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Several researchers have noted how children's whole number schemes can interfere with their efforts to learn fractions. An Australian study found that children who were successful with the solution of rational number tasks exhibited greater whole number knowledge and more flexible solution strategies. Behr and Post (1988) indicated that children needed to be competent in the four operations of whole numbers, along with an understanding of measurement, for them to understand rational numbers. This paper describes a "hands on" approach developed by researchers that focuses on the use of paper folding, fraction walls and number lines to develop an understanding of fractions using a measurement model. (Contains 8 figures.)

Middle year students often do not see the mathematics in the real world whereas the "Australian Curriculum: Mathematics" aims for students to be "confident and creative users and communicators of mathematics" (Australian Curriculum Assessment and Reporting Authority [ACARA] 2012). Using authentic and real mathematics tasks can address this situation. This paper is an account of how, working within a Knowledge Producing Schools' framework, a group of middle year students addressed a real community issue, the problem of the lack of a teenage safe space using mathematics and technology. Data were collected for this case study via journal observations and reflections, semi-structured interviews, samples of the students' work and videos of students working. The data were analysed by identifying the mathematics the students used determining the function and location of the space and focused on problem negotiation, formulation and solving through the statistical investigation cycle. The paper will identify the mathematics and statistics these students used as they addressed a real problem in their local community.

A recent international study of pre-service teachers identified that proportional reasoning was problematic for pre-service teachers. Proportional reasoning is an important topic in the middle years of schooling and therefore it is critical that teachers understand this topic and can rely on their Mathematical Content Knowledge (MCK) when teaching. The focus of this paper is second-year Australian primary pre-service teachers' MCK of real number items related to ratio, rate, proportion and proportional reasoning. This paper reports on strengths and weakness of pre-service teachers' MCK when responding to test items; including a method suitable for analysing responses to five items and ranked by three levels of difficulty. The results revealed insights into their correct methods of solutions and common incorrect responses, identifying difficulty, where multiplication and division were required. The method of coding test items by difficulty ranking may assist with developing an appropriate learning trajectory, which will assist pre-service teachers develop their MCK of this and other difficult topics. (Contains 7 tables and 4 figures.)

To date, the way that academic misconduct is manifested in undergraduate mathematics coursework has been unexamined in the literature, with the consequence that policy and preventive education can fail to address it appropriately. This paper describes the particular features of the responses expected in mathematical tasks and provides concrete examples of what lapses of integrity look like in this context. The intent is to lay groundwork for discussion and a response to this issue from within our discipline.

A critical incident is a situation or event that holds significance for learning, both for the students and teachers. This paper presents four examples of critical incidents from a Year 7 teacher's lesson excerpts in Indonesia involving teaching of fractions, to show how they shaped classroom situation, brought forward elements of conflict, and created learning opportunities. Three examples are drawn from the lesson using a web-based applet (Examples 1, 2 and 3). The illustration of these critical incidents will be followed by a discussion on how to harness them in order to develop students' understanding or be used as a challenge as well as a learning process for teachers. This paper highlights the effectiveness of a web-based applet for displaying pictorial representations in an interactive manner.

Since the 1990s, computer algebra systems (CAS) have been available in Australia as hand-held devices designed for students with the expectation that they will be used in the mathematics classroom. The data discussed in this paper was collected as part of a pilot study that investigated first year university mathematics and statistics students' understanding of functions and variables, as well as the use of technology in their last year of school (Year 12). Did their teachers discourage the use of CAS for algebra? Did the students actually learn how to use CAS to support their work in algebra or to support their learning of algebra? Did they find that, given the level of algebra, it was faster to work with pen-and-paper than to correctly enter algebraic expressions? The results reported in this paper are based on items included in a pilot survey. They raise questions rather than provide answers. The results do however tell us that, at least from these first year university students' recollection of their Year 12 experience, most or their VCE mathematics teachers made little use of CAS as a pedagogical tool in their classes, despite the institutional approval and encouragement indicated by both the State's curriculum and assessment for the past decade. A better understanding of the barriers to teachers using CAS technology to enhance their pedagogy is needed and then perhaps more effective professional learning programs can be provided for teachers.

