# BSRLM Proceedings: Vol 35 No 1 at St Patrick’s College, Dublin, Feb 2015

**Proceedings of the Day Conference held at St Patrick’s College, Dublin on 28 Feb 2015**

## Contents

#### 01 A good foundation for early years mathematics education in England?

Sue Gifford

University of Roehampton

The most recent number curricula for three to seven year olds in England have raised expectations which are not supported by research. A focus on number sense would provide a better foundation, especially if supported by clear exemplification of progression, and this could be appropriately integrated with creative processes identified in the curricula.

#### 02 Developing mathematics learning and teaching in the context of curriculum renewal

Gerry Shiel^{a}, Thérèse Dooley^{b}, and Elizabeth Dunphy^{b}

^{a}Educational Research Centre, Dublin; ^{b}St Patrick’s College, Dublin

The National Council for Curriculum and Assessment (NCCA) in Ireland recently published two research reports to support the review and redevelopment of the primary school mathematics curriculum for 3 – 8 year olds. The first report (Dunphy, Dooley, Shiel, et al., 2014) focuses on theoretical aspects underpinning mathematics education for young children. The second report (Dooley, Dunphy, Shiel, et al., 2014) is concerned with pedagogical implications. In this paper the authors of the reports synopsize the background to this renewal and the key themes of an emerging curriculum model.

#### 03 Stimulating productive mathematical noticing: developing a framework for exploring the affordances of task and talk

Nancy Barclay

University of Brighton, UK

This small scale action research doctoral study aims to deepen understanding of how children can be supported to notice and use mathematically relevant ideas in the course of their class-based mathematical activity. The focus is on supporting and encouraging ways to look rather than dictating what to see. I employ the theoretical lens of ecological psychology to develop a framework to analyse the affordances of children’s tasks and of classroom dialogue in stimulating mathematical noticing. In this study, noticing is positioned as a particular type of mathematical engagement; this paper focuses on the development and early use of an analytic tool.

#### 04 Transition through mathematical tasks

Sinead Breen^{a} and Ann O’Shea^{b}

^{a}CASTeL, St Patrick’s College Drumcondra; ^{b}Maynooth University

The transition to university level mathematics is often problematic for students. Clark & Lovric (2008) have written about some of the differences between mathematics at school and at university, including the type of mathematics taught and the way mathematics is taught. Students at this stage also have to contend with social and cultural changes. As part of a project on task design, ten first year students at two different universities in Ireland were interviewed. In this paper, we will discuss their experiences of mathematics at school and university. In particular, we will consider the differences in the types of mathematical tasks encountered at both levels and the students’ views of the influences of such tasks.

#### 05 The challenge of collecting useful qualitative data on students’ visits to a Mathematics Support Centre at a university in Ireland

Nuala Curley and Maria Meehan

University College Dublin

Since September 2009, the Mathematics Support Centre (MSC) in University College Dublin (UCD) has kept an electronic record of each student visit to the Centre. By September 2013 there had been 21,200 visits and an analysis of the qualitative data, specifically the tutors’ comments on students’ difficulties, was planned to identify the mathematical topics and concepts that were causing persistent difficulties. However we found the nature of the data collected lacked the detail to allow the analysis to take place. We realized that in order to identify the mathematical topics with which students experience difficulty, firstly we needed to identify the nature of the data we required and to do this rigorously and secondly, to work with the tutors to find ways in which they could identify this data and record it efficiently. We describe our efforts, and those of the tutors, over the last eighteen months to collect this data. In September 2014, we commenced our data recording proper. This involved eight weeks of intensive collaborative work with the tutors to ensure the quality and authenticity of the data collected. During this period there were 2,012 visits to the MSC. We also present a preliminary analysis of the most prevalent mathematical topics that are emerging from this eight-week data collection.

#### 06 Using Modified Lesson Study with Mathematics Post-Graduate Teaching Assistants

Jessica M. Deshler

West Virginia University, United States

This paper describes an effort on the part of seven post-graduate instructors, or teaching assistants (TAs), to work together to develop, implement and critique two lessons for a Calculus I course. The TAs were asked to develop lessons for a fifty minute Calculus I class on topics of their choice. As a group, the TAs decided to focus their lessons on procedural topics instead of conceptual ones. One TA was chosen at random to teach the group-developed lessons during a calculus class while the other TAs observed. Upon teaching and observing each lesson, TAs were asked to provide a short reflective statement about what they saw during the lesson and about the overall experience of planning the lesson and either implementing or observing it. Their written responses were analyzed using an open and axial coding method (Corbin & Strauss, 2008). A preliminary analysis reveals the reflections of the TAs be focused superficially, supporting previous work on K12 pre-service teacher reflective abilities. TAs can be considered pre-service faculty with the same low level of reflective abilities.

