Measuring conceptual understanding: the case of fractions

Developing measures of the quality of understanding of a given mathematical concept has traditionally been a difficult and resource-intensive process. We tested an alternative approach, called Comparative Judgement (CJ), that is based not on psychometric instruments or clinical interviews but collective expertise. Eight mathematics education experts used CJ to assess 25 student responses to a test designed to probe conceptual understanding of fractions. Analysis revealed the CJ assessment process yielded high internal consistency, inter-rater reliability and validity. We discuss the implications of the results for using CJ to measure mathematical understanding in a variety of domains and contexts

Introduction to the papers of TWG21: Assessment in mathematics education

TWG21 met for the second time in Utrecht at CERME11 and in this conference we sought to continue the work started at CERME10. The aim of the previous meeting was to ascertain where the interest of our community is when thinking about assessment, and to maintain the focus firmly on mathematics. At CERME11 we discussed 14 papers and 3 posters which helped defining such interest. We noticed again a variety of focal points: from validation of large-scale assessment instruments, to the affordances and drawbacks of online assessment – especially in the university context - to the details of construction of individualised feedback. As in the previous meeting the papers also presented a variety of methodologies: from large quantitative studies to more nuanced qualitative investigations. Among the submissions we also received papers related to students’ perspectives and teachers’ perspectives on assessment. These themes were not prominent in the past meeting of the group and we welcomed the new perspectives they brought. Finally, we decided to group papers together that indicated the role that mathematics has in the assessment: this is to say papers that focus on the specifics of mathematics, such as assessing proof

Challenges in mathematical cognition: a collaboratively-derived research agenda

This paper reports on a collaborative exercise designed to generate a coherent agenda for research on mathematical cognition. Following an established method, the exercise brought together 16 mathematical cognition researchers from across the fields of mathematics education, psychology and neuroscience. These participants engaged in a process in which they generated an initial list of research questions with the potential to significantly advance understanding of mathematical cognition, winnowed this list to a smaller set of priority questions, and refined the eventual questions to meet criteria related to clarity, specificity and practicability. The resulting list comprises 26 questions divided into six broad topic areas: elucidating the nature of mathematical thinking, mapping predictors and processes of competence development, charting developmental trajectories and their interactions, fostering conceptual understanding and procedural skill, designing effective interventions, and developing valid and reliable measures. In presenting these questions in this paper, we intend to support greater coherence in both investigation and reporting, to build a stronger base of information for consideration by policymakers, and to encourage researchers to take a consilient approach to addressing important challenges in mathematical cognition

Integrating ‘just-in-time’ learning in the design of mathematics professional development

This paper describes a professional development (PD) programme design integrating just-in-time learning (JITL) to support teachers to learn about and enact inquiry teaching. Through JITL, teachers are provided with support that is responsive and applicable to their needs. This case study reports on the professional journeys of three teachers who received JITL via online resources, face-to-face meetings, opportunities for community building within and across schools, and reflections on pupils’ reactions to learning mathematics through inquiry. Findings indicate that JITL embedded within PD facilitated teacher learning about inquiry enactment, since the PD was responsive to teachers’ immediate and contextual needs. We suggest that explicit attention to JITL is given in the design of teacher PD, providing support that is made readily available for teachers to access and utilise.</div

Low attainment in mathematics: An analysis of 60 years of policy discourse in England

The problem of low attainment in mathematics has been an increasingly prominent feature of the policy discourse in England over the last 60 years; however, evidence from comparative studies indicates that little progress has been made in finding a solution. In this paper, we analyse the changing policy discourse of low attainment in mathematics through the main reports and speeches published in England, beginning with the Newsom Report, Half Our Future, in 1963, and continuing to the present day. We chart the evolving perspectives on the nature of ability, expectations, curriculum ideology and frame of reference through the changing language used in these documents, noting tensions and inconsistencies which arise through continuing lack of clarity about definitions and assumptions

Teachers’ structuring of mathematical inquiry lessons: Shifting from ‘task-first’ to ‘scaffolded inquiry’

A three-phase ‘task-first’ lesson structure is frequently suggested when teaching mathematics through inquiry. We investigate how secondary school teachers of mathematics structure their inquiry lessons and examine how and why they deviate from a ‘task-first’ structure. We present detailed lesson observation data from three teachers participating in a year-long professional development programme focused on inquiry teaching. We track the developing structure of these teachers’ inquiry lessons through minute-by-minute lesson analysis, describe how their lesson structures altered over time and explore why. Our data show that contextual constraints may explain why teachers departed from the ‘task-first’ lesson structure. In their inquiry teaching, two of the teachers adopted more scaffolded approaches, including the use of a sequence of smaller sub-tasks and teacher interventions. We argue that these modifications to a ‘task-first’ lesson structure are legitimate ways to support student learning through inquiry; indeed, that they may offer some advantages for inquiry teaching.</p

Teachers’ structuring of mathematical inquiry lessons: Shifting from ‘task-first’ to ‘scaffolded inquiry’

A three-phase ‘task-first’ lesson structure is frequently suggested when teaching mathematics through inquiry. We investigate how secondary school teachers of mathematics structure their inquiry lessons and examine how and why they deviate from a ‘task-first’ structure. We present detailed lesson observation data from three teachers participating in a year-long professional development programme focused on inquiry teaching. We track the developing structure of these teachers’ inquiry lessons through minute-by-minute lesson analysis, describe how their lesson structures altered over time and explore why. Our data show that contextual constraints may explain why teachers departed from the ‘task-first’ lesson structure. In their inquiry teaching, two of the teachers adopted more scaffolded approaches, including the use of a sequence of smaller sub-tasks and teacher interventions. We argue that these modifications to a ‘task-first’ lesson structure are legitimate ways to support student learning through inquiry; indeed, that they may offer some advantages for inquiry teaching.</p

Low-attaining secondary school mathematics students’ perspectives on recommended teaching strategies

Recent research syntheses have identified several potentially high-leverage teaching strategies for improving low-attaining secondary school students’ learning of mathematics. These strategies include the structured use of representations and manipulatives and an emphasis on derived facts and estimation. This paper reports on 70 semi-structured interviews conducted with low-attaining students in Years 9-10 (ages 13-15) in England. The interviews addressed the students’ perceptions of learning mathematics and the teaching strategies that they experienced and believed were most helpful. Many students reported rarely using number lines, not spontaneously estimating answers and being unfamiliar with derived facts. During the interviews, with minimal direction, students often showed that they were well able to make use of these strategies; however, they did not report making spontaneous use of them independently. We conclude that many of the most well-evidenced and recommended strategies to support low-attaining students in mathematics appear to be unfamiliar and unvalued, and we discuss how this might be addressed.</p

Low-attaining secondary school mathematics students’ perspectives on recommended teaching strategies

Recent research syntheses have identified several potentially high-leverage teaching strategies for improving low-attaining secondary school students’ learning of mathematics. These strategies include the structured use of representations and manipulatives and an emphasis on derived facts and estimation. This paper reports on 70 semi-structured interviews conducted with low-attaining students in Years 9-10 (ages 13-15) in England. The interviews addressed the students’ perceptions of learning mathematics and the teaching strategies that they experienced and believed were most helpful. Many students reported rarely using number lines, not spontaneously estimating answers and being unfamiliar with derived facts. During the interviews, with minimal direction, students often showed that they were well able to make use of these strategies; however, they did not report making spontaneous use of them independently. We conclude that many of the most well-evidenced and recommended strategies to support low-attaining students in mathematics appear to be unfamiliar and unvalued, and we discuss how this might be addressed.</p

Low attainment in mathematics: an investigation focusing on Year 9 students in England

This project investigated low attainment in mathematics by focusing on the lowest attaining 40% of pupils in Year 9 in England and addressing the following research questions:x What mathematics do low attaining secondary pupils understand, and what are their particular strengths and weaknesses in number, multiplicative reasoning and algebra?x Can low attainment be characterised simply as delay? If not, to what extent and in what ways do low attaining pupils understand mathematics in qualitatively different ways to high attaining pupils?x To what extent do low attaining pupils' prior understandings of mathematics, and of particular mathematical topics, help to explain the existence of the attainment gap? What is the relative contribution of these mathematical understandings in comparison to socio-economic status and other demographic factors?x What is currently known about the effectiveness of teaching strategies and approaches that address low attainment in secondary mathematics?x To what extent is mathematics currently taught in appropriate ways for low attainers? </div