The study of higher-level secondary mathematics is considered essential for national economic growth, competitiveness in research and innovation, and further education opportunities. Yet the reported trend within Australian secondary schools is that enrolments in higher-level mathematics are declining and have been in a state of decline for over a decade. The little available and recent literature published on this phenomenon has looked at why secondary students elect to study higher-level mathematics courses, both from the perspective of teachers and students. This research paper presents findings as to why Heads of Learning Area: Mathematics (HOLAMs) believe capable secondary students elect not to enrol in those courses. Data were collected from 50 secondary schools across the three sectors (Government, Catholic, Independent) in Western Australia. The key findings are that capable students do not enrol in higher-level mathematics courses because these courses are not required for university entrance, other courses appear to be less rigorous and more viable, and the Australian Tertiary Admissions Ranking (ATAR) score can be maximised by taking one mathematics course instead of two courses.

In this paper, we report on an experimental activity for discussing the concepts of speed, instantaneous speed and acceleration, generally introduced in first year university courses of calculus or physics. Rather than developing the ideas of calculus and using them to explain these basic concepts for the study of motion, we led 82 first year university students through Galileo's experiments designed to investigate the motion of falling bodies, and his geometrical explanation of his results, via simple dynamic geometric applets designed with GeoGebra. Our goal was to enhance the students' development of mathematical thinking. Through a scholarship of teaching and learning study design, we captured data from students before, during and after the activity. Findings suggest that the historical development presented to the students helped to show the growth and evolution of the ideas and made visible authentic ways of thinking mathematically. Importantly, the activity prompted students to question and rethink what they knew about speed and acceleration, and also to appreciate the novel concepts of instantaneous speed and acceleration at which Galileo arrived.

School autonomy has been identified as having an impact on a school's performance, yet less has been reported about the effect this has on students' goal orientations and engagement with mathematics. In a national study conducted in schools across Australia, measures of school autonomy were collected from teachers and school leaders, along with students' perceptions of the mastery and performance goal orientations of their classrooms and personally using surveys. Schools were identified as having high or low levels of autonomy on the basis of school leaders' responses. For the study discussed in this paper, a subset of 14 schools for which matched student and teacher data were available provided students' responses to a variety of variables including goal orientations. The findings suggested students in high-autonomy schools were less likely to hold a personal performance approach and avoidance goals than their peers in low-autonomy schools. Fifty-five case studies conducted in 52 schools provided evidence of some of the practical aspects of these findings, which have implications for systems, schools and teachers.

There is not just one mathematics taught at university level, nor is there one group of students. Mathematics is taught differently depending on the discipline and the perceived background of the student. There is engineering mathematics for the students heading towards engineering degrees, life science mathematics for those heading towards biology degrees and so on. This paper considers the phases of transitions that students experience as they embark on a course of study and then go on to professional life. We make inferences about the ways the curriculum should be designed to alleviate the difficulties of these phases as well as to take account of the capabilities that graduates will require in the workplace. It is not only where students are coming from that affects their learning but where they are heading to, in combination with their perceptions of that destination. (Contains 1 table and 1 figure.)

