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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.)

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.

This paper reports on an investigation into pre-service teachers' mathematical content knowledge and their ability to interpret students' responses to a multi-digit multiplication task and make subsequent appropriate teaching decisions. Using a combination of quantitative and qualitative methods, the researchers tested aspects of the mathematical knowledge held by a volunteer group of twenty final year preservice primary teachers. A volunteer sample of seven pre-service teachers were involved in a follow-up interview, where they were provided with hypothetical student work samples, including one using the long multiplication algorithm, and asked to analyse the student's mathematical thinking and make suggestions as to appropriate teaching approaches. The results indicated that the pre-service teachers in the study had an instrumental understanding of the long multiplication process that impacted on their ability to both recognise and address students' mathematical errors. This study provides an insight into the lack of content knowledge of a small sample of pre-service teachers with respect to multiplication of two and three digit numbers and subsequent lack of pedagogical content knowledge for teaching this topic.

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 outlines the practical aspects of a project aimed to offer access to numerate knowledge for preschool children by providing them with take-home numeracy kits. A Koori preschool in an urban regional area of New South Wales, Australia, was involved in the project. The centre catered for 18 four- and five-year-old children. The two main resources for the project were numeracy-related activities, and interactive adult and peer support. In the first part of the paper observational transcripts of the children playing with the kits at the preschool are analysed according to Foundation and Transition Level outcomes and indicators for number (NSW Department of School Education, 1994). The second part of the paper clarifies and synthesises key aspects of culturally-situated learning: the children's mathematical language and problem-solving; the responsiveness of the teaching strategies; and the congruence between early childhood education philosophies and national numeracy policies.

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.

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.

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?"

This paper explores how a secondary school in western Sydney used educational research as an impetus to change its mathematical education culture over a three year period. Key changes occurred in four areas: leadership; pedagogy; structures for teaching and learning; and mathematical environments. These included increased professional conversations, adoption of a numeracy lesson structure, regular use of manipulatives and open ended tasks and a structured intervention program for mathematically vulnerable students. Critical to the development of these changes were partnerships with a university academic and the CEDP system leadership team as well as school leadership participation in professional learning.

In writing this paper we draw considerably on the work of Jo Boaler and Leone Burton. Boaler's studies of classrooms have been particularly poignant in alerting the mathematics education community to a number of key features of successful classrooms, and how such features can turn around the successes for students who traditionally perform poorly in school mathematics. This is supplemented by the recent work of Leone Burton who worked extensively with research mathematicians in order to understand their communities and ways of working. Collectively these two seminal works provide valuable insights into potential ways to move the field of school mathematics forward. In times when there is international recognition of the plight of school mathematics, there is a need for new teaching practices that overcome the hiatus of contemporary school mathematics. (Contains 4 figures.)

In the teaching of calculus, the algebraic derivation of the derivative (gradient function) enables the student to obtain an analytic "global" gradient function. However, to the best of this author's knowledge, all current technology-based approaches require the student to obtain the derivative (gradient) at a single point by implementing differentiation using first principles. This paper shows that the ability of spreadsheets to fit a polynomial to a set of discrete (x,y) points enables students to not just evaluate a gradient at a single point, but at a whole family of points, thus generating the analytic global gradient function of secants without doing any algebraic manipulations. Students can then perform "numerical experiments" to see the effect of taking the limit as the secants tend to tangents of the original function. Finally, students can derive the rules for differentiation through exploration and experimentation, again, without doing any algebraic manipulations. This approach enables the class to focus on the concepts being taught, rather than being hindered by the mechanics of (for example) trying to factorize a cubic polynomial. (Contains 4 figures and 1 table.)

The concept of a function is fundamental to virtually every aspect of mathematics and every branch of quantitative science. Presently this type of thinking is carolled at the secondary level, and yet it has many benefits for deepening the understanding of early arithmetic. This is particularly so in the way that operations can be considered as "changing" and how it explicitly illustrates the way in which addition and subtraction are inverse operations, with each "undoing" the other. With the move to introduce algebraic thinking into the elementary classrooms, this paper explores activities that exemplify this thinking with 6 year-old children. The authors collaboratively planned and implemented a series of hands-on activities over an eight-lesson program. The aim of these learning activities was to assist young children in understanding the key ideas in this area of mathematics. The activities not only encouraged active learning but also reflected the principles of socio-constructivist learning. (Contains 3 figures.)

Mathematical computation consists of both written computation and mental computation. The strategies for mental computation can be used to check the reasonableness of written computations. Mental computation has two distinguishing characteristics: "it produces an exact answer, and the procedure is performed mentally, without using external devices such as pencil and paper" (Reys, 1984, p. 548). It is important to consider the accuracy and efficiency of various mental computation strategies and this can be useful to discuss with students. Discussion about different strategies will enable students to consider different approaches and work towards development of efficient mental strategies for computation. Utilising a variety of strategies and methods for computation will enable students to develop a better understanding of computation processes and number sense. In this article, the authors put forth a strong case for an emphasis on developing mental calculation strategies with students, and suggest helpful teaching approaches to achieve this. (Contains 4 figures and 2 resources.)

