Search

The multi-faceted nature of mathematics knowledge for teaching, including pedagogical content knowledge (PCK), has been studied widely in elementary classrooms, but little research has focused on senior secondary mathematics teaching. This study utilised the Knowledge Quartet (Rowland et al., Research in Mathematics Education, 17(2), 74–91, 2005) to analyse mathematics teaching at the senior secondary level using excerpts from a lesson on differential calculus and another on discrete probability distributions. The findings reveal that, at this level, there is a complex interplay among aspects of the Knowledge Quartet, including the impact of foundational knowledge on contingent moments. Horizon content knowledge is shown to play an important role in teaching decisions, as do perceived constraints. This has implications for future research into how teachers’ horizon knowledge might be expanded and into teachers’ perceptions of mathematics course constraints on the enactment and development of their mathematics knowledge for teaching.

This theoretical paper examines the concept of pedagogical content knowledge (PCK) and explores how it might contribute to conversations around quality teaching and learning in outdoor education. This paper begins by summarizing the historical and contemporary literature, including issues of definitions, curriculum, content, and pedagogy in outdoor education. We then review the concept of PCK, its history, and contributions to other subject areas, including mathematics. We present a framework for PCK from the field of mathematics education and propose a 'modified' PCK framework for outdoor education. We postulate that this framework might help articulate knowledge areas needed by a teacher of outdoor education, and how these differ from and are similar to those required in other subject areas. We conclude by exploring how the idea of PCK and the modified framework might add to existing understandings of what it means to provide high quality outdoor education teaching and learning experiences.

The notion of pedagogical content knowledge (PCK) was posited in the context of school teaching and the knowledge used by teachers teaching school students. It has been examined for a number of discipline areas, notably mathematics. There are, however, other teaching contexts, including those of teacher educators, whose students are pre-service teachers (PSTs). The content these teacher educators teach is not subject discipline knowledge (or not solely), but the PCK for teaching a subject discipline. What knowledge do teacher educators use as they teach PCK? This paper presents a framework for the PCK required of mathematics teacher educators as they work to develop PSTs’ PCK for teaching mathematics. The framework builds on existing research into PCK and categorises aspects of the work of teacher education. The framework’s usefulness is examined by studying the PCK used by the first author in building PSTs’ understanding of mathematics teacher PCK.

With increasing ethical issues and global corporate scandals, many organisations are now looking to employ well-rounded professionals, who take ownership of their workplace while leading with their heart and soul. These organisations seem to be more concerned with relationship building and future employability (Cunha, Rego, & D’Oliveira, 2006) and are interested in the concept of spirituality with the hope that it could address ethical issues influencing their businesses. ‘Spirituality and ethics are core values that have shaped human life from time immemorial’ (Mahadevan, 2013, p. 91). Ethics and spirituality are interrelated but different as ethics is about customs and habits, while spirituality is concerned with personal meaningful experiences and differs from person to person, making it hard to define. Organisations moving towards spirituality require leadership that can develop a spiritual climate and their learning and development has to be top priority (Pawar, 2009). This requires management education to appreciate the concept of spirituality and like some universities globally, incorporate it within their programmes (Harris & Crossman, 2005). To explore whether spirituality could be incorporated within the higher education curriculum, my PhD researched academic’s viewpoints in selected faculties within a regional university in Australia. This paper reports some of its findings from the data gathered through semi-structured interviews, with a focus on leadership, its relevance to ethics and the teaching of spirituality. Results indicate that academics support the inclusion of spirituality but the programmes need to be carefully designed.

