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We demonstrate the power of Variation Theory as an analytical tool used to understand the underlying conceptual structure of mathematics lessons taught by English primary school teachers. We study excerpts of three lessons that are posted on a professional website. We show how lesson analysis using variation allows us to focus on what is made available to be learnt in the lesson excerpts. We identify some differences in their use of dimensions of variation and the associated ranges of change and discuss how suitable patterns of variation and invariance might differ according to the nature of the learning focus. We reflect on the value of our analytical approach.

In this paper, we provide an elaboration of the notion of mathematical structure -- a term agreed upon as valuable but difficult to define. We pull apart the terminology surrounding the notion of structure in mathematics: relationship, generalising/generalisation and properties, and offer an architecture of structure that distinguishes and connects these terms. The terminology is then applied to teaching and learning phenomena related to multiplicative reasoning, illustrating how the demarcation of terms can offer a more precise language of description of learning about structure. We present 'formatting activity' as a key pedagogic action for accomplishing a focus on structure in teaching.

We take "mathematical structure" to mean the identification of general properties which are instantiated in particular situations as relationships between elements or subsets of elements of a set. Because we take the view that appreciating structure is powerfully productive, attention to structure should be an essential part of mathematical teaching and learning. This is not to be confused with teaching children mathematical structure. We observe that children from quite early ages are able to appreciate structure to a greater extent than some authors have imagined. Initiating students to appreciate structure implies, of course, that their appreciation of it needs to be cultivated in order to deepen and to become more mature. We first consider some recent research that supports this view and then go on to argue that unless students are encouraged to attend to structure and to engage in structural thinking they will be blocked from thinking productively and deeply about mathematics. We provide several illustrative cases in which structural thinking helps to bridge the mythical chasm between conceptual and procedural approaches to teaching and learning mathematics. Finally we place our proposals in the context of how several writers in the past have attempted to explore the importance of structure in mathematics teaching and learning. (Contains 1 figure and 3 footnotes.)

This paper arises from a study of how concepts related to understanding functions develop for students across the years of secondary/high school, using small samples from two different curricula systems: England and Israel. We used a survey consisting of function tasks developed in collaboration with teachers from both curriculum systems. We report on 120 higher achieving students, 10 from each of English and Israeli, 12-18 years old. Iterative and comparative analysis identified similarities and differences in students' responses and we conjecture links between curriculum, enactment, task design, and students' responses. Towards the end of school, students from both curriculum backgrounds performed similarly on most tasks but approached these by different routes, such as intuitive or formal and with different understandings, including correspondence and covariational approaches to functions.

This article, by drawing on data collected from qualitative case-study research, raises questions about discourses that claim male teachers are better equipped to improve boys' literacy achievement on the basis of gender affiliation. It reports on interview and observational data collected from a number of boys attending five different secondary-schools in Ontario, Canada. It strategically draws on Connell's critical masculinities theoretical perspective and Foucault's analytic framework for power/knowledge, subjectivity and discourse, as a basis for producing a more nuanced understanding of commonly-accepted beliefs about the potential influence of male teachers to improve boys' engagement and achievement as literacy learners. This research found that only four of the 29 boys who participated in this study understand the gender of their teachers to be a critical factor for their engagement and achievement. Their narratives draw attention, not only to the complex and multiple factors influencing engagement and achievement, but also to the way in which power/knowledge and gender regimes influence their articulation of teacher influence. The findings of this research suggest that building more nuanced theories about teacher influence that move beyond so-called "common-sense" assumptions about role modeling and gender affiliation is critical if effective school reforms to address boys' literacy achievement are to be implemented.

The Marketing column features essays about a variety of creative and innovative marketing strategies used to highlight collections or promote services in all types of libraries. Its purpose is to provide real examples of how libraries are using marketing and outreach techniques in interesting ways. This article describes the social media content management strategy used at Red Deer College Library (Alberta, Canada), which includes an evolving and adaptable social media calendar, targeted timing of social media posts, using a social media management tool, and scheduling content in advance. The author also discusses creating content that is relevant to one's users and evaluating one's social media activities to constantly improve.

This study aims to explore the conceptualisations of function that some students express when they are responding to fictitious students' statements about functions. We also asked them what is meant by "function" and many voluntarily used examples in their responses. The task was developed in collaboration with teachers from two curriculum systems, England and Israel. It was given to 10 high-achieving English students from each of the years 10-13 and to 10 high-achieving Israeli students from comparable years (total of 80 students). Data analysis included identifying students' dominant ideas for functions as expressed in their responses, and analysing the types of examples that students used to explain their responses. Differences found between the samples from the countries led to conjectures about the influence of curriculum and teaching, and in particular, about the role of word, in this case "function", in concept image development. Whereas most students showed that they had a meaning for the word, those students whose relevant experience of earlier concepts had been organised around the word "function" generally showed stronger understanding of function as object.

This study examines expressions of reasoning by some higher achieving 11 to 18 year-old English students responding to a survey consisting of function tasks developed in collaboration with their teachers. We report on 70 students, 10 from each of English years 7-13. Iterative and comparative analysis identified capabilities and difficulties of students and suggested conjectures concerning links between the affordances of the tasks, the curriculum, and students' responses. The paper focuses on five of the survey tasks and highlights connections between informal and formal expressions of reasoning about variables in learning. We introduce the notion of "schooled" expressions of reasoning, neither formal nor informal, to emphasise the role of the formatting tools introduced in school that shape future understanding and reasoning.