Traditionally, primary schools place minimal emphasis upon relations and transformations as objects of study. In their research, the authors have found the young children can engage in conversations about equivalence and equations (Warren & Cooper, 2005a) and functional thinking (Warren & Cooper, 2005b). Fundamental to relations and transformations is the concept of the function, that is, how the value of certain quantities relate to the value of other quantities (Chazan, 1996), or how values are changed or mapped to other quantities, referred to in the literature as co-variational thinking. This paper reports on some recent classroom teaching that attempts to examine repeating patterns and use children's understandings of repeating pattern to begin to explore concepts related to functional thinking.

On her death in 1910, Florence Nightingale left a vast collection of reports, letters, notes and other written material. There are numerous publications that make use of this material, often highlighting Florence's attitude to a particular issue. In this paper we gather a set of quotations and construct a dialogue with Florence Nightingale on the subject of statistics. Our dialogue draws attention to strong points of connection between Florence Nightingale's use of statistics and modern evidence-based approaches to medicine and public health. We offer our dialogue as a memorable way to draw the attention of students to the key role of data-based evidence in medicine and in the conduct of public affairs.

Aim and objectives: To examine the factors that influence nursing students' mathematics self‐efficacy, the effect of numeracy instruction on self‐efficacy, and the association between self‐efficacy and numeracy test performance. Background: Medication administration errors, including administering incorrect dosages or infusion rates, can result in serious harm to patients. Hence, it is essential that nursing students are adequately prepared with the necessary numeracy skills during their nursing program. Design: This quasi‐experimental cohort study used a pre‐ and post‐test survey design. The study complied with the STROBE checklist for cohort research. Methods: In total, n = 715 undergraduate first year nursing students participated in the study from June to October 2017 at a single multi‐campus university in the Western Sydney region of Australia. Data were collected at three time‐points: (a) baseline, including assessing pre‐instruction mathematics self‐efficacy (NSE‐Math scale); (b) 6‐week follow‐up; including assessing post‐instruction mathematics self‐efficacy; and (c) numeracy test performance was collected at 7‐week follow‐up. Findings: At baseline, those with high NSE‐Math scale scores were more likely to be male and have at least high school advanced mathematics level education. Following structured numeracy instruction, NSE‐Math scale scores increased significantly, and those who obtained a satisfactory grade in their numeracy assessment were more likely to have high NSE‐Math scale scores and high academic performance in the previous semester. Conclusion: The study shows that structured numeracy instruction improved mathematics self‐efficacy, which in turn influenced numeracy test performance. Relevance to clinical practice: Using a structured medication numeracy pedagogical approach, to teach skills in nursing undergraduate programs, provides students with the foundations to improve mathematics self‐efficacy and to be successful and safe with medication numeracy calculations and administration in clinical practice. [ABSTRACT FROM AUTHOR]

The practices of effective primary school teachers including students with Down syndrome in their mathematics classes are largely unexplored and many teachers feel unprepared to teach students with an intellectual disability. The authors sought to find out what these specific professional learning (PL) needs were and if they changed over time. A mixed methods study was undertaken to investigate the practices of primary mathematics teachers including students with Down syndrome in regular class lessons in two Australia educational jurisdictions. A variety of approaches was used to gain an understanding of their practices. Findings are based on responses to a questionnaire administered to teaching teams at the beginning and end of the school year. There was a strong endorsement of inclusion as an appropriate practice in primary mathematics. The PL needs of teaching teams changed across the teaching year. Early requirements concerned understanding Down syndrome and its impact on learning and behavior. As the school year progressed, PL needs changed toward a greater focus on the need for specific resources and strategies for teaching mathematics. The PL requirements of teachers included the need for 'in-time' support. The authors conclude that even if it may be infrequent when a teacher has a student with Down syndrome in a mathematics class, teachers still need access to PL that evolves across the school year, from less focus on syndrome-specific information to greater emphasis on teaching mathematics through adjusting learning outcomes and teaching approaches. [ABSTRACT FROM AUTHOR]

ICT (information and communication technology) creates numerous opportunities for teachers to re-think their pedagogies. In subjects like mathematics which draws upon abstract concepts, ICT creates such an opportunity. Instead of a mimetic pedagogical approach, suitably designed activities with ICT can enable learners to engage more proactively with their learning. In this quasi-experimental designed study, ICT was used in teaching mathematics to a group of first year high school students (N = 25) in Australia. The control group was taught predominantly through traditional pedagogies (N = 22). Most of the variables that had previously impacted on the design of such studies were suitably controlled in this year-long investigation. Quantitative and qualitative results showed that students who were taught by ICT driven pedagogies benefitted from the experience. Pre- and post-test means showed that there was a difference between the treatment and control groups. Of greater significance was that the students (in the treatment group) believed that the technology enabled them to engage more with their learning. [ABSTRACT FROM AUTHOR]