is linked to the use of mathematical talk and language (Goos, 2004). The purpose of this observational study is to describe the type of talk (Mercer, 2004) that children use to solve, reason and explain mathematical ideas whilst using manipulatives in mathematics. The setting is an inner city 2 form entry primary school with students engaged in their normal classroom behaviour. Two types of qualitative data are collected for this study: video recording of the mathematics activity within the classroom and audio recordings of the follow up semi-structured stimulated-recall interviews. Disputational, cumulative and exploratory talk types are present to varying amounts when students used manipulatives. This study demonstrates that schools and teachers need to consistently create opportunities for students to demonstrate and explain their thinking and that scaffolding is required to ensure that mathematical reasoning occurs (Mason, 2000).
Caroline Rickard and Lorna Earle
University of Chichester
The learning of times tables (the collection of multiplication facts up to 12×12) is currently in the spotlight in the UK, with the planned introduction of a times tables check for all children in Y4 (children aged 8-9 years) from 2020. Whilst not disputing the benefit of having well-embedded known facts, we were keen to establish the extent to which children saw times tables as a connected body of knowledge as opposed to 144 isolated facts. This led to a small project undertaken in two primary schools where, after establishing the children’s existing understanding, we gave them the opportunity to explore and reason about multiplication facts over four teaching sessions. Here the implications of this research are shared.
İpek Saralar, Shaaron Ainsworth and Geoff Wake
University of Nottingham, UK
The challenge of teaching polycubical shapes has received considerable attention, especially when considering how to use technology to support students’ learning. Although there exists a plethora of tools to use in teaching the topic, relatively little is known about how to use these well. If students are to realize the benefits of such tools, it is imperative that lessons be specifically designed that best integrate what is known about the affordances offered to students by use of digital and traditional tools. Consequently, a 6-lesson course was designed and tested with an initial sample of 8 and then 30 students, aged 13-14. The findings showed that the lessons whose content was enriched with real-life videos, worked examples, concrete manipulatives, and dynamic geometry tools did enhance students’ learning of polycubical shapes and this has paved the way for wider adoption.