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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 Refereed/Peer-reviewed

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. Refereed/Peer-reviewed

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 Refereed/Peer-reviewed

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 funtioning 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. Refereed/Peer-reviewed

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. Refereed/Peer-reviewed

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. Recent reform agendas in mathematical teaching and learning have been underpinned by the need to foster robust and meaningful learning, one that aids in the long-term retention of core concepts and principles, and application in a wide variety of contexts. This latter shift has been instrumental in the development and characterisations of notions of numeracy as the utilisation of mathematics in understanding real-life contexts and solution of problems. As a consequence, an important goal of mathematics teaching is now seen as developing students’ structural understanding of concepts and the embedding of concepts in realistic contexts. Determining and utilising effective and appropriate strategies to promote enduring understandings is crucial to quality learning outcomes in mathematics. A principal element of such understandings would be the establishment of links among concepts, facts, conventions and procedures. This need to examine connections that students construct has been endorsed by major curriculum reform documents (Board of Studies, 2002; National Council of Teachers of Mathematics, 2000). How can we elucidate these relations that are central to connected learning? The mathematics community has taken this issue by adopting a model or modelling perspective in analysing effective mathematical practices.

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. Refereed/Peer-reviewed

There is international concern that the standard of mathematics education requires improvement. In order to facilitate this improvement, preservice teacher education needs to meet the demands of our technological age. As well, an international trend in mathematics curriculum development should focus on problem solving and modeling. This trend is evident in Australia, where the first national curriculum is currently being implemented. The need for attention to changing curriculum and introduction of reform-oriented pedagogies has ramifications for programs that provide preservice teacher education. New ways of thinking about mathematics and the teaching of mathematics is an important theme in such reforms. One offshoot of this development is focusing on teacher knowledge and its potential impact on students' performance. Our research is set against this background and the aim of this exploratory study is to identify teacher knowledge that is activated in the context of designing mathematical problems. The methodological approach is qualitative and quantitative. The participants of the study are twenty-six preservice teachers at an Australian university. Findings show that having preservice teachers design rich learning activities which incorporate modeling of problems provides opportunities for them to demonstrate a rich repertoire of mathematical knowledge for teaching. This mathematical knowledge is complex and its development can be captured by mapping the trajectory of their learning. A key finding is that our approach provided opportunities for participants to construct subject-matter and pedagogical-content knowledge in a blended manner. An implication for preservice teacher education is that program coursework needs to support the development and activation of different types of knowledge for teaching. Refereed/Peer-reviewed

A current concern is student learning outcomes and these are largely a function of teachers' knowledge and their practice. This position paper is premised on the notion that certain knowledge is required for the teaching of mathematics. An exploration of literature demonstrates that such professional knowledge development can be supported by Learning Trajectories (LT). We propose to use LT as theoretical lens to examine pre-service teachers' Content and Pedagogical Content knowledge and advance a research design.

Statistical literacy has been argued to constitute a key aspect of students' numeracy. In the present study we examine statistical literacy of two Year 7 students and their abilities to extract and interpret information from a familiar real-life context. Analysis of data indicate that while both of the students developed facility in extracting information and drawing graphs, the low-achieving student tended to process the graphical information somewhat superficially in comparison to the high-achiever. Implications for teaching and learning are discussed.

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.

The shift in the way we visualise the nature of mathematics and mathematics learning has presented educational technologists with new challenges in the design of rich and powerful learning environments.Against this background, the design and use of learning objects in supporting meaningful mathematical learning assumes increased significance. I argue that learning objects need to be sufficiently pliable such that both teachers and learners could engage in knowledge construction that provides further avenues for growth and sophistication of mathematical schemas. In this chapter, the author aims to show the limitations of current views about mathematical learning objects and the need to reconceptualise these in terms of generic meta-schemas. A meta-schematic framework would provide the mathematics community with powerful pedagogical tools to support and assess mathematics learning. Two examples ofthese meta-schemas for geometry are described.

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™, 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™ and similar online course management systems to support teacher reflections on best practice are discussed.

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’.