Previous research into the knowledge required for teaching has focused primarily on pre-service and in-service teachers’ knowledge. What is less researched, however, is the role of the teacher educator in helping pre-service teachers (PSTs) develop the knowledge needed in order to teach mathematics to students. The focus thus shifts from examining school teachers’ knowledge for teaching mathematics to school students, to studying teacher educators’ knowledge for teaching teachers. This raises the question of what is the nature of this knowledge as required by teacher educators, and how evident is it in their practice? This paper documents the interactions among two teacher educators and two cohorts of PSTs enrolled in a unit designed to teach mathematics pedagogy to early childhood and primary PSTs. Over one semester, two teacher educators observed each other’s classes, engaged in reflective professional conversations, and surveyed PSTs about lesson material and delivery. The results indicated there were a number of issues faced by the teacher educators that could be interpreted through the use of a teacher knowledge framework, with examples for this study focussing on a representative lesson. The findings add to the field of research into teacher educator knowledge and have implications for mathematics teacher educators and the pre-service teachers they teach.

As a result of the growing use of state and national testing of literacy and numeracy among school students, there are increasing demands for teachers to interpret assessment data. In light of this, there is a need to provide benchmarks or a framework that identifies critical aspects of teachers’ understanding that are needed to interpret data effectively. In this study, 24 items from the Attitudes and Statistical Literacy Instrument are used with 704 teachers to provide a hierarchical scale of teacher ability to interpret these assessment data. Using an item response theory model for partial credit scoring, three levels of ability are identified, related to reading values, comparing values, and analyzing a data set as a single entity. Teacher ability is then compared across various demographic variables, such as number of years of teaching, main teaching grade levels, previous professional learning experience, last time statistics was studied, and gender. Implications are drawn for professional learning for teachers and for further research.

In Australia, as in other countries, school students participate in national literacy and numeracy testing with the resulting reports being sent to teachers and school administrators. In this study, the Theory of Planned Behaviour provides a framework for examining teachers’ perceptions of factors influencing their intention to engage with these data. Most teachers perceived the data to be useful, but there were some negatively held views. For both primary and secondary teachers, males were more positive and had weaker perceptions of barriers to their use of data from system reports compared to females. Teachers who had studied statistics at the post-secondary level and/or attended relevant professional learning generally felt more capable of using the data, and senior teachers and principals were more favourably disposed to using these kinds of statistical reports. Many teachers had concerns about the timeliness of the data’s release and the effort required to interpret them.

Some of mathematics teaching is routine, like an exercise from a textbook for which you have received instruction and already know what to do. On other occasions, however, teaching mathematics is challenging, involving problems of teaching for which the solutions may not be readily apparent. These situations require the application of mathematical knowledge in concert with other types of knowledge for teaching. In this article, we explore the idea that teachers of mathematics act as applied mathematicians in applying mathematical knowledge to the resolution of teaching problems. This task involves the complex interplay of mathematical and teaching knowledge and processes of problem solving with success judged according to how well students learn. The article discusses these ideas through an examination of seven scenarios.

Anne was on a crowded bus looking over the shoulder of a young student who was trying to solve a geometry problem: to select, from several choices, the ratio of the radii of the inner and outer circles of a regular hexagon. The student was clearly stuck and going along irrelevant directions. When Anne asked, "How are you tackling this?" the young woman looked surprised and then took out her earphones and said she had let the radius of the inner circle be 1. Anne said, "Good choice" (while thinking it would have been much more use to label the outer radius), and asked if that had been helpful. A discussion of possible approaches ensued, with Anne trying not to give any direct clues. One of the possible answers had √3 in it and Anne asked if that suggested anything. The student talked about angles of 60 degrees, and Anne said, "You have told me everything you need to know to resolve this." Before she left the bus, Anne said, "Whenever you label anything in a geometry diagram, choose a label that applies to as many aspects as possible."

As a consequence of the increased use of data in workplace environments, there is a need to understand the demands that are placed on users to make sense of such data. In education, teachers are being increasingly expected to interpret and apply complex data about student and school performance, and, yet it is not clear that they always have the appropriate knowledge and experience to interpret the graphs, tables and other data that they receive. This study examined the statistical literacy demands placed on teachers, with a particular focus on box plot representations. Although box plots summarise the data in a way that makes visual comparisons possible across sets of data, this study showed that teachers do not always have the necessary fluency with the representation to describe correctly how the data are distributed in the representation. In particular, a significant number perceived the size of the regions of the box plot to be depicting frequencies rather than density, and there were misconceptions associated with outlying data that were not displayed on the plot. As well, teachers' perceptions of box plots were found to relate to three themes: attitudes, perceived value and misconceptions.

