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This paper reports on concept images of 38 secondary school mathematics prospective teachers, regarding the evenness of numbers. Written assignments, individual interviews, and lesson transcripts uncover salient, erroneous concept images of even numbers as numbers that are two times "something" (i.e., 2i is an even number), or to reject the evenness of zero. The notion of concept image serves in the analysis of the findings, and the findings serve in offering two refinement notions: mis-in concept images that mistakenly grant non-examples the status of examples (e.g., 2i is an even number), and mis-out concept images that mistakenly regard examples as non-examples (e.g., zero is not an even number). We discuss possible benefits in distinguishing between these two refinement notions in mathematics education. [ABSTRACT FROM AUTHOR]

This paper reports from a case study which explores kindergarten children's mathematical abstraction in a teaching–learning activity about reflection symmetry. From a dialectical perspective, abstraction is here conceived as a process, as a genuine part of human activity, where the learner establishes "a point of view from which the concrete can be seen as meaningfully related" (van Oers & Poland Mathematics Education Research Journal, 19(2), 10–22, 2007, p. 13–14). A cultural-historical semiotic perspective to embodiment is used to explore the characteristics of kindergarten children's mathematical abstraction. In the selected segment, two 5-year-old boys explore the concept of reflection symmetry using a doll pram. In the activity, the two boys first point to concrete features of the sensory manifold, then one of the boys' awareness gradually moves to the imagined and finally to grasping a general and establishing a new point of view. The findings illustrate the essential role of gestures, bodily actions, and rhythm, in conjunction with spoken words, in the two boys' gradual process of grasping a general. The study advances our knowledge about the nature of mathematical abstraction and challenges the traditional view on abstraction as a sort of decontextualised higher order thinking. This study argues that abstraction is not a matter of going from the concrete to the abstract, rather it is an emergent and context-bound process, as a genuine part of children's concrete embodied activities. [ABSTRACT FROM AUTHOR]

Girls' underrepresentation in high-level post-compulsory mathematics is a longstanding issue of concern in many Western nations, with innumerable efforts to increase their participation producing little impact. In this paper, we shed new light on girls' underrepresentation through a post-structural feminist investigation of mathematics teachers' discursive constructions of high-level senior secondary mathematics students. Our analysis of semi-structured interviews with 22 Australian mathematics teachers revealed gendered views that serve to exclude many students from the high-level mathematics student category. Most concerning was their recurring naturalised construction of successful high-level mathematics students as endowed with the right, invariably male, brain. In so doing, teachers repeatedly closed off the possibility of success to those lacking such a 'mathematics gift', effectively 'counting girls out'. We argue that increasing girls' participation in mathematics requires moving beyond current efforts to raise female interest and confidence to, more profoundly, disrupt enduring discourses of male superiority in mathematics. [ABSTRACT FROM AUTHOR]

The work that mathematics teachers do is frequently mathematical in nature and different from other professions. Understanding and describing common ways that teachers draw upon their content knowledge in the practice of teaching is important. Building on the descriptions by McCrory et al. (Journal for Research in Mathematics Education 43(5) 584-615, ) of two primary ways teachers use their content knowledge-decompressing and trimming-this paper differentiates between micro- and macro-levels of mathematical complexity as a way to extend and further conceptualize the moves that teachers make at a more nuanced grain size. This paper explains and portrays their counterparts, micro-level trimming and macro-level decompressing, which we discuss as concealing and foreshadowing complexity, respectively. Support for their use in teachers' work is provided through specific examples. Implications for the mathematical preparation and professional development of teachers are discussed. [ABSTRACT FROM AUTHOR]

Technology is available and accessible in many mathematics classrooms. Adopting technology to support teaching and learning requires teachers to change their teaching practices. This paper reports the responses of a diverse cohort of 92 secondary mathematics teachers who chose to respond to an Australian state-wide survey (Mathematics with Technology Perceptions Survey) developed using a Theory of Planned Behaviour framework. The items discussed in this paper targeted mathematics teachers’ perceptions of possible barriers and enablers to their intention to use technology in their teaching. The responses are varied but, overall, strength of agreement with enablers outweighed agreement with perceived barriers. However, it is clear that despite an overall positive attitude towards the use of technology for teaching mathematics, some perceived barriers to change are notable. It is, therefore, helpful if those responsible for professional development, promoting the use of technology, recognise and address these barriers as well as working to strengthening enablers. [ABSTRACT FROM AUTHOR]

Building on the papers in this special issue as well as on our own experience and research, we try to shed light on the construct of example spaces and on how it can inform research and practice in the teaching and learning of mathematical concepts. Consistent with our way of working, we delay definition until after appropriate reader experience has been brought to the surface and several ‘examples’ have been discussed. Of special interest is the notion of accessibility of examples: an individual’s access to example spaces depends on conditions and is a valuable window on a deep, personal, situated structure. Through the notions of dimensions of possible variation and range of permissible change, we consider ways in which examples exemplify and how attention needs to be directed so as to emphasise examplehood (generality) rather than particularity of mathematical objects. The paper ends with some remarks about example spaces in mathematics education itself. [ABSTRACT FROM AUTHOR]

This paper indicates that prospective teachers’ familiarity with theoretical models of students’ ways of thinking may contribute to their mathematical subject matter knowledge. This study introduces the intuitive rules theory to address the intuitive, same sides-same angles solutions that prospective teachers of secondary school mathematics come up with, and the proficiency they acquired during the course “Psychological aspects of mathematics education”. The paper illustrates how drawing participants’ attention to their own erroneous applications of same sides-same angles ideas to hexagons, challenged and developed their mathematical knowledge. [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]

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]

In this paper we examine the possibility of differentiating between two types of nonexamples. The first type, intuitive nonexamples, consists of nonexamples which are intuitively accepted as such. That is, children immediately identify them as nonexamples. The second type, non-intuitive nonexamples, consists of nonexamples that bear a significant similarity to valid examples of the concept, and consequently are more often mistakenly identified as examples. We describe and discuss these notions and present a study regarding kindergarten children’s grasp of nonexamples of triangles. [ABSTRACT FROM AUTHOR]

This study is grounded in the theoretical position that solving problems in different ways creates mathematical connections when learning and teaching mathematics. It acknowledges the central role teachers play in providing students with learning opportunities, and it is based on the empirical finding that mathematics teachers are reluctant to solve problems in different ways in the classroom. In this paper we address the contradiction between theory-based recommendations and school mathematics practice. Based on analysis of individual interviews and two group meetings with 12 Israeli secondary school mathematics teachers, we demonstrate that in the context of multiple-solution connecting tasks this discrepancy is caused by the situated nature of the teachers’ knowledge. We also reveal the complex relationship between different types of teacher knowledge and argue the significance of developing a common language between members of the mathematics education community, including teacher educators and researchers. [ABSTRACT FROM AUTHOR]