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This study investigates students' ways of attending to linear sequential data in two tasks, and conjectures possible relationships between those ways and elements of the task design. Drawing on the substantial literature about such situations, we focus for this paper on linear rate of change, and on covariation and correspondence approaches to linear data. Data sources included a survey instrument of six tasks that was developed in collaboration with a group of teachers, and the tasks for this paper are two concerned with linear functions. The whole survey was given to 20 students from each of UK years 7-11 and 10 students from each year 12-13 (total of 120 students). Our analytical approach was to identify what all students appear to do, not how correct they were or what pre-determined methods they might use. Our analysis uses theories of dual-process and dynamic graded continuum to suggest conjectures about how students' capabilities in acting with sequential data depend to some extent on task features, as well as on curriculum and pedagogy.

We aim to open up discussion about the intertwined roles of teachers and tasks that involve students communicating about mathematics when working in groups. Over many years we have observed, researched and ourselves have taught students working on mathematics in groups and find that it is often easier to pay attention to the forms of communication rather than its mathematical content. The notion of 'groupworthy' tasks has become a popular focus and communication an explicit component of curricula in many countries; it is timely to consider further the mathematical content of communication while working in groups on such tasks.

We investigate whether and how personal mathematical knowledge at an advanced level impacts on teaching at a lower school level. We study this in the context of functions because understanding them permeates secondary and advanced mathematics. Textbook treatment of these can be patchy, implying a need for knowledgeable teachers to rectify weaknesses. A small observational study of two mathematically well-qualified teachers, teaching at a range of levels, leads to a deeper understanding of how their personal mathematical knowledge is manifested in their teaching. Finally, we present a theory of pedagogy combining dual development, intellectual necessity, and repeated experience of reasoning, for which these teachers' practices provide illustrations.

This article presents an e-mail conversation between two teachers discussing how to have a "rich task" lesson in which they get to the heart of mathematical modeling and in which students are motivated into working on mathematics. One teacher emphasizes that the power of maths is in developing mathematical descriptions of situations by looking at them mathematically, and using such models as tools to create, conjecture, develop new objects, and to reach new mathematical understandings. The work of the Freudenthal Institute is important in this discussion because they talk about the difference between a "model-of" and a "model-for." The power of maths is in developing mathematical descriptions of situations by looking at them mathematically, that is the "model-of," and using such models as tools to create, conjecture, develop new objects, and to reach new mathematical understandings, those are the "models-for." The other teacher shares that the idea of "models-for" has to be taught with the use of real-life contexts such as those offered by Bowland maths resource. (Contains 1 endnote.)

In this theoretical paper the informal assessment practices of two experienced teachers are used as cases for generating questions about future developments in formative assessment practice. Both teachers maintain a consistent formative assessment focus on the development of their students as enquirers, and one of them supplements this with explicit self-assessment activities. However, there are subject-specific gaps in the ways in which they assess and describe their students and these are not addressed in widely promulgated advice about formative assessment. Questions are raised about how teachers might be supported to develop their assessment of subject-specific behaviour. (Contains 3 notes.)

This paper contributes to knowledge about principled action which makes a difference to learners' attainment. We report on the Improving Attainment in Mathematics Project, a project focusing on low-attaining secondary students. The purpose of the project was to introduce innovations in practice through action research with 10 teachers over 2 years, and evaluate the effect on students' learning using national test scores, teachers' reports, non-routine tasks and other performance indicators. However, this is not a study which shows how certain methods lead to better results. While it was found that learning improved, the methods and strategies the teachers used were not always generalisable across the project, indeed some were contradictory. Continued searching led to the identification of common underlying principles of teaching which different teachers manifested in different ways. Overt methods were less important than the collection of beliefs and commitments which underpinned teachers' choices. There was, however, a convergence of practice around a focus on long-term development, the process of becoming a learner of mathematics, rather than short-term gains. In addition, we had to deal with some of the realities of authentic collaborative research with practitioners.

The Smith Report has generated central questions about the mathematics education of UK adolescents. This paper highlights the close match between the goals of school mathematics, adolescence and exploratory pedagogy. This is contrasted with the prescriptive nature of current regimes. In particular, without careful attention to pedagogy it is possible that the introduction of different pathways may lead to a failure to achieve the outcomes desired by employers and universities, and to inequity in provision for students.

