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Understanding children's attitudes towards mathematics provides insights into their lived mathematical experience and engagement. Despite the considerable amount of research into students' attitudes toward mathematics, limited research has been conducted into "young children's" attitudes toward mathematics (YCATM). Within this limited research, investigating YCATM has certain challenges. From a methodological perspective, limitations exist regarding the type of research techniques that can be employed to study the nuances of the issue. The foci of this paper are to present and evaluate a methodological approach that used children's drawings (N = 106) and interview responses as the primary sources of data. Findings indicate that the strategy of "Draw yourself doing mathematics", when used with other research methods, generated rich attitudinal data in the form of personal stories about YCATM.

A deep understanding of fraction concepts and operations is necessary if pre-service teachers (PSTs) are to present the concepts in multiple forms to learners. Such an understanding needs to be grounded in rich conceptual knowledge. In the present study, we explore the development of this understanding by supporting a cohort of 103 PSTs, who had previously demonstrated poor conceptual understanding of fraction concepts and operations, with a Representational Reasoning in Teaching and Learning (RRTL) approach aimed at strengthening their conceptual knowledge. A comparison of pre- and post-test results indicated that participants showed a significant improvement in shifting the balance of their fraction knowledge to the conceptual end of the procedural-conceptual spectrum. Insights into how this approach assisted in developing PSTs' conceptual understanding were explored through interviews with four participants and an analysis of their pre- and post-test responses. We suggest that the use of teaching strategies such as RRTL are necessary in order to assist PSTs develop strong conceptual knowledge of fractions. [Note: The volume number (1) shown on the PDF is incorrect. The correct citation is v21 n1.]

Students' performances in TIMSS have featured strongly in recent discussions and debates about the quality of mathematical learning outcomes both from teachers and policy makers. Findings of TIMSS trends showed that most high school students in Malaysia continue to perform at less than satisfactory levels, particularly, in tasks that are cognitively demanding. In this article, we present a critical analysis of selected TIMSS Tasks and demonstrate how to support students better in the use of Higher Order Thinking Skills ("HOTS") in making progress with such tasks. In so doing we present analyses of TIMSS tasks and a model of these tasks that relate "HOTS," representation and concepts which can be utilised by teachers to understand the role of "HOTS" better in empowering students shift to higher levels of cognitive functioning in the context of tackling TIMSS and similarly demanding tasks. Our analyses and model provide an important starting point for the design of future professional development programs for Malaysian mathematics teachers in reconceptualising "HOTS" and implementing them in regular classrooms.

Reforms in Australia about the education of future teachers has placed a high degree of emphasis on the development of knowledge and skills that are necessary for practitioners who will ply their trade in culturally rich and diverse classrooms (Ramsey, 2000). There is now a broad consensus from key stakeholders (Australian Institute for Teaching and School Leadership, 2012) that pre-service teachers need to be provided with a range of opportunities that are grounded in classroom practices including exposure to teaching students overseas. The aim of the study that is reported here is to better understand the skills and knowledge that pre-service teachers need in order to function in and gain from their overseas professional experiences (OSPEX). In this study, we compared a cohort of pre-service teachers' perceptions of OSPEX before and after their completed their professional experiences in Fiji. Results indicate that pre-service teachers need to be better prepared locally before attempting an OSPEX visit. (Contains 4 tables.)

Contemporary debates on effective pedagogies for K-12 mathematics have called for shifts in the way teachers and teacher educators conceptualise mathematics as a subject and how it should be taught. This is reflected by changes in the curriculum including the inclusion of a strand called Working Mathematically within K-12 mathematics curriculum documents not only in New South Wales but also across Australia. This strand brings focus to mental processes that underpin students' ability to acquire mathematical principles, concepts, and conventions, and the use of this knowledge in the solution of problems. The focus on cognitive processes that support mathematical learning and problem solving is a welcome change. However, there is a paucity of information about the nature of links that need to be made between process and mathematics content, and how students might be assisted to construct the links. In this article, the authors outline results of research about cognitive load that is associated with mental processes, the management of this load so that students can be better supported in the construction of connected mathematical information, and the interpretation of that information in making sense of worked examples. They attempt to show that worked examples can be effective in promoting useful and powerful mathematics schemas. (Contains 3 figures.)

In the present study, I examine a modelling strategy as employed by a teacher in the context of an algebra lesson. The actions of this teacher suggest that a modelling approach will have a greater impact on enriching student learning if we do not lose sight of the need to manage associated cognitive loads that could either aid or hinder the integration of core concepts with processes that are at play. Results here also show that modelling a problem that is set within an authentic context helps learners develop a better appreciation of variables and relations that constitute the model. The teacher's scaffolding actions revealed the use of strategies that foster the development of connected, meaningful and more useable algebraic knowledge. (Contains 1 figure.)