#### 07 Motivating young people to seek careers in STEM: Research conclusions from interviews and observations in Ireland and the U.K.

Jim Freemyer^{a}, Patrick Johnson^{b} and Olivia Fitzmaurice^{b}

^{a}Indiana Wesleyan University; ^{b}University of Limerick

The purpose of this research was to determine if Irish and U.K. mathematics teachers would support or deny what Indiana mathematics teachers would say are the most important skills necessary to successfully engage second-level students in maths. These European teachers would assess effective teaching strategies initially identified by some of Indiana’s most effective maths teachers, providing a multi-national perspective on effective teaching practices. It is hoped that this research would allow Irish, U.K. and U.S. maths teachers to learn from each other. Principals in these countries could then use the results to help all mathematics teachers improve and excel.

#### 08 How good at mathematics do students need to be on entry to primary school initial teacher education?

Lorraine Harbison and Joseph Harbison

Church of Ireland College of Education, Dublin & Trinity College, Dublin.

There is a momentum in Ireland towards recruiting students who are competent in mathematics into primary school teaching. It is hoped that by doing so, standards of teaching, learning and assessment will improve. The initiative instigated was to revisit current minimum entry requirements for mathematics with a view to increasing threshold levels. This paper attempts to ascertain if there is a correlation between attainment at Leaving Certificate (LC) level, a state run examination taken by students aged approximately 18, and competency in primary school mathematics. It goes further to determine if it is possible to establish a minimum threshold level. Finally it looks at the potential effects on enrolment into initial teacher education (ITE) should entry grades be increased. 95 first year ITE students completed a standardised attainment test that is typically taken by children in their final year in primary school. The results were compared with their LC mathematics grades. There was a moderate correlation between the two scores. A possible revised threshold level would have excluded 40% of students who are competent in mathematics from entering into ITE. Therefore, mathematics grades at LC appear to be an unsuitable measure for establishing a threshold entry requirement for ITE.

#### 09 Investigating expected progress in mathematics in an English secondary school

Edmund Lowe^{a} and Sue Pope^{b}

^{a}University of Manchester, ^{b}Manchester Metropolitan University

England is notorious for its high stakes performativity culture and a school inspection regime that strikes fear and dread into the heart of many teachers and school leaders. A key driver in any school inspection is academic outcomes (exam results) and, more recently, progress. Children are tested at the end of primary schooling (age 11) and awarded a level, level 4 is the national expectation. Secondary schools are expected to secure three levels of progress for all learners by the end of the subsequent five years of schooling. Children who achieve level 4 at the end of primary are expected to progress to achieve GCSE grade C which has been matched to level 7. Drawing on data for two cohorts of students from a school in a relatively deprived area of the country, this research found that three levels of progress was unlikely to be achieved by students who failed to meet the national expectations at the end of primary, and was probably insufficiently challenging for students who had exceeded national expectations at the end of primary. Only prior attainment, eligibility for free school meals and being on the school’s SEN register were found to produce statistically significant progress outcomes.

#### 10 Developing Mathematical Knowledge for Teaching (MKT) for pre-service teachers: a study of students’ developing thinking in relation to the teaching of mathematics

Brien Nolan^{a}, Majella Dempsey^{b}, James Lovatt^{a} and Ann O’Shea^{b}

^{a}CASTeL, Dublin City University; ^{b}Maynooth University

The concept of Mathematical Knowledge for Teaching (MKT) was introduced by Ball and colleagues (Ball, Thames & Phelps, 2008), building on Shulman’s (1986) notion of Pedagogical Content Knowledge. MKT is ‘the mathematical knowledge needed to carry out the work of teaching mathematics’. In this project, a team of researchers at two Irish universities studied the development of MKT in two groups of pre-service teachers. The project aimed to help students develop their own MKT, and to develop a richer conception of the role of mathematics content knowledge in teaching, through a series of workshops designed and delivered by the authors. The students’ awareness and level of MKT was investigated using pre- and post-intervention questionnaires. We describe the intervention and present the findings from the analysis of the data collected. In particular, we describe how the group’s view of the mathematical work of a teacher changed over the course of the project.

