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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.

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 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 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.

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.

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 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.)

The author has worked for three days on a range of mathematical tasks in order to gain a deeper understanding of the role of notation in mathematics education. The general conclusions were that while mathematical notation can be powerful it can also be obstructive. In this article, the author explores the importance on notation in supporting children's understanding of mathematics. (Contains 1 footnote.)

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.

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.)

In this theoretical article, we take an exercise to be a collection of procedural questions or tasks. It can be useful to treat such an exercise as a single object, with individual questions seen as elements in a mathematically and pedagogically structured set. We use the notions of dimensions of possible variation and range of permissible change, derived from Ference Marton, to discuss affordances and constraints of some sample exercises. This gives insight into the potential pedagogical role of exercises, and shows how exercise analysis and design might contribute to hypotheses about learning trajectories. We argue that learners' response to an exercise has something in common with modeling that we might call micromodeling, but we resort to a more inclusive description of mathematical thinking to describe learners' possible responses to a well-planned exercise. Finally we indicate how dimensions of possible variation inform the design and use of an exercise.

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)

Of the four mathematical operators, division seems to not sit easily for many learners. Division is often described as "the odd one out". Pupils develop coping strategies that enable them to "get away with it". So, problems, misunderstandings, and misconceptions go unresolved perhaps for a lifetime. Why is this? Is it a case of "out of sight out of mind?", or "least said soonest mended?", or maybe it is a case of "let sleeping dogs, or lions even, lie". However, to ignore this sense of unease with "division", is not a viable long term option. Here, the authors offer a more considered approach to issues that have the potential to blight mathematics for many learners.

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.]

Defines 'thenwhat' as the moment when strategy that is supposed to be about mathematical thinking has no clear path forward as a result. Discusses how strategies designed to stimulate mathematical learning and thinking go nowhere if teachers do not think about where to go next or how to use what happens as a result of the strategy. (KHR)

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 article is the first in a series which draws on findings from Nunes, Watson and Bryant (2009): "Key understandings in school mathematics: a report to the Nuffield Foundation." The Nuffield report is soundly based on research about how children learn some of the concepts involved in mathematics. In this series of articles the author takes key ideas from the report and shows how they impact on school mathematics for all years, beyond those in which particular aspects of number are taught. The understanding of number which school children eventually need to develop comes from four threads of development: (1) counting discrete objects; (2) knowing about quantities of all kinds; (3) comparing quantities and collections of object; and (4) using notations for these. In this article, the author considers the importance of relations among quantities. (Contains 2 notes.)

The author recounts how he had students in his physics high school class identify energy-related problems within their own community and then try and figure out a solution to solve them. It is the author's belief that a physics teacher should not teach about energy unless energy problems are also discussed within a class.