The benefit of homework (HW) has been the subject of ongoing debate among various stakeholders. Within this debate, and in relation to teacher development, prospective teachers' views of HW have received limited attention. In this study, we survey primary pre-service teachers' ('PSTs') views of HW use (n = 45 teaching grades 2-5; n = 39 teaching grades 6 and 7) in a single discipline (mathematics) at one large Australian university.The literature suggests students in upper primary and secondary grades will benefit from the use of mathematics HW. Results of the study, in contrast, suggest our PSTs teaching early primary grades planned to give proportionally (but not significantly) more mathematics HW than those teaching later grades. Reasons for this disparity are discussed. We argue there is a need in teacher development to equip PSTs with a better understanding of how classroom and home-based learning work together in the development of mathematical understanding.

The use of fully online instruction has been proliferating in higher education. In undergraduate mathematics, arguments about the effectiveness of fully online instruction have been met with contrasting claims about its potential. Some assert that fully online instruction can promote a deeper and richer conceptual understanding of mathematical ideas while others claim it is difficult to teach mathematics effectively in a fully online context. The aim of this study is to examine these claims by documenting relevant empirical evidence, the findings of which we hope will inform current efforts, by both faculty and administrators, to improve fully online mathematics instruction.

Recent developments about cognitions underlying mathematical learning are beginning to suggest that the activation and appropriate use of prior knowledge by students is, to a large measure, controlled by the quality of organisation of that knowledge. Thus, teaching needs to support the construction of well-connected mathematical knowledge. An important assumption here is that teachers need to construct a repertoire of subject-matter knowledge that is rich and well connected before they can help their students build similar mathematical knowledge. Thus, mathematics knowledge building is an important issue in teacher preparation programs. This paper reports on a study about the knowledge state of a pre-service teacher who planned to use computers in the teaching of linear functions. The results of the study indicate the existence of gaps in the student teacher's subject-matter knowledge. Significantly, there was also a lack of important connections between his understanding of linear functions and the instructional use of a computer software. Knowledge gaps and implications for classroom students' acquisition of mathematical schemas and mathematics teacher education programs are examined and discussed. (Contains 2 tables and 1 figure.)

Knowledge that teachers bring to the teaching context is of interest to key stakeholders in improving levels of numeracy attained by learners. In this regard, the centrality of, and the need to investigate, the quality of teachers' mathematical knowledge for teaching mathematics has been gaining momentum in recent years. There is a general consensus that teachers need a robust body of content and pedagogical knowledge related to mathematics and that one impacts on the other. However, in current debates about this interconnection between content knowledge and pedagogical content knowledge, there is limited analysis about the procedural-conceptual nature of content knowledge that, we argue, has significant impact on the development of pedagogical content knowledge. In this report, this issue is investigated by examining the state of procedural and conceptual knowledge of two cohorts of pre-service teachers and analyzing the impact of a representational reasoning teaching and learning (RRTL) approach aimed at supporting a balanced development of these two dimensions of Content Knowledge.

Numeracy in schools is becoming an increasingly important part of mathematics learning and teaching. This is because educators want students to engage with mathematical concepts more deeply, use mathematics to make sense of their environment and make decisions that are based on the analysis of mathematical information. In order to be numerate, students must be able to acquire mathematical concepts and procedures, and apply these flexibly in a range of real life contexts. The school mathematics curriculum provides a number of strands of mathematics from which students can draw from, such as geometry and algebra in order to exhibit their numeracy skills. In the present study, numeracy is investigated from the perspective of students' abilities to gather, display and interpret data--an area of numeracy that has been broadly referred to as statistical literacy. A statistically literate student can be expected to demonstrate an ability to use statistical concepts to make sense of his or her immediate environment. This area of students' numeracy continues to be challenging for many students. In this paper the authors draw on their recent research that focuses on the interpretative aspects of real-life data and generating ideas for activities that would better engage children in the complex and somewhat more demanding area of statistical literacy. They do this firstly by advancing a model of phases that they suggest students go through from being able to draw a graph to being able to interpret a graph and make decisions. Secondly, they present findings from an authentic real-life context that they investigated in order to examine the usefulness of the phases outlined in their model. Finally the authors examine possible strategies for classroom practice.

Within the domain of geometry, proof and proof development continues to be a problematic area for students. Battista (2007) suggested that the investigation of knowledge components that students bring to understanding and constructing geometry proofs could provide important insights into the above issue. This issue also features prominently in the deliberations of the 2009 International Commission on Mathematics Instruction Study on the learning and teaching of proofs in mathematics, in general, and geometry, in particular. In the study reported here, we consider knowledge use by a cohort of 166 Sri Lankan students during the construction of geometry proofs. Three knowledge components were hypothesised to influence the students' attempts at proof development: "geometry content knowledge," "general problem-solving skills" and "geometry reasoning skills." Regression analyses supported our conjecture that all 3 knowledge components played important functions in developing proofs. We suggest that whilst students have to acquire a robust body of geometric content knowledge, the activation and the utilisation of this knowledge during the construction of proof need to be guided by general problem-solving and reasoning skills.