#### 11 Teaching postgraduate research students in mathematics about teaching mathematics to undergraduates: education or training?

Melissa Rodd

UCL Institute of Education

Postgraduate research students in university mathematics departments often teach undergraduates mathematics while they pursue their own research. This paper reports on a mathematics-specific 10 learning hour introduction to teaching for post-graduate mathematics research students. There are two strands to the paper: (1) presentation of the course design, delivery and data; (2) a discussion of the claim that such a course could not ‘train’ them but that it did offer them opportunities for ‘education’.

#### 12 How can we get more (good) teachers of mathematics – in our primary schools, secondary schools and F.E. colleges?

Naomi Sani

Plymouth University

One way to get more teachers of mathematics in schools and colleges is to ‘re-train’ the teachers we already have. Re-training teachers, of other subjects and from other phases, to teach mathematics has been happening in the UK for some years. Originally known as Post Initial Teacher Training (Post-ITT) Subject Knowledge Enhancement (SKE), the programme has recently been renamed Teacher Subject Specialism Training (TSST). The UK government has pledged £67 million for new programmes to train up to 17,500 teachers of mathematics and physics over the next Parliament. With the reformed GCSE and the expectation that most post-16 students will engage with some mathematics – retaking GCSE, studying for a Core Maths qualification as well as A and AS levels, many more mathematics teachers will certainly be needed. How viable is TSST? My research centres on case studies from the 2014 cohort of students with a view to tracking them for four years.The Post-ITT SKE course combined 100 hours each of face-to-face tuition and e-learning provision. Participants are from primary, secondary and F.E. I focus on one case study: a science teacher who is now Head of Mathematics; she claims the course has changed her life.

#### 13 Raising girls’ participation in A-level mathematics: initial findings from ‘good practice’ case studies.

Cathy Smith and Jennie Golding

UCL Institute of Education

Fewer girls than boys in England participate in post-compulsory mathematics and the recent increase in popularity of Mathematics and Further Mathematics (FM) at age 16 has not changed the gender balance. Previous studies have shown the significance to girls of their mathematics lessons and teachers, of discursive co-constructions of masculinity and mathematics, of the range of careers associated with mathematics and science, and family ‘science capital’. This study identified four case-study schools and one Further Education (FE) college with unusually high participation by girls in mathematics A-level. Focus groups and lesson observations were used to explore factors relevant to girls’ participation. Common factors were: preparation for demanding mathematics during key stage 4, a departmental ethos which encouraged student-teacher interactions in and out of lessons, teachers who explicitly and repeatedly confirmed that girls would succeed at mathematics A-level, appreciation of mathematics as opening doors to many careers. Messages about FM were more restrictive but emphasised interest over unusual ability.

#### 14 Ratio and proportional thinking: a study in an Irish context

Patsy Stafford, Elizabeth Oldham and Valerie O’Dowd

Maynooth University, Trinity College Dublin and Marino Institute of Education, Dublin

The concepts of ratio and proportional thinking can be problematic for students, teachers and also prospective teachers. As part of a project examining the meanings ascribed to ratio and the representations offered for it especially by the latter group, data were collected from prospective primary teachers in Irish teacher education institutions. In this paper, graduates taking an eighteen-month teacher education course are considered. A curriculum and textbook analysis was undertaken to examine the treatment of ratio-related concepts for primary classes. Findings show that curricular and textbook support for teaching ratio are too sparse to support the prospective teachers’ rather patchy knowledge adequately; further work in initial education is indicated.

#### 15 Revisiting students’ perceptions of feedback on one Mathematics Enhancement Course

Jayne Stansfield

University of Bristol and Bath Spa University

A Mathematics Subject Knowledge Enhancement Course (MEC or SKE) is designed for non-mathematics specialists to bring their subject knowledge to the required level prior to commencing training as a mathematics teacher. Having previously found, with one cohort, that over the length of the course students’ reliance on teachers and correct answers decreased whilst reliance on feedback increased, I was left with unanswered questions as to what it means to rely on either correct answers or feedback. I am now reporting on my research with another cohort, which was undertaken to discover if the same patterns occurred and the same questions raised. This second cohort also appeared to move away from correct answers. The outstanding questions are whether correct answers and feedback are viewed by the students as external influences on their decisions or whether they are taking responsibility for next steps themselves.