In recent years the educational policy environment has emphasised data-driven change. This has increased the expectation for school personnel to use statistical information to inform their programs and to improve teaching practices. Such data include system reports of student achievement tests and socio-economic profiles provided to schools by various state education departments’ data services. This paper reports on a pilot study that explored factors affecting Mathematics and English teachers’ intentions to engage with the statistical data their schools receive and to consider these data when making decisions about their teaching practices. It was found that most teachers perceived that such data identify weak students and some teachers (mostly mathematics teachers) thought that they can help to identify curriculum topics that need attention. Most teachers felt that the reports were not easy to understand. Confidence in dealing with statistical data was a problem for many teachers, but especially for English teachers.

This paper examines the anecdotal claim of “Not enough time” made by teachers when expressing their struggle to cover a stipulated syllabus. The study focuses on the actual experiences of a teacher teaching mathematics to a Year 7 class in Singapore according to a designated time schedule. The demands of fulfilling multiple instructional goals within a limited time frame gave rise to numerous junctures where time pressure was felt. The interactions between ongoing time consciousness and instructional decisions will be discussed. An examination of the role played by instructional goals sheds light on the nature and causes of time pressure situations.

It is widely held that context is important in teaching mathematics and statistics. Consideration of context is central to statistical thinking, and any teaching of statistics must incorporate this aspect. Indeed, it has been advocated that real-world data sets can motivate the learning of statistical principles. It is not, however, a straightforward task to take a real-world example and incorporate it into a lesson that will teach important statistical principles. This paper considers issues involved in using real data to exemplify statistical ideas and examines pre-service teachers’ attempts to design teaching activities using such data. Pre-service teachers were supplied with a topical data set and asked to plan lessons that would teach some key statistical idea to year 6 students. The lessons were analysed using a hierarchy for teaching statistical literacy, and great variation was found in the level of statistical thinking demanded in the planned lessons. Teachers who had completed a preliminary activity helping them to think carefully about what might be taught from real data in general produced lessons with stronger statistical content. A key requirement for having lessons with deep consideration of statistical ideas is to identify the actual affordances for teaching contained within a data set; the planning process then benefits from explicit attention to making that content evident in the teaching activities.

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, 2002), 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. © FIZ Karlsruhe 2008.

This study presents a continuing investigation of influences on outcomes achieved by students working in groups of three on tasks related to chance and data. Earlier research described final mathematical outcomes and identified 17 factors influencing three types of short-term outcomes for groups working in an isolated setting. The current report documents the 17 factors for groups working in a classroom setting and 1654 events for all groups in both settings are identified and each associated with a factor and a short-term outcome. Consideration is then given to variables that have the potential to influence the factors, the short-term outcomes, and their interaction. The overarching variable is the setting within which the collaboration took place. Within each setting, however, two other variables operated: the task carried out, the age/grade of students, gender balance, or collaborative characteristics. The influences of these variables are described within the two settings before consideration is given...

