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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 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.

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 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.

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.

We report on the work of 3 schools which set out to make a difference for their previously low-attaining students. Through naturalistic enquiry over 3 years, we built a picture of their practices. Against a national trend, students reported positive attitudes to mathematics and, in 2 out of 3 schools, showed improvement in attainment, so we probe more deeply into the teaching. The lessons broadly conformed to an approach combining inclusive participation with complex and coherent development of mathematical ideas. To understand how the teachers orchestrated this, we developed lesson analysis techniques that focus on how thinking, repertoire and understanding are scaffolded by teachers through sequences of microtasks.

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.)

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.)

This article discusses how sixth through twelfth grade science teachers can engage their students in the design and implementation of sustainable energy projects as part of a unit of study on energy. The project challenges students to engage in an energy project that gives them the opportunity to make a difference in their local community and the world. Student projects include a project description, an approved protocol, project design, community experts as a resource, and a discussion of obstacles and problems. (Contains 2 tables and 2 figures.)

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.

Presents anecdotal data from informal discussions with 12 secondary teachers about types of evidence of students' mathematics achievements that were used by the teachers. Discusses oral evidence, written work, observed actions, voluntary use of mathematics, knowledge of the child, teachers' views of mathematics, and behavior and body language. (MKR)

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 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

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]

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. [ABSTRACT FROM AUTHOR]

This theoretical paper relates key features of the mathematics adolescents are expected to learn in school to other aspects of adolescent development. Difficulties in mathematical learning at that age include changes in perspective and in the actions that are mathematically productive. Commonly-recommended methods of trying to engage adolescents in mathematics do not necessarily enable students to shift to new perceptions and new ways of constructing mathematical understandings, yet the shifts students need to make are in accord with other aspects of adolescent development. [ABSTRACT FROM AUTHOR]

This article discusses how sixth through twelfth grade science teachers can engage their students in the design and implementation of sustainable energy projects as part of a unit of study on energy. The project challenges students to engage in an energy project that gives them the opportunity to make a difference in their local community and the world. Student projects include a project description, an approved protocol, project design, community experts as a resource, and a discussion of obstacles and problems. [ABSTRACT FROM AUTHOR]

In this paper we describe learners being asked to generate examples of new mathematical concepts, thus developing and exploring example spaces. First we elaborate the theoretical background for learner generated examples (LGEs) in learning new concepts. The data we then present provides evidence of the possibility of learning new concepts through a symbiosis of induction and abduction from experience and deduction from the relationships generated in exemplification. In other words, experience can be organised in such a way that shifts of understanding take place as a result of learners’ own actions. Actions, in this context, include mental acts of organisational reflection. [ABSTRACT FROM AUTHOR]