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This article reports on the design and findings of the first iteration of a classroom-based design research project which endeavours to design a professional development intervention for teachers' mathematical problem-solving pedagogy. The major outcome of this study is the generation of design principles that can be used by other researchers developing a professional development (PD) intervention for mathematical problem-solving pedagogy. This study contributes to the mathematical problem-solving pedagogy and PD body of knowledge by working with teachers in an under-researched environment (an informal settlement in Gauteng, South Africa). In this iteration, two experienced Grade 9 mathematics teachers and their learners at a public secondary school in Gauteng, South Africa, participated in a 6-month intervention. Findings from the data are discussed in light of their implications for the next cycle and other PD studies.

Writing assessment is a key feature of most education systems, yet there are limitations with traditional methods of assessing writing involving rubrics. In contrast, comparative judgement appears to overcome the reliability issues that beset the assessment of performance assessment tasks. The approach presented here extends previous work on comparative judgement by directly involving teachers in a large number of schools in the judging of young pupils' writing. To ensure quality control the process incorporated a process of 'anchoring' that ensured that teachers could not artificially inflate their own pupils' scores. The approach was used to assess the writing of 55,599 primary pupils in England in 2017-2018. Overall, the results showed that a comparative judgement approach to writing incorporating anchoring shows promise in providing a fair and robust large-scale method to assess writing.

The South African curriculum emphasizes the teaching of problem solving in mathematics. However, little is known about South African teachers' views on the teaching of mathematical problem solving (MPS). The purpose of this study was to establish Grade 9 South African teachers' views, teaching strategies and the support required in their teaching of MPS. We worked with teachers who teach MPS to large, under-resourced and multilingual classes. Data for the study were collected using an open-ended questionnaire and classroom observations. The questionnaire sample consisted of 31 teachers from 20 schools in a district in Gauteng, South Africa. Data was analysed through grounded theory techniques and the findings indicate that overcrowded classes and learners' problems with English as the language of learning and teaching compel teachers to retain strict ownership of the MPS process. The findings also uncovered that teachers' adoption of the MPS as a step-by-step guided sense-making strategy is a response to learners' perceived lack of mastery of basic algorithms. All participants in the study stated that they required support on the teaching of MPS that included practical demonstration of MPS pedagogy in the context of their classes where discipline is a challenge and learners are under-prepared, lack basic MPS skills and struggle with the language of learning and teaching.

This study describes a comparison of how worked examples in selected textbooks from England and Shanghai presented possible learning trajectories towards understanding linear function. Six selected English textbooks and one Shanghai compulsory textbook were analysed with regards to the understanding required for pure mathematics knowledge in linear function. Understanding was defined as being at five levels: Dependent Relationship, Connecting Representations, Local Properties Noticing, Object Analysis and Inventising. These levels were developed by examining the most prominent theories from the existing literature on understanding function. Findings suggested that the English textbooks constrained the structural aspect of understanding linear function due to a point-to-point view of function, while the Shanghai textbook which focussed on a variable view of function overemphasised the algebraic approach. The discussion explored the drawbacks to each approach and what teachers or textbook writers could do to balance these two approaches in order to facilitate students' understanding towards a structural view of linear function.

This study investigates how students in England and Shanghai understand linear function. Understanding is defined theoretically in terms of five hierarchical levels: Dependent Relationship; Connecting Representations; Property Noticing; Object Analysis; and Inventising. A pilot study instrument presented a set of problems to both cohorts, showing that the English cohort generally operated at the levels of Property Noticing and Object Analysis, whereas the Shanghai cohort reached the higher level of Inventising. The main study explored understanding levels and students' errors within each cohort in detail, in order to gain insights into reasons for apparent differences. The instrument used in the main study included two overlapping items, which were the same for both cohorts, while others were pitched at levels of understanding revealed in the pilot. Analysis of students' solutions revealed that the English students' errors were manifested in a lack of basic skills including dealing with negative numbers, while the Shanghai students showed weaknesses in their ability to use graphs. The discussion highlights different views of understanding as a possible background reason for the contrasts observed. Errors and apparent difficulties suggest implications for teaching linear function in each context.

