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In this study, we identify ways in which a sample of 18 graduates with mathematics-related first degrees found the nth term for quadratic sequences from the first values of a sequence of data, presented on a computer screen in various formats: tabular, scattered data pairs and sequential. Participants' approaches to identifying the nth term were recorded with eye-tracking technology. Our aims are to identify their strategies and to explore whether and how format influences these strategies. Qualitative analysis of eye-tracking data offers several strategies: Sequence of Differences, Building a Relationship, Known Formula, Linear Recursive and Initial Conjecture. Sequence of Differences was the most common strategy, but Building a Relationship was more likely to lead to the right formula. Building from Square and Factor Search were the most successful methods of Building a Relationship. Findings about the influence of format on the range of strategies were inconclusive but analysis indicated sporadic evidence of possible influences. [ABSTRACT FROM AUTHOR]

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 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. [ABSTRACT FROM AUTHOR]

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 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. [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]

This paper explores the important role the media play in informing young people about masturbation. A pilot study of focus groups with twenty-two young Australians aged between 14 and 16 explored what they know about sex and sexuality, and where they have found that knowledge. This paper reports on their knowledge about masturbation. Although researchers agree that masturbation can be a positive part of healthy sexual development, most young people reported that they received very little positive information about it from their parents or in formal sex education in school. These young people's discussions around this topic were largely ambivalent, but also highly complex due to the varying levels and types of information that they receive. In this context the media play a vital role in providing information about masturbation through books and magazines for young women, and television comedies for young men. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

The article introduces a 2013 special issue of the "Canadian Journal of Science, Mathematics and Technology Education" which focuses on the nature of advanced knowledge and the nature of using that knowledge in mathematics teaching.

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. [ABSTRACT FROM AUTHOR]

In this paper, we theorise about the different kinds of relationship between examples and the classes of mathematical objects that they exemplify as they arise in mathematical activity and teaching. We ground our theorising in direct experience of creating a polynomial that fits certain constraints to develop our understanding of engagement with examples. We then relate insights about exemplification arising from this experience to a sequence of lessons. Through these cases, we indicate the variety of fluent uses of examples made by mathematicians and experienced teachers. Following Thompson's concept of 'didactic object' (Symbolizing, modeling, and tool use in mathematics education. Kluwer, Dordrecht, The Netherlands, pp 191-212, ), we talk about 'didacticising' an example and observe that the nature of students' engagement is important, as well as the teacher's intentions and actions (Thompson avoids using a verb with the root 'didact'. We use the verb 'didacticise' but without implying any connection to particular theoretical approaches which use the same verb.). The qualities of examples depend as much on human agency, such as pedagogical intent or mathematical curiosity or what is noticed, as on their mathematical relation to generalities. [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]

Clear-fell logging around small headwater streams in Tasmanian wet eucalypt forests was predicted to affect both the retention of leaf litter and the composition and size of leaf packs. Retention structures were surveyed in six natural streams and six streams in forest regenerated 3–5 years after clear-fell and burn logging. Logged streams had more wood, but retained less leaves than natural streams, and consequently had fewer and smaller leaf packs. Leaf packs from natural streams contained 200% more leaves, bark and twigs than packs from logged streams. The effect of buoyancy on leaf retention was assessed with release and recapture of marked Eucalyptus obliqua and Nothofagus cunninghamii leaves. Eucalypt leaves were more likely to be trapped by retention structures on the bed of the stream, while smaller, more buoyant N. cunninghamii leaves were mainly trapped by leaf packs. Leaf packs in natural streams were formed on a matrix of small twigs and long strips of bark, shed from the upper branches of mature stringybark eucalypts, while leaf retention was reduced in logged streams because there are no mature trees to provide effective retention structures. Changes to the channel form increase both discharge and sedimentation. These factors have strong implications for downstream nutrient processing and riverine food webs. [ABSTRACT FROM AUTHOR]

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. [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]

The article discusses several reports published within the issue including one by Caleb Gattegno on behavioral training for students and another by Mary Stein on the teaching of mathematics.

This paper contributes to knowledge about principled action which makes a difference to learners’ attainment. We report on the Improving Attainment in Mathematics Project,1 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. [ABSTRACT FROM AUTHOR]

Examples, illustrations, occurrences, and instances play a central role in the learning of mathematics. Deliberately getting students to generate examples is a particularly powerful tool in teaching, but, while some of the possibilities they offer are used by many teachers, their full potential is rarely exploited. The authors came to this notion through reflection on their experience of observing teachers in classrooms, of getting students to take initiatives, and of creating examples while working on their own mathematics. They examine the roles played by examples constructed and generated by students, give illustrations and analyses of the use of this tool, and develop a theory for the act of exemplification as an act of cognition. A framework is developed for researchers and practitioners to further this work. [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]

The article discusses various reports published within the issue including one about the role, selection, design and use of examples in mathematics education and another about the concept of advanced mathematics.

Invites authors to submit papers for a special issue of “Journal of Mathematics Teacher Education” on the nature and role of tasks in mathematics teachers' education.