A critical component in the development of students' statistical thinking and reasoning is transnumerative thinking; that is, changing representations of data to engender an understanding of observed phenomena. Examples from Years 6 to 9 New Zealand students' and Australian students' representations of data from a given multivariate dataset are described. Their representations are discussed in terms of their developing abilities to explore data and unlock the stories contained therein. The implications of changing the focus of statistics instruction and the curriculum from merely teaching students how to construct graphs to exploring and representing patterns and relationships in data are presented. Introduction What do you think the graph in Figure 1 is telling us? Is it helpful to know that the children on the left are girls and that the group on the right is made up of boys? [FIGURE 1 OMITTED] Does Table 1 contain the same data? Does it tell the same story? Flow does it differ from the graph? Which representation better tells the story: the graph or the table? Why? What might the data have looked like before they were turned into a graph or a table? Are there other ways of showing the data? Would these tell the same story? How can we tell the story clearly? Collecting and exploring data in order to answer questions of interest is an important component of statistical learning. Given a dataset, then, what can we do with it in order to reveal the answers or stories that are hidden within it? Messages are not always easy to see in raw data, and so strategies for making those messages visible are important. Such strategies involve analysing and representing the data in ways that show the outcomes clearly. The two examples above demonstrate that there are different ways of representing data, but that some approaches may be better than others for revealing the stories within them. The graph, for instance, allows a visual comparison of the two groups--the striking contrast between its two halves clearly shows the difference between the girls' and the boys' fast food consumption. The table, on the other hand, also presents the contrast between the boys' and girls' data, but here this contrast is not as visually obvious as in the graph. Nevertheless, the table summarises the data better, and would be well suited to displaying larger datasets. The process of deciding what to do with a dataset in order to represent it is critical. There has been considerable emphasis on ensuring that students can interpret data in an already existing representation, often focusing on students' ability to read data and read beyond the data, as suggested by Curcio (2001). In contrast, it appears to be more difficult to create successful representations that reveal stories within data (Chick & Watson, 2001). Even for adults, producing good representations of data is difficult. There are numerous examples in the media of poorly designed representations, including some that are actually wrong or misleading. The process of going from a raw dataset to a representation that reveals and provides evidence for the "story within" is, apparently, challenging. Part of the problem is that the curriculum has emphasised univariate datasets and the construction of conventional statistical graphs, but without emphasising the actual purpose of statistical exploration. For many students, graphs are illustrations rather than reasoning tools to detect patterns and unlock the information contained in the data. Furthermore, the emphasis on univariate datasets has prevented students from observing differences and relationships between variables and realising that the purpose of statistical investigations is to seek explanations, to make predictions, and to explore new contextual knowledge. Instruction has focused on how to draw graphs. It now needs to focus on how to represent, explore, and think with data. …

There has been very little provision of professional development for teachers of mathematics in the technical and further education (TAFE) sector and even less evaluation and reporting on such programs. This paper presents a model for a professional development program in mathematics that was designed for TAFE teachers, as well as those from the K-12 sector. It is a case study that describes the implementation of the model with a group of TAFE teachers, considering such issues as the professional development needs perceived by the teachers, the components included in the program, and the evidence for the short- and long-term effectiveness of the program. The flexible structure of the program and the qualitative research base for the evaluation could be used as a model for professional development in other TAFE curriculum areas.

The purpose of this case study is to examine how collaboration affects the emergent statistical thinking of a group of three Grade 6 boys. Results of previous studies of students in Grades 3, 6, and 9 suggested that (a) when finding and justifying associations in data sets students working in groups may produce higher level outcomes than those working individually, and (b) there are numerous factors that influence the success or otherwise of collaborative activity. The current study, based on detailed analysis of video tape and transcripts of a group working collaboratively on a data handling task, documents various factors that affect collaboration and how these contribute to the attainment of desirable cognitive outcomes in terms of the task set. These outcomes are classified by emergent statistical themes and insight is gained into how naive statistical thinking begins to develop during the collaborative process. Implications for educators and researchers are considered.

Given the deluge of data that school principals and teachers receive as a result of student assessment, it has become essential for them to have statistical literacy skills and understanding. Earlier work with primary and secondary teachers in Victoria revealed that, although most saw school statistical reports as valuable for planning and thought that they could adequately interpret them, their confidence was often not well founded, with some fundamental misconceptions evident in their statistical understanding. Based on these results, a workshop was developed to target key aspects of statistical literacy particularly relevant to the education context. The workshop incorporated simple hands-on activities to develop understanding of box plot representations, critiquing descriptions of distributions and applying the newly learned principles to participants’ own school reports. Although principals and teachers responded favourably to the activities, delayed post-testing indicated limited retention of the relevant aspects of statistical literacy. These results suggest that when teachers are dealing with data on only one or two occasions in a year, it may be important to provide timely and efficient access to reminders of basic concepts.