Identifying and expressing relations between quantities is a key aspect of understanding and using functions. We are aiming to understand the development of functions understanding throughout school years in Israel. A survey instrument was developed with teachers and given to 20 high and average achieving students from each of years 7-11 and to 10 high achieving students from year 12, a total of 110 students. Our analytical approach is to identify qualitatively what students appeared to do and whether their approaches led to complete solutions. We look for progress in understanding variables and relations between them, and we found that there does not appear to be a strong link between curriculum and informal understandings of variables and covariation, but there are other strengths.

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.

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.

Despite rising levels of safe-sex knowledge in Australia, sexually transmitted infection notifications continue to increase. A culture-centred approach suggests it is useful in attempting to reach a target population first to understand their perspective on the issues. Twenty focus groups were conducted with 89 young people between the ages of 14 and 16 years. Key findings suggest that scientific information does not articulate closely with everyday practice, that young people get the message that sex is bad and they should not be preparing for it and that it is not appropriate to talk about sex. Understanding how young people think about these issues is particularly important because the focus groups also found that young people disengage from sources of information that do not match their own experiences.

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.

It is a well-established norm in England that secondary school mathematics is taught in groups categorised by prior attainment. It is therefore worthwhile to report alternative practices of all-attainment teaching--but these are rare. In this article, we report aspects of all-attainment mathematics teaching in a secondary school that has maintained this practice as its norm over a considerable time, including in recent years, when a hierarchical approach to measuring mathematics learning has become the norm for accountability purposes. The teaching described here takes account of the needs and progress of different students within a common curriculum focus, and we identify key principles behind it. The article is intended to contribute to a record of all-attainment grouping practices in mathematics in England, so that these practices are not lost.

This paper examines literacy under-achievement and the limitations of gender-based literacy reforms grounded in essentialist notions of masculinity. It draws on qualitative case-study research conducted in one Ontario secondary school in a working-class community. It focuses on two grade 9 students and their teacher who participated in a larger study which examines how the norms and values associated with school-based literacy practices contribute to under-achievement. The cases highlighted in this paper are employed to raise critical questions about the way in which literacy under-achievement continues to be articulated as a "boy problem". This paper also illustrates how the complex and situated nature of the students' gendered and classed identities, interwoven with contextual and pedagogical factors, contribute to literacy under-achievement for some boys and some girls. In addition, it argues that the disjuncture between in- and out-of-school literacy practices warrants further study. (Contains 1 note.)

This article revisits methods and debates about teaching mathematics that were common in the 1980s and early 1990s, and then moves up to date with the findings from three mathematics departments that set out to make a difference for their lowest attaining students. The methods they used were distinctly focused on core mathematical ideas, and how all students could work with these. This Vygotskian approach supports students' development towards thinking in new-to-them mathematical ways, rather than accessing mathematical enquiry through particular social structures or non-mathematical modes of engagement. The author claims that any school which does not take seriously the mathematical understanding of the lowest achieving students is not truly comprehensive. (Contains 1 note and 1 table.)

We draw on two studies of mathematics departments in 11-18 comprehensive maintained schools in England to compare and contrast the insights provided by differing theoretical perspectives. In one study, activity theory was used to describe common features of the work of three departments. In the other, a mathematics department was viewed and analysed as a complex system. In both cases, it was the learning of the departments as systems rather than of individuals that was of interest. The affordances and limitations of the analytical perspectives are discussed. Taken as a whole, this paper opens up the workings of school mathematics departments in a country which has a strong department culture.

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.)

Boys' literacy underachievement continues to garner significant attention and has been identified by journalists, educational policymakers, and scholars in the field as the cause for much concern. It has been established that boys perform less well than girls on literacy benchmark or standardized tests. According to the National Assessment of Education Progress (2009), female students consistently score higher than boys on average in both reading and writing. This trend is supported by the Programme for International Student Assessment (PISA) test results. In 2006, the largest gender gap was found in reading. Girls on average outperformed boys in this area in all of the Organisation for Economic Co-operation and Development (OECD) countries. Test scores and achievement gaps such as the ones described here have been used to create a sense of "moral panic" concerning boys' literacy skills and engagement. In this paper, the authors express some concerns about the ways in which boys' literacy underachievement is defined and taken up within a context that continues to represent all boys as victims or as the "new disadvantaged." The authors argue for the need to engage with literature and analytic perspectives that are capable of addressing the complex interplay between various social, cultural, and institutional factors--such as gender, social class, race, ethnicity, and sexuality--that affect both boys' and girls' engagement with literacy. They also draw attention to the implications associated with movements to reclaim schooling as a masculine domain and suggest pedagogical rather than structural reforms.

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.

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)

Discusses fairness and equity in mathematics assessment. Illustrates these issues with two studies of teachers' informal classroom assessment practices and their interpretation and evaluation of students' formal written mathematical texts. Indicates that different teachers could interpret similar work differently. Discusses and critiques approaches for improving the quality and equity of teachers' assessment. (Author/KHR)

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.]