This paper presents a study about using scientific simulations to enhance the process of mathematical modelling. The main component of the study is a lesson whose major objective is to have students mathematise a trajectory of a projected object and then apply the model to formulate other trajectories by using the properties of function transformations. It was hypothesised that situating the lesson in a modelling environment would enhance the meaning of transformations that are not often conceptualised in mathematics textbooks. The lesson is guided by inductive reasoning. As a medium of data gathering, a free simulation called "Projectile Motion" was used (available at http://phet. colorado.edu/sims/projectile-motion/projectile-motion_en.html). The inductively organised stages of the activity described in this paper were conducted with a group of (N = 22) mathematics students in a high school in Texas. The students' verbal reflections upon this type of novel learning environment supported the study hypothesis. Their perception of the process of studying function transformations has evolved into a meaningful and purposeful experience. Although, the unit was developed for high school math curriculum in the US, its objectives reflect the aims and scope of Australian math curriculum. The Victorian Certificate of Education Study Design (VCAA, 2010) states that students should model investigate and solve problems in unfamiliar situations. The proposed lesson supports this aim.

Teaching and learning of mathematics is challenging when lecturer and students are separated geographically. While student engagement and interaction with the course, with other students and with the lecturer is vital to mathematics learning, it is difficult to facilitate this electronically, because of the nature of mathematics. With tablet technology now becoming ubiquitous and many new and inexpensive models entering the market, it is timely to investigate how the distance student experience in mathematics can be impacted by the use of tablet technologies. This paper reports on a case study of a first year mathematics course at a regional Australian university, where distance students were provided with affordable tablet PCs. An investigation of the impact of this technology on engagement and interaction is at the centre of this study. Evidence from journals, students' assessment submissions, screen snippets, student communication and formal student evaluations is analysed. It was found that distance students acknowledged the value of tablets for communicating mathematics, particularly for assignment submission and feedback, but they also recognized the potential for easier interaction with content and the lecturer. This paper highlights the specific benefits and challenges tablet PCs present to the learning experiences in mathematics within the distance context. (Contains 10 figures.)

This paper focuses on very young students' ability to engage in repeating pattern tasks and identifying strategies that assist them to ascertain the structure of the pattern. It describes results of a study which is part of the Early Years Generalising Project (EYGP) and involves Australian students in Years 1 to 4 (ages 5-10). This paper reports on the results from the early years' cohort (Year 1 and 2 students). Clinical interviews were used to collect data concerning students' ability to determine elements in different positions when two units of a repeating pattern were shown. This meant that students were required to identify the multiplicative structure of the pattern. Results indicate there are particular strategies that assist students to predict these elements, and there appears to be a hierarchy of pattern activities that help students to understand the structure of repeating patterns. (Contains 9 tables.)

The importance of actively engaging in mathematics discourse in order to learn mathematics is well recognized. In this paper, I use Basil Bernstein's concepts of pedagogic discourse to document and analyse academic learning time of students in Years 8 and 9 at a suburban lower secondary school: in particular, for what proportion of class time students reported being academically engaged, their explanations for this engagement and how they felt about the discourse. It was found that many students had disengaged from mathematical endeavour as a result of the failure of the instructional discourse either to engage students or to serve the purpose of developing discipline-specific content knowledge. The reasons for this relate to the overemphasis on mundane mathematics resulting in some students lacking the cognitive tools to engage with the concepts and having neither the intrinsic nor instrumental motivation to persist with secondary school esoteric mathematics. The implications for mathematics curriculum development are discussed.

Closed captioning of instructional videos is a topic that has not seen much discussion despite its importance for hearing-impaired students and recent legal ramifications if videos are not appropriately captioned. In particular, it is unclear what best practice in captioning videos should be to benefit all learners in disciplines such as mathematics with a reliance on the development of visual explanation while providing audio narration. In this paper, we report on a study undertaken at an Australian university, to investigate the perceived level of usefulness of captions and their automatic translations in a mathematics course. We discovered that students broadly agreed that captions are a useful learning feature: to allow flexibility of where and when a video is watched, but also to help understand speaker accents, and clarify explanations that are difficult to hear in the recording. Due to the high levels of use and perceived educational benefits of closed captions in online video but limited literature, there is a significant need for new research in this area. An urgent discussion is needed to explore how students engage with closed captions, how they may support learning, and to investigate implications on instructional design of mathematical videos.