Twenty-seven Grade 5/6 students working in triads considered a supplied data set. They were asked to hypothesise about associations in the data and to represent these. Each student was classified according to the level of interpreting the information, the level of representing the chosen data, and the type of collaboration observed in the group. Levels of interpretation and representation skills were related and there was some indication of a possible association with the type of collaboration. There was no association of type of collaboration and students’ views on group work. Implications for future research and the classroom are considered.

We investigate the role played by torsion properties in determining whether or not a commutative quasiregular ring has its additive and circle composition (or adjoint) groups isomorphic. We clarify and extend some results for nil rings, showing, in particular, that an arbitrary torsion nil ring has the isomorphic groups property if and only if the components from its primary decomposition into p-rings do too. We look at the more specific case of finite rings, extending the work of others to show that a non-trivial ring with the isomorphic groups property can be constructed if the additive group has one of the following groups in its decomposition into cyclic groups: Z2 n (for n ≥ 3), Z2 ⊕ Z2 ⊕ Z2, Z2 ⊕ Z4, Z4 ⊕ Z4, Z p ⊕ Z p (for odd primes, p), or Z p n (for odd primes, p, and n ≥ 2). We consider, also, an example of a ring constructed on an infinite torsion group and use a specific case of this to show that the isomorphic groups property is not hereditary.

Mathematics researchers put considerable cognitive effort into trying to expand the body of mathematical knowledge. In so doing, is their cognitive behaviour different from those who work on more standard mathematical problems? This paper attempts to examine some aspects of mathematical cognition at the highest level of formal functioning. It illustrates how the structure of a mathematician’s output—and, to a certain extent, its cognitive complexity—can be characterised by the SOLO taxonomy. A number of cognitive and philosophical issues concerning mathematical functioning at the research level will also be discussed.

We show that any commutative nil ring which is an algebra over the rationals has its additive group isomorphic to its circle composition (or adjoint) group and demonstrate the importance of divisibility in establishing this result. We conclude by showing that if a ring has isomorphic additive and circle composition groups then this property need not be inherited by its quasiregular subrings.

Beliefs have long been known to affect teaching and learning. In statistics education, little research has been conducted on the nature of teachers’ beliefs, despite the likely impact these beliefs have on teachers’ activities. This chapter first considers content-focused beliefs about statistics, its relationship with mathematics, and its place in the curriculum, before addressing beliefs associated with teaching and learning statistics. Influences on beliefs and the impact of beliefs on teaching are considered, and suggestions for further research are proposed.

En los ultimos anos, ha habido una expansion y renovacion del contenido de estadistica en los planes de estudio de matematicas en muchos paises a traves de todos los niveles escolares desde el nivel primario hasta el secundario. Sin embargo, no se ha prestado una atencion similar a la preparacion del profesor de matematicas para ensenar estadistica en estos niveles. Este libro presenta los resultados del estudio conjunto ICMI / IASE, Ensenanza de las estadisticas en las matematicas escolares. Desafios para la ensenanza y la formacion del profesorado que tenia como objetivo abordar la falta de atencion a la ensenanza de la estadistica mediante la promocion de la investigacion colaborativa internacional especificamente centrada en la educacion y el desarrollo profesional de los profesores para ensenar estadistica. El volumen cubre un campo muy amplio, incluidos ejemplos de planes de estudio de estadistica. y programas de formacion docente en todo el mundo; analisis de los fundamentos de la ensenanza de la estadistica; examinar capitulos de investigacion relacionados con las actitudes, creencias y conocimientos de los profesores relacionados con las ideas fundamentales de la estadistica y su ensenanza; y analisis de desafios y experiencias relacionados con la formacion de profesores para ensenar estadistica. El libro esta disenado para ser util a los investigadores en educacion matematica y a los formadores de profesores de educacion estadistica, y a las personas involucradas en el desarrollo curricular en estadistica, con la esperanza de que fomente una mayor investigacion sobre los problemas relacionados con la educacion de los profesores para ensenar estadistica en los diferentes niveles escolares. Podria ser de interes para los propios profesores, ya que las ideas basicas para la ensenanza de la estadistica y la investigacion resumida en el libro tanto en dificultades de aprendizaje como en estrategias de ensenanza es aplicable tanto en la formacion de alumnos como de profesores.