Background: It has been shown that mathematical representations can aid children's understanding of mathematical concepts but that children can sometimes have difficulty in interpreting them correctly. New advances in eye-tracking technology can help in this respect because it allows data to be gathered concerning children's focus of attention and so indicate on what aspects of the representations they are focussing. However, recent eye-tracking technology has not been used to any great degree in investigating the way children view and interpret mathematical representations. Purpose: This research explored the use of new advances in eye-tracking technology in investigating how young children view and interpret mathematical representations of multiplication. Sample: Nine Year 5 children (four boys, five girls, aged 9-10 years of age) from a local primary (elementary) school in the North-East of England were asked to complete the test during school time. The children represented a range of attainment levels across the mathematical domain (three higher-, three middle- and three lower-attaining children) and were selected accordingly by their class teacher. We recognise that this study was only based on a small sample of children, however, this number still allowed us to make meaningful comparisons in particular between the different types of representations presented. Design and methods: The study consisted of each child looking at 18 static slides, one after the other, with each slide presenting a symbolic and a picture representation of multiplication problems. The data that was captured by the eye tracker and recorded was then analysed quantitatively (e.g. time on each slide, time on each area of interest specified within the software) and qualitatively (video recordings of each child's gaze trajectory during each representation was carried out, thereby allowing a categorisation of the different approaches adopted). Results: The study showed that (a) the particular form of the number line representation used in this study was less successful than the other picture representations used (equal groups, array) in promoting multiplicative thinking in children, and (b) the success of children to think multiplicatively with the "groups" and the array representation was related to their general mathematics attainment levels. Conclusion: These findings have implications for teacher practice in that teachers need to be clear about the possible drawbacks of particular representations. Even in using more successful representations, for lower-attaining children, the progression in their understanding of the representation needs to be taken into account by the teacher. The study also highlighted that the eye-tracking technology does have some limitations but is useful in investigating young children's focus of attention whilst undertaking a mathematics assessment task.

Background: The research on diagrammatic representations highlights their importance for the teaching and learning of mathematics. However, the empirical evidence to support their use in the classroom is mixed and somewhat lacking. Purpose: The aim of this study was to develop the use of diagrammatic representations of mathematical concepts in primary classrooms, through introducing primary teachers to the research literature in this area, and researching the subsequent impact on children and teachers. A professional development programme was designed, involving three one-day training sessions for mathematics co-ordinators. They were asked to implement the ideas from the training back in their schools. Sample: Mathematics co-ordinators from eight primary schools attended the professional development programme. The study focussed on Year 3 pupils (aged seven- eight) and Year 5 pupils (aged nine-10). Design and methods: In this paper, we report the qualitative findings from the larger project looking at the overall impact of the professional development programme. The paper focuses on semi-structured interviews carried out with the mathematics co-ordinators attending the professional development sessions, and the Year 3 and Year 5 class teachers who subsequently worked with the co-ordinators on their use of diagrammatic representations in their teaching of mathematics. Lesson observations involving the class teachers were also carried out in order to explore further the possible impact of the project on classroom practice. Results: The qualitative results identified the impact of the project on mathematics co-ordinators and class teachers' knowledge and practice. However, the nature of this impact was complex, with a variety of facilitating and hindering factors identified for the transfer of the professional development ideas on the use of diagrammatic representations. In addition, different levels of sophistication of class teachers' use of diagrammatic representations were identified. Conclusions: Implications for the development of professional development programmes to facilitate the transfer of research into practice were identified. Recommendations for the use of diagrammatic representations are also put forward.

Background: Successfully selecting students for undergraduate courses and for teaching is notoriously difficult. Traditionally, in England, A level grades and interviews have been used in conjunction with a variety of other approaches. Purpose: This paper examines the predictive validity of students' A level grades and of scores from a structured interview on their subsequent degree classification as well as the ratings from their final teaching placements. Sample: A cohort of 71 students who started studying for a Bachelor of Arts (BA) degree in Primary Education in 2007 and graduated in 2010. Design and methods: The A level grades and the ratings from structured interviews, carried out prior to the start of the course, were linked to the final degree classification and to the teaching placement grades. Results: A-level scores correlated significantly with degree classification but not with the placement grade whereas the interview grades correlated significantly with both outcome measures. Conclusions: These results are discussed in the light of the restricted range available for A-level grades. (Contains 8 tables and 2 notes.)

We examine whether the array representation can support children's understanding and reasoning in multiplication. To begin, we define what we mean by understanding and reasoning. We adopt a "representational-reasoning" model of understanding, where understanding is seen as connections being made between mental representations of concepts, with reasoning linking together the different parts of the understanding. We examine in detail the implications of this model, drawing upon the wider literature on assessing understanding, multiple representations, self explanations and key developmental understandings. Having also established theoretically why the array representation might support children's understanding and reasoning, we describe the results of a study which looked at children using the array for multiplication calculations. Children worked in pairs on laptop computers, using Flash Macromedia programs with the array representation to carry out multiplication calculations. In using this approach, we were able to record all the actions carried out by children on the computer, using a recording program called Camtasia. The analysis of the obtained audiovisual data identified ways in which the array representation helped children, and also problems that children had with using the array. Based on these results, implications for using the array in the classroom are considered.