This paper is presented in two parts. Through an example the first part takes up the issue of applying mathematics to situations that form part of the life context of students--the priority expressed in three curriculum statements presented. Then, noting the particular point in time--development of a National Curriculum for Mathematics--the second part goes on to address broader curriculum issues that a purely illustrative exercise in real world problem solving might not normally engage. The chosen example relates to a real world question that is located within the domain of Australian Rules Football, and it is recognized that while this provides a familiar, and often an emotionally engaged context in the majority of states, it may not do so for all. The specific mathematical and modelling issues raised in this particular problem have no essential connection with the discussion in the final part of the paper (other than in providing illustrations), where issues regarding the place of modelling and applications in curricula are considered. For that purpose, the football example can be replaced by any authentic modelling problem. (Contains 1 figure and 2 tables.)

Trigonometry is a well known branch of Mathematics. The study of trigonometry is of great importance in surveying, astronomy, navigation, engineering, and in different branches of science. This paper reports on the discovery of flaws in the traditional definitions of trigonometric ratios of an angle, which (in most cases) make use of the most unrealistic concept of negative length (or distance). With a view to getting rid of the misleading concept of negative length (distance), the definitions of novel trigonometric ratios (falling within the purview of Year 9 to Year 10A in the "Australian Curriculum: Mathematics") have been offered first with the help of vector algebra and then subsequently employed to derive some important formulae of trigonometry. This paper first considers the traditional branch of trigonometry, examines it to see that it is very much unrealistic at its root level and finally it gives birth to the definitions of novel trigonometric ratios of an angle with the help of vector algebra so as to uproot the most unrealistic concept of negative length. It deals with a debatable issue regarding the misleading concept of negative length (distance) prevailing at the basic level of defining trigonometric ratios of an angle in the traditional literature and in fact reflects a discovery of real truth. (Contains 3 figures.)

This article is intended to assist people who have never written for a journal, or who perhaps have never even thought about doing so. In this article, the authors provide some advice for budding Australian Primary Mathematics Classroom (APMC) authors. Information on where to start, what's already been done, what will be in the paper, who will read the paper, what happens after the paper is submitted, and the technical aspects of submitting your work is included.

In this paper, the authors describe the initial stage of developing a framework designed to support out-of-field, less experiences or isolated mathematics and science teachers to make decisions about the use of resources in their teaching. The process highlighted the complexity and extent of the knowledge on which expert teachers draw in making such decisions and thus underscored the enormity of the task of teaching out-of-field. The eventual product, the Science, Technology, Engineering and Mathematics: Critical Appraisal for Teachers (STEMCrAfT) framework has proven useful not only for the target audience, but also as a tool for colleagues who take on a mentoring role. The authors begin with a brief description of teacher knowledge before describing the project and then presenting what they unearthed about expert teachers' thinking and knowledge.

Two decades ago, in an award-winning paper, Dan Kennedy (1995) likened learning mathematics to climbing a tree, for which there was only one way to climb: up a large and solid trunk. In the limited time that is available, many students give up the climb, impede others, fall off the trunk, or fail to climb the tree sufficiently well. In the case of integration, the solid trunk seems to be heavily laden with algebraic manipulation. Kennedy suggested that technology might provide help in the form of ladders to climb the tree in other ways. Just as the use of technology allowed us to bypass the numerical requirements to calculate square roots (and other aspects of basic mathematics), it now seems time to look carefully at the use of computer algebra to reconsider how much of the algebraic trunk is really needed to help students climb the tree, look around and start to explore the branches of the tree that look interesting to them.