This chapter draws together the major themes emerging from the 12th ICMI Study on The Future of the Teaching and Learning of Algebra and serves as an introduction to this book. The chapter begins with a short description of the major challenges that the teaching of algebra presents to researchers, curriculum writers and teachers. There follows a brief introduction to each chapter which surveys some key ideas presented, after which the significant suggestions for future algebra teaching and learning from that chapter are highlighted. The chapter finishes by drawing together the major themes that offer guidelines for making the future of the teaching and learning of algebra brighter than the past.

Solving the Problem with Algebra.- The Core of Algebra: Reflections on its Main Activities.- Responses to 'The Core of Algebra'.- The Early Development of Algebraic Reasoning: The Current State of the Field.- A Toolkit for Analysing Approaches to Algebra.- Research on the Role of Technological Environments in Algebra Learning and Teaching.- Computer Algebra Systems and Algebra: Curriculum, Assessment, Teaching, and Learning.- The History of Algebra in Mathematics Education.- Symbols and Language.- Teachers' Knowledge and the Teaching of Algebra.- The Teaching and Learning of Tertiary Algebra.- Goals and Content of an Algebra Curriculum for the Compulsory Years of Schooling.- Algebra: A World of Difference.

In this thesis we will investigate some of the properties of the circle composition (or adjoint) operation in rings, where the operation o is defined by aob = a + b + ab. In arbitrary rings, R, the properties of addition and multiplication imply that (R, o) is a semigroup; in certain classes of rings this semigroup has additional properties and we shall examine a few of these. Our main concern will be commutative quasiregular (Jacobson radical) rings. In such rings (R, o) is an abelian group, giving R a second such structure besides (R,+). It seems a natural question to ask if these group structures can ever be isomorphic. The zero rings, in which multiplication is trivial, obviously have this property since the additive and circle composition groups coincide; thus the class, K, of rings having isomorphic additive and circle composition groups is non-empty. There are also non-trivial examples and we illustrate the construction of some, including the so-called quasifields which are constructed on partially ordered sets, and examples which use finite groups for addition. It might be suspected that for these less trivial examples the isomorphism between addition and circle composition will still force multiplication to behave in a nearly trivial way, so that perhaps such rings are nil or nilpotent. This need not be the case as there is a ring in K which has no zero divisors. In fact, we show that there exist rings in K which are nilpotent but not zero rings, nil but not nilpotent, and quasiregular without being nil. We will also consider the algebraic properties of the class K, including the question of its inheritance under ring theoretic constructions. In particular, we show that K is not a radical class, that it is closed under direct products, but that it is not hereditary and that it is not closed under homomorphisms nor taking quasiregular subrings. There are, however, certain subclasses of K which are better behaved, including, for example, rings which are algebras over Zp or Q and the rings constructed on certain finite groups. For commutative nilpotent rings we prove the existence of a polynomial homomorphism between the additive and circle composition groups, which in certain circumstances will be an isomorphism. We show, too, that all finitely generated nilpotent Q-algebras and Z-algebras are in K. The former result allows us to demonstrate that all commutative nil Q-algebras are in K. We conclude by considering a family of ring examples in which the circle composition semigroup is regular. Our construction is developed from the idea behind the quasifield construction and also generalised power series rings. We investigate the existence of nilpotence in such rings, and show that, like K, the class of rings in which (R,o) is a regular semigroup is not a radical class. This result also holds for the stronger property that (R,o) is a union of groups.

A reader from outside mathematics education research may be surprised at the diversity within the chapters in this section. There is a long tradition of educational research into the teaching and learning of mathematics, beginning with cognitive issues associated with mathematical content and its impact on the learning of different topics, and moving towards a consideration of broader issues in an effort to understand better the myriad factors that influence what takes place in mathematics learning environments.