This study, carried out in England, examined the variation of attitudes towards science over the first three years of secondary schooling and with gender. The study in question was part of an evaluation of the "Lab in a Lorry" project, and involved 932 pupils completing a pre-measure questionnaire containing items looking at six separate attitude constructs. From these data, two main patterns emerged; pupils' attitudes towards science declined as they progressed through secondary school, and this decline was more pronounced for female pupils. These conclusions are largely in agreement with previous studies in this field. However, in examining separate attitude constructs, we were also able to identify that the sharpest decline occurred specifically for pupils' attitude towards learning science in school. Furthermore, using linear regression, we identified that, as pupils progress through school, this construct becomes a greater influence on attitudes towards future participation in science. Therefore, we also concluded that learning science in school is a particular area that needs to be concentrated upon, if we are to improve children's attitudes towards science. In the final part of the paper, we drew on interview data obtained from 44 pupils involved in the Lab in a Lorry study. Pupils' comments in these interviews provided further insight into why pupils are "switched off" by school science. We drew out the most prevalent themes that emerged in the interviews, in order to provide further insight into why pupils do not enjoy science in school. (Contains 6 tables, 5 figures, and 3 notes.)

In this study, the authors wish to explore the use of visual representations in facilitating the understanding of multiplication. In doing so, they examine the different aspects of multiplication that they can access through different representations. In addition, they draw on a study that they have been carrying out looking at pupils' actual use of one of the representations--the array. By bringing together their theoretical views and actual experiences in the classroom, the authors put forward suggestions for teaching multiplication and also future directions for research. (Contains 8 figures.)

This paper describes the analysis of data collected by Durham University's YELLIS project, over the period of 1999 to 2004. Included in this data was the degree to which pupils in England at the end of their secondary education "liked" or "disliked" different subjects, and their expected examination grades in these subjects. The authors' study focused on the perceptions of pupils in the science subjects of biology, chemistry and physics. Using the available data, they were able to analyse the perceptions of a large number of pupils (e.g. 9827 pupils in 2004) who took examinations in the separate sciences. The study found that physics was perceived as the least popular science, particularly by female pupils. We also found that the expected grade in a particular science subject correlated quite strongly (Spearman's rho of around 0.5) with the liking of that subject. These expected grades were found to be the lowest in physics, again particularly for female pupils. The authors therefore concluded that in order to redress the gender imbalance in physics, they need to tackle this problem that physics is perceived as difficult by female pupils. (Contains 5 tables and 13 figures.)

Background: The shortage of teachers in England and Wales continues to be a high profile area of scrutiny. Particular subjects (including mathematics, science and English) are categorized by the Training and Development Agency (TDA) for schools as priority or shortage subjects, and London especially has experienced particular shortages in teacher numbers over recent years. Purpose: This paper reports on the findings from a telephone survey with teachers, examining the issues of teacher recruitment and retention. The paper highlights the important factors, as perceived by teachers, that impact on why teachers enter the profession, and also why they choose to leave. Sample: The survey involved 246 teachers in England and Wales, teaching the shortage subjects of English, maths and science. The sample was chosen such that 25% of the teachers taking part in the study would be working in London. Our intention was to have sample numbers for teachers inside and outside London from which meaningful comparisons could be made regarding their views, and so highlight particular issues that were important for teachers in London. The 75% sample not from London was geographically representative of the rest of England and Wales. Design and methods: Prior to the survey, a provisional examination of recent research findings was carried out, leading us to categorize relevant issues into three areas: (1) reasons for wanting to enter teaching; (2) reasons for not wanting to enter teaching; and (3) reasons for wanting to leave teaching. The telephone survey with teachers therefore looked at these three broad areas. A pilot study was carried out with 22 teachers, trialling the survey questions to be used in the telephone interviews. With the agreement of the teachers, the telephone interviews were recorded to aid the data gathering process. Copies of the interview script were sent out beforehand to the teachers being interviewed, in order to try to reassure teachers that no "unexpected" questions would be asked of them, and also to give teachers the opportunity to provide considered responses during the interviews. Results: The survey found that although more "intrinsic" and "altruistic" reasons were given by teachers for going into teaching, the issues of workload and pupil behaviour were found to be most important in dissuading teachers from entering the profession or possibly causing them to leave teaching. Conclusions: The study concluded that in order to have an impact on teacher numbers, the two issues regarding workload and pupil behaviour should be seen to be tackled from the perspectives of teachers. (Contains 8 tables and 2 notes.)

Mathematics in Education and Industry (MEI), with funding from Gatsby Technical Education Projects (GTEP), ran the "Enabling Access to Further Mathematics" project from 2000 to 2003. This pilot project was designed to provide AS- and A-level students with additional opportunities to study Further Mathematics, through the use of distance-learning materials. This was in response to the dramatic decline in the numbers of students taking Further Mathematics over the last four decades. An independent evaluation of the project was commissioned by GTEP, and this paper describes the methodology of the evaluation and its findings. In addition to highlighting the benefits of the project for the students and institutions involved, the paper discusses how the project could be developed in the future, based on the issues and difficulties raised in the evaluation.