Traditionally, "z" is assumed to be a complex number and the roots are usually determined by using de Moivre's theorem adapted for fractional indices. The roots are represented in the Argand plane by points that lie equally pitched around a circle of unit radius. The "n"-th roots of unity always include the real number 1, and also include the real number -1 if "n" is even. The non-real "n"-th roots of unity always form complex conjugate pairs. This topic is taught to students studying a mathematics specialism (ACARA, n.d., Unit 3, Topic 1: Complex Numbers) as an application of de Moivre's theorem with the understanding that the roots occur in the complex domain. Meanwhile, in the Cartesian plane, a closely related topic deals with the solution of polynomials (ACARA, n.d., Unit 2, Topic 3: Real and Complex Numbers). The aim of this paper is to demonstrate visually the connection between the reduced polynomial "y" = "x"[superscript "n"] - 1 in the Cartesian plane and the resulting n-roots which invariably appear in the Argand plane. There is no contradiction here: the reader will find a three-dimensional surface representation of Equation (2) provides the full link between both the Cartesian and Argand planes, and illustrates not only the location of the roots in relation to the original equation but also shows why they occur with conjugate pairings. Examples will be provided for the cases "n" = 3, "n" = 5 and "n" = 8 which will be sufficient to illustrate the general pattern that emerges. The approach adopted here is a natural extension of the surface visualisation techniques first presented by Bardell (2012) for quadratic equations.

This article aims to illustrate a process of making connections, not between mathematics and other activities, but within mathematics itself--between diverse parts of the subject. Novel connections are still possible in previously explored mathematics when the material happens to be unfamiliar, as may be the case for a learner at any career stage. The geometrical configuration explored in this paper, now known as "Ford circles" after Lester R. Ford, Sr. (1886-1967), is related to ideas about mutually tangent circles that were studied by, among others, Apollonius of Perga in the third century BC and by René Descartes in the 17th century. This exposition is intended to conjure the thoughts of a hypothetical mathematician attempting to find and explain some connections, in the process exploring some lines that turn out to be unproductive, and making observations that are really non sequiturs, before eventually achieving some success. The author suggests that seemingly innocent mathematical fragments can have connections to many related ideas. If a teacher is in possession of a broad subject knowledge, then the likelihood seems high that it is possible to draw out useful connections in the classroom or in well-designed projects and assignments. For this reason, the author claims that an ever-widening subject knowledge is of utmost importance in a teacher's program of professional development.

School mathematics tends to have developed from the major cultures of Asia, the Mediterranean and Europe. However, indigenous cultures in particular may have distinctly different systematic ways of referring to space and thinking mathematically about spatial activity. Their approaches are based on the close link between the environment and cultural activity. The affinity to place strengthens the efficient, abstract, mathematical system behind the reference and its connection to the real world of place and a holistic worldview. This paper sets out to present an overview of various approaches to aspects of space and geometry by drawing on linguistic and cultural literature, my collaborative research in Papua New Guinea, and from personal communications with indigenous colleagues in Australia and elsewhere. This diversity provides a challenge by which teachers can deconstruct their thinking about mathematics and subsequently to review the content of teaching and to be more responsive to the diversity of cultural backgrounds of students. To assist with recognising ecocultural mathematics on space and geometry, 4 principles are established and discussed on language structures, reference lines and points, measures of space and worldviews and interpretations of space as place.

This paper is a natural extension of the root visualisation techniques first presented by Bardell (2012) for quadratic equations with real coefficients. Consideration is now given to the familiar quadratic equation "y = ax[superscript 2] + bx + c" in which the coefficients "a," "b," "c" are generally complex, as shown explicitly in Equation (1), which is presented in the article, with the usual notation.

The learning sequence described in this article was developed to provide students with a demonstration of the development of box plots from authentic data as an illustration of the advantages gained from using multiple forms of data representation. The sequence follows an authentic process that starts with a problem to which data representations provide the solution. The advantage of using box plots is that they allow clear and efficient comparison of related data sets. In this case, students are given a maze on paper and timed while they complete it. This produces the first set of data. They then attempt the maze again, expecting that their time to do this will decrease. The need to compare these two data sets arises from the question, "Did the group improve their maze times on their second attempt?"