The teaching of mathematics for the development of numeracy is emerging to be an important goal of recent reforms. In broad terms, numeracy involves the application of mathematical concepts and procedures in the understanding and solution of a range of problems including real-life problems. Teaching for numeracy, thus calls for skills in translating abstract concepts in mathematics into appropriate real-life contexts and developing an understanding of children's difficulties in this area. In this study, we examined this issue by analysing knowledge of teachers as they attempted to contextualise an abstract fraction problem involving division. Data analyses showed that the participating teachers (n = 4) have developed a conceptually weak schema that girds the comprehension and subsequent contextualisation of the given problem. This was evidenced by not only their failure to provide a correct real-life problem representation but also limited knowledge about how they could help students who might have difficulty with similar problems. However, the teachers have developed a robust body of general pedagogical knowledge that was relevant to the teaching of fractions. Taken together the data suggested that their content knowledge for teaching mathematics and pedagogical content knowledge were both weak in this specific area of fractions. (Contains 5 tables and 1 figure.)

Two developments have contributed to the convergence of views about the benefits of real-life and inquiry-based pedagogies in mathematics learning. First, the mathematics teaching community is increasingly focused on the learning of mathematics that involves the transfer of prior knowledge to novel problem-solving situations, a key element in recent characterizations of the notion of "Numeracy." Second, research about human cognition in domain-specific learning suggests that problem-solving activity provides better contexts in which to observe the construction of creative connections of disparate information. The question is how can we examine the above cognitive processes, as these are played out in Numeracy contexts? (Tan, "Educ Res Policy Pract" 6:101-114, 2007) identified "dialogue" and "inquiry" as important themes of the psychology that girds problem-based learning. In this report, we take up Tan's suggestion that research needs to make learner's cognition more visible by immersing a cohort of Malaysian and Australian students in arithmetic word problems. Students were required to explain and justify their solutions. The frameworks of "dialogue" representation and schema guided our analyses that focused on students' readings, explanations, representations and reflections about a given set of problem contexts. Results showed that both the groups experienced difficulty in representing far-transfer numeracy problems. However, Australian children tended to develop longer and more varied explanations in comparison to their Malaysian counterparts. Potential implications for classroom practices, policy-making and future research directions are explored.

Productive pedagogies (PP) is an influential framework for teaching and learning that has featured well in current reforms of teacher education. The present study was designed to examine principles of PP adopted by a cohort of beginning mathematics teachers. A networked online learning environment, WebCT[TM], facilitated the teachers' deliberations about PP. The uptake of PP by these new teachers was examined in the context of a collaborative activity in which the teachers were required to reflect on, critique, and share views with peers about a lesson on the teaching of multiplication. These online interactions are argued to provide opportunities for shared learning and activity. Qualitative analyses of the results show that participants espoused and embraced dimension of PP in varying degrees. The implications for using WebCT[TM] and similar online course management systems to support teacher reflections on best practice are discussed. (Contains 1 table.)

Current reform-driven mathematics documents stress the need for teachers to provide learning environments in which students will be challenged to engage with mathematics concepts and extend their understandings in meaningful ways (e.g., National Council of Teachers of Mathematics, 2000, "Curriculum and evaluation standards for school mathematics." Reston, VA: The Council). The type of rich learning contexts that are envisaged by such reforms are predicated on a number of factors, not the least of which is the quality of teachers' experience and knowledge in the domain of mathematics. Although the study of teacher knowledge has received considerable attention, there is less information about the teachers' content knowledge that impacts on classroom practice. Ball (2000, "Journal of Teacher Education," 51(3), 241-247) suggested that teachers' need to "deconstruct" their content knowledge into more visible forms that would help children make connections with their previous understandings and experiences. The documenting of teachers' content knowledge for teaching has received little attention in debates about teacher knowledge. In particular, there is limited information about how we might go about systematically characterising the key dimensions of quality of teachers' mathematics knowledge for teaching and connections among these dimensions. In this paper we describe a framework for describing and analysing the quality of teachers' content knowledge for teaching in one area within the domain of geometry. An example of use of this framework is then developed for the case of two teachers' knowledge of the concept "square."

The quality of knowledge underlying students' algebraic thinking is a major concern of K-12 mathematics teaching and reform-driven curriculum documents. In this paper the author examines this issue by characterizing a group of high school students' algebraic knowledge and patterns of use of that knowledge during the solution of selected problems. Results show that these students tended to show acceptable levels of proficiency with problems that involved substitution of values and simplification of equations. Students experienced difficulties with the interpretation of algebraic relations. Multiple regression analysis showed that knowledge about equations and pattern formation was a predictor of students' success at interpretation of graphs of linear functions. These results are interpreted as suggesting that understanding and analysis of graphs could be facilitated by teaching approaches that focus on the construction of robust schemas about patterns and symbolic relations in algebra. (Contains 7 tables.)

Examines a group of preservice primary mathematics teachers' understanding of fractions using software called JavaBars. Suggests that while teachers had built up robust knowledge about fractions, they experienced difficulty in translating this knowledge in the JavaBars environment. (Contains 26 references.) (Author/ASK)

Examines the nature of prior mathematical knowledge that facilitates the construction of useful problem representations in the domain of geometry. Indicates that high achievers build schemas that are qualitatively more sophisticated than low achievers, which in turn helps them construct representations that are conducive to understanding the structure of geometry problems. Contains 31 references. (Author/ASK)