Points out the effectiveness of student-generated examples as a teaching tool. Examines the roles played by examples constructed and generated by students, illustrates and analyzes the use of this tool, and develops a theory for the act of exemplification as an act of cognition. Introduces a framework developed for this work. (Contains 39 references.) (Author/YDS)

Interviews with 30 British primary and secondary teachers about informal math-assessment practices revealed differences in perceptions of how students can change, teaching practices, learning styles, desired outcomes, and views of math. These differences can lead to inequities in the way students are treated in educational systems. (SK)

This is the second of three articles that draw on findings from Nunes, Watson and Bryant (2009): "Key understandings in school mathematics: a report to the Nuffield Foundation". The report was soundly based on research about how children learn mathematics, much of it done in the UK in the '80s. Most of the findings about algebra are very well-known, but the author uses them to emphasise challenges for teaching. At school level, algebra consists of: (1) manipulation and transformation of symbolic statements about relationships; (2) rules for transforming and solving equations; (3) generalisations of laws about patterns and number; (4) modelling situations; (5) the study of structures and systems abstracted from computations and relations; and (6) learning about variables, functions and expressing change and relationships. Students have to understand the nature of arithmetical operations and inverse operations and understand that algebra represents relations between variables, and structures which combine operations. These understandings are essential in other areas of mathematics. Learning algebra is a mixture of learning to use the notations and learning the underlying meaning. In this article, the author describes what learners already do that can help them learn algebra and what they do naturally that hinders the learning of algebra. She concludes the article with some recommendations for teaching. (Contains 1 footnote.) [For Part 1, see EJ889866.]

This article is the third in a series which draws on findings from Nunes, Watson and Bryant (2009): "Key understandings in school mathematics: a report to the Nuffield Foundation". In this article the author focuses on what learners have to understand and learn in order to do secondary mathematics well in general terms. She is assuming a general educational context that encourages thinking and problem-solving across subjects. A key difference about mathematics is that empirical approaches may solve individual problems, and offer directions for reasoning, but do not themselves lead to new mathematical knowledge or mathematical reasoning. Nor do they lead to the empowerment that comes from identifying key abstract ideas and tools as they are manifested in particular situations. For this reason general problem-solving initiatives in schools may not make a difference to students' ability to learn abstract and formal ideas, although they may change motivation, effort, and resilience. (Contains 1 endnote.) [For Part 2, see EJ972354.]

This paper reports the activity of three secondary school mathematics departments in England in self-initiated states of change that led to overall improvements in students' achievements when compared to previous cohorts. This took place without intervention and without their participation in external projects. They provide examples of departments that can work effectively on their own development, and hence, their work adds to our knowledge of the potential for development through collaboration. The departments were monitored over 3 years, and data were analysed using the lens of activity theory. In contrast to departments in many studies, these departments worked overtly on mathematics pedagogy through the shared production and discussion of resources, shared planning and task design. Also in contrast to several other studies, they developed distinct ways to handle differences of subject knowledge among the teachers in the department. Their focus changed during the study from developing resource banks to supporting students' learning through hybrid teaching. [ABSTRACT FROM AUTHOR]

This article reproduces and discusses a series of blog posts posted by academics in anticipation of the report on commercialisation, sexualisation and childhood, ‘Letting Children Be Children’ by Reg Bailey for the UK Department of Education in June 2011. The article discusses the difficulty of ‘translating’ scholarly work for the public in a context where ‘impact’ is increasingly important and the challenges that academics face in finding new ways of speaking about sex in public. [ABSTRACT FROM AUTHOR]

The purpose of this paper is initially to present findings which identify components of the practices of teachers acting as assessors of students' mathematics in the normal course of classroom work. At an informal level such practice is found to be complex and intimately related to every aspect of teaching and learning. It is found that even teachers who have undergone some assessment training may underestimate the role of interpretation of evidence, and questions about equity in the uses of teachers' judgements are raised in relation to awareness and practice. It is suggested that more care needs to be taken over the formation and use of `professional judgements' within systems of assessment. [ABSTRACT FROM AUTHOR]