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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 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.

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.

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 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.)

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.)

While Socrates and Meno are discussing the nature of knowledge through considering the square-root of two with one of Meno's slaves, their wives are discussing the nature of knowledge through considering the square-root of three with one of the slave girls, using a diagram based on Raphael's "The School of Athens" being contemplated by Euclid.

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.

Can teachers contact the inner coherence of mathematics while working in a context fragmented by always-new objectives, criteria, and initiatives? How, more importantly, can learners experience the inner coherence of mathematics while working in a context fragmented by testing, modular curricular, short-term learning objectives, and lessons that jump from one topic to another because of coverage pressures, textbook design, staffing problems, and so on? In this article, the author presents these questions in two mathematical contexts: (1) trigonometry; and (2) multiplication. (Contains 1 note.)

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.

In this paper I illustrate how kinaesthetic experiences associated with dance might be used in teaching to promote engagement and learning in spatial, rhythmic, structural and symbolic aspects of mathematics.

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 findings that identify components of the practices of teachers acting as assessors of students' mathematics in the normal course of classroom work. Indicates that even teachers who have undergone some assessment training may underestimate the role of evidence interpretation, and questions about equity in the uses of teachers' judgements are raised in relation to awareness and practice. (Contains 51 references.) (Author/ASK)

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)

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 authors see mathematics as an endless source of surprise, which excites and motivates them. Many writers have exploited this characteristic of the subject to nurture learning. The challenge is to create conditions for learners so that they too will experience a surprise. The authors tend to see surprise as a positive emotion; mathematics as full of philosophical and cognitive surprises; and surprise as motivating curiosity and effort. They have collected and developed tasks which seem to generate discussion about how surprise can be created, generated, nurtured and harnessed during mathematics teaching and learning. This article offers some of them. (Contains 1